Big Data Conference 2023
2023-08-31 09:00 - 2023-09-01 17:00 (America/New_York) | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
On August 31-Sep 1, 2023 the CMSA hosted the ninth annual Conference on Big Data. The Big Data Conference features speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
Speakers:

Jacob Andreas, MIT
Morgane Austern, Harvard
Albert-László Barabási, Northeastern
Rachel Cummings, Columbia
Melissa Dell, Harvard
Jianqing Fan, Princeton
Tommi Jaakkola, MIT
Ankur Moitra, MIT
Mark Sellke, Harvard
Marinka Zitnik, Harvard Medical School

Organizers:

Michael Douglas, CMSA, Harvard University
Yannai Gonczarowski, Economics and Computer Science, Harvard University
Lucas Janson, Statistics and Computer Science, Harvard University
Tracy Ke, Statistics, Harvard University
Horng-Tzer Yau, Mathematics and CMSA, Harvard University
Yue Lu, Electrical Engineering and Applied Mathematics, Harvard University

Schedule
(PDF download)
Thursday, August 31, 2023

9:00 AM
Breakfast

9:30 AM
Introductions

9:45–10:45 AM
Albert-László Barabási (Northeastern, Harvard)
Title: From Network Medicine to the Foodome: The Dark Matter of Nutrition
Abstract: A disease is rarely a consequence of an abnormality in a single gene but reflects perturbations to the complex intracellular network. Network medicine offer a platform to explore systematically not only the molecular complexity of a particular disease, leading to the identification of disease modules and pathways, but also the molecular relationships between apparently distinct (patho) phenotypes. As an application, I will explore how we use network medicine to uncover the role individual food molecules in our health. Indeed, our current understanding of how diet affects our health is limited to the role of 150 key nutritional components systematically tracked by the USDA and other national databases in all foods. Yet, these nutritional components represent only a tiny fraction of the over 135,000 distinct, definable biochemicals present in our food. While many of these biochemicals have documented effects on health, they remain unquantified in any systematic fashion across different individual foods. Their invisibility to experimental, clinical, and epidemiological studies defines them as the ‘Dark Matter of Nutrition.’ I will speak about our efforts to develop a high-resolution library of this nutritional dark matter, and efforts to understand the role of these molecules on health, opening novel avenues by which to understand, avoid, and control disease.
https://youtu.be/UmgzUwi6K3E

10:45–11:00 AM
Break

11:00 AM–12:00 PM
Rachel Cummings (Columbia)
Title: Differentially Private Algorithms for Statistical Estimation Problems
Abstract: Differential privacy (DP) is widely regarded as a gold standard for privacy-preserving computation over users’ data. It is a parameterized notion of database privacy that gives a rigorous worst-case bound on the information that can be learned about any one individual from the result of a data analysis task. Algorithmically it is achieved by injecting carefully calibrated randomness into the analysis to balance privacy protections with accuracy of the results.
In this talk, we will survey recent developments in the development of DP algorithms for three important statistical problems, namely online learning with bandit feedback, causal interference, and learning from imbalanced data. For the first problem, we will show that Thompson sampling — a standard bandit algorithm developed in the 1930s — already satisfies DP due to the inherent randomness of the algorithm. For the second problem of causal inference and counterfactual estimation, we develop the first DP algorithms for synthetic control, which has been used non-privately for this task for decades. Finally, for the problem of imbalanced learning, where one class is severely underrepresented in the training data, we show that combining existing techniques such as minority oversampling perform very poorly when applied as pre-processing before a DP learning algorithm; instead we propose novel approaches for privately generating synthetic minority points.
Based on joint works with Marco Avella Medina, Vishal Misra, Yuliia Lut, Tingting Ou, Saeyoung Rho, and Ethan Turok.
https://youtu.be/0cPE6rb1Roo

12:00–1:30 PM
Lunch

1:30–2:30 PM
Morgane Austern (Harvard)
Title: To split or not to split that is the question: From cross validation to debiased machine learning
Abstract: Data splitting is a ubiquitous method in statistics with examples ranging from cross-validation to cross-fitting. However, despite its prevalence, theoretical guidance regarding its use is still lacking. In this talk, we will explore two examples and establish an asymptotic theory for it. In the first part of this talk, we study the cross-validation method, a ubiquitous method for risk estimation, and establish its asymptotic properties for a large class of models and with an arbitrary number of folds. Under stability conditions, we establish a central limit theorem and Berry-Esseen bounds for the cross-validated risk, which enable us to compute asymptotically accurate confidence intervals. Using our results, we study the statistical speed-up offered by cross-validation compared to a train-test split procedure. We reveal some surprising behavior of the cross-validated risk and establish the statistically optimal choice for the number of folds. In the second part of this talk, we study the role of cross-fitting in the generalized method of moments with moments that also depend on some auxiliary functions. Recent lines of work show how one can use generic machine learning estimators for these auxiliary problems, while maintaining asymptotic normality and root-n consistency of the target parameter of interest. The literature typically requires that these auxiliary problems are fitted on a separate sample or in a cross-fitting manner. We show that when these auxiliary estimation algorithms satisfy natural leave-one-out stability properties, then sample splitting is not required. This allows for sample reuse, which can be beneficial in moderately sized sample regimes.
https://youtu.be/L_pHxgoQSgU

2:30–2:45 PM
Break

2:45–3:45 PM
Ankur Moitra (MIT)
Title: Learning from Dynamics
Abstract: Linear dynamical systems are the canonical model for time series data. They have wide-ranging applications and there is a vast literature on learning their parameters from input-output sequences. Moreover they have received renewed interest because of their connections to recurrent neural networks.
But there are wide gaps in our understanding. Existing works have only asymptotic guarantees or else make restrictive assumptions, e.g. that preclude having any long-range correlations. In this work, we give a new algorithm based on the method of moments that is computationally efficient and works under essentially minimal assumptions. Our work points to several missed connections, whereby tools from theoretical machine learning including tensor methods, can be used in non-stationary settings.
https://youtu.be/UmgzUwi6K3E

3:45–4:00 PM
Break

4:00–5:00 PM
Mark Sellke (Harvard)
Title: Algorithmic Thresholds for Spherical Spin Glasses
Abstract: High-dimensional optimization plays a crucial role in modern statistics and machine learning. I will present recent progress on non-convex optimization problems with random objectives, focusing on the spherical p-spin glass. This model is related to spiked tensor estimation and has been studied in probability and physics for decades. We will see that a natural class of “stable” optimization algorithms gets stuck at an algorithmic threshold related to geometric properties of the landscape. The algorithmic threshold value is efficiently attained via Langevin dynamics or by a second-order ascent method of Subag. Much of this picture extends to other models, such as random constraint satisfaction problems at high clause density.
https://youtu.be/JoghiwiIbT8

6:00 – 8:00 PM
Banquet for organizers and speakers

Friday, September 1, 2023

9:00 AM
Breakfast

9:30 AM
Introductions

9:45–10:45 AM
Jacob Andreas (MIT)
Title: What Learning Algorithm is In-Context Learning?
Abstract: Neural sequence models, especially transformers, exhibit a remarkable capacity for “in-context” learning. They can construct new predictors from sequences of labeled examples (x,f(x)) presented in the input without further parameter updates. I’ll present recent findings suggesting that transformer-based in-context learners implement standard learning algorithms implicitly, by encoding smaller models in their activations, and updating these implicit models as new examples appear in the context, using in-context linear regression as a model problem. First, I’ll show by construction that transformers can implement learning algorithms for linear models based on gradient descent and closed-form ridge regression. Second, I’ll show that trained in-context learners closely match the predictors computed by gradient descent, ridge regression, and exact least-squares regression, transitioning between different predictors as transformer depth and dataset noise vary, and converging to Bayesian estimators for large widths and depths. Finally, we present preliminary evidence that in-context learners share algorithmic features with these predictors: learners’ late layers non-linearly encode weight vectors and moment matrices. These results suggest that in-context learning is understandable in algorithmic terms, and that (at least in the linear case) learners may rediscover standard estimation algorithms. This work is joint with Ekin Akyürek at MIT, and Dale Schuurmans, Tengyu Ma and Denny Zhou at Stanford.
https://youtu.be/UNVl64G3BzA

10:45–11:00 AM
Break

11:00 AM–12:00 PM
Tommi Jaakkola (MIT)
Title: Generative modeling and physical processes
Abstract: Rapidly advancing deep distributional modeling techniques offer a number of opportunities for complex generative tasks, from natural sciences such as molecules and materials to engineering. I will discuss generative approaches inspired from physical processes including diffusion models and more recent electrostatic models (Poisson flow), and how they relate to each other in terms of embedding dimension. From the point of view of applications, I will highlight our recent work on SE(3) invariant distributional modeling over backbone 3D structures with ability to generate designable monomers without relying on pre-trained protein structure prediction methods as well as state of the art image generation capabilities (Poisson flow). Time permitting, I will also discuss recent analysis of efficiency of sample generation in such models.
https://youtu.be/GLEwQAWQ85E

12:00–1:30 PM
Lunch

1:30–2:30 PM
Marinka Zitnik (Harvard Medical School)
Title: Multimodal Learning on Graphs
Abstract: Understanding biological and natural systems requires modeling data with underlying geometric relationships across scales and modalities such as biological sequences, chemical constraints, and graphs of 3D spatial or biological interactions. I will discuss unique challenges for learning from multimodal datasets that are due to varying inductive biases across modalities and the potential absence of explicit graphs in the input. I will describe a framework for structure-inducing pretraining that allows for a comprehensive study of how relational structure can be induced in pretrained language models. We use the framework to explore new graph pretraining objectives that impose relational structure in the induced latent spaces—i.e., pretraining objectives that explicitly impose structural constraints on the distance or geometry of pretrained models. Applications in genomic medicine and therapeutic science will be discussed. These include TxGNN, an AI model enabling zero-shot prediction of therapeutic use across over 17,000 diseases, and PINNACLE, a contextual graph AI model dynamically adjusting its outputs to contexts in which it operates. PINNACLE enhances 3D protein structure representations and predicts the effects of drugs at single-cell resolution.
https://youtu.be/hjt4nsN_8iM

2:30–2:45 PM
Break

2:45–3:45 PM
Jianqing Fan (Princeton)
Title: UTOPIA: Universally Trainable Optimal Prediction Intervals Aggregation
Abstract: Uncertainty quantification for prediction is an intriguing problem with significant applications in various fields, such as biomedical science, economic studies, and weather forecasts. Numerous methods are available for constructing prediction intervals, such as quantile regression and conformal predictions, among others. Nevertheless, model misspecification (especially in high-dimension) or sub-optimal constructions can frequently result in biased or unnecessarily-wide prediction intervals. In this work, we propose a novel and widely applicable technique for aggregating multiple prediction intervals to minimize the average width of the prediction band along with coverage guarantee, called Universally Trainable Optimal Predictive Intervals Aggregation (UTOPIA). The method also allows us to directly construct predictive bands based on elementary basis functions. Our approach is based on linear or convex programming which is easy to implement. All of our proposed methodologies are supported by theoretical guarantees on the coverage probability and optimal average length, which are detailed in this paper. The effectiveness of our approach is convincingly demonstrated by applying it to synthetic data and two real datasets on finance and macroeconomics. (Joint work Jiawei Ge and Debarghya Mukherjee).
https://youtu.be/WY6dr1oEOrk

3:45–4:00 PM
Break

4:00–5:00 PM
Melissa Dell (Harvard)
Title: Efficient OCR for Building a Diverse Digital History
Abstract: Many users consult digital archives daily, but the information they can access is unrepresentative of the diversity of documentary history. The sequence-to-sequence architecture typically used for optical character recognition (OCR) – which jointly learns a vision and language model – is poorly extensible to low-resource document collections, as learning a language-vision model requires extensive labeled sequences and compute. This study models OCR as a character-level image retrieval problem, using a contrastively trained vision encoder. Because the model only learns characters’ visual features, it is more sample-efficient and extensible than existing architectures, enabling accurate OCR in settings where existing solutions fail. Crucially, it opens new avenues for community engagement in making digital history more representative of documentary history.
https://youtu.be/u0JY9vURUAs

Information about the 2022 Big Data Conference can be found here.

CMSA/Math Fall Gathering
2023-09-22 16:00 - 18:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Friday, Sep 22, 2023

4:00 pm

All CMSA and Math affiliates are invited.

Mathematics in Science: Perspectives and Prospects
2023-10-27 - 2023-10-28 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Mathematics in Science: Perspectives and Prospects
A showcase of mathematics in interaction with physics, computer science, biology, and beyond.
October 27–28, 2023
Location: Harvard University Science Center Hall D & via Zoom.
Directions and Recommended Lodging
Mathematics in Science: Perspectives and Prospects Youtube Playlist

Speakers

Nima Arkani-Hamed (IAS)
Constantinos Daskalakis (MIT)
Alison Etheridge (Oxford)
Mike Freedman (Harvard CMSA)
Greg Moore (Rutgers)
Bernd Sturmfels (MPI Leipzig)

Organizers

Michael R. Douglas (Harvard CMSA)
Dan Freed (Harvard Math & CMSA)
Mike Hopkins (Harvard Math)
Cumrun Vafa (Harvard Physics)
Horng-Tzer Yau (Harvard Math)

Schedule
Friday, October 27, 2023

2:00–3:15 pm

Greg Moore (Rutgers)
Title: Remarks on Physical Mathematics
Abstract: I will describe some examples of the vigorous modern dialogue between mathematics and theoretical physics (especially high energy and condensed matter physics). I will begin by recalling Stokes’ phenomenon and explain how it is related to some notable developments in quantum field theory from the past 30 years. Time permitting, I might also say something about the dialogue between mathematicians working on the differential topology of four-manifolds and physicists working on supersymmetric quantum field theories. But I haven’t finished writing the talk yet, so I don’t know how it will end any more than you do.
Slides (PDF)

3:15–3:45 pm
Break

3:45–5:00 pm

Bernd Sturmfels (MPI Leipzig)
Title: Algebraic Varieties in Quantum Chemistry
Abstract: We discuss the algebraic geometry behind coupled cluster (CC) theory of quantum many-body systems. The high-dimensional eigenvalue problems that encode the electronic Schroedinger equation are approximated by a hierarchy of polynomial systems at various levels of truncation. The exponential parametrization of the eigenstates gives rise to truncation varieties. These generalize Grassmannians in their Pluecker embedding. We explain how to derive Hamiltonians, we offer a detailed study of truncation varieties and their CC degrees, and we present the state of the art in solving the CC equations. This is joint work with Fabian Faulstich and Svala Sverrisdóttir.
Slides (PDF)

Saturday, October 28, 2023

9:00 am
Breakfast

9:30–10:45 am

Mike Freedman (Harvard CMSA)
Title: ML, QML, and Dynamics: What mathematics can help us understand and advance machine learning?
Abstract: Vannila deep neural nets DNN repeatedly stretch and fold. They are reminiscent of the logistic map and the Smale horseshoe. What kind of dynamics is responsible for their expressivity and trainability. Is chaos playing a role? Is the Kolmogorov Arnold representation theorem relevant? Large language models are full of linear maps. Might we look for emergent tensor structures in these highly trained maps in analogy with emergent tensor structures at local minima of certain loss functions in high-energy physics.
Slides (PDF)

10:45–11:15 am
Break

11:15 am–12:30 pmvia Zoom

Nima Arkani-Hamed (IAS)
Title: All-Loop Scattering as A Counting Problem
Abstract: I will describe a new understanding of scattering amplitudes based on fundamentally combinatorial ideas in the kinematic space of the scattering data. I first discuss a toy model, the simplest theory of colored scalar particles with cubic interactions, at all loop orders and to all orders in the topological ‘t Hooft expansion. I will present a novel formula for loop-integrated amplitudes, with no trace of the conventional sum over Feynman diagrams, but instead determined by a beautifully simple counting problem attached to any order of the topological expansion. A surprisingly simple shift of kinematic variables converts this apparent toy model into the realistic physics of pions and Yang-Mills theory. These results represent a significant step forward in the decade-long quest to formulate the fundamental physics of the real world in a new language, where the rules of spacetime and quantum mechanics, as reflected in the principles of locality and unitarity, are seen to emerge from deeper mathematical structures.

12:30–2:00 pm
Lunch break

2:00–3:15 pm

Constantinos Daskalakis (MIT)
Title: How to train deep neural nets to think strategically
Abstract: Many outstanding challenges in Deep Learning lie at its interface with Game Theory: from playing difficult games like Go to robustifying classifiers against adversarial attacks, training deep generative models, and training DNN-based models to interact with each other and with humans. In these applications, the utilities that the agents aim to optimize are non-concave in the parameters of the underlying DNNs; as a result, Nash equilibria fail to exist, and standard equilibrium analysis is inapplicable. So how can one train DNNs to be strategic? What is even the goal of the training? We shed light on these challenges through a combination of learning-theoretic, complexity-theoretic, game-theoretic and topological techniques, presenting obstacles and opportunities for Deep Learning and Game Theory going forward.
Slides (PDF)

3:15–3:45 pm
Break

3:45–5:00 pm

Alison Etheridge (Oxford)
Title: Modelling hybrid zones
Abstract: Mathematical models play a fundamental role in theoretical population genetics and, in turn, population genetics provides a wealth of mathematical challenges. In this lecture we investigate the interplay between a particular (ubiquitous) form of natural selection, spatial structure, and, if time permits, so-called genetic drift. A simple mathematical caricature will uncover the importance of the shape of the domain inhabited by a species for the effectiveness of natural selection.
Slides (PDF)

Limited funding to help defray travel expenses is available for graduate students and recent PhDs. If you are a graduate student or postdoc and would like to apply for support, please register above and send an email to mathsci2023@cmsa.fas.harvard.edu no later than October 9, 2023.
Please include your name, address, current status, university affiliation, citizenship, and area of study. F1 visa holders are eligible to apply for support. If you are a graduate student, please send a brief letter of recommendation from a faculty member to explain the relevance of the conference to your studies or research. If you are a postdoc, please include a copy of your CV.

CMSA/Tsinghua Math-Science Literature Lecture: Scott Kominers
2023-11-20 09:00 - 10:30 | Virtual
CMSA/Tsinghua Math-Science Literature Lecture

Prof. Scott Kominers will present a lecture in the CMSA/Tsinghua Math-Science Literature Lecture Series.
Date: Monday, November 20, 2023
Time: 9:00 – 10:30 am ET
Location: Via Zoom Webinar
Title: 60 Years of Matching: From Gale and Shapley to Trading Networks
Abstract: Gale and Shapley’s 1962 American Mathematical Monthly paper, “College Admissions and the Stability of Marriage,” is by now one of the most cited articles in the journal’s history, having served as the foundation for an entire branch of the field of market design. This success owes in large part to the beautiful, applicable, and surprisingly general theory of matching mechanisms uncovered in Gale and Shapley’s work. This talk traces the history and evolution of matching theory from that paper forward to the present day, along the way touching on real-world applications to everything from medical residency matching to electricity markets.
Moderator: Sergiy Verstyuk

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

CMSA COVID-19 Policies

CMSA/MATH Bi-Annual Gathering
2024-01-26 16:00 - 18:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
On Friday, Jan. 26, 2024 the CMSA will host the CMSA/MATH Bi-Annual Gathering for Harvard CMSA and Math affiliates in the CMSA Common Room at 20 Garden Street, Cambridge MA 02138.

Arithmetic Quantum Field Theory Program
2024-02-05 09:00 - 2024-03-29 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Arithmetic Quantum Field Theory Program
Dates: Feb. 5–Mar. 29, 2024
Location: Harvard CMSA, 20 Garden Street, Cambridge MA 02138
Arithmetic Quantum Field Theory Program Youtube Playlist
Organizers:

David Ben-Zvi (University of Texas Austin)
Solomon Friedberg (Boston College)
Natalie Paquette (University of Washington Seattle)
Brian Williams (Boston University)

This program features a weekly seminar series, workshops, and a conference.
The object of the program is to develop and disseminate exciting new connections emerging between quantum field theory and algebraic number theory, and in particular between the fundamental invariants of each: partition functions and L-functions.
On one hand, there has been tremendous progress in the past decade in our understanding of the algebraic structures underlying quantum field theory as expressed in terms of the geometry and topology of low-dimensional manifolds, both on the level of states (via the Atiyah-Segal / Baez-Dolan / Lurie formalism of extended, functorial field theory) and on the level of observables (via the Beilinson–Drinfeld / Costello–Gwilliam formalism of factorization algebras). On the other hand, Weil’s Rosetta Stone and the Mazur–Morishita–Kapranov–Reznikov arithmetic topology (the “knots and primes” dictionary) provide a sturdy bridge between the topology of 2- and 3-manifolds and the arithmetic of number fields. Thus, one can now port over quantum field theoretic ideas to number theory, as first proposed by Minhyong Kim with his arithmetic counterpart of Chern-Simons theory. Most recently, the work of Ben-Zvi–Sakellaridis–Venkatesh applies an understanding of the Langlands program as an arithmetic avatar of electric-magnetic duality in four-dimensional gauge theory to reveal a hidden quantum mechanical nature of the theory of $L$-functions.
The program will bring together a wide range of mathematicians and physicists working on adjacent areas to explore the emerging notion of arithmetic quantum field theory as a tool to bring quantum physics to bear on questions of interest for the theory of automorphic forms, harmonic analysis and L-functions. Conversely, we will explore potential geometric and physical consequences of arithmetic ideas, for example, the Langlands correspondence theory of L-functions for 3-manifolds.

Schedule
The first week of the program will feature several lecture series aimed at a broad local community of mathematicians and physicists, aiming to introduce the main ideas underlying our program and help establish a common reference point.
The program will host a weekly seminar series on Fridays.
The speakers will be selected with the aim of covering a wide panorama of the subjects over the course of the program.
The program will conclude with a week-long Conference on Arithmetic Quantum Field Theory March 25–29, 2024.

AQFT Youtube Playlist
Lecture series
All lectures take place in Room G10, Harvard CMSA, 20 Garden Street Cambridge.
Week 1: Feb. 5–9, 2024
Abstract: In this lecture series we will introduce some of the themes underlying the CMSA program on Arithmetic Quantum Field Theory taking place this winter and the upcoming conference March 25-29, 2024.
Some of the themes we plan to discuss include:
Structures in QFT (like factorization for observables and functorial QFT for states and their relation to geometric / deformation quantization) that are sufficiently algebraic and formal to allow for arithmetic analogs.
The setup of arithmetic topology as a bridge between the background of QFT to that of arithmetic (both “global” and “local”), including the “middle realm” of positive characteristic function fields.
Questions and structures in arithmetic that have been / might be amenable to inspiration from QFT, in particular the theory of L-functions and the Langlands program.
Schedule

Monday, Feb. 5, 2024

11:00 am – 12:00 pm
Minhyong Kim
Arithmetic topology and field theory
Video
(Slides part 1 pdf)

1:30 – 2:30 pm
Brian Williams
Algebraic quantum field theory
Video
(Lecture Notes)

2:30 – 3:30 pm
David Ben-Zvi
The Langlands program via arithmetic QFT
Video

Wednesday, Feb. 7, 2024

11:00 am – 12:00 pm
Minhyong Kim
Arithmetic topology and field theory
Video
(Slides part 2 pdf)

2:30 – 3:30 pm
Brian Williams
Algebraic quantum field theory
Video
(Lecture Notes)

Thursday, Feb.8, 2024

2:30 – 3:30 pm
Minhyong Kim
Arithmetic topology and field theory
Video
(Slides part 3 pdf)

4:00 – 5:00 pm
David Ben-Zvi
The Langlands program via arithmetic QFT
Video

Friday, Feb. 9, 2024

1:00 – 2:00 pm
Brian Williams
Algebraic quantum field theory
Video
(Lecture Notes)

2:00 – 3:00 pm
David Ben-Zvi
The Langlands program via arithmetic QFT 1
Video

3:30 – 4:30 pm
David Ben-Zvi
The Langlands program via arithmetic QFT 2
Video

Monday, Feb. 26, 2024

1:00 – 2:00 pm
Omer Offen (Brandeis)
Period integrals of automorphic forms and the residue method
Video

Tuesday, Feb. 27, 2024

2:00 – 3:00 pm
Wei Zhang (MIT)
Shtuka special cycles and their generating series
Video

Friday, March 1, 2024

11:00 am – 12:00 pm
Chen Wan (Rutgers Newark)
Some examples of the relative Langlands duality
Video

2:00 – 3:00 pm
Peng Shan (Tsinghua)
Skein algebras and quantized Coulomb branches
Video

Thursday, March 7, 2024

1:30 – 2:30 pm
An Huang (Brandeis)
Tate’s thesis and p-adic strings
Video

3:00 – 4:00 pm
John Francis (Northwestern)
Integrating braided categories over 3-manifolds
Video

Friday, March 8, 2024

1:00 – 2:00 pm
Dihua Jiang (U Minnesota)
Shalika Periods: Functoriality and Arithmetic
Video

Friday, March 15, 2024

11:45 – 1:00 pm
Baiying Liu (Purdue)
Recent progress on certain problems related to local Arthur packets of classical groups
Video

2:15 – 3:30 pm
Tasho Kaletha (Michigan)
Covers of reductive groups and functoriality
Video

Monday, March 18, 2024

1:00 – 3:00 pm
Xinwen Zhu (Stanford)
The tame categorical local Langlands correspondence
Video

4:30 – 5:30 pm
Natalie Paquette (U Washington)
Koszul duality & twisted holography for asymptotically flat spacetimes

Wednesday, March 20, 2024

11:00 – 12:15 pm
Stephen D. Miller (Rutgers)
What 4-graviton scattering amplitudes had to say about the unitary dual

Friday, March 22, 2024

1:45 – 3:00 pm
Jayce Getz (Duke)
The Poisson summation conjecture and the fiber bundle method
Video

Program Visitors

Mina Aganagic, University of California, Berkeley
Anne-Marie Aubert, Institut de Mathématiques de Jussieu-Paris Rive Gauche, March 15-29
Clark Barwick, University of Edinburgh, February 19-March 15
Alexander Braverman, Perimeter Institute
Alejandra Castro, Cambridge University, March 25-29
YoungJu Choie, Pohang University of Science and Technology, February 12-16; March 22-28
John Francis, Northwestern University, March 1-14
David Gaiotto, Perimeter Institute, March 25-29
Jayce Getz, Duke University, March 18-22
Ezra Getzler, Northwestern University, March 11-22
Sam Gunningham, Montana State University, February 9-12
Sarah Harrison, Northeastern University
Dihua Jiang, University of Minnesota, February 29-March 9
Tasho Kaletha, University of Michigan, March 12-20
Minhyong Kim, University of Edinburgh, February 1-29
Axel Kleinschmidt, Max Planck Institute for Gravitational Physics, Potsdam, March 18-28
Kim Klinger-Logan, Kansas State University, March 25-29
Kobi Kremnitzer, Oxford University, March 25-29

Baiying Liu, Purdue University, March 13-16

Steven Miller, Rutgers University

Greg Moore, Rutgers University, February 5-9
David Nadler, University of California, Berkeley, March 17-30
Bảo Châu Ngô, University of Chicago, March 25-29
George Pappas, Michigan State University, March 25-29
Daniel Persson, Chalmers Institute of Technology, March 25-29
Sam Raskin, Yale University, March 26-29
Yiannis Sakellaridis, Johns Hopkins University, March 18-22
Peng Shan, Tsinghua University, February 12-April 14
Akshay Venkatesh, Institute for Advanced Study
Roberto Volpato, University of Padova, February 4-10
Chen Wan, Rutgers University, February 29-March 9
Fei Yan, Brookhaven National Laboratory, March 18-29
Xinwen Zhu, Stanford University

CMSA/Tsinghua Math-Science Literature Lecture: Amie Wilkinson
2024-02-07 09:00 - 10:30 | Virtual
CMSA/Tsinghua Math-Science Literature Lecture
Prof. Amie Wilkinson gave a lecture in the CMSA/Tsinghua Math-Science Literature Lecture Series.
Date: Wednesday, February 7, 2024
Time: 9:00–10:30 am ET
Title: Stretching and shrinking: 85 years of the Hopf argument for ergodicity
Abstract: The early 20th century witnessed an explosion of activity, much of it centered at Harvard, on rigorizing the property of ergodicity first proposed by Boltzmann in his 1898 Ergodic Hypothesis for ideal gases. Earlier, in the 1880’s, Henri Poincaré and Felix Klein had also initiated a study of discrete groups of hyperbolic isometries. The geodesics in hyperbolic manifolds were discovered to carry a rich structure, first investigated from a topological perspective by Emil Artin and Marston Morse. The time was ripe to investigate geodesics in hyperbolic manifolds from an ergodic theoretic (i.e., statistical) perspective, and indeed Gustav Hedlund proved in 1934 that the geodesic flow for closed hyperbolic surfaces is ergodic.

In 1939, Eberhard Hopf published a proof of the ergodicity of geodesic flows for negatively curved surfaces containing a novel method, now known as the Hopf argument. The Hopf argument, a “soft” argument for ergodicity of systems with some hyperbolicity (the “stretching and shrinking” in the title) has since seen wide application in geometry, representation theory and dynamics. I will discuss three results relying on the Hopf argument:

Theorem (E. Hopf, 1939, D. Anosov, 1967): In a closed manifold of negative sectional curvatures, almost every geodesic is directionally equidistributed.

Theorem (G. Mostow, 1968) Let M and N be closed hyperbolic manifolds of dimension at least 3, and let f:M->N be a homotopy equivalence. Then f is homotopic to a unique isometry.

Theorem (R. Mañé, 1983, A. Avila- S. Crovisier- A.W., 2022) The C^1 generic symplectomorphism of a closed symplectic manifold with positive entropy is ergodic.

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

Math Science Lectures in Honor of Raoul Bott: Maggie Miller: Fibered ribbon knots vs. major 4D conjectures
2024-02-20 16:00 - 17:30 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Fibered ribbon knots vs. major 4D conjectures
Location: Harvard University Science Center Hall A & via Zoom webinar
Dates: Feb 20 & 22, 2024
Time: 4:00-5:30 pm
Maggie Miller is an assistant professor in the mathematics department at the University of Texas at Austin and a Clay Research Fellow.
This is the fourth annual Math Science Lecture Series held in Honor of Raoul Bott.
Talk topic: Fibered ribbon knots vs. major 4D conjectures

Feb. 20, 2024
Title: Fibered ribbon knots and the Poincaré conjecture
Abstract: A knot is “fibered” if its complement in S^3 is the total space of a bundle over the circle, and ribbon if it bounds a smooth disk into B^4 with no local maxima with respect to radial height. A theorem of Casson-Gordon from 1983 implies that if a fibered ribbon knot does not bound any fibered disk in B^4, then the smooth 4D Poincaré conjecture is false. I’ll show that unfortunately (?) many ribbon disks bounded by fibered knots are fibered, giving some criteria for extending fibrations and discuss how one might search for non-fibered examples.

Feb. 22, 2024
Title: Fibered knots and the slice-ribbon conjecture
Abstract: The slice-ribbon conjecture (Fox, 1962) posits that if a knot bounds any smooth disk into B^4, it also bounds a ribbon disk. The previously discussed work of Casson-Gordon yields an obstruction to many fibered knots being ribbon, yielding many interesting potential counterexamples to this conjecture — if any happy to bound a non-ribbon disk. In 2022, Dai-Kong-Mallick-Park-Stoffregen showed that unfortunately( ?) many of these knots don’t bound a smooth disk into B^4 and thus can’t disprove the conjecture. I’ll show a simple alternate proof that a certain interesting knot (the (2,1)-cable of the figure eight) isn’t slice and discuss remaining open questions. This talk is joint with Paolo Aceto, Nickolas Castro, JungHwan Park, and Andras Stipsicz.

Talk Chair: Cliff Taubes (Harvard Mathematics)
Moderator: Freid Tong (Harvard CMSA)

Raoul Bott (9/24/1923 – 12/20/2005) is known for the Bott periodicity theorem, the Morse–Bott functions, and the Borel–Bott–Weil theorem.

Fourth Annual Yip Lecture | Josh Tenenbaum | How to grow a mind from a brain: From guessing and betting to thinking and talking
2024-02-29 16:00 - 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Josh Tenenbaum gave the Fourth Annual Yip Lecture on February 29, 2024.
Title: How to grow a mind from a brain: From guessing and betting to thinking and talking
Time: 4:00-5:00 pm ET
Location: Harvard Science Center
The Yip Lecture takes place thanks to the support of Dr. Shing-Yiu Yip.

The previous Yip Lecture featured Andrew Strominger (Harvard), who spoke on Black Holes.

CMSA/Tsinghua Math-Science Literature Lecture: Cameron Gordon
2024-03-20 09:00 - 10:30 | Hybrid
CMSA/Tsinghua Math-Science Literature Lecture
Prof. Cameron Gordon presented a lecture in the CMSA/Tsinghua Math-Science Literature Lecture Series.

Date: Wednesday, March 20, 2024
Time: 9:00–10:30 am ET
Location: Room G10, CMSA, 20 Garden Street, Cambridge MA and via Zoom Webinar
Title: The Unknotting Number of a Knot
Abstract: One of the oldest and most natural knot invariants is the unknotting number, which is the minimum number of times a knot must be allowed to pass through itself in order to unknot it. Although this invariant was discussed by Tait almost 150 years ago, it is still poorly understood. For instance it is not known if it is algorithmically computable, and indeed there is an 8-crossing knot whose unknotting number is unknown. Nevertheless, the many developments in knot theory since Tait have led to some understanding of unknotting number, for example through its connection with 4-dimensional topology. We will give a historical account of this progress, and discuss some of the questions that are still open.

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

CMSA COVID-19 Policies

Arithmetic Quantum Field Theory Conference
2024-03-25 09:00 - 2024-03-29 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Arithmetic Quantum Field Theory Conference
Dates: March 25-29, 2024
Location: Room G10, Harvard CMSA, 20 Garden Street, Cambridge MA 02138
Arithmetic Quantum Field Theory Conference Youtube Playlist
Organizers:

David Ben-Zvi (University of Texas Austin)
Solomon Friedberg (Boston College)
Natalie Paquette (University of Washington Seattle)
Brian Williams (Boston University)

Scientific Goals: On one hand, there has been tremendous progress in the past decade in our understanding of the algebraic structures underlying quantum field theory as expressed in terms of the geometry and topology of low-dimensional manifolds, both on the level of states (via the formalism of extended, functorial field theory) and on the level of observables (via the formalism of factorization algebras). On the other hand, the arithmetic topology (or “knots and primes”) dictionary provides a sturdy bridge between the topology of 2- and 3-manifolds and the arithmetic of number fields. Thus, one can now port over quantum field theoretic ideas to number theory, as first proposed by Minhyong Kim with his arithmetic counterpart of Chern-Simons theory. Moreover, automorphic objects appear in string theory where they play a role in the study of graviton scattering. Most recently, the work of Kapustin-Witten has been extended towards an understanding of the Langlands program as an arithmetic avatar of electric-magnetic duality in four-dimensional gauge theory to reveal a hidden quantum mechanical nature of the theory of L-functions.
The conference will bring together a wide range of mathematicians and physicists working on adjacent areas to explore the emerging notion of arithmetic quantum field theory as a tool to bring quantum physics to bear on questions of interest for the theory of automorphic forms, representation theory, harmonic analysis and L-functions. Conversely, we will explore potential geometric and physical consequences of arithmetic ideas. Our program will also build on the significant interactions between number theorists and physicists arising from the frequent appearance of modular and automorphic forms in partition functions, scattering amplitudes, and other quantities of interest in quantum field theory and quantum gravity.
Monday, March 25: Connections for Women in Mathematics and Physics
Speakers

Charlotte Chan (U Michigan)
Kim Klinger-Logan (Kansas State)
Sarah Harrison (Northeastern)
Melanie Matchett Wood (Harvard)
Fei Yan (Brookhaven National Lab)

Tuesday, March 26–Friday, March 29: Arithmetic Quantum Field Theory
Speakers

Anne-Marie Aubert (IMJ-PRG)
Roman Bezrukavnikov (MIT)
Sasha Braverman (Toronto / Perimeter)
Alejandra Castro (Cambridge)
YoungJu Choie (POSTECH)
Pavel Etingof (MIT)
Davide Gaiotto (Perimeter)
Axel Kleinschmidt (Max Planck Institute for Gravitational Physics)
Kobi Kremnitzer (Oxford)
Spencer Leslie (Boston College)
David Nadler (Berkeley)
Bảo Châu Ngô (U Chicago)
George Pappas (Michigan State)
Sam Raskin (Yale)
Peng Shan (Tsinghua)
Zhiwei Yun (MIT)

Conference Schedule
Arithmetic Quantum Field Theory Conference
March 25–29, 2024
Download Program (pdf)

Monday, March 25, 2024 – Women in Math and Physics

8:30 – 9:00 am

Breakfast

9:00 – 10:00 am

Melanie Matchett Wood (Harvard)
Title: Statistics of Number fields, function fields, and 3-manifolds
Abstract: Motivated by conjectures of Cohen, Lenstra, and Martinet on the distribution of class groups of number fields, we describe the analogous questions of understanding distributions of class groups and fundamental groups of curves over finite fields, and the distribution of fundamental groups of 3-manifolds. We describe results on these distributions in the cases of curves over finite fields and 3-manifolds, joint with Liu, Zureick-Brown, and Sawin, and discuss how ideas have passed back and forth between the number field, curves over finite fields, and 3-manifold settings.

10:00 – 10:20 am

Coffee break

10:20 – 11:20 am

Charlotte Chan (U Michigan)
Title: Generic character sheaves on parahoric subgroups
Abstract: Lusztig’s theory of character sheaves for connected reductive groups is one of the most important developments in representation theory in the last few decades. I will give an overview of this theory and explain the need, from the perspective of the representation theory of p-adic groups, of a theory of character sheaves on jet schemes. Recently, R. Bezrukavnikov and I have developed the “generic” part of this desired theory. In the simplest nontrivial case, this resolves a conjecture of Lusztig and produces perverse sheaves on jet schemes compatible with parahoric Deligne–Lusztig induction. This talk is intended to describe in broad strokes what we know about these generic character sheaves, especially within the context of the Langlands program.

11:30 – 12:30 pm

Kim Klinger–Logan (Kansas State)
Title: Connections between special values of L-functions and scattering amplitudes
Abstract: In this talk we will attempt make a connection between zeros and special values of L-functions and scattering amplitudes. The connection is best seen through solutions to differential equations of the form $(Delta-lambda)f = S$ on $X=SL(2,Z)SL(2,R)/SO(2,R)$ for $Delta=y^2(partial_x^2+partial_y^2)$ and $H^{-infty}(X)cup M$ where $M$ is the space of moderate growth functions. Recently, Bombieri and Garrett (following work of Hass, Hejhal, and Colin de Verdiere) laid out the possibly connection with eigenvalue solutions to equations of this form with zeros of L-functions. On the other hand, physicists such as Green, Russo, Vanhove found that eigenfunction solutions to equations of this form give coefficients of the 4-graviton scattering amplitude. We will elaborate on these connections and discuss some recent work on finding solutions for such equations. This work is in collaboration with Ksenia Fedosova, Stephen D. Miller, Danylo Radchenko and Don Zagier.
Slides (pdf)

12:30 – 2:15 pm

Lunch

2:15 – 3:15 pm

Fei Yan (Brookhaven National Lab)
Title: Topological defects on the lattice
Abstract: Topological defects, endowed with a rich mathematical structure, play important roles in condensed matter physics, high energy theory and quantum information science. Realization of such defects on the lattice not only has interesting theoretical consequences, but also opens the pathway to quantum simulation of physical systems. In this talk, I will discuss lattice realizations of topological defects in simple (1+1)-d systems, taking the transverse field Ising model and the three-state Potts model as examples. Time permitting, I will also briefly comment on topological defects in non-equilibrium systems, such as periodically-driven Floquet systems.

3:15 – 3:30 pm

Coffee break

3:30 – 4:30 pm

Sarah Harrison (Northeastern)
Title: Liouville Theory and Weil-Petersson Geometry
Abstract: Two-dimensional conformal field theory is a powerful tool to understand the geometry of surfaces. Liouville conformal field theory in the classical (large central charge) limit encodes the geometry of the moduli space of Riemann surfaces. I describe an efficient algorithm to compute the Weil–Petersson metric to arbitrary accuracy using Zamolodchikov’s recursion relation for conformal blocks, focusing on examples of a sphere with four punctures and generalizations to other one-complex-dimensional moduli spaces. Comparison with analytic results for volumes and geodesic lengths finds excellent agreement. In the case of M_{0,4}, I discuss numerical results for eigenvalues of the Weil-Petersson Laplacian and connections with random matrix theory. Based on work with K. Coleville, A. Maloney, K. Namjou, and T. Numasawa.
Slides (pdf)

Tuesday, March 26, 2024

9:00 – 9:30 am

Breakfast

9:30 – 10:30 am

Roman Bezrukavnikov (MIT)
Title: From affine Hecke category to invariant distributions
Abstract: By a result of Ben-Zvi, Nadler and Preygel the cocenter of the affine Hecke category can be identified with coherent sheaves on the appropriate stack of commuting pairs in the Langlands dual group. In a joint work (in progress) with Ciubotaru, Kazhdan and Varshavsky we recover the space of unipotent invariant distributions on the p-adic group from that category and develop applications to harmonic analysis, including endoscopic property of unipotent L-packets. Time permitting, I will explain how a part of this result can be recovered from a geometric realization of Lusztig’s asymptotic affine Hecke algebra J (joint with Karpov and Krylov), and present a conjecture generalizing the story to other depth zero representations; another special case of this generalization appears in a joint work with Varshavsky.

10:30 – 11:00am

Coffee break

11:00 – 12:00 pm

Sasha Braverman (Toronto/Perimeter)
Title: Hecke operators for algebraic curves over local non-archimedian fields: a survey of some recent results
Abstract: The main goal of this talk is to discuss Hecke operators and Hecke eigen-functions for the moduli space of G-bundles on a smooth projective algebraic curve X defined over a local non-archimedian field K (possibly with level structures at finitely many points). The plan is to discuss the following subjects: 1) Definition of Hecke operators and the space on which they act 2) Relation to “classical story” – i.e. eigen-functions of Hecke operators for curves over a finite field. 3) Detailed discussion of the examples when X has genus zero and we consider bundles with trivialization at two points. In this case we’ll discuss the relation to classical representation theory of p-adic groups and two representation theory of Cherednik algebras. Based on joint works with P. Etingof, D.Kazhdan, and A. Polishchuk.

12:00 – 12:15 pm

Group photo.

12:15 – 1:30 pm

Lunch

1:30 – 2:30 pm

Peng Shan (Tsinghua)
Title: Modularity for W-algebras, affine Springer fibres and associated variety
Abstract: I will explain a bijection between admissible representations of affine Kac-Moody algebras and fixed points in affine Springer fibres. I will also explain how to match the modular group action on the characters of representations with the one defined by Cherednik in terms of double affine Hecke algebras, and extensions of these relations to representations of W-algebras. If time permits, I will discuss some extension of these results to non-admissible levels and some conjectures about their associated varieties. This is based on joint work with Dan Xie, Wenbin Yan, and Qixian Zhao.

2:30 – 3:00 pm

Coffee break

3:00 – 4:00 pm

Bảo Châu Ngô (U Chicago)
Title: On the nonabelian Fourier kernel and the Lafforgue transform
Abstract: In the case of SL2, we present an analytic formula for the nonabelian Fourier kernel responsible for the functional equation of automorphic L-functions. We use the Gelfand-Graev formula for Langlands’ stable transfer factor and a linear map between the Bernstein center and the cocenter that we call the Lafforgue transform. This is a joint work with Zhilin Luo.

Wednesday, March 27, 2024

9:00 – 9:30 am

Breakfast

9:30 – 10:30 am

YoungJu Choie (POSTECH)
Title: Schubert Eisenstein series and Poisson summation for Schubert varieties
Abstract: Schubert Eisenstein series by restricting the summation in a degenerate Eisenstein series to a particular Schubert variety has been studied. In the case of GL3 over Q it was proved that these Schubert Eisenstein series have meromorphic continuations in all parameters and conjectured the same is true in general. We revisit the conjecture and relate it to the program of Braverman, Kazhdan, Lafforgue, Ngˆo, and Sakellaridis aimed at establishing generalizations of the Poisson summation formula. This is a joint work with Jayce Getz.
Slides (pdf)

10:30 – 11:00 am

Coffee break

11:00 – 12:00 pm

Axel Kleinschmidt (MPI)
Title: Automorphic representations in string amplitudes
Abstract: I will review how automorphic representations arise in the low-energy expansion of string scattering amplitudes, highlighting the connection found by Green/Miller/Vanhove between wavefront sets and BPS conditions. To study the wavefront sets I will present reduction principles for the calculation of Fourier coefficients. String theory also predicts new types of automorphic objects that are characterised by lacking finiteness under the center of the universal enveloping algebra.
Slides (pdf)

12:00 – 1:30 pm

Lunch

1:30 – 2:30 pm

Pavel Etingof (MIT)
Title: Analytic Langlands correspondence over C and R
Abstract: I will review the analytic component of the geometric Langlands correspondence, developed recently in my joint work with E. Frenkel and D. Kazhdan (based on previous works by other authors), with a special focus on archimedian local fields, especially R. This is based on our work with E. Frenkel and D. Kazhdan and insights shared by D. Gaiotto and E. Witten.
Slides (pdf)

2:30 – 3:00 pm

Coffee break

3:00 – 4:00 pm

Davide Gaiotto (Perimeter)
Title: Unexpected Unitarity
Abstract: Much of the mathematical content of Supersymmetric Quantum Field Theories can be extracted through “twisted theories”: simplified QFTs which are topological (or holomorphic) in a derived sense and often amenable of a rigorous mathematical treatment. The twisting procedure destroys or obfuscates certain properties of the underlying SQFTs, including unitarity. I will discuss a variety of situations where some form of unitarity can be restored, endowing the twisted theories with unexpected structures. This includes the recently developed Analytic Langlands program, an analytic version of Symplectic Duality, an A-model description of quantization (as opposed to deformation quantization) and other constructions of Hodge-theoretic or twistorial flavour.

Thursday, March 28, 2024

8:30 – 9:00 am

Breakfast

9:00 – 10:00 am

Spencer Leslie (Boston College)
Title: Relative Langlands and endoscopy
Abstract: Spherical varieties play an important role in the study of periods of automorphic forms. But very closely related varieties can lead to very distinct arithmetic problems. Motivated by applications to relative trace formulas, we discuss the natural question of distinguishing different forms of a given spherical variety in arithmetic settings, giving a solution for symmetric varieties. It turns out that the answer is intimately connected with the construction of the dual Hamiltonian variety associated with the symmetric variety by Ben-Zvi, Sakellaridis, and Venkatesh. I will explain the source of these questions in the theory of endoscopy for symmetric varieties, with application to the (pre)-stabilization of relative trace formulas.

10:00 – 10:30 am

Coffee break

10:30 – 11:30 am

Anne-Marie Aubert (IMJ–PRG)
Title: The Local Langlands correspondence: from extended quotients to affine Hecke algebras
Abstract: We will introduce the notion of extended quotient, illustrate it on examples, and show how it can be used to construct the local Langlands correspondence in the nonarchimedean case. Next, we will connect extended quotients, that are attached to the Bernstein decomposition of the category of smooth representations of p-adic groups, and their Langlands duals, to representations of affine Hecke algebras in order to get a description of the LLC as a correspondence between the categories of modules of two collections of such algebras.
Slides (pdf)

11:45 – 12:45 pm

Kobi Kremnitzer (Oxford)
Title: Functional analysis over the integers, L-functions and global Hodge theory
Abstract: In this talk I will explain how using bornological methods one can develop functional analysis over the integers unifying Archimedean and non-Archimedean analysis. I will give examples of algebras of functions and distributions defined over the integers which base change to the usual algebras over the reals and p-adics. Using these it is possible to write some version of L-functions over the integers. I will then introduce an analytic stack over the integers for which the category of quasi-coherent sheaves gives global Hodge structures. I will relate the integral L-functions to trivialisations of line bundles on this stack. I will also explain how to define a cohomology theory for schemes valued in global Hodge structures (possibly related to q-deRham) and speculate on the relation between the determinant of cohomology and L-functions. This is work in progress joint with Federico Bambozzi and Jack Kelly.

12:45 – 2:00 pm

Lunch

2:00 – 3:00 pm

David Nadler (Berkeley)
Title: Going to the boundary
Abstract: I’ll describe several situations where degenerating a marked smooth curve to a marked nodal curve leads to interesting structures on automorphic moduli spaces. In particular, I’ll discuss its implications for the cocenter of the affine Hecke category, real-symmetric duality in relative Langlands, and some other conjectural situations. The talk will borrow from joint work with various authors including D. Ben-Zvi, T.-H. Chen, P. Li, and Z. Yun.

3:00 – 3:30 pm

Coffee break

Friday, March 29, 2024

9:00 – 9:30 am

Breakfast

9:30 – 10:30 am

George Pappas (Michigan State)
Title: Finite and p-adic Chern-Simons type invariants
Abstract: We will define arithmetic invariants of Galois covers and of ‘etale local systems which are inspired by the classical constructions of Dijkgraaf-Witten and Chern-Simons. We will discuss various conjectures and recent results about these invariants.

10:30 – 11:00 am

Coffee break

11:00 – 12:00 pm

Sam Raskin (Yale)
Title: The geometric Langlands conjecture
Abstract: I will describe the main ideas that go into the proof of the (unramified, global) geometric Langlands conjecture. All of this work is joint with Gaitsgory and some parts are joint with Arinkin, Beraldo, Chen, Faergeman, Lin, and Rozenblyum.

12:00 – 1:30 pm

Lunch

1:30 – 2:30 pm

Alejandra Castro (Cambridge)
Title: The light we can see: Extracting black holes from weak Jacobi forms
Abstract: Modular forms play a pivotal role in the counting of black hole microstates. The underlying modular symmetry of counting formulae was key in the precise match between the Bekenstein-Hawking entropy of supersymmetric black holes and Cardy’s formula for the asymptotic growth of states. The goal of this talk is to revisit the connection between modular forms and black hole entropy, and tie it with other consistency conditions of AdS/CFT. We will focus our attention on weak Jacobi forms.
I will quantify how constraints on polar states affect the asymptotic growth of non-polar states in weak Jacobi forms. The constraints I’ll consider are sparseness conditions on the Fourier coefficients of these forms, which are necessary to interpret them as gravitational path integrals. In short, the constraints will leave an imprint on the subleading corrections to the asymptotic growth of heavy states. With this we will revisit the UV/IR connection that relates black hole microstate counting to modular forms. In particular, I’ll provide a microscopic interpretation of the logarithmic corrections to the entropy of supersymmetric black holes and tie it to consistency conditions in AdS_3/CFT_2.

2:30 – 3:00 pm

Coffee break

3:00 – 4:00 pm

Zhiwei Yun (MIT)
Title: Theta correspondence and relative Langlands
Abstract: A reductive dual pair (such as a symplectic group and an orthogonal group) acting on the tensor product of their standard representations is an example of hyperspherical varieties, and is the geometric avatar for theta correspondence. I will explain two geometric results/constructions motivated by the theta correspondence over finite fields. The first one describes how principal series representations behave under theta correspondence using Springer correspondence (joint with Jiajun Ma, Congling Qiu and Jialiang Zou). The second one is a definition of character sheaves in the setup of theta correspondence (joint with Shamgar Gurevich). I will speculate how the first result fits into relative Langlands duality.

Limited funding to help defray travel expenses is available for graduate students and recent PhDs. If you are a graduate student or postdoc and would like to apply for support, please register above and send an email to cstillman@math.harvard.eduno later than Sunday, February 25, 2024.
Please include your name, address, current status, university affiliation, citizenship, and area of study. F1 visa holders are eligible to apply for support. If you are a graduate student, please send a brief letter of recommendation from a faculty member to explain the relevance of the conference to your studies or research. If you are a postdoc, please include a copy of your CV.

This event will be co-funded by the National Science Foundation.
The conference is part of the Arithmetic Quantum Field Theory Program, Feb 4-March 29, 2024.

2024 Ding Shum Lecture: Yann LeCun: Objective-Driven AI: Towards AI systems that can learn, remember, reason, and plan
2024-03-28 16:30 - 17:30 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
LECTURE SLIDES (pdf)
On March 28, 2024, the CMSA will host the fifth annual Ding Shum Lecture, given by Yann LeCun.
Time: 4:30–5:30 pm ET
Speaker: Yann Lecun, New York University & META
Location: Harvard Science Center Hall A & via Zoom Webinar
Title: Objective-Driven AI: Towards AI systems that can learn, remember, reason, and plan

Abstract:
How could machines learn as efficiently as humans and animals?
How could machines learn how the world works and acquire common sense?
How could machines learn to reason and plan?
Current AI architectures, such as Auto-Regressive Large Language Models fall short. I will propose a modular cognitive architecture that may constitute a path towards answering these questions. The centerpiece of the architecture is a predictive world model that allows the system to predict the consequences of its actions and to plan a sequence of actions that optimize a set of objectives. The objectives include guardrails that guarantee the system’s controllability and safety. The world model employs a Hierarchical Joint Embedding Predictive Architecture (H-JEPA) trained with self-supervised learning. The JEPA learns abstract representations of the percepts that are simultaneously maximally informative and maximally predictable. The corresponding working paper is available here: https://openreview.net/forum?id=BZ5a1r-kVsf

This event is made possible by the generous funding of Ding Lei and Harry Shum.

Current Developments in Mathematics Conference 2024
2024-04-05 14:00 - 2024-04-06 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
CURRENT DEVELOPMENTS IN MATHEMATICS 2024
APRIL 5-6, 2024
HARVARD UNIVERSITY SCIENCE CENTER
LECTURE HALL C
https://www.math.harvard.edu/event/current-developments-in-mathematics-2024/

Speakers:
Daniel Cristofaro-Gardiner – University of Maryland
Samit Dasgupta – Duke University
Jiaoyang Huang – University of Pennsylvania
Daniel Litt – University of Toronto
Lisa Piccirillo – MIT/University of Texas

Download PDF for a detailed schedule of lectures and events.

Friday, April 5

Saturday, April 6

1:30 p.m. – 2:20 p.m. Part 1
2:20 p.m. – 2:30 p.m. Break
2:30 p.m. – 3:20 p.m. Part 2

Jiaoyang Huang
Random Matrix Statistics and Airy Line Ensembles

9:05 a.m. – 9:55 a.m. Part 1
9:55 a.m. – 10:05 a.m. Break
10:05 a.m. – 10:55 a.m. Part 2

Daniel Litt
Motives, mapping class groups, and monodromy

3:20 p.m. – 3:35 p.m.
Break

10:55 a.m. – 11:10 a.m.
Break

3:35 p.m. – 4:25 p.m. Part 1
4:25 p.m. – 4:35 p.m. Break
4:35 p.m. – 5:25 p.m. Part 2

Lisa Piccirillo
Exotic phenomena in dimension 4

11:10 a.m. – 12 p.m. Part 1
12 p.m. – 1:30 p.m. Lunch
1:30 p.m. – 2:20 p.m. Part 2

Samit Dasgupta
Stark’s conjectures and explicit class field theory

2:20 p.m. – 2:35 p.m.
Break

2:35 p.m. – 3:25 p.m. Part 1
3:25 p.m. – 3:35 p.m. Break
3:35 p.m. – 4:25 p.m. Part 2

Dan Cristofaro-Gardiner
Low-dimensional topology and dynamics

Organizers: David Jerison, Paul Seidel, Nike Sun (MIT); Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau, Shing-Tung Yau (Harvard).
Sponsored by the National Science Foundation, Harvard University Mathematics, and the Massachusetts Institute of Technology.
Harvard University is committed to maintaining a safe and healthy educational and work environment in which no member of the University community is, on the basis of sex, sexual orientation, or gender identity, excluded from participation in, denied the benefits of, or subjected to discrimination in any University program or activity. More information can be found here.

Program on Mathematical Aspects of Scattering Amplitudes
2024-04-15 09:00 - 2024-05-24 17:00 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Aspects of Scattering Amplitudes Program
Dates: April 15 – May 24, 2024
Location: Harvard CMSA, 20 Garden Street, Cambridge, MA 02138
This program will bring together and foster interaction between theoretical physicists and mathematicians working on various topics connected to recent developments in our understanding of scattering amplitudes in quantum field theory. The field has advanced considerably since 2019 when the CMSA hosted the program “Spacetime and Quantum Mechanics, Total Positivity and Motives.” Recent developments have primed this area for further significant advances, which will be facilitated by bringing together many of the relevant experts for intensive discussion and collaboration.
The program will feature a weekly seminar series.
Tuesday, April 16, 2024
4:15 pm
Sabrina Pasterski, Perimeter Institute
Title: Radiation in Holography
[Physics Talk]
Wednesday, April 17, 2024
4:30 pm – Cruft 309
Ana-Maria Raclariu, King’s College London
Thursday, April 18, 2024
4:15 pm
Lionel Mason, University of Oxford
Title: Hidden symmetries of SD Poincare Einstein metrics in split signature
[Physics Talk]
Tuesday, April 23, 2024
4:30 pm – Jefferson 256
Juan Maldacena, Institute for Advanced Study
Thursday, April 25, 2024
4:15 pm
Tomasz Taylor, Northeastern University
Title: Progress in Yang-Mills-Liouville Theory
[CMSA Colloquium]
Monday, April 29, 2024
4:30 – 5:30 pm
Lance Dixon, Stanford
Title: The DNA of Particle Scattering
Tuesday, April 30, 2024
9:00 am- Jefferson 453
Nima Arkani-Hamed, IAS
Title: Surfaceology and the Real World Part 1
1:00 pm – Jefferson 453
Nima Arkani-Hamed, IAS
Title: Surfaceology and the Real World Part 2
4:00 pm – Jefferson 453
Nima Arkani-Hamed, IAS
Title: Surfaceology and the Real World Part 3
Wednesday, May 1, 2024
11:00 am – Science Center 507
Jaroslav Trnka, UC Davis
Title: Loops of loops expansion in the Amplituhedron
3:00 pm
Yu-tin Huang, National Taiwan University
Title: Loop in trees: Chambers in amplitudes and correlation functions
4:00 pm
Livia Ferro, University of Hertfordshire
Title: Scattering Amplitudes from Null-cone Geometry
5:00 pm
Stephan Stieberger, Max Planck Institute
Title: One-loop Double Copy Relation in String Theory and Twisted (Co)homology
Thursday, May 2, 2024
11:00 am – Science Center 310
Daniil Rudenko, Chicago
Title: Introduction to Cluster Polylogarithms
Friday, May 3, 2024
11:00 am
Andrew McLeod, Edinburgh
Title: Genealogical Constraints on Feynman Integrals
Tuesday, May 7, 2024
3:00 pm
Jacob Bourjaily, Penn State
Title: The Algebraic and Transcendental Structure of Perturbative QFT
Wednesday, May 8, 2024
3:00 pm
Ruth Britto, Trinity
Title: Cuts and Symbols
Tuesday, May 14, 2024
3:00 pm
James Drummond, University of Southampton
Title: Multiple light-like Wilson loops in N=4 super Yang-Mills theory
Wednesday, May 15, 2024
3:00 pm
Matteo Parisi, Harvard CMSA
Title: The amplituhedron and cluster algebras
Tuesday, May 21, 2024
11:00 am
Michael Borinsky, ETH Zurich
Title: On the Euler characteristic of the commutative graph complex and the top-weight cohomology of the moduli space of curves
Wednesday, May 22, 2024
11:00 am
Chaim Even-Zohar, Technion
Title: Amplituhedron tiles and twistor polynomials

Organizers:

Nima Arkani-Hamed (Institute for Advanced Study)
Marcus Spradlin (Brown University)
Andrew Strominger (Harvard University)
Anastasia Volovich (Brown University)
Lauren Williams (Harvard University)

Participants:

Michael Borinsky, ETH Zurich
Jacob Bourjaily, Pennsylvania State University
Ruth Britto, Trinity College
Lance Dixon, Stanford Linear Accelerator Center
James Drummond, University of Southampton
Chaim Even-Zohar, Technion
Livia Ferro, University of Hertfordshire
Carolina Figueiredo, Princeton University
Hadleigh Frost, Oxford University
Bruno Gimenez, University of Southampton
Omer Gurdogan, University of Southampton
Xuhua He, Chinese University of Hong Kong
Paul Heslop, Durham University
Yu-Tin Huang, National Taiwan University
Dani Kaufman, University of Copenhagen
Jianrong Li, University of Vienna
Tomasz Lukowski, University of Hertfordshire
Yelena Mandelshtam, University of California, Berkeley
Lionel Mason, University of Oxford
Andrew McLeod, University of Edinburgh
Natalie Paquette, University of Washington
Matteo Parisi, Harvard University
Sabrina Pasterski, Perimeter Institute
Dmitri Pavlov, Max Planck Institute for Mathematics in the Sciences, Leipzig
Lizzie Pratt, University of California, Berkeley
Claudia Rella, University of Geneva
Daniil Rudenko, University of Chicago
Giulio Salvatori, Max Planck Institute for Physics
Melissa Sherman-Bennett, Massachusetts Institute of Technology
Jonah Stalknecht, University of Hertfordshire
Stephan Stieberger, Max Planck Institute
Tomasz Taylor, Northeastern University
Ran Tessler, Weizmann Institute of Science
Hugh Thomas, Université du Québec à Montréal
Jaroslav Trnka, University of California, Davis
Cristian Vergu, Pennsylvania State University

Workshop on Global Categorical Symmetries
2024-04-29 09:00 - 2024-05-03 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
The CMSA will be hosting a Workshop on Global Categorical Symmetries from April 29–May 3, 2024.
Participation in the workshop is by invitation.
The workshop will hold three Symmetry Colloquia open to the community on Thursday, May 2, 2024.
Location: Room G-10, CMSA, 20 Garden Street, Cambridge MA 02138
Time: 2:00 – 2:50 pm
Speaker: Clay Còrdova, University of Chicago
Title: Particle-Soliton Degeneracies from Spontaneously Broken Non-Invertible Symmetry
Abstract: We study non-invertible topological symmetry operators in massive quantum field theories in (1+1) dimensions. In phases where this symmetry is spontaneously broken we show that the particle spectrum often has degeneracies dictated by the non-invertible symmetry and we deduce a procedure to determine the allowed multiplets. These degeneracies are robust predictions and do not require integrability or other special features of renormalization group flows. We exhibit these conclusions in examples where the spectrum is known, recovering soliton and particle degeneracies. For instance, the Tricritical Ising model deformed by the subleading Z2 odd operator flows to a gapped phase with two degenerate vacua. This flow enjoys a Fibonacci fusion category symmetry which implies a threefold degeneracy of its particle states, relating the mass of solitons interpolating between vacua and particles supported in a single vacuum.

Location: Room G-10, CMSA, 20 Garden Street, Cambridge MA 02138
Time: 3:00 – 3:50 pm
Speaker: Thomas Dumitrescu, UCLA
Title: Symmetries, Invertible Field Theories, and Gauge Theory Phases
Abstract: I will start with a brief overview of gauge theory phases in 3+1 dimensions through the lens of higher symmetries — in particular the realization of 1-form symmetries acting on loop order parameters. I will then review recent progress in refining this characterization using invertible field theories, or equivalently symmetry protected topological phases (SPTs). This refinement leads to new results in gauge theories with fundamental matter, such as quantum chromodynamics (QCD), which do not possess 1-form symmetries. I will explain why these theories must sometimes undergo a phase transition between their confining and Higgs regimes, despite the fact that classic results and standard lore say they should be continuously connected.

Location: Room G-10, CMSA, 20 Garden Street, Cambridge MA 02138
Time: 4:30 – 5:20 pm
Speaker: Theo Johnson-Freyd, Dalhousie University and Perimeter Institute
Title: The Universal Target Category
Abstract: Hilbert’s Nullstellensatz says that the complex numbers C satisfy a universal property among all R-algebras: every not-too-large nonzero commutative R-algebra maps to C. Deligne proved a similar statement in categorical dimension 1: every not-too-large symmetric monoidal category over R maps to the category sVec of super vector spaces. In other words, sVec (and not Vec!) is “algebraically closed”. These statements help explain why quantum field theory requires imaginary numbers and fermions. I will describe the universal symmetric monoidal higher category that extends the sequence C, sVec, …. This is joint work in progress with David Reutter, and builds on closely-related work by GCS collaborators Freed, Scheimbauer, and Teleman and Schlank et al.

Organizers:
Dan Freed (Harvard CMSA & Math)
Constantin Teleman (UC Berkeley)
This event is co-sponsored by the Simons Foundation.

Symmetry Colloquia - Global Categorical Symmetries
2024-05-02 14:00 - 14:50 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Symmetry Colloquia – Global Categorical Symmetries
May 2, 2024
Location: Room G-10, CMSA, 20 Garden Street, Cambridge MA 02138
Speaker: Clay Còrdova, University of Chicago
Title: Particle-Soliton Degeneracies from Spontaneously Broken Non-Invertible Symmetry
Abstract: We study non-invertible topological symmetry operators in massive quantum field theories in (1+1) dimensions. In phases where this symmetry is spontaneously broken we show that the particle spectrum often has degeneracies dictated by the non-invertible symmetry and we deduce a procedure to determine the allowed multiplets. These degeneracies are robust predictions and do not require integrability or other special features of renormalization group flows. We exhibit these conclusions in examples where the spectrum is known, recovering soliton and particle degeneracies. For instance, the Tricritical Ising model deformed by the subleading Z2 odd operator flows to a gapped phase with two degenerate vacua. This flow enjoys a Fibonacci fusion category symmetry which implies a threefold degeneracy of its particle states, relating the mass of solitons interpolating between vacua and particles supported in a single vacuum.

Symmetry Colloquia - Global Categorical Symmetries
2024-05-02 15:00 - 15:50 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Symmetry Colloquia – Global Categorical Symmetries
May 2, 2024
Location: Room G-10, CMSA, 20 Garden Street, Cambridge MA 02138
Speaker: Thomas Dumitrescu, UCLA
Title: Symmetries, Invertible Field Theories, and Gauge Theory Phases
Abstract: I will start with a brief overview of gauge theory phases in 3+1 dimensions through the lens of higher symmetries — in particular the realization of 1-form symmetries acting on loop order parameters. I will then review recent progress in refining this characterization using invertible field theories, or equivalently symmetry protected topological phases (SPTs). This refinement leads to new results in gauge theories with fundamental matter, such as quantum chromodynamics (QCD), which do not possess 1-form symmetries. I will explain why these theories must sometimes undergo a phase transition between their confining and Higgs regimes, despite the fact that classic results and standard lore say they should be continuously connected.

Symmetry Colloquia - Global Categorical Symmetries
2024-05-02 16:30 - 17:20 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Symmetry Colloquia – Global Categorical Symmetries
May 2, 2024
Location: Room G-10, CMSA, 20 Garden Street, Cambridge MA 02138
Speaker: Theo Johnson-Freyd, Dalhousie University and Perimeter Institute
Title: The Universal Target Category
Abstract: Hilbert’s Nullstellensatz says that the complex numbers C satisfy a universal property among all R-algebras: every not-too-large nonzero commutative R-algebra maps to C. Deligne proved a similar statement in categorical dimension 1: every not-too-large symmetric monoidal category over R maps to the category sVec of super vector spaces. In other words, sVec (and not Vec!) is “algebraically closed”. These statements help explain why quantum field theory requires imaginary numbers and fermions. I will describe the universal symmetric monoidal higher category that extends the sequence C, sVec, …. This is joint work in progress with David Reutter, and builds on closely-related work by GCS collaborators Freed, Scheimbauer, and Teleman and Schlank et al.

Amplituhedra, Cluster Algebras, and Positive Geometry
2024-05-29 09:00 - 2024-05-31 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Amplituhedra, Cluster Algebras, and Positive Geometry
Dates: May 29-31, 2024
Location: Harvard CMSA, 20 Garden Street, Cambridge MA 02138 & via Zoom
In recent years, a remarkable paradigm shift has occurred in understanding quantum observables in particle physics and cosmology, revealing their emergence from underlying novel mathematical objects known as positive geometries. The conference will center on the amplituhedron—the first and major example of a positive geometry. Building on the work of Lusztig and Postnikov on the positive Grassmannian, the physicists Arkani-Hamed and Trnka introduced the amplituhedron in 2013 as a geometric object that “explains” the so-called BCFW recurrence for scattering amplitudes in N = 4 super Yang Mills theory (SYM). Simultaneously, cluster algebras, originally introduced by Fomin and Zelevinsky to study total positivity, have been revealed to have a crucial role in describing singularities of N = 4 SYM scattering amplitudes. Thus, one can use ideas from quantum field theory (QFT) to connect cluster algebras to positive geometries, and in particular to the amplituhedron. Additionally, QFT can also be used to discover new examples of positive geometries. The conference will bring together a wide range of mathematicians and physicists both to draw new connections within algebraic combinatorics and geometry and to advance our physical understanding of scattering amplitudes and QFT.
The conference features: Introductory Lectures, an Open Problems Forum, Emerging Scholars Talks, and talks by experts in the fields.

Conference Videos (Youtube Playlist)

Confirmed Speakers:

Evgeniya Akhmedova, Weizmann Institute of Science
Nima Arkani-Hamed, IAS
Paolo Benincasa, MPI
Nick Early, Weizmann Institute of Science
Carolina Figueiredo, Princeton University
Yu-tin Huang, National Taiwan University
Dani Kaufman, University of Copenhagen
Chia-Kai Kuo, National Taiwan University
Thomas Lam, University of Michigan
Yelena Mandelshtam, UC Berkeley
Shruti Paranjape, UC Davis
Lizzie Pratt, UC Berkeley
Lecheng Ren, Brown University
Sebastian Seemann, KU Leuven
Khrystyna Serhiyenko, University of Kentucky
Melissa Sherman-Bennett, MIT & UC Davis
Marcus Spradlin, Brown University
Ran Tessler, Weizmann Institute of Science
Hugh Thomas, Université du Québec à Montréal
Jaroslav Trnka, UC Davis
Anastasia Volovich, Brown University

Organizers:

Matteo Parisi, Harvard CMSA
Lauren Williams, Harvard Mathematics

Participants (PDF)
This event is co-funded by the National Science Foundation.
Limited funding to help defray travel expenses is available for graduate students and recent PhDs. If you are a graduate student or postdoc and would like to apply for support, please register above and send an email to amplituhedra@cmsa.fas.harvard.edu no later than Friday, April 19, 2024.
Please include your name, address, current status, university affiliation, citizenship, and area of study. F1 visa holders are eligible to apply for support. If you are a graduate student, please send a brief letter of recommendation from a faculty member to explain the relevance of the conference to your studies or research. If you are a postdoc, please include a copy of your CV.

SCHEDULE (pdf download)
Wednesday, May 29, 2024
8:30 – 9:00 am
Registration and Breakfast
9:00 – 10:00 am
Jaroslav Trnka, UC Davis
Title: Amplituhedron
Abstract: I will review basics of the Amplituhedron, connection to the positive Grassmannian on the mathematical side, and the scattering amplitudes on the physics side.
10:00 – 10:15 am
Coffee Break
10:15 – 11:15 am
via Zoom
Khrystyna Serhiyenko, University of Kentucky
Title: Introduction to Cluster Algebras
Abstract: Cluster algebras is a class of commutative rings with an intricate combinatorial structure. They were introduced by Fomin and Zelevinsky in 2002 to study total positivity and canonical basis in Lie Theory, but quickly evolved into a highly active research area with surprising connections to numerous other areas of mathematics and physics.
In this course we will introduce cluster algebras and discuss their basic properties including positivity and Laurent phenomenon. We will also review cluster structures coming from coordinate rings of Grassmannians and the combinatorics of plabic graphs.
11:15 – 11:30 am
Coffee Break
11:30 – 12:30 pm
Thomas Lam, University of Michigan
Title: Introductory Lecture on Positive Geometries
Abstract: Positive geometries are semialgebraic spaces that appear in the study of scattering amplitudes. Examples include polytopes, totally nonnegative parts of flag varieties, and conjecturally, the amplituhedron. We will give a broad introduction to positive geometries, and to their canonical forms.
12:30 – 2:00 pm
Lunch Break
2:00 – 2:50 pm
Anastasia Volovich, Brown University
Title: Scattering Amplitudes and Cluster Algebras
Abstract: I will review some of the deep connections between cluster algebras and the (loop level) scattering amplitudes in N=4 super Yang-Mills theory, focusing on the cases of n=6 and 7 particles where the corresponding Grassmannian cluster algebras Gr(4,n) are finite and certain features of the amplitudes are known or believed to be true to all loop order.
2:50 – 3:00 pm
Coffee Break
3:00 – 3:50 pm
Marcus Spradlin, Brown University
Title: Scattering Amplitudes, Positive Geometry and the Amplituhedron
Abstract: I will review the status of (loop level) scattering amplitudes in N=4 super Yang-Mills theory for n>7, where the corresponding Grassmannian cluster algebras Gr(4,n) are infinite and novel features emerge, notably the appearance of certain “marginally positive” algebraic functions of cluster variables.
3:50 – 4:00 pm
Coffee Break
4:00 – 4:30 pm
Carolina Figueiredo, Princeton University
Title: All-order splits and multi-soft limits for particle and string amplitude
Abstract: The most important aspects of scattering amplitudes have long been thought to be associated with their poles. Recently a very different sort of “split” factorizations for a wide range of particle and string tree amplitudes have been discovered away from poles. In this talk, I will explain how natural properties of the binary geometry of the curve integral formulation for scattering amplitudes for Tr$(Phi^3)$ theory give a simple, conceptual origin for these splits, that generalizes them to all orders in the topological expansion. I will also explain how the splits allow us to access and compute loop-integrated multi-soft limits for particle and string amplitudes in Tr$(Phi^3)$ theory, the non-linear sigma model and Yang-Mills theory.
4:30 – 5:15 pm
Yelena Mandelshtam, UC Berkeley
Title: Combinatorics of m=1 Grasstopes
Abstract: A Grasstope is a linear projection of the totally nonnegative Grassmannian to a smaller Grassmannian. This is a generalization of the amplituhedron, a geometric object of great importance to calculating scattering amplitudes in physics. The amplituhedron is a Grasstope arising from a totally positive linear map. While amplituhedra are relatively well-studied, much less is known about general Grasstopes. In this talk, I will discuss combinatorics and geometry of Grasstopes in the m=1 case. In particular, I will show that they can be characterized as unions of cells of a hyperplane arrangement satisfying a certain sign variation condition and argue that amplituhedra are (in a certain sense) minimal Grasstopes. This is based on joint work with Dmitrii Pavlov and Lizzie Pratt.
5:30 – 6:30 pm
Welcome Reception

Thursday, May 30, 2024
8:30 – 9:00 am
Breakfast
9:00 – 10:00 am
Nima Arkani-Hamed, IAS
Title: Surface Kinematics and THE all-loop integrand for gluon amplitudes
10:00 – 10:30 am
Coffee Break
10:30 – 11:20 am
Hugh Thomas, Université du Québec à Montréal
Title: u-equations from finite dimensional algebras
Abstract: In this talk, I will explain how to write down and solve a system of u-equations associated to any finite dimensional algebra with finitely many indecomposable representations. These vastly generalize the system of equations written down by Koba and Nielsen in 1969, which from our point of view are associated to the representation theory of a Dynkin type A quiver. I will discuss features of the resulting solution spaces, including connections to tau-tilting theory, and the relationships that exist among different spaces of solutions. I will also say something about how different choices of finite-dimensional algebra put us in (i) the setting of cluster algebras, (ii) the Grassmannian combinatorics of non-kissing complexes, or (iii) the curves-on-surfaces model directly relevant to amplitudes. This talk reports on joint work with Nima Arkani-Hamed, Hadleigh Frost, Pierre-Guy Plamondon, and Giulio Salvatori.
11:20 – 11:30 am
Coffee Break
11:30 – 12:20 pm
Dani Kaufman, University of Copenhagen
Title: Affine Cluster Algebras
Abstract: Affine cluster algebras form the simplest examples of non-finite type cluster algebras. While they have infinitely many clusters, they are still mutation finite and have essentially one mutation sequence which produces infinitely many clusters. I will give an introduction to these cluster algebras by comparing them with finite cluster algebras. I will also show how some structures similar to finite type cluster algebras appear “in the limit” along this infinite mutation sequence. If time I will also mention how the “infinite cluster variables” which live in the limit are related to the algebraic letters appearing in the symbol alphabet for 8 particle N=4 SYM amplitudes.
12:30-12:45 pm
Group Photo, 20 Garden Street, front entrance stairs outside building
12:45 – 2:00 pm
Lunch Break
2:00 – 2:50 pm
via Zoom
Ran Tessler, Weizmann Institute of Science
Title: The magic number for the m=2 amplituhedron
Abstract: We will start by reviewing the amplituhedron and its tilings.
We will then show that all tilings of the m=2 amplituhedron have the same cardinality (“the magic number”), proving the m=2 case of a conjecture that the same holds for all even-m amplituhedra. If time permits we will discuss related results and consequences.
Based on a joint work with Parisi, Sherman-Bennett and Williams.
2:50 – 3:00 pm
Coffee Break
3:00 – 3:50 pm
Melissa Sherman-Bennett, MIT & UC Davis
Title: Cluster algebras and tilings of amplituhedra
Abstract: Physicists Arkani-Hamed and Trnka introduced the amplituhedron to better understand scattering amplitudes in N=4 super Yang-Mills theory. The amplituhedron is the image of the totally nonnegative Grassmannian under the “amplituhedron map”. Examples of amplituhedra include cyclic polytopes, the totally nonnegative Grassmannian itself, and cyclic hyperplane arrangements. Of primary interest to physics are tilings of amplituhedra, which are roughly analogous to subdivisions of polytopes. I will discuss joint work with Even-Zohar, Lakrec, Parisi, Tessler and Williams on BCFW tilings of m=4 amplituhedra and the surprising connection between these tilings and the cluster algebra structure of the Grassmannian.
3:50 – 4:00 pm
Coffee Break
4:00 – 5:30 pm
Open Problems Forum
6:00 – 8:00 pm
Conference Dinner (by invitation)

Friday, May 31, 2024
8:30 – 9:00 am
Breakfast
9:00 – 10:00 am
Yu-tin Huang, National Taiwan University
Title: Chambers and all loop geometry for four-point correlators
Abstract: The all loop amplituhedron for N=4 SYM (and ABJM theory) can be recast into the notion of loop fibration over tree geometry. This leads to a further dissection of the tree geometry into “chambers”, whose boundaries represents when the associated loop-form changes. In this talk I will present a new geometry associated with the all loop four-point correlator of N=4 SYM, where similar description is present. Interestingly, at four-loops, this gives a first example where the chamber form is rational even though it’s loop form contains elliptic integrals.
10:00 – 10:15 am
Coffee Break
10:15 – 12:30 am
Emerging Scholar Talks
10:15 – 10:40 am
Evgeniya Akhmedova, Weizmann Institute of Science
Title: The tropical amplituhedron
Abstract: The Amplituhedron is a geometric object discovered recently by Arkani-Hamed and Trnka, that provides a completely new direction for calculating scattering amplitudes in quantum field theory. We define a tropical analogue of this object, the tropicial amplituhedron and study its structure and boundaries. It can be considered as both the tropical limit of the amplituhedron and a generalization of the tropical positive Grassmannian.
10:40 – 11:10 am
Lizzie Pratt, UC Berkeley
Title: The Chow-Lam Form
Abstract: The classical Chow form encodes any projective variety by one equation. We introduce the Chow-Lam form for subvarieties of a Grassmannian. By evaluating the Chow-Lam form at twistor coordinates, we obtain universal projection formulas, which were pioneered by Thomas Lam for positroid varieties in the study of amplituhedra. This is joint work with Bernd Sturmfels.
11:10– 11:30 am
Sebastian Seemann, KU Leuven
Title: Vandermonde cells as positive geometries
Abstract: Vandermonde cells represent semialgebraic subsets of R^n, characterized as the image of a simplex under the Vandermonde map. However, within the realm of positive geometry, several challenges arise in establishing canonical forms for these cells. These include issues such as non-normal boundaries, non-transversal intersections, and singularities of boundary curves. Even more difficulties appear when considing the limiting Vandermonde cell, which is not semi-algebraic and thus doesn’t fit within the standard framework of positive geometries. In this presentation, I will first review the notion of Polypols and their canonical forms, examining the complexities encountered when dealing with Vandermonde cells. In particular, I will explain what goes wrong in the case of Vandermonde cells and which obstructions we can deal with.
11:30 – 11:40 am
Coffee break
11:40 – 12:10 pm
Chia-Kai Kuo, National Taiwan University
Title: Geometric transition from maximal SYM to ABJM
Abstract: Recently, the ABJM amplituhedron has been proposed, encoding all-loop and all-multiplicity ABJM amplitudes. It is constructed by slightly modifying the original definition. In this talk, I will explore the significance of these modifications in transitioning theoretical models from super Yang-Mills theory to ABJM theory. A key focus will be on how symplectic reduction and the overall sign change in the positivity conditions ensure the consistency of ABJM amplitudes. Additionally, I will discuss some distinct features of this geometry.
12:10– 12:30 pm
Lecheng Ren, Brown University
Title: Symbol alphabets from tensor diagrams
Abstract: We propose to use tensor diagrams and the Fomin-Pylyavskyy conjectures to explore the connection between symbol alphabets of n-particle amplitudes in planar N= 4 Yang-Mills theory and certain polytopes associated to the Grassmannian Gr(4, n). We show how to assign a web (a planar tensor diagram) to each facet of these polytopes. Webs with no inner loops are associated to cluster variables (rational symbol letters). For webs with a single inner loop we propose and explicitly evaluate an associated web series that contains information about algebraic symbol letters. In this manner we reproduce the results of previous analyses of n ≤ 8, and find that the polytope C(4,9) encodes all rational letters, and all square roots of the algebraic letters, of known nine-particle amplitudes.
12:30 – 2:00 pm
Lunch Break
2:00 – 2:50 pm
via Zoom
Paolo Benincasa, MPI
Title: Cosmological Polytopes & Beyond
Abstract: Together with being the source of the most profound questions in fundamental physics, cosmology turns out to be an arena from where novel combinatorial structures emerge. In this talk, I will give a gentle introduction to the cosmological polytopes, describing the so-called Bunch-Davies wavefunction for a large class of scalar theories, and how it can be used to define and characterize less conventional objects, named optical polytopes and weighted cosmological polytopes, which provide examples of non-convex and weighted geometries respectively.
2:50 – 3:00 pm
Coffee Break
3:00 – 3:45 pm
Shruti Paranjape, UC Davis
Title: Loops in a loop expansion
Abstract: In a paper by Arkani-Hamed, Henn and Trnka, it was shown that the amplituhedron construction of N=4 SYM can be recast in terms of negative geometries with a certain hierarchy of loops (closed cycles) in the space of loop momentum twistors. Furthermore, using differential equation methods, it was possible to calculate and resum integrated expressions and obtain strong coupling results. In this talk, we provide a more general framework for the loops of loops expansion and outline a powerful method for the determination of differential forms for higher-order geometries. In particular, we will focus on the case of 1 closed cycle in loop space and select integrated results.
3:45 – 4:30 pm
Nick Early, Weizmann Institute of Science
Title: Minimal Kinematics on $mathcal{M}_{0,n}$, and beyond
Abstract: Minimal Kinematics (MK) identifies kinematic degenerations of the CHY scattering potential where the critical points are given by rational formulas. These rest on the Horn uniformization of Kapranov-Huh; they are specified combinatorially by 2-trees. On the other hand, Planar Kinematics (PK) identifies the locus in $M_{0,n}$ which is fixed by cyclic permutation. Combining MK and PK realizes a maximally thin relative of the associahedron known as the PK polytope; it is a reflexive polytope, and its polar dual, the root polytope, has volume a Catalan number. In this talk, we start by exploring MK and PK on the moduli space $M_{0,n}$. We explain how this story generalizes to moduli spaces $X(k,n)$ of points in projective space $mathbb{P}^{k-1}$, to CEGM amplitudes and beyond.
4:30 – 5:00 pm
Coffee and Farewell

About the image:

Left: the 3-dimensional associahedron, Fomin and Zelevinsky

Center: artistic depiction of the amplituhedron, Gilmore
Right: Schlegel diagram of a hypersimplex, Ziegler

Workshop on Fibration and Degeneration in Calabi-Yau Geometry
2024-06-24 08:00 - 2024-06-26 17:00 | 20 Garden Street, Cambridge, MA 02138, MA, MA, 02138, United States
Workshop on Fibration and Degeneration in Calabi-Yau Geometry
Dates: June 24-26, 2024
Location: Harvard CMSA, 20 Garden Street, Cambridge, MA 02138
Organizer: Chuck Doran, Harvard CMSA

Calabi-Yau manifolds occupy a central place in geometry. Their critical role as the cut-case between basic Fano building blocks and the zoo of General Type manifolds is key to the wide variety of important applications of Calabi-Yau geometry to theoretical physics. In turn, ideas from theoretical physics, such as Mirror Symmetry, help shape investigations in Calabi-Yau geometry
This workshop focuses on a structural feature of Calabi-Yau geometry identified a decade ago by Doran, Harder, and Thompson. It is an organizing principle that conjecturally underlies any and all constructions of mirror pairs of Calabi-Yau manifolds. Put simply, the DHT Mirror Symmetry slogan is: “Degeneration is mirror to fibration.”

Confirmed Speakers:

David Favero (University of Minnesota)
Andrew Harder (Lehigh University)
Jesse Huang (University of Alberta)
Mohsen Karkheiran* (University of Alberta)
Matt Kerr* (Washington University in St. Louis)
Thorsten Schimannek* (Utrecht University)
Michael Schultz (Virginia Tech)
Alan Thompson (Loughborough University)
Fenglong You (University of Nottingham & ETH Zurich)

*= via Zoom

Schedule
Monday, June 24, 2024
9:30 – 10:00 am: Breakfast
10:00 – 11:00 am
Speaker: Alan Thompson, Loughborough University
Title: Mirror symmetry for fibrations and degenerations of K3 surfaces
Abstract: I will describe recent progress, joint with Luca Giovenzana, on the DHT problem for K3 surfaces. I will give an lattice-theoretic definition for when a Tyurin degeneration of K3 surfaces and an elliptically-fibred K3 surface, with an appropriate splitting of the base, form a mirror pair. I will then explain how this definition is compatible with lattice polarised mirror symmetry for K3 surfaces and with Fano-LG mirror symmetry for (quasi) del Pezzo surfaces. The upshot will be a concrete statement of the DHT conjecture for K3 surfaces.
12:00 – 1:00: Lunch
1:00 – 2:00 pm
Speaker: David Favero, University of Minnesota
Title: Homotopy Path Algebras and Resolutions
Abstract: A homotopy path algebra is like a directed version of the group ring on a fundamental group. One can imagine a directed graph (quiver) embedded in a topological space and considering the path algebra up to homotopy. Alternatively, one can think of homotopy classes of directed paths in a stratified topological space. I will introduce homotopy path algebras and describe their connections to mirror symmetry and resolutions of coherent sheaves on toric varieties.
3:00 – 4:00 pm
Speaker: Andrew Harder, Lehigh University
Title: Tropical Hodge theory for hypersurfaces and Clarke duality
Abstract: Results of Itenberg, Katzarkov, Mikhalkin, and Zharkov (IKMZ) show that if a projective variety admits a smooth tropicalization, then there is a collection of sheaves on its tropicalization that can be used to compute its Hodge numbers. However, smooth tropicalizations fail to exist even in the case of toric hypersurfaces. In work with Sukjoo Lee, we show that for any toric hypersurface, an analogue of IKMZ’s result holds. I’ll discuss this sheaf, and how this allows us to prove that Clarke dual pairs of Landau-Ginzburg models satisfy a particular Hodge number duality. This is a vast generalization of work of Batyrev and Borisov from the 90s.
4:00 – 4:30 pm: Coffee/Tea

Tuesday, June 25, 2024
9:30 – 10:00 am: Breakfast
10:00 – 11:00 am
Speaker: Matt Kerr, Washington University in St. Louis
Title: Hypergeometric families and Beilinson’s conjectures
Abstract: I will describe the construction of motivic cohomology classes on hypergeometric families of Calabi-Yau 3-folds using Hadamard convolutions. These are analogous to elements of the Mordell-Weil group for families of elliptic curves, and produce solutions to certain inhomogeneous Picard-Fuchs equations. This is part of a joint project with Vasily Golyshev in which we numerically verify Beilinson’s conjectures in some new cases.
12:00 – 1:00: Lunch
1:00 – 2:00 pm
Speaker: Fenglong You, University of Nottingham & ETH Zurich
Title: Theta functions in mirror symmetry
Abstract: To obtain a mirror of a Calabi—Yau manifold using Gross—Siebert’s intrinsic mirror symmetry, one considers a maximally unipotent monodromy degeneration of the Calabi—Yau and take proj of the degree zero part of a relative quantum cohomology ring associated with the degeneration. Theta functions form a canonical basis of the degree zero part of the relative quantum cohomology ring. Theta functions can also be defined in terms of punctured invariants of the broken line type. I will explain a variant of intrinsic mirror symmetry using orbifold invariants, theta functions for general snc pairs and a relation with the DHT conjecture.
3:00 – 4:00 pm
Speaker: Mohsen Karkheiran, University of Alberta
Title: Emergence of Heterotic-Type II duality from DHT conjecture
Abstract: The duality between Heterotic and Type IIA strings was conjectured in mid-90’s based on the properties of 4D N=2 field theories and solitonic strings in 6D. Here, we show that this duality can also emerge from the DHT conjecture. We assume both IIA and IIB strings are compactified over toric Calabi-Yau threefolds which admit K3-fibrations with arbitrary polarizations. Then by applying the Hori-Vafa mirror symmetry to the “pieces” of these Calabi-Yau manifolds, we will be able to derive the defining data for Heterotic strings. This approach works for any gauge group on the Heterotic side, and we will show how it can be practically useful to derive the Heterotic dual for any toric Calabi-Yau threefolds in Type IIA or F-theory.
4:00 – 4:30 pm: Coffee/Tea

Wednesday, June 26, 2024
9:30 – 10:00 am: Breakfast
10:00 – 11:00 am
Speaker: Thorsten Schimannek, Utrecht University
Title: Enumerative geometry and modularity in two-modulus K3-fibered Calabi-Yau threefolds
Abstract: Smooth M_m-polarized K3-fibered Calabi-Yau (CY) 3-folds have been classified in [DHNT] and [KT] in terms of the choice of a generalized functional invariant (GFI) and, in the case m=1, a generalized homological invariant (GHI). The resulting geometries generally exhibit a small number of complex structure moduli greater or equal to two. I will start my talk by discussing a concrete choice of these invariants that realizes (almost all of) the geometries with exactly two moduli and describe the structure of the corresponding moduli spaces. The corresponding variations of Hodge structure are entirely determined by the regular periods, for which we obtain a generic expression in terms of m and three integers i,j,s. Using the form of this period and Batyrev-Borisov mirror symmetry I will then explicitly construct the corresponding mirror CY 3-folds with two Kaehler moduli and show consistency with the DHT conjecture. In the cases with s=0, the mirror CY 3-folds are again K3-fibered but with a 2m-polarization. The generic form of the periods allows us to derive generic modular expressions for the A-model topological string free energies and we argue that those are a consequence of a Tyurin degeneration of the GFI with the central fiber being an M_m-polarized K3.
The talk is based on work in progress with Charles Doran and Boris Pioline.
12:00 – 1:00: Lunch
1:00 – 2:00 pm
Speaker: Michael Schultz, Virginia Tech
Title: Mirror Symmetry from Irrationality Proofs and a Proposal for Local Invariants
Abstract: While Apéry’s original proof of the irrationality of ζ(3) stunned the mathematics community in 1978, subsequent generations of mathematicians (including a number of those at this workshop) have discovered geometric and modular structures underlying these irrationality proofs that are arguably even more striking. One such well known example are connections to modular pencils of elliptic curves and K3 surfaces and their Picard-Fuchs operators, which exhibit maximally unipotent monodromy. These objects are respectively mirror dual to anticanonical divisors in certain del Pezzo surfaces and Fano threefolds, and their Picard-Fuchs operators to the A-side connection on small quantum cohomology for these varieties. Although the Yukawa couplings calculated in classical mirror symmetry for elliptic curves and K3 surfaces are trivial, I will show in this talk how a blend of the perspectives above allows one to define “virtual” Yukawa couplings for these families that are not trivial. It will be proposed that the utility of this perspective is in computing local invariants related to the mirror, which recovers some known results in the literature and utilizes connections to work on the DHT conjecture and the twist construction of Doran & Malmendier.
3:00 – 4:00 pm
Speaker: Jesse Huang, University of Alberta
Title: An invitation to global toric mirror symmetry
4:00 – 4:30 pm: Coffee/Tea

Advances in Probability Theory and Interacting Particle Systems
2024-08-26 09:00 - 2024-08-28 17:00 | Harvard Geological Lecture Hall, 24 Oxford St, Cambridge, 02138, United States
Advances in Probability Theory and Interacting Particle Systems

A conference in honor of S. R. Srinivasa Varadhan.

August 26 – August 28, 2024

Harvard Geological Lecture Hall

Conference Website: www.math.harvard.edu/event/math-conference-honoring-srinivasa-varadhan

Speakers

Ines Armendariz, Universidad de Buenos Aires
Yuri Bakhtin, Courant Institute
Gérard Ben Arous, Courant Institute
Sourav Chatterjee, Stanford University
Amir Dembo, Stanford University
Peter K. Friz, TU-Berlin
Nina Holden, Courant Institute
Jiaoyang Huang, University of Pennsylvania
Elena Kosygina, City University of New York
Claudio Landim, IMPA
Eyal Lubetzky, Courant Institute
Chiranjib Mukherjee, Uni Münster
Stefano Olla, Université Paris Dauphine
Jeremy Quastel, University of Toronto
Kavita Ramanan, Brown University
Alejandro Ramirez, NYU Shanghai
Fraydoun Rezakhanlou, Berkeley
Sunder Sethuraman, University of Arizona
Scott Sheffield, MIT
Ofer Zeitouni, Weizmann Institute

Organizers: Paul Bourgade (New York University, Courant Institute) and Horng-Tzer Yau (Harvard University).

Sponsored by Harvard University Department of Mathematics and the Center of Mathematical Studies and Applications (CMSA).

Harvard University is committed to maintaining a safe and healthy educational and work environment in which no member of the University community is, on the basis of sex, sexual orientation, or gender identity, excluded from participation in, denied the benefits of, or subjected to discrimination in any University program or activity. More information can be found here.

Mathematics and Machine Learning Program
2024-09-03 09:00 - 2024-11-01 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematics and Machine Learning Program
Dates: September 3 – November 1, 2024
Location: Harvard CMSA, 20 Garden Street, Cambridge, MA 0213
Machine learning and AI are increasingly important tools in all fields of research. Recent milestones in machine learning for mathematics include data-driven discovery of theorems in knot theory and representation theory, the discovery and proof of new singular solutions of the Euler equations, new counterexamples and lower bounds in graph theory, and more. Rigorous numerical methods and interactive theorem proving are playing an important part in obtaining these results. Conversely, much of the spectacular progress in AI has a surprising simplicity at its core. Surely there are remarkable mathematical structures behind this, yet to be elucidated.
The program will begin and end with two week-long workshops, and will feature focus weeks on number theory, knot theory, graph theory, rigorous numerics in PDE, and interactive theorem proving, as well as a course on geometric aspects of deep learning.

September 3–5, 2024: Opening Workshop: AI for Mathematicians, with Leon Bottou, François Charton, David McAllester, Adam Wagner and Geordie Williamson. A series of six lectures covering logic and theorem proving, AI methods, theory of machine learning, two lectures on case studies in math-AI, and a lecture and discussion on open problems and the ethics of AI in science.
Opening Workshop Youtube Playlist

September 6–7, 2024: Big Data Conference

September 9–13, 2024: Applying Machine Learning to Math, with François Charton and Geordie Williamson
Public Lecture September 12, 2024: Geordie Williamson, University of Sydney: Can AI help with hard mathematics? (Youtube link)
The focus of this week will be on practical examples and techniques for the mathematics researcher keen to explore or deepen their use of AI techniques. We will have talks showcasing easily stated problems, on which machine learning techniques can be employed profitably. These provide excellent toy examples for generating intuition. We will also have expert talks on some of the technical subtleties which arise. There are several instances where the accepted heuristics emerging from the study of large language models (LLM) and image recognition don’t appear to apply on mathematics problems, and we will try to highlight these subtleties.
Applying Machine Learning to Math Youtube Playlist

September 16–20, 2024: Number theory, with Drew Sutherland
The focus of this week will be on the use of ML as a tool for finding and understanding statistical patterns in number-theoretic datasets, using the recently discovered (and still largely unexplained) “murmurations” in the distribution of Frobenius traces in families of elliptic curves and other arithmetic L-functions as a motivating example.
Number Theory Youtube Playlist

September 23–27, 2024: Knot theory, with Sergei Gukov
Knot theory is a great source of labeled data that can be synthetically generated. Moreover, many outstanding problems in knot theory and low-dimensional topology can be formulated as decision and classification tasks, e.g. “Is the knot 123_45 slice?” or “Can two given Kirby diagrams be related by a sequence of Kirby moves?” During this focus week we will explore various ways in which AI can be applied to problems in knot theory and how, based on these applications, mathematical reasoning can advance development of AI algorithms. Another goal will be to develop formal knot theory libraries (e.g. contributions to mathlib) and to apply AI models to formal proof systems, in particular in the context of knot theory.
Knot Theory Youtube Playlist

September 30: Teaching and Machine Learning Panel Discussion, 3:30-5:30 pm ET

September 30–October 4, 2024: Graph theory and combinatorics, with Adam Wagner
This week, we will consider how machine learning can help us solve problems in combinatorics and graph theory, broadly interpreted, in practice. The advantage of these fields is that they deal with finite objects that are simple to set up using computers, and programs that work for one problem can often be adapted to work for several other related problems as well. Many times, the best constructions for a problem are easy to interpret, making it simpler to judge how well a particular algorithm is performing. On the other hand, there are lots of open conjectures that are simple to state, for which the best-known constructions are counterintuitive, making it perhaps more likely that machine learning methods can spot patterns that are difficult to understand otherwise.
Graph Theory and Combinatorics Youtube Playlist

October 7–11, 2024: More number theory, with Drew Sutherland
The focus of this week will be on the use of AI as a tool to search for and/or construct interesting or extremal examples in number theory and arithmetic geometry, using LLM-based genetic algorithms, generative adversarial networks, game-theoretic methods, and heuristic tree pruning as alternatives to conventional local search strategies.
More Number Theory Youtube Playlist

October 14 –18, 2024: Interactive theorem proving
This week we will discuss the use of interactive theorem proving systems such as Lean, Coq and Isabelle in mathematical research, and AI systems which prove theorems and translate between informal and formal mathematics.
Interactive Theorem Proving Youtube Playlist

October 21–25, 2024: Numerical Partial Differential Equations (PDE), with Tristan Buckmaster and Javier Gomez-Serrano
The focus of this week will be on constructing solutions to partial differential equations and dynamical systems (finite and infinite dimensional) more broadly defined. We will discuss several toy problems and comment on issues like sampling strategies, optimization algorithms, ill-posedness, or convergence. We will also outline strategies about further developing machine-learning findings and turn them into mathematical theorems via computer-assisted approaches.
Numerical PDEs Youtube Playlist

October 28–Nov. 1, 2024: Closing Workshop: The closing workshop will provide a forum for discussing the most current research in these areas, including work in progress and recent results from program participants.
Math and Machine Learning Closing Workshop Youtube Playlist

September 3–Nov. 1: Graduate topics in deep learning theory (Boston College) taught by Eli Grigsby, held at the CMSA Tuesdays and Thursdays 2:30–3:45 pm Eastern Time. Course website (link).
Graduate Topics in Deep Learning Youtube Playlist
Course description: This is a course on geometric aspects of deep learning theory. Broadly speaking, we’ll investigate the question: How might human-interpretable concepts be expressed in the geometry of their data encodings, and how does this geometry interact with the computational units and higher-level algebraic structures in various parameterized function classes, especially neural network classes? During the portion of the course Sep. 3-Nov. 1, the course will be presented as part of the Math and Machine Learning program at the CMSA in Cambridge. During that portion, we will focus on the current state of research on mechanistic interpretability of transformers, the architecture underlying large language models like Chat-GPT.

Prerequisites: This course is targeted to graduate students and advanced undergraduates in mathematics and theoretical computer science. No prior background in machine learning or learning theory will be assumed, but I will assume a degree of mathematical maturity (at the level of–say—the standard undergraduate math curriculum+ first-year graduate geometry/topology sequence)

Program Organizers

Francois Charton (Meta AI)
Michael R. Douglas (Harvard CMSA)
Michael Freedman (Harvard CMSA)
Fabian Ruehle (Northeastern)
Geordie Williamson (Univ. of Sydney)

Program Schedule
Monday
10:30–noon
Open Discussion
Room G10
12:00–1:30 pm
Group lunch
CMSA Common Room
Tuesday
2:30–3:45 pm
Topics in deep learning theory
Room G10
4:00–5:00 pm
Open Discussion/Tea
CMSA Common Room
Wednesday
10:30 am–12:00 pm
Open Discussion
Room G10
2:00–3:00 pm
New Technologies in Mathematics Seminar
Room G10
Thursday
2:30–3:45 pm
Topics in deep learning theory
Room G10
Friday
10:30 am–12:00 pm
Open Discussion
Room G10

Harvard CMSA thanks Mistral AI for a generous donation of computing credit.

Mathematics and Machine Learning Program Opening Workshop
2024-09-03 10:00 - 2024-09-05 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematics and Machine Learning Program Opening Workshop
Dates: September 3 – 5, 2024
Location: Room G10, CMSA, 20 Garden Street, Cambridge MA & via Zoom Webinar
AI for Mathematicians, with Leon Bottou, François Charton, David McAllester, Adam Wagner, Boris Hanin, and Geordie Williamson. A series of 6 tutorial lectures introducing concepts of AI and of theorem proving, with many case studies of AI applied to mathematics, and including lectures and discussion sessions on open questions, future prospects, and ethical questions.

Speakers

Leon Bottou (Meta AI)
François Charton (Meta AI)
Michael R. Douglas (Harvard CMSA)
Boris Hanin (Princeton)
David McAllester (TTIC)
Adam Wagner (WPI)
Geordie Williamson (University of Sydney)

Schedule (link to downloadable pdf)
Videos from the Opening Workshop (Youtube Link)

Tuesday Sep. 3, 2024

9:30–10:00 am
Morning refreshments

10:00–11:30 am
Mike Douglas: Overview of AI for mathematics
Slides (pdf)

11:30 am–12:00 pm
Discussion

12:00–1:30 pm
Break

1:30–2:30 pm
David McAllester: Logic and formal methods
Slides (pdf)

2:30–3:00 pm
Coffee break

3:00–4:00 pm
Panel Discussion: Automated mathematical discovery

Wednesday Sep. 4, 2024

9:30–10:00 am
Morning refreshments

10:00–11:30 am
Boris Hanin: Theory of Machine Learning
Slides (pdf)

11:30 am–12:00 pm
Discussion

12:00–1:30 pm
Break

1:30–2:30 pm
Adam Wagner: Case studies I: Reinforcement learning and pattern finding
Slides (pdf)

2:30–3:00 pm
Coffee break

Thursday Sep. 5, 2024

9:30–10:00 am
Morning refreshments

10:00–11:30 am
François Charton (slides-pdf)
and Geordie Williamson: Case studies II

11:30 am–12:00 pm
Discussion

12:00–1:30 pm
Break

1:30–2:30 pm
Leon Bottou: Open questions in AI
Slides (pdf)

2:30–3:00 pm
Coffee break

3:00–4:00 pm
Panel Discussion: How might AI change mathematics?

CMSA/Math Fall Gathering
2024-09-05 16:00 - 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
September 5, 2024
4:00 pm
CMSA Common Room, 20 Garden Street, Cambridge MA
All CMSA and Math affiliates are invited.

Big Data Conference 2024
2024-09-06 09:00 - 2024-09-07 17:00 | 20 Garden Street, Cambridge, MA 02138, MA, MA, 02138, United States
Youtube Playlist
On September 6-7, 2024, the CMSA hosted the tenth annual Conference on Big Data. The Big Data Conference features speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
Location: Harvard University CMSA, 20 Garden Street, Cambridge & via Zoom

Speakers:

Tianxi Cai, Harvard Chan School
Raj Chetty, Harvard
Bianca Dumitrascu, Columbia
Boris Hanin, Princeton
Peter Hull, Brown
Jamie Morgenstern, U Washington
Kavita Ramanan, Brown
Neil Thompson, MIT
Melanie Weber, Harvard
Kun-Hsing Yu, Harvard Medical School

Organizers:

Rediet Abebe, Harvard Society of Fellows
Morgane Austern, Harvard University Statistics
Michael R. Douglas, Harvard CMSA
Yannai Gonczarowski, Harvard University Economics and Computer Science
Sam Kou, Harvard University Statistics

SCHEDULE (downloadable pdf)
Friday, Sep. 6, 2024
9:00 am: Breakfast
9:30 am: Introductions
9:45–10:45 am
Speaker: Peter Hull, Brown University
Title: Measuring Discrimination in Multi-Phase Systems, with an Application to Child Protection
Abstract: Large racial disparities have been documented in many high-stakes settings—such as employment, health care, housing, and criminal justice—raising concerns of discrimination by individual decision-makers. At the same time, there is growing understanding that a focus on individual decisions can yield an incomplete view of discrimination; an extensive theoretical literature shows how discrimination can arise and compound across multiple decision-makers in interconnected systems. We develop new empirical tools for studying discrimination in such multi-phase systems and apply them to the setting of foster care placement by child protective services. Leveraging the quasi-random assignment of two sets of decision-makers—initial hotline call screeners and subsequent investigators—we study how unwarranted racial disparities arise and propagate through this system. Using a sample of over 200,000 maltreatment allegations, we find that calls involving Black children are 55% more likely to result in foster care placement than calls involving white children with the same potential for future maltreatment in the home. Call screeners account for up to 19% of this unwarranted disparity, with the remainder due to investigators. Unwarranted disparity is concentrated in cases with potential for future maltreatment, suggesting that white children may be harmed by “underplacement” in high-risk situations.
10:45–11:00 am: Break
11:00 am –12:00 pm
Speaker: Jamie Morgenstern, U Washington
Title: What governs predictive disparity in modern machine learning applications?
Abstract: The deployment of statistical models in impactful environments is far from new—simple correlations have been used to guide decisions throughout the sciences, health care, political campaigns, and in pricing financial instruments and other products for decades. Many such models, and the decisions they supported, were known to have different degrees of predictive power for different demographic groups. These differences had numerous sources, including: limited expressiveness of the statistical models; limited availability of data from marginalized populations; noisier measurements of both features and targets from certain populations; and features with less mutual information about the prediction target for some populations than others.
Modern decision systems which use machine learning are more ubiquitous than ever, as are their differences in performance for different populations of people. In this talk, I will discuss some similarities and differences in the sources of differing performance in contemporary ML systems including facial recognition systems and those incorporating generative AI.
12:00–1:30 pm: Lunch Break
1:30–2:30 pm
Speaker: Kavita Ramanan, Brown University
Title: Understanding High-dimensional Stochastic Dynamics on Realistic Networks
Abstract: Large collections of randomly evolving particles that interact locally with respect to an underlying network model a variety of phenomena ranging from magnetism, the spread of diseases, neural and neuronal networks, opinion dynamics and load balancing on computer networks. Due to their high-dimensional nature, these systems are typically intractable to analyze exactly. Classical work, falling under the rubric of mean-field approximations, has mostly focused on the case when this interaction graph is dense. However, most real-world networks are sparse and often random. We describe a new approach to develop principled approximations for dynamics on realistic networks that beats the curse of dimensionality, and illustrate its efficacy on a class of epidemiological models. This is based on joint works with Michel Davydov, Ankan Ganguly and Juniper Cocomello.
2:30–2:45 pm: Break
2:45–3:45 pm
Speaker: Raj Chetty, Harvard University
Title: The Science of Economic Opportunity: New Insights from Big Data
Abstract: How can we improve economic opportunities for children growing up in low-income families? This talk will present findings from a recent set of studies that use various sources of big data — ranging from anonymized tax records to social network data — to understand the science of economic opportunity. Among other topics, the talk will discuss how and why children’s chances of climbing the income ladder vary across neighborhoods, the drivers of racial disparities in economic mobility, how highly selective colleges may amplify the persistence of privilege, and the role of social capital as a driver of upward mobility. The talk will conclude by giving examples of how academic research using big data is informing policy decisions from the local to federal level to expand opportunities for all.
3:45–4:00 pm: Break
4:00–5:00 pm
Speaker: Neil Thompson
Title: How Algorithmic Progress is driving progress in Big Data and AI
Abstract: Algorithm improvement is one of the purest forms of innovation: it allows the same computational task to be achieved with far fewer resources by proposing clever new ways to do that computation. In this talk, I will discuss the work that my lab has done tracking and quantifying progress across decades of algorithm research and practice. As I will show, this algorithmic progress has often outpaced hardware improvement as the most important driver of progress in Big Data and AI.

Saturday, Sep. 7, 2024
9:00 am: Breakfast
9:30 am: Introductions
9:45–10:45 am
Speaker: Tianxi Cai, Harvard Chan School
Title: Crowdsourcing with Multi-institutional EHR to Improve Reliability of Real World Evidence – Opportunities and Challenges
Abstract: The wide adoption of electronic health records (EHR) systems has led to the availability of large clinical datasets available for discovery research. EHR data, linked with bio- repository, is a valuable new source for deriving real-word, data-driven prediction models of disease risk and progression. Yet, they also bring analytical difficulties especially when aiming to leverage multi-institutional EHR data. Synthesizing information across healthcare systems is challenging due to heterogeneity and privacy. Statistical challenges also arise due to high dimensionality in the feature space. In this talk, I’ll discuss analytical approaches for mining EHR data to improve the reliability and generalizability of real world evidence generated from the analyses. These methods will be illustrated using EHR data from Mass General Brigham and Veteran Health Administration.
10:45–11:00 am: Break
11:00 am–12:00 pm
Speaker: Bianca Dumitrascu, Columbia Data Science Institute
Title: Statistical machine learning for learning representations of embryonic development
Abstract: During embryonic development, single cells read in local information from their environments and use this information to move, divide and specialize. As a result, the environments themselves change. However, it remains unclear how gene expression programs interact with cell morphology and mechanical forces to orchestrate organogenesis in early embryos. Recent advances in single cell techniques and in toto imaging enable unique venues in exploring this link between genomics and biophysics, which dynamically maps cells to organisms.
In this talk, I will describe statistical machine learning frameworks aimed at understanding how tissue level mechanical and morphometric information impact gene expression patterns in spatio-temporal contexts. We use these tools to understand boundary formation in the early development of mouse embryos and to align data from light sheet recordings of pre-gastrulation development.
12:00–1:30 pm: Lunch Break
1:30–2:30 pm
Speaker: Melanie Weber, Harvard Mathematics
Title: Data and Model Geometry in Deep Learning
Abstract: Data with geometric structure is ubiquitous in machine learning. Often such structure arises from fundamental symmetries in the domain, such as permutation-invariance in graphs and sets, and translation-invariance in images. In this talk we discuss implications of this structure on the design and complexity of neural networks. Equivariant architectures, which encode symmetries as inductive bias, have shown great success in applications with geometric data, but can suffer from instabilities as their depths increases. We propose a new architecture based on unitary group convolutions, which allows for deeper networks with less instability. In the second part of the talk we discuss the impact of data and model geometry on the learnability of neural networks. We discuss learnability in several geometric settings, including equivariant neural networks, as well as learnability with respect to the geometry of the input data manifold.
2:30–2:45 pm: Break
2:45–3:45 pm
Speaker: Boris Hanin, Princeton University
Title: Scaling Limits of Neural Networks
Abstract: Neural networks are often studied analytically through scaling limits: regimes in which taking some structural network parameters (e.g. depth, width, number of training datapoints, and so on) to infinity results in simplified models of learning. I will motivative and discuss recent results using several such approaches. I will emphasize both new theoretical insights into how model, training data, and optimizer impact learning and their practical implications for hyperparameter transfer.
3:45–4:00 pm: Break
4:00–5:00 pm
Speaker: Kun-Hsing Yu, Harvard Medical School
Title: Foundation Models for Real-Time Cancer Diagnosis
Abstract: Artificial intelligence (AI) is transforming the landscape of medical research and practice. Recent advances in microscopic image digitization, foundation models, and scalable computing infrastructure have opened new avenues for AI-enhanced cancer diagnosis. In this talk, I will highlight recent breakthroughs in multi-modal AI systems for cancer pathology evaluation, discuss integrative biomedical informatics methods that link cell morphology with molecular profiles, and outline critical challenges in developing robust medical AI systems.

Information about the 2023 Big Data Conference can be found here.

Math and Machine Learning Lecture: Can AI help with hard mathematics?
2024-09-12 16:00 - 17:00 | Hybrid
Math and Machine Learning Lecture
Date: Thursday, Sep. 12, 2024
Time: 4:00 – 5:00 pm
Location: Science Center & via Zoom Webinar

Speaker: Geordie Williamson, University of Sydney
Title: Can AI help with hard mathematics?
Abstract: The last years have seen remarkable advances in what AI can do. It is perhaps surprising that its impact on research in pure mathematics has been modest. Reasoning, which is so quintessential to the mathematical process, remains a major challenge for current AI systems. I will survey some exciting recent applications of AI in mathematics research, trying to highlight what AI can do and the challenges that remain.

CMSA/Tsinghua Math-Science Literature Lecture: Marc Lackenby
2024-09-18 09:00 - 10:30 | Virtual
CMSA/Tsinghua Math-Science Literature Lecture
Date: Wednesday, September 18, 2024
Time: 9:00 – 10:30 am ET
Location: Via Zoom Webinar
Speaker: Marc Lackenby, University of Oxford
Title: The complexity of knots
Abstract: In his final paper in 1954, Alan Turing wrote `No systematic method is yet known by which one can tell whether two knots are the same.’ Within the next 20 years, Wolfgang Haken and Geoffrey Hemion had discovered such a method. However, the computational complexity of this problem remains unknown. In my talk, I will give a survey on this area, that draws on the work of many low-dimensional topologists and geometers. Unfortunately, the current upper bounds on the computational complexity of the knot equivalence problem remain quite poor. However, there are some recent results indicating that, perhaps, knots are more tractable than they first seem. Specifically, I will explain a theorem that provides, for each knot type K, a polynomial p_K with the property that any two diagrams of K with n_1 and n_2 crossings differ by at most p_K(n_1) + p_K(n_2) Reidemeister moves.

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

Machine Learning in Science Education Panel Discussion
2024-09-30 15:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Machine Learning in Science Education Panel Discussion
Monday, Sep. 30, 2024
3:30-5:30 pm ET
Machine Learning is rapidly influencing many spheres of human activity. As part of the CMSA Mathematics and Machine Learning Program, this panel discussion will explore current and future uses of Machine Learning in science education. Panelists will make brief presentations, which will be followed by discussion and audience questions.
Gregory Kestin (Harvard University)
AI-Supported Activities: Design Principles and Impact on Student Learning
Logan McCarty (Harvard University)
Surveying the Landscape: Teaching and Learning with AI
Alexis Ross (Massachusetts Institute of Technology)
Adaptive Teaching towards Misconceptions with LLMs
Ilia Sucholutsky (New York University)
Why should machines have human-like representations? Towards student-centric AI tutors

Organizers:

Dan Freed (Harvard University and CMSA)
Michael Douglas (CMSA)

Math Science Lectures in Honor of Raoul Bott: Andrew Neitzke
2024-10-16 16:00 - 17:00, 2024-10-17 16:00 - 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Speaker: Andrew Neitzke, Yale University
Location: Harvard University Science Center Hall D & via Zoom webinar
Dates: October 16 & 17, 2024
Time: 4:00 pm

Wednesday, Oct. 16, 2024
Title: Abelianization in analysis of ODEs
Abstract: I will describe the exact WKB method for asymptotic analysis of families of ODEs in one variable, and its interpretation as a kind of abelianization procedure, which replaces GL(N)-connections over a Riemann surface by GL(1)-connections over an N-fold branched cover. This abelianization procedure connects exact WKB to various subjects in geometry (cluster algebras, moduli of Higgs bundles, enumerative geometry). One application is a conjectural description of Hitchin’s hyperkahler metric on the moduli of Higgs bundles; I will review some recent progress on these conjectures.

Thursday, Oct. 17, 2024
Title: Abelianization in quantum topology
Abstract: I will describe new applications of abelianization to various related subjects: perturbative Chern-Simons invariants, skein algebras, and conformal blocks. The aim is to explain how abelianization gives a unifying perspective on constructions familiar in each of these subjects (e.g. dilogarithmic formulas for Chern-Simons invariants, vertex models for computing quantum invariants of links, and iterated-fusion constructions of conformal blocks for the Virasoro algebra), and also suggests various extensions, which are just beginning to be explored.

Raoul Bott (9/24/1923 – 12/20/2005) is known for the Bott periodicity theorem, the Morse–Bott functions, and the Borel–Bott–Weil theorem.
