CMSA

12-6-2017 Mathematical Physics Seminar
2017-12-06 14:17 (America/New_York)

11-15-17 RM & PT Seminar
2017-12-15 13:55

Simons Collaboration on Homological Mirror Symmetry
2018-01-02 09:00 - 2018-05-18 17:00
The Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation.
Mirror symmetry, which emerged in the late 1980s as an unexpected physical duality between quantum field theories, has been a major source of progress in mathematics. At the 1994 ICM, Kontsevich reinterpreted mirror symmetry as a deep categorical duality: the HMS conjecture states that the derived category of coherent sheaves of a smooth projective variety is equivalent to the Fukaya category of a mirror symplectic manifold (or Landau-Ginzburg model).
We envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:

categorical algebraic geometry and non-commutative (nc) spaces: in this language, homological mirror symmetry is the statement that the same nc-spaces can arise either from algebraic geometry or from symplectic geometry.
the Strominger-Yau-Zaslow (SYZ) approach, which provides a global geometric prescription for the construction of mirror pairs.
Lagrangian Floer theory and family Floer cohomology, which provide a concrete path from symplectic geometry near a given Lagrangian submanifold to an open domain in a mirror analytic space.

The Center of Mathematical Sciences and Applications is hosting the following short-term visitors for an HMS focused semester:

Jacob Bourjaily (Neils Bohr Institute) 4/1/2018 – 4/14/2018
Colin Diemer (IHES) 2/25/2018 – 3/10/2018
Charles Doran (University of Alberta) 5/13/2018 – 5/25/2018
Baohua Fu (Chinese Academy of Sciences) 4/15/2018 – 4/28/2018
Andrew Harder (University of Miami) 4/15/2018 – 4/28/2018
Shinobu Hosono (Gakushuin University) 2/25/2018 – 3/10/2018
Adam Jacob (UC Davis) 3/5/2018 – 3/16/2018
Tsung-Ju Lee (National Taiwan University) 4/18/2018 – 5/13/2018
Ivan Loseu (Northeastern University) 1/21/2018 – 2/3/2018
Cheuk-Yu Mak (Cambridge University) 4/1/2018 – 4/15/2018
Daniel Pomerleano (Imperial College) 3/19/2018 – 3/23/2018
Mauricio Romo (Tsinghua University) 4/1/2018 – 4/18/ 2018
Emanuel Scheidegger (Albert Ludwigs University of Freiburg) 2/22/2018 – 3/22/2018
Dmytro Shklyarov (Technische Universität Chemnitz) 3/4/2018 – 3/17/2018
Alan Thompson (University of Cambridge) 4/15/2018 – 4/21/2018
Weiwei Wu (University of Georgia) 4/27/2018 – 5/6/2018
Matt Young (Chinese University of Hong Kong) 1/15/2018 – 2/9/2018
Jeng-Daw Yu (National Taiwan University) 4/2/2018 – 4/6/2018
Minxian Zhu (Yau Mathematical Sciences Center, Tsinghua University) 1/ 22/2018 – 2/25/2018

As part of their CMSA visitation, HMS focused visitors will be giving lectures on various topics related to Homological Mirror Symmetry throughout the Spring 2018 Semester. Click here for information.

The Collaboration will include two workshops hosted by The Center. The workshops will take place January 10-13, 2018 and April 5-7, 2018 at CMSA. Click here for more information.

Simons Collaboration Workshop
2018-01-10 09:00 - 2018-01-13 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
The CMSA will be hosting a four-day Simons Collaboration Workshop on Homological Mirror Symmetry and Hodge Theory on January 10-13, 2018. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.

Confirmed Participants:

Mohammed Abouzaid (Columbia University)
Sergueï Barannikov (Paris Diderot University)
Cheol-Hyun Cho (Seoul National University)
Young-Hoon Kiem (Seoul National University)
Thomas Lam (University of Michigan)
Siu-Cheong Lau (Boston University)
Radu Laza (Stony Brook University)
Si Li (Tsinghua University)
Kaoru Ono (Kyoto University)
Tony Pantev (University of Pennsylvania)
Colleen Robles (Duke University)
Yan Soibelman (Kansas State University)
Kazushi Ueda (University of Tokyo)
Chenglong Yu (Harvard University)
Eric Zaslow (Northwestern University)

2020-2021 Colloquium, Wednesdays
2018-01-13 15:25
During the Spring 2021 semester, and until further notice, all seminars will take place virtually.
The 2020-2021 Colloquium will take place every Wednesday from 9:00 to 10:00am ET virtually, using zoom. All CMSA postdocs/members are required to attend the weekly CMSA Members’ Seminars, as well as the weekly CMSA Colloquium series. Please email the seminar organizers to obtain a link. This year’s colloquium will be organized by Wei Gu and Sergiy Verstyuk. The schedule below will be updated as speakers are confirmed.
To learn how to attend, please fill out this form.
Information on previous colloquia can be found here.

Spring 2021:

Date
Speaker
Title/Abstract

1/27/2021
Evelyn Tang (Max Planck Institute for Dynamics and Self-Organization)
Slides

Video
Title: Topology protects chiral edge currents in stochastic systems
Abstract: Living systems can exhibit time-scales much longer than those of the underlying components, as well as collective dynamical behavior. How such global behavior is subserved by stochastic constituents remains unclear. I will present two-dimensional stochastic networks that consist of out-of-equilibrium cycles at the molecular scale and support chiral edge currents in configuration space. I will discuss the topological properties of these networks and their uniquely non-Hermitian features such as exceptional points and vorticity. As these emergent edge currents are associated to macroscopic timescales and length scales, simply tuning a small number of parameters enables varied dynamical phenomena including a global clock, stochastic growth and shrinkage, and synchronization.

2/3/2021
André Luiz de Gouvêa (Northwestern)
Video
Title: The Brave Nu World
Abstract: Neutrinos are the least understood of the fundamental particles that make up the so-called Standard Model of Particle Physics. Measuring neutrino properties and identifying how they inform our understanding of nature at the smallest distant scales is among the highest priorities of particle physics research today. I will discuss our current understanding of neutrinos, concentrating on the observation of neutrino oscillations and neutrino masses, along with all the open questions that came of these discoveries from the end of the 20th century.

2/10/2021
Mykhaylo Shkolnikov (Princeton)
Video
Title: Probabilistic approach to free boundary problems and applications
Abstract: We will discuss a recently developed probabilistic approach to (singular) free boundary problems, such as the supercooled Stefan problem. The approach is based on a new notion of solution, referred to as probabilistic, which arises naturally in the context of large system limits of interacting particle systems. In the talk, I will give an example of how such interacting particle systems arise in applications (e.g., finance), then obtain a solution of a free boundary problem in the large system limit, and discuss how this solution can be analyzed mathematically (thereby answering natural questions about the systemic risk in financial systems and neural synchronization in the brain). The talk is based on recent and ongoing joint works with Sergey Nadtochiy, Francois Delarue, Jiacheng Zhang and Xiling Zhang

2/17/2021
9:00 – 10:00PM ET
C. Seshadhri (UC Santa Cruz)
Video
Title: Studying the (in)effectiveness of low dimensional graph embeddings
Abstract: Low dimensional graph embeddings are a fundamental and popular tool used for machine learning on graphs. Given a graph, the basic idea is to produce a low-dimensional vector for each vertex, such that “similarity” in geometric space corresponds to “proximity” in the graph. These vectors can then be used as features in a plethora of machine learning tasks, such as link prediction, community labeling, recommendations, etc. Despite many results emerging in this area over the past few years, there is less study on the core premise of these embeddings. Can such low-dimensional embeddings effectively capture the structure of real-world (such as social) networks? Contrary to common wisdom, we mathematically prove and empirically demonstrate that popular low-dimensional graph embeddings do not capture salient properties of real-world networks. We mathematically prove that common low-dimensional embeddings cannot generate graphs with both low average degree and large clustering coefficients, which have been widely established to be empirically true for real-world networks. Empirically, we observe that the embeddings generated by popular methods fail to recreate the triangle structure of real-world networks, and do not perform well on certain community labeling tasks. (Joint work with Ashish Goel, Caleb Levy, Aneesh Sharma, and Andrew Stolman.)

2/24/2021
David Ben-Zvi (U Texas)
Video
Title: Electric-Magnetic Duality for Periods and L-functions
Abstract: I will describe joint work with Yiannis Sakellaridis and Akshay Venkatesh, in which ideas originating in quantum field theory are applied to a problem in number theory.
A fundamental aspect of the Langlands correspondence — the relative Langlands program — studies the representation of L-functions of Galois representations as integrals of automorphic forms. However, the data that naturally index the period integrals (spherical varieties for G) and the L-functions (representations of the dual group G^) don’t seem to line up.
We present an approach to this problem via the Kapustin-Witten interpretation of the [geometric] Langlands correspondence as electric-magnetic duality for 4-dimensional supersymmetric Yang-Mills theory. Namely, we rewrite the relative Langlands program as duality in the presence of supersymmetric boundary conditions. As a result the partial correspondence between periods and L-functions is embedded in a natural duality between Hamiltonian actions of the dual groups.

3/3/2021
9:00pm ET
Omer Tamuz (Caltech)
Title: Monotone Additive Statistics
Abstract: How should a random quantity be summarized by a single number? We study mappings from random variables to real numbers, focussing on those with the following two properties: (1) monotonicity with respect to first-order stochastic dominance, and (2) additivity for sums of independent random variables. This problem turns out to be connected to the following question: Under what conditions on the random variables X and Y does there exist an independent Z so that X + Z first-order stochastically dominates Y + Z?
(Joint work with Tobias Fritz, Xiaosheng Mu, Luciano Pomatto and Philipp Strack.)

3/10/2021
9:00pm ET
Piotr Indyk (MIT)
Title: Learning-Based Sampling and Streaming
Abstract: Classical algorithms typically provide “one size fits all” performance, and do not leverage properties or patterns in their inputs. A recent line of work aims to address this issue by developing algorithms that use machine learning predictions to improve their performance. In this talk I will present two examples of this type, in the context of streaming and sampling algorithms. In particular, I will show how to use machine learning predictions to improve the performance of (a) low-memory streaming algorithms for frequency estimation (ICLR’19), and (b) sampling algorithms for estimating the support size of a distribution (ICLR’21). Both algorithms use an ML-based predictor that, given a data item, estimates the number of times the item occurs in the input data set. (The talk will cover material from papers co-authored with T Eden, CY Hsu, D Katabi, S Narayanan, R Rubinfeld, S Silwal, T Wagner and A Vakilian.

3/17/2021
9:00pm ET
Chiu-Chu Melissa Liu (Columbia)
Title: Topological Recursion and Crepant Transformation Conjecture
Abstract: The Crepant Transformation Conjecture (CTC), first proposed by Yongbin Ruan and later refined/generalized by others, relates Gromov-Witten (GW) invariants of K-equivalent smooth varieties or smooth Deligne-Mumford stacks. We will outline a proof of all-genus open and closed CTC for symplectic toric Calabi-Yau 3-orbifolds based on joint work with Bohan Fang, Song Yu, and Zhengyu Zong. Our proof relies on the Remodeling Conjecture (proposed by Bouchard-Klemm-Marino-Pasquetti and proved in full generality by Fang, Zong and the speaker) relating open and closed GW invariants of a symplectic toric Calabi-Yau 3-orbifold to invariants of its mirror curve defined by Chekhov-Eynard-Orantin Topological Recursion.

3/24/2021
Weinan E (Princeton)
Video
Title: Machine Learning and PDEs
Abstract: I will discuss two topics:
(1) Machine learning-based algorithms and “regularity” theory for very high dimensional PDEs;
(2) Formulating machine learning as PDE (more precisely, integral-differental equation) problems.

3/31/2021
Thore Graepel (DeepMind/UCL)
Video
Title: From AlphaGo to MuZero – Mastering Atari, Go, Chess and Shogi by Planning with a Learned Model
Abstract: Constructing agents with planning capabilities has long been one of the main challenges in the pursuit of artificial intelligence. Tree-based planning methods have enjoyed huge success in challenging domains, such as chess and Go, where a perfect simulator is available. However, in real-world problems the dynamics governing the environment are often complex and unknown. In this work we present the MuZero algorithm which, by combining a tree-based search with a learned model, achieves superhuman performance in a range of challenging and visually complex domains, without any knowledge of their underlying dynamics. MuZero learns a model that, when applied iteratively, predicts the quantities most directly relevant to planning: the reward, the action-selection policy, and the value function. When evaluated on 57 different Atari games – the canonical video game environment for testing AI techniques, in which model-based planning approaches have historically struggled – our new algorithm achieved a new state of the art. When evaluated on Go, chess and shogi, without any knowledge of the game rules, MuZero matched the superhuman performance of the AlphaZero algorithm that was supplied with the game rules.

4/7/2021
Kui Ren (Columbia)
Title: Inversion via Optimization: Revisiting the Classical Least-Squares Formulation of Inverse Problems
Abstract: The classical least-squares formulation of inverse problems has provided a successful framework for the computational solutions of those problems. In recent years, modifications and alternatives have been proposed to overcome some of the disadvantages of this classical formulation in dealing with new applications. This talk intends to provide an (likely biased) overview of the recent development in constructing new least-squares formulations for model and data-driven solutions of inverse problems.

4/14/2021
Siu-Cheong Lau (Boston U)
Title: An algebro-geometric formulation of computing machines
Abstract: Neural network in machine learning has obvious similarity with quiver representation theory. The main gap between the two subjects is that network functions produced from two isomorphic quiver representations are not equal, due to the presence of non-linear activation functions which are not equivariant under the automorphism group. This violates the important math/physics principle that isomorphic objects should produce the same results. In this talk, I will introduce a general formulation using moduli spaces of framed modules of (noncommutative) algebra and fix this gap. Metrics over the moduli space are crucial. I will also explain uniformization between spherical, Euclidean and hyperbolic moduli.

4/21/2021
Vasco Carvalho (Cambridge)
Title: The Economy as a Complex Production Network
Abstract: A modern economy is an intricately linked web of specialized production units, each relying on the flow of inputs from their suppliers to produce their own output, which in turn is routed towards other downstream units. From this production network vantage point we: (i) present the theoretical foundations for the role of such input linkages as a shock propagation channel and as a mechanism for transforming micro-level shocks into macroeconomic, economy-wide fluctuations (ii) selectively survey both empirical and simulation-based studies that attempt to ascertain the relevance and quantitative bite of this argument and (time permitting) (iii) discuss a range of domains where this networked production view is currently being extended to.

4/28/2021
9:00 – 10:00pm ET
Shamit Kachru (Stanford)
Slides
Title: K3 Metrics from String Theory
Abstract: Calabi-Yau manifolds have played a central role in important developments in string theory and mathematical physics. Famously, they admit Ricci flat metrics — but the proof of that fact is not constructive, and the metrics remain mysterious. K3 is perhaps the simplest non-trivial compact Calabi-Yau space. In this talk, I describe two different methods of constructing (smooth, Ricci flat) K3 metrics, and a string theory duality which relates them. The duality re-sums infinite towers of disc instanton corrections via a purely classical infinite-dimensional hyperkahler quotient construction, which can be practically implemented.

Fall 2020:

Date
Speaker
Title/Abstract

9/23/2020
David Kazhdan (Hebrew University)
Title: On Applications of Algebraic Combinatorics to Algebraic Geometry
Abstract: I present a derivation of a number of results on morphisms of a high Schmidt’s rank from a result in Algebraic Combinatorics. In particular will explain the flatness of such morphisms and show their fibers have rational singularities.

10/7/2020
10:00am
Mariangela Lisanti (Princeton University)
Video
Title: Mapping the Milky Way’s Dark Matter Halo with Gaia
Abstract: The Gaia mission is in the process of mapping nearly 1% of the Milky Way’s stars—-nearly a billion in total. This data set is unprecedented and provides a unique view into the formation history of our Galaxy and its associated dark matter halo. I will review results based on the most recent Gaia data release, demonstrating how the evolution of the Galaxy can be deciphered from the stellar remnants of massive satellite galaxies that merged with the Milky Way early on. This analysis is an inherently “big data” problem, and I will discuss how we are leveraging machine learning techniques to advance our understanding of the Galaxy’s evolution. Our results indicate that the local dark matter is not in equilibrium, as typically assumed, and instead exhibits distinctive dynamics tied to the disruption of satellite galaxies. The updated dark matter map built from the Gaia data has ramifications for direct detection experiments, which search for the interactions of these particles in terrestrial targets.

10/14/2020
Gil Kalai (Hebrew University and IDC Herzliya)
Video
Title: Statistical, mathematical, and computational aspects of noisy intermediate-scale quantum computers
Abstract: Noisy intermediate-scale quantum (NISQ) Computers hold the key for important theoretical and experimental questions regarding quantum computers. In the lecture I will describe some questions about mathematics, statistics and computational complexity which arose in my study of NISQ systems and are related to
a) My general argument “against” quantum computers,
b) My analysis (with Yosi Rinott and Tomer Shoham) of the Google 2019 “quantum supremacy” experiment.
Relevant papers:
Yosef Rinott, Tomer Shoham and Gil Kalai, Statistical aspects of the quantum supremacy demonstration, https://gilkalai.files.
wordpress.com/2019/11/stat-quantum2.pdf
Gil Kalai, The Argument against Quantum Computers, the Quantum Laws of Nature, and Google’s Supremacy Claims, https://gilkalai.files.
wordpress.com/2020/08/laws-blog2.pdf
Gil Kalai, Three puzzles on mathematics, computations, and games, https://gilkalai.files.
wordpress.com/2019/09/main-pr.pdf

10/21/2020
Marta Lewicka (University of Pittsburgh)
Video
Title: Quantitative immersability of Riemann metrics and the infinite hierarchy of prestrained shell models
Abstract: We propose results that relate the following two contexts:
(i) Given a Riemann metric G on a thin plate, we study the question of what is its closest isometric immersion, with respect to the distance measured by energies E^h which are modifications of the classical nonlinear three-dimensional elasticity.
(ii) We perform the full scaling analysis of E^h, in the context of dimension reduction as the plate’s thickness h goes to 0, and derive the Gamma-limits of h^{-2n}E^h for all n. We show the energy quantization, in the sense that the even powers 2n of h are the only possible ones (all of them are also attained).
For each n, we identify conditions for the validity of the corresponding scaling, in terms of the vanishing of Riemann curvatures of G up to appropriate orders, and in terms of the matched isometry expansions. Problems that we discuss arise from the description of elastic materials displaying heterogeneous incompatibilities of strains that may be associated with growth, swelling, shrinkage, plasticity, etc. Our results display the interaction of calculus of variations,
geometry and mechanics of materials in the prediction of patterns and shape formation.

10/28/2020
Jonathan Heckman (University of Pennsylvania)
Video
Title: Top Down Approach to Quantum Fields
Abstract: Quantum Field theory (QFT) is the common language of particle physicists, cosmologists, and condensed matter physicists. Even so, many fundamental aspects of QFT remain poorly understood. I discuss some of the recent progress made in understanding QFT using the geometry of extra dimensions predicted by string theory, highlighting in particular the special role of seemingly “exotic” higher-dimensional supersymmetric QFTs with no length scales known as six-dimensional superconformal field theories (6D SCFTs). We have recently classified all examples of such 6D SCFTs, and are now using this to extra observables from strongly correlated systems in theories with more than four spacetime dimensions, as well as in spacetimes with four or fewer spacetime dimensions. Along the way, I will also highlight the remarkable interplay between physical and mathematical structures in the study of such systems

11/4/2020
9:00pm ET
Surya Ganguli (Stanford)
Video
Title: Weaving together machine learning, theoretical physics, and neuroscience through mathematics
Abstract: An exciting area of intellectual activity in this century may well revolve around a synthesis of machine learning, theoretical physics, and neuroscience. The unification of these fields will likely enable us to exploit the power of complex systems analysis, developed in theoretical physics and applied mathematics, to elucidate the design principles governing neural systems, both biological and artificial, and deploy these principles to develop better algorithms in machine learning. We will give several vignettes in this direction, including: (1) determining the best optimization problem to solve in order to perform regression in high dimensions; (2) finding exact solutions to the dynamics of generalization error in deep linear networks; (3) developing interpretable machine learning to derive and understand state of the art models of the retina; (4) analyzing and explaining the origins of hexagonal firing patterns in recurrent neural networks trained to path-integrate; (5) delineating fundamental theoretical limits on the energy, speed and accuracy with which non-equilibrium sensors can detect signals
Selected References:
M. Advani and S. Ganguli, Statistical mechanics of optimal convex inference in high dimensions, Physical Review X, 6, 031034, 2016.
M. Advani and S. Ganguli, An equivalence between high dimensional Bayes optimal inference and M-estimation, NeurIPS, 2016.
A.K. Lampinen and S. Ganguli, An analytic theory of generalization dynamics and transfer learning in deep linear networks, International Conference on Learning Representations (ICLR), 2019.
H. Tanaka, A. Nayebi, N. Maheswaranathan, L.M. McIntosh, S. Baccus, S. Ganguli, From deep learning to mechanistic understanding in neuroscience: the structure of retinal prediction, NeurIPS 2019.
S. Deny, J. Lindsey, S. Ganguli, S. Ocko, The emergence of multiple retinal cell types through efficient coding of natural movies, Neural Information Processing Systems (NeurIPS) 2018.
B. Sorscher, G. Mel, S. Ganguli, S. Ocko, A unified theory for the origin of grid cells through the lens of pattern formation, NeurIPS 2019.
Y. Bahri, J. Kadmon, J. Pennington, S. Schoenholz, J. Sohl-Dickstein, and S. Ganguli, Statistical mechanics of deep learning, Annual Reviews of Condensed Matter Physics, 2020.
S.E. Harvey, S. Lahiri, and S. Ganguli, A universal energy accuracy tradeoff in nonequilibrium cellular sensing, https://arxiv.org/abs/2002.10567

11/11/2020
Kevin Buzzard (Imperial College London)
Video
Title: Teaching proofs to computers
Abstract: A mathematical proof is a sequence of logical statements in a precise language, obeying some well-defined rules. In that sense it is very much like a computer program. Various computer tools have appeared over the last 50 years which take advantage of this analogy by turning the mathematical puzzle of constructing a proof of a theorem into a computer game. The newest tools are now capable of understanding some parts of modern research mathematics. In spite of this, these tools are not used in mathematics departments, perhaps because they are not yet capable of telling mathematicians *something new*.
I will give an overview of the Lean theorem prover, showing what it can currently do. I will also talk about one of our goals: using Lean to make practical tools which will be helpful for future researchers in pure mathematics.

11/18/2020
Jose A. Scheinkman (Columbia)
Video
Title: Re-pricing avalanches
Abstract: Monthly aggregate price changes exhibit chronic fluctuations but the aggregate shocks that drive these fluctuations are often elusive. Macroeconomic models often add stochastic macro-level shocks such as technology shocks or monetary policy shocks to produce these aggregate fluctuations. In this paper, we show that a state-dependent pricing model with a large but finite number of firms is capable of generating large fluctuations in the number of firms that adjust prices in response to an idiosyncratic shock to a firm’s cost of price adjustment. These fluctuations, in turn, cause fluctuations in aggregate price changes even in the absence of aggregate shocks. (Joint work with Makoto Nirei.)

11/25/2020
10:45am
Eric J. Heller (Harvard)
Video
Title: Branched Flow
Abstract: In classical and quantum phase space flow, there exists a regime of great physical relevance that is belatedly but rapidly generating a new field. In evolution under smooth, random, weakly deflecting but persistent perturbations, a remarkable regime develops, called branched flow. Lying between the first cusp catastrophes at the outset, leading to fully chaotic statistical flow much later, lies the visually beautiful regime of branched flow. It applies to tsunami wave propagation, freak wave formation, light propagation, cosmic microwaves arriving from pulsars, electron flow in metals and devices, sound propagation in the atmosphere and oceans, the large scale structure of the universe, and much more. The mathematical structure of this flow is only partially understood, involving exponential instability coexisting with “accidental” stability. The flow is qualitatively universal, but this has not been quantified. Many questions arise, including the scale(s) of the random medium, and the time evolution of manifolds and “fuzzy” manifolds in phase space. The classical-quantum (ray-wave) correspondence in this flow is only partially understood. This talk will be an introduction to the phenomenon, both visual and mathematical, emphasizing unanswered questions

12/2/2020
Douglas Arnold (U of Minnesota)
Video
Title: Preserving geometry in numerical discretization
Abstract: An important design principle for numerical methods for differential equations is that the discretizations preserve key geometric, topological, and algebraic structures of the original differential system. For ordinary differential equations, such geometric integrators were developed at the end of the last century, enabling stunning computations in celestial mechanics and other applications that would have been impossible without them. Since then, structure-preserving discretizations have been developed for partial differential equations. One of the prime examples has been the finite element exterior calculus or FEEC, in which the structures to preserve are related to Hilbert complexes underlying the PDEs, the de Rham complex being a canonical example. FEEC has led to highly successful new numerical methods for problems in fluid mechanics, electromagnetism, and other applications which relate to the de Rham complex. More recently, new tools have been developed which extend the applications of FEEC far beyond the de Rham complex, leading to progress in discretizations of problems from solid mechanics, materials science, and general relativity.

12/9/2020
Manuel Blum and Lenore Blum (Carnegie Mellon)
Video
Title: What can Theoretical Computer Science Contribute to the Discussion of Consciousness?
Abstract: The quest to understand consciousness, once the purview of philosophers and theologians, is now actively pursued by scientists of many stripes. We study consciousness from the perspective of theoretical computer science. This is done by formalizing the Global Workspace Theory (GWT) originated by cognitive neuroscientist Bernard Baars and further developed by him, Stanislas Dehaene, and others. We give a precise formal definition of a Conscious Turing Machine (CTM), also called Conscious AI, in the spirit of Alan Turing’s simple yet powerful definition of a computer. We are not looking for a complex model of the brain nor of cognition but for a simple model of (the admittedly complex concept of) consciousness.
After formally defining CTM, we give a formal definition of consciousness in CTM. We then suggest why the CTM has the feeling of consciousness. The reasonableness of the definitions and explanations can be judged by how well they agree with commonly accepted intuitive concepts of human consciousness, the range of related concepts that the model explains easily and naturally, and the extent of the theory’s agreement with scientific evidence

2018 HMS Focused Lecture Series
2018-01-23 17:00
As part of their CMSA visitation, HMS focused visitors will be giving lectures on various topics related to Homological Mirror Symmetry throughout the Spring 2018 Semester. The lectures will take place on Tuesdays and Thursdays in the CMSA Building, 20 Garden Street, Room G10.
The schedule will be updated below.

Date
Speaker
Title/Abstract

January 23, 25, 30 and February 1
3-5pm
*Room G10*
Ivan Losev
(Northeastern)
Title: BGG category O: towards symplectic duality
Abstract: We will discuss a very classical topic in the representation theory of semisimple Lie algebras: the Bernstein-Gelfand-Gelfand (BGG) category O. Our aim will be to motivate and state a celebrated result of Beilinson, Ginzburg and Soergel on the Koszul duality for such categories, explaining how to compute characters of simple modules (the Kazhdan-Lusztig theory) along the way. The Koszul duality admits a conjectural generalization (Symplectic duality) that is a Mathematical manifestation of 3D Mirror symmetry. We will discuss that time permitting.
Approximate (optimistic) plan of the lectures:
1) Preliminaries and BGG category O.
2) Kazhdan-Lusztig bases. Beilinson-Bernstein localization theorem.
3) Localization theorem continued. Soergel modules.
4) Koszul algebras and Koszul duality for categories O.
Time permitting: other instances of Symplectic duality.
Prerequisites:
Semi-simple Lie algebras and their finite dimensional representation theory.
Some Algebraic geometry. No prior knowledge of category O/ Geometric
Representation theory is assumed.
Scanned from a Xerox Multifunction Device

February 27,
and March 1
3-5pm
Colin Diemer
(IHES)
Title: Moduli spaces of Landau-Ginzburg models and (mostly Fano) HMS.
Abstract: Mirror symmetry as a general phenomenon is understood to take place near the large complex structure limit resp. large radius limit, and so implicitly involves degenerations of the spaces under consideration. Underlying most mirror theorems is thus a mirror map which gives a local identification of respective A-model and B-model moduli spaces. When dealing with mirror symmetry for Calabi-Yau’s the role of the mirror map is well-appreciated. In these talks I’ll discuss the role of moduli in mirror symmetry of Fano varieties (where the mirror is a Landau-Ginzburg (LG) model). Some topics I expect to cover are a general structure theory of moduli of LG models (follows Katzarkov, Kontsevich, Pantev), the interplay of the topology of LG models with autoequivalence relations in the Calabi-Yau setting, and the relationship between Mori theory in the B-model and degenerations of the LG A-model. For the latter topic we’ll focus on the case of del Pezzo surfaces (due to unpublished work of Pantev) and the toric case (due to the speaker with Katzarkov and G. Kerr). Time permitting, we may make some speculations on the role of LG moduli in the work of Gross-Hacking-Keel (in progress work of the speaker with T. Foster).

March 6 and 8
4-5pm
Adam Jacob
(UC Davis)
Title: The deformed Hermitian-Yang-Mills equation
Abstract: In this series I will discuss the deformed Hermitian-Yang-Mills equation, which is a complex analogue of the special Lagrangian graph equation of Harvey-Lawson. I will describe its derivation in relation to the semi-flat setup of SYZ mirror symmetry, followed by some basic properties of solutions. Later I will discuss methods for constructing solutions, and relate the solvability to certain geometric obstructions. Both talks will be widely accessible, and cover joint work with T.C. Collins and S.-T. Yau.

March 6, 8, 13, 15
3-4pm
Dmytro Shklyarov
(TU Chemnitz)
Title: On categories of matrix factorizations and their homological invariants
Abstract: The talks will cover the following topics:
1. Matrix factorizations as D-branes. According to physicists, the matrix factorizations of an isolated hypersurface singularity describe D-branes in the Landau-Ginzburg (LG) B-model associated with the singularity. The talk is devoted to some mathematical implications of this observation. I will start with a review of open-closed topological field theories underlying the LG B-models and then talk about their refinements.
2. Semi-infinite Hodge theory of dg categories. Homological mirror symmetry asserts that the “classical” mirror correspondence relating the number of rational curves in a CY threefold to period integrals of its mirror should follow from the equivalence of the derived Fukaya category of the first manifold and the derived category of coherent sheaves on the second one. The classical mirror correspondence can be upgraded to an isomorphism of certain Hodge-like data attached to both manifolds, and a natural first step towards proving the assertion would be to try to attach similar Hodge-like data to abstract derived categories. I will talk about some recent results in this direction and illustrate the approach in the context of the LG B-models.
3. Hochschild cohomology of LG orbifolds. The scope of applications of the LG mod- els in mirror symmetry is significantly expanded once we include one extra piece of data, namely, finite symmetry groups of singularities. The resulting models are called orbifold LG models or LG orbifolds. LG orbifolds with abelian symmetry groups appear in mir- ror symmetry as mirror partners of varieties of general type, open varieties, or other LG orbifolds. Associated with singularities with symmetries there are equivariant versions of the matrix factorization categories which, just as their non-equivariant cousins, describe D-branes in the corresponding orbifold LG B-models. The Hochschild cohomology of these categories should then be isomorphic to the closed string algebra of the models. I will talk about an explicit description of the Hochschild cohomology of abelian LG orbifolds.

April 10 & 12
3-4pm
Mauricio Romo
(IAS)
Title: Gauged Linear Sigma Models, Supersymmetric Localization and Applications
Abstract: In this series of lectures I will review various results on connections between gauged linear sigma models (GLSM) and mathematics. I will start with a brief introduction on the basic concepts about GLSMs, and their connections to quantum geometry of Calabi-Yaus (CY). In the first lecture I will focus on nonperturbative results on GLSMs on closed 2-manifolds, which provide a way to extract enumerative invariants and the elliptic genus of some classes of CYs. In the second lecture I will move to nonperturbative results in the case where the worldsheet is a disk, in this case nonperturbative results provide interesting connections with derived categories and stability conditions. We will review those and provide applications to derived functors and local systems associated with CYs. If time allows we will also review some applications to non-CY cases (in physics terms, anomalous GLSMs).
Lecture notes

April 17, 19, 26
3-5pm
Andrew Harder
(University of Miami)
Title: Perverse sheaves of categories on surfaces
Abstract: Perverse sheaves of categories on a Riemann surface S are systems of categories and functors which are encoded by a graphs on S, and which satisfy conditions that resemble the classical characterization of perverse sheaves on a disc.
I’ll review the basic ideas behind Kapranov and Schechtman’s notion of a perverse schober and generalize this to perverse sheaves of categories on a punctured Riemann surface. Then I will give several examples of perverse sheaves of categories in both algebraic geometry, symplectic geometry, and category theory. Finally, I will describe how one should be able to use related ideas to prove homological mirror symmetry for certain noncommutative deformations of projective 3-space.

May 15, 17
1-3pm
Charles Doran
(University of Alberta)

Lecture One:
Title: Picard-Fuchs uniformization and Calabi-Yau geometry
Abstract:

Part 1: We introduce the notion of the Picard-Fuchs equations annihilating periods in families of varieties, with emphasis on Calabi-Yau manifolds. Specializing to the case of K3 surfaces, we explore general results on “Picard-Fuchs uniformization” of the moduli spaces of lattice-polarized K3 surfaces and the interplay with various algebro-geometric normal forms for these surfaces. As an application, we obtain a universal differential-algebraic characterization of Picard rank jump loci in these moduli spaces.

Part 2: We next consider families with one natural complex structure modulus, (e.g., elliptic curves, rank 19 K3 surfaces, b_1=4 Calabi-Yau threefolds, …), where the Picard-Fuchs equations are ODEs. What do the Picard-Fuchs ODEs for such families tell us about the geometry of their total spaces? Using Hodge theory and parabolic cohomology, we relate the monodromy of the Picard-Fuchs ODE to the Hodge numbers of the total space. In particular, we produce criteria for when the total space of a family of rank 19 polarized K3 surfaces can be Calabi-Yau.

Lecture Two:
Title: Calabi-Yau fibrations: construction and classification
Abstract:
Part 1: Codimension one Calabi-Yau submanifolds induce fibrations, with the periods of the total space relating to those of the fibers and the structure of the fibration. We describe a method of iteratively constructing Calabi-Yau manifolds in tandem with their Picard-Fuchs equations. Applications include the tower of mirrors to degree n+1 hypersurfaces in P^n and a tower of Calabi-Yau hypersurfaces encoding the n-sunset Feynman integrals.
Part 2: We develop the necessary theory to both construct and classify threefolds fibered by lattice polarized K3 surfaces. The resulting theory is a complete generalization to threefolds of that of Kodaira for elliptic surfaces. When the total space of the fibration is a Calabi-Yau threefold, we conjecture a unification of CY/CY mirror symmetry and LG/Fano mirror symmetry by mirroring fibrations as Tyurin degenerations. The detailed classification of Calabi-Yau threefolds with certain rank 19 polarized fibrations provides strong evidence for this conjecture by matching geometric characteristics of the fibrations with features of smooth Fano threefolds of Picard rank 1.

Blockchain Conference
2018-01-24 09:00 - 2018-01-25 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
On January 24-25, 2019 the Center of Mathematical Sciences will be hosting a conference on distributed-ledger (blockchain) technology. The conference is intended to cover a broad range of topics, from abstract mathematical aspects (cryptography, game theory, graph theory, theoretical computer science) to concrete applications (in accounting, government, economics, finance, management, medicine). The talks will take place in Science Center, Hall D.
https://youtu.be/FyKCCutxMYo
Photos

Speakers:

Joseph Abadi, Princeton University
Benedikt Bunz, Stanford University
Jake Cacciapaglia, Nebula Genomics/Harvard Medical School
Eduardo Castello, Massachusetts Institute of Technology
Alisa DiCaprio, R3
Zhiguo He, University of Chicago
Steven Kou, Boston University
Anne Lafarre, Tilburg University
Jacob Leshno, University of Chicago
Bruce Schneier, Harvard Kennedy School
David Schwartz, Ripple
Elaine Shi, Cornell University/Thunder Research
Hong Wan, NCSU

Quantum Cohomology, Nakajima Varieties and Quantum groups
2018-01-25 14:21
During the Spring 2018 Semester Artan Sheshmani (QGM/CMSA) will be teaching a CMSA special lecture series on Quantum Cohomology, Nakajima Vareties and Quantum groups. The lectures will be held Tuesdays and Thursdays beginning January 25th, from 1:00 to 3:00pm in room G10, CMSA Building.
You can watch Prof. Sheshmani describe the series here.
The Syllabus is as follows:

Date………..
Topic
Video/Audio

1-25-2018
Gromov-Witten invariants
Definition, examples via algebraic geometry I
Video / Audio / Combined

*due to technical difficulties the audio and video are split for this lecture.

2-01-2018
Gromov-Witten invariants
Virtual Fundamental Class I (definition)
Video / Audio / Combined

*due to technical difficulties the audio and video are split for this lecture

2-13-2018
Gromov-Witten invariants
Virtual Fundamental Class II (computation in some cases)

2-15-2018
Computing GW invariants
Three level GW classes
Genus zero invariants of the projective plane

2-20-2018
Quantum Cohomology
Small Quantum Cohomology (Definition and Properties) I

2-22-2018
Quantum Cohomology
Small Quantum Cohomology (Definition and Properties) II

2-27-2018
Quantum Cohomology
Big Quantum Cohomology I

3-1-2018
Quantum Cohomology
Big Quantum Cohomology II
GW potential
WDVV equation

3-6-2018
GW invariants via Quantum Cohomology
The Quintic threefold case
The P^2 case

GW invariants via Quantum Cohomology
Dubrovin (quantum) connection

Nakajima varieties
-Algebraic and symplectic reduction

Nakajima varieties
Quasi maps to Nakajima varieties

Quantum cohomology of Nakajima varieties
Small Quantum Cohomology of Hilb^n (C2) I

Quantum cohomology of Nakajima varieties
Small Quantum Cohomology of Hilb^n (C2) II

Quantum cohomology of Nakajima varieties
Small Quantum Cohomology of Hilb^n (C2) III

Quantum cohomology of Nakajima varieties
Big Quantum Cohomology of Hilb^n (C2) I

Quantum cohomology of Nakajima varieties
Big Quantum Cohomology of Hilb^n (C2) II

Quantum cohomology of Nakajima varieties
Big Quantum Cohomology of Hilb^n (C2) III

Quantum cohomology of Nakajima varieties
Big Quantum Cohomology of Hilb^n (C2) IV

01-26-2018 Mirror Symmetry Seminar
2018-01-26 14:37

1-29-2018 Mathematical Physics Seminar
2018-01-29 14:42

2-2-2018 Mirror Symmetry Seminar
2018-02-02 14:44

Workshop on Probabilistic and Extremal Combinatorics
2018-02-05 09:00 - 2018-02-09 17:00 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
The workshop on Probabilistic and Extremal Combinatorics will take place February 5-9, 2018 at the Center of Mathematical Sciences and Applications, located at 20 Garden Street, Cambridge, MA.
Extremal and Probabilistic Combinatorics are two of the most central branches of modern combinatorial theory. Extremal Combinatorics deals with problems of determining or estimating the maximum or minimum possible cardinality of a collection of finite objects satisfying certain requirements. Such problems are often related to other areas including Computer Science, Information Theory, Number Theory and Geometry. This branch of Combinatorics has developed spectacularly over the last few decades. Probabilistic Combinatorics can be described informally as a (very successful) hybrid between Combinatorics and Probability, whose main object of study is probability distributions on discrete structures.
There are many points of interaction between these fields. There are deep similarities in methodology. Both subjects are mostly asymptotic in nature. Quite a few important results from Extremal Combinatorics have been proven applying probabilistic methods, and vice versa. Such emerging subjects as Extremal Problems in Random Graphs or the theory of graph limits stand explicitly at the intersection of the two fields and indicate their natural symbiosis.
The symposia will focus on the interactions between the above areas. These topics include Extremal Problems for Graphs and Set Systems, Ramsey Theory, Combinatorial Number Theory, Combinatorial Geometry, Random Graphs, Probabilistic Methods and Graph Limits.
Participation: The workshop is open to participation by all interested researchers, subject to capacity.
Confirmed participants include:

Jozsef Balogh, University of Illinois, Urbana
Fan Chung (Graham), University of California, San Diego
Asaf Ferber, Massachusetts Institute of Technology
Jacob Fox, Stanford Unviersity
David Gamarnik, Massachusetts Institute of Technology
Penny Haxell, University of Waterloo
Hao Huang, Emory University
Jeff Kahn, Rutgers University
Peter Keevash, Oxford University
Michael Krivelevich, Tel Aviv University
Daniela Kühn, University of Birmingham
Shoham Letzer, ITS Zürich
Shachar Lovett, University of California, San Diego
Eyal Lubetzky, Courant Institute
Rob Morris, IMPA
Bhargav Narayanan, Rutgers University
Deryk Osthus, University of Birmingham
Janos Pach, NYU
Yuval Peres, Microsoft Redmond
Alexey Pokryovskyi, ETH Zürich
Wojciech Samotij, Tel Aviv University
Lisa Sauermann, Stanford University
Mathias Schacht, University of Hamburg
Alexander Scott, University of Oxford
Asaf Shapira, Tel Aviv University
Jozef Skokan, London School of Economics
Joel Spencer, New York University
Angelika Steger, ETH Zurich
Jacques Verstraete, University of California, San Diego
Yufei Zhao, Massachusetts Institute of Technology
David Zuckerman, University of Texas at Austin

Co-organizers of this workshop include Benny Sudakov and David Conlon. More details about this event, including participants, will be updated soon.

1-5-2018 Mathematical Physics Seminar
2018-02-05 12:00 - 13:00

Sequences: random, structured or something in between
2018-02-08 17:00 - 18:00
Speaker: Fan Chung (University of California, San Diego)
Title: Sequences: random, structured or something in between
Abstract: There are many fundamental problems concerning sequences that arise in many areas of mathematics and computation. Typical problems include finding or avoiding patterns; testing or validating various ‘random-like’ behavior; analyzing or comparing different statistics, etc. In this talk, we will examine various notions of regularity or irregularity for sequences and mention numerous open problems.

02-09-2018 Mirror Symmetry Seminar
2018-02-09 11:00

A new program on quantum subgroups
2018-02-14 16:30 - 17:30
Speaker: Zhengwei Liu (Harvard Physics)
Title: A new program on quantum subgroups
Abstract: Quantum subgroups have been studied since the 1980s. The A, D, E classification of subgroups of quantum SU(2) is a quantum analogue of the McKay correspondence. It turns out to be related to various areas in mathematics and physics. Inspired by the quantum McKay correspondence, we introduce a new program that our group at Harvard is developing.

02-16-2018 Special Seminar
2018-02-16 09:30

2-16-2018 Mirror Symmetry Seminar
2018-02-16 11:00

2-16-2018 RM & PT Seminar
2018-02-16 15:09 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States

Essential concepts of Causal inference—a remarkable history
2018-02-21 16:30 - 17:30 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Don Rubin (Harvard Statistics)
Title: Essential concepts of Causal inference—a remarkable history
Abstract: I believe that a deep understanding of cause and effect, and how to estimate causal effects from data, complete with the associated mathematical notation and expressions, only evolved in the twentieth century. The crucial idea of randomized experiments was apparently first proposed in 1925 in the context of agricultural field trails but quickly moved to be applied also in studies of animal breeding and then in industrial manufacturing. The conceptual understanding seemed to be tied to ideas that were developing in quantum mechanics. The key ideas of randomized experiments evidently were not applied to studies of human beings until the 1950s, when such experiments began to be used in controlled medical trials, and then in social science — in education and economics. Humans are more complex than plants and animals, however, and with such trials came the attendant complexities of non-compliance with assigned treatment and the occurrence of “hawthorne” and placebo effects. The formal application of the insights from earlier simpler experimental settings to more complex ones dealing with people, started in the 1970s and continue to this day, and include the bridging of classical mathematical ideas of experimentation, including fractional replication and geometrical formulations from the early twentieth century, with modern ideas that rely on powerful computing to implement aspects of design and analysis.

2-23-2018 Mirror Symmetry Seminar
2018-02-23 10:00 - 11:00

2-23-2018 RM & PT Seminar
2018-02-23 15:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States

2-26-2018 Mathematical Physics Seminar
2018-02-26 15:42 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States

Computer-assisted analysis of singularity formation of a regularized 3D Euler equation
2018-02-26 16:30 - 17:30
Speaker: Tom Hou, Caltech
Title: Computer-assisted analysis of singularity formation of a regularized 3D Euler equation

2-27-2018 HMS Lecture
2018-02-27 15:00 - 16:00

3-2-2018 Mirror Symmetry Seminar
2018-03-02 11:00 - 12:00

3-5-2018 Mathematical Physics Seminar
2018-03-05 15:27 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States

Harmonic functions and the chromatic polynomial
2018-03-07 16:30 - 17:30
Speaker: Richard Kenyon, Brown
Title: Harmonic functions and the chromatic polynomial

Dmytro Shklyrov HMS Focused Lecture Series
2018-03-15 15:00 - 16:00

3/19/2018 Mathematical Physics Seminar
2018-03-19 12:00 - 13:30
