The Extended Vertex Algebra of 4d N = 2 SCFTs and their Higher Products
2025-04-14 15:00 - 16:00 (America/New_York) | Hybrid – G10
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Mitch Weaver, KAIST
Title: The Extended Vertex Algebra of 4d N = 2 SCFTs and their Higher Products
Abstract: Every 4d N=2 superconformal field theory contains a BPS protected sub-algebra of local operators that has the structure of a vertex operator algebra (VOA). This VOA is identified by passing to the cohomology of a nilpotent supercharge, T, whose local operator cohomology is represented by twist-translated Schur operators with support in a Euclidean two-plane. When working in 4d Minkowski space, this cohomology admits a web of three extended operators (called descent operators) that are constructed from each Schur operator in the VOA, have worldvolume support in the Lorentzian two-plane that is transverse to the Euclidean plane supporting the VOA, and behave as point-like insertions in the plane of the VOA, i.e. as new chiral operators. The combined result is the extended vertex algebra (EVA): a universal extension of the VOA that canonically has the structure of a quasi-VOA, i.e. a vertex algebra (VA) with no conformal vector but which still possesses a representation of sl(2). After reviewing the VOA of Schur operators, I will explain the origin of the descent operators and present the OPEs for a subsector of the EVA in the free hyper SCFT.
Time permitting, I will also describe the construction and basic properties of a set of higher products that are associated to each descent operator. Such products function as higher dimensional versions of 2d chiral algebra λ-brackets, i.e. positive mode operators: they are defined on the EVA and map to the operators appearing in the singular terms of OPEs involving descent operators. Their existence offers a route toward sl(2) symmetry enhancement of the EVA and suggests the latter has structural properties that are common to the higher dimensional chiral algebras describing the minimal twists of 3d N = 2 and 4d N = 1 SQFTs. This talk is based on [2211.04410] and forthcoming work.

Quantum K-theory at roots of unity
2025-04-14 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Andrey Smirnov, University of North Carolina at Chapel Hill
Title: Quantum K-theory at roots of unity
Abstract: In this talk, I will discuss a version of quantum K-theory introduced by A.Okounkov, which can be defined through quasimap counts. In this framework, the quantum K-theory ring is obtained as a specialization of the equivariant quasimap count at $q=1$, where $q$ is the equivariant parameter associated with the torus action on the source of the quasimaps. A related, but less explored, structure emerges when $q$ is specialized at the roots of unity. I will outline the key ideas behind this construction and its implications. As an application, I’ll also describe the spectrum of $p$-curvature for the quantum connection, which offers a new proof of a recent result by P.Etingof and A.Varchenko. This talk is based on joint work with P. Koroteev.

Positive mass theorem for ALE(AE) and ALF(AF) Toric 4-Manifolds
2025-04-15 11:00 - 12:00 | CMSA G102, 20 Garden Street, Cambridge, MA, 02138
General Relativity Seminar
Speaker: Aghil Alaee, Clark University
Title: Positive mass theorem for ALE(AE) and ALF(AF) Toric 4-Manifolds
Abstract: One of the fundamental conjectures in mathematical relativity is the positivity of total mass (if it is defined!) for complete non-compact Riemannian manifolds assuming appropriate lower bounds on scalar curvature. This conjecture has been proved for AE manifolds using several techniques, starting with the celebrated results of Schoen-Yau and Witten. There are counter-examples to this conjecture in the AF, ALF, and ALE cases. In this talk, we will refine this conjecture and prove it for toric 4-manifolds. The proof is robust and can be extended to higher dimensions if additional assumptions are added. This is a joint work with Marcus Khuri and Hari Kunduri.

Instantons on Taub-NUT space and Bow Construction
2025-04-15 16:15 - 18:15 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Anadil Saeed Rao, Northeastern University
Title: Instantons on Taub-NUT space and Bow Construction
Abstract: In this talk I will discuss Yang-Mills Instantons in a very special geometric setting: the Taub-NUT space, which itself is a Gravitational Instanton. In the first part of the talk I will review the essentials of gravitational instanton theory from the differential geometry/physics perspective and discuss the geometry of Taub-NUT space in detail. Then I will review the essentials of the ADHM-Nahm construction which describe Instantons and Monopoles in Euclidean R^4 and R^3 respectively. In the second part of my talk I will extend the ADHM-Nahm description to Instantons on Taub-NUT space and show how the ADHM-Nahm data may be presented in terms of combinatorial diagrams called Bows and their representations.

CMSA Q&A Seminar: Noah Golowich
2025-04-16 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Noah Golowich (MIT)
Topic: What is length generalization in large language models?

Fifth Annual Yip Lecture | Scott Aaronson (UT Austin): How Much Math Is Knowable?
2025-04-17 16:00 - 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Speaker: Scott Aaronson, Department of Computer Science, University of Texas, Austin
Scott Aaronson is the founding director at the Quantum Information Center at the University of Texas at Austin.
Date: April 17, 2025
Time: 4:00-5:00 pm ET (Reception following in the Math Common Room)
Location: Harvard Science Center Hall A

Title: How Much Math Is Knowable?
Abstract: Theoretical computer science has over the years sought more and more refined answers to the question of which mathematical truths are knowable by finite beings like ourselves, bounded in time and space and subject to physical laws. I’ll tell a story that starts with Gödel’s Incompleteness Theorem and Turing’s discovery of uncomputability. I’ll then introduce the spectacular Busy Beaver function, which grows faster than any computable function. Work by me and Yedidia, along with recent improvements by O’Rear and Riebel, has shown that the value of BB(745) is independent of the axioms of set theory; on the other end, an international collaboration proved last year that BB(5) = 47,176,870. I’ll speculate on whether BB(6) will ever be known, by us or our AI successors. I’ll next discuss the P!=NP conjecture and what it does and doesn’t mean for the limits of machine intelligence. As my own specialty is quantum computing, I’ll summarize what we know about how scalable quantum computers, assuming we get them, will expand the boundary of what’s mathematically knowable. I’ll end by talking about hypothetical models even beyond quantum computers, which might expand the boundary of knowability still further, if one is able (for example) to jump into a black hole, create a closed timelike curve, or project oneself onto the holographic boundary of the universe.

The Yip Lecture takes place thanks to the support of Dr. Shing-Yiu Yip.

The previous Yip Lecture featured Josh Tenenbaum (MIT) who spoke on How to grow a mind from a brain: From guessing and betting to thinking and talking

Member Seminar
2025-04-18 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Han Shao

Modeling the emergence of complex cortical structure from simple precursors in the brain: maps, hierarchies, and modules
2025-04-21 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Ila Fiete, MIT
Title: Modeling the emergence of complex cortical structure from simple precursors in the brain: maps, hierarchies, and modules
Abstract: Modular and hierarchical structures are ubiquitous in the brain. Two distinct hypotheses for such morphogenesis involve genetic specification (the positional information hypothesis) or spontaneous structure emergence from symmetry breaking (the pattern formation hypothesis). Indeed, there is rich evidence supporting both hypotheses in different systems, and more recently evidence that both systems might interact, for instance with genetic specification providing an initial but relatively low-information scaffold of positional guidance and pattern formation constructing sharper structures by bootstrapping from this guidance. In this talk, I will consider the emergence of two systems in the brain: the visual processing hierarchy with topographic structure, and a modular cognitive circuit consisting of functionally independent grid cell networks that compute spatial location from velocity cues as animals move and navigate the world. I will describe how simple activity-driven growth and competition rules can lead to the emergence of topographically ordered sensory processing hierarchies, and how genetically specified smooth gradients with purely local recurrent interactions on two scales can lead to global module emergence. In sum, simple growth rules, local interactions and smooth gradients can interact to produce rich emergent order on multiple scales in the form of maps, modules, and hierarchies, with predictions that bridge scales from genes to connectivity to function.

Hyperbolic equations in a double null gauge
2025-04-22 11:00 - 12:00 | CMSA G102, 20 Garden Street, Cambridge, MA, 02138
General Relativity Seminar
Speaker: Christopher Stith, University of Michigan
Title: Hyperbolic equations in a double null gauge
Abstract: The hyperbolic nature of the Einstein equations is well-known and has been used in many different contexts. More recently, the double null gauge has proven to be a powerful tool for quantitative analysis of the Einstein equations. It has the advantage of reducing the equations for many dynamical quantities to ODEs along null curves, and the Bianchi equations to a first-order hyperbolic system. The double null gauge has been used extensively and to great effect in analyzing the structure of spacetime for many purposes, including (for instance) stability problems and trapped surface formation. However, the local existence problem for the Einstein equations in a double null gauge has never been treated in full in its own right. In this talk, we discuss how to formulate a general procedure for solving the linearized problem, namely, the local existence theory for systems of first-order hyperbolic equations in a double null gauge.

Classifying Space for Phases of Matrix Product States
2025-04-22 16:15 - 18:15 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speakers: Bryan Wang Peng Jun (Harvard) and Bowen Yang (CMSA)
Title: Classifying Space for Phases of Matrix Product States
Abstract: In this talk we will introduce recent work of Beaudry-Hermele-Pflaum-Qi-Spiegel-Stephen (arXiv:2501.14241), on constructing a classifying space for phases of matrix product states (MPS).

CMSA Q&A Seminar: Alexei Borodin
2025-04-23 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Alexei Borodin (MIT)
Topic: Connections between physics and probability

Machine learning for analytic calculations in theoretical physics
2025-04-23 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Matthias Wilhelm (University of Southern Denmark)
Title: Machine learning for analytic calculations in theoretical physics
Abstract: In this talk, we will present recent progress on applying machine-learning techniques to improve calculations in theoretical physics, in which we desire exact and analytic results. One example are so-called integration-by-parts reductions of Feynman integrals, which pose a frequent bottleneck in state-of-the-art calculations in theoretical particle and gravitational-wave physics. These reductions rely on heuristic approaches for selecting a finite set of linear equations to solve, and the quality of the heuristics heavily influences the performance. In this talk, we investigate the use of machine-learning techniques to find improved heuristics. We use funsearch, a genetic programming variant based on code generation by a Large Language Model, in order to explore possible approaches, then use strongly typed genetic programming to zero in on useful solutions. Both approaches manage to re-discover the state-of-the-art heuristics recently incorporated into integration-by-parts solvers, and in one example find a small advance on this state of the art.

Mass gap in AdS space
2025-04-24 10:00 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Ziming Ji, Northeastern University
Title: Mass gap in AdS space
Abstract: AdS space can be used as an IR regulator of QFT. The asymptotic conformal boundary in AdS space provides rich, unique observables. We study asymptotic free theories in two-dimensional AdS space. By changing the AdS curvature scale Lambda L, we observe boundary signals of quantum phase transitions where mass gaps are dynamically generated in the bulk. We also utilize supersymmetry to study gauge theories in AdS4. We argue a connection between the AdS partition function and the prepotential and use the F-maximization of the Nekrasov partition function to study supersymmetric boundary conditions and its connection to the Seiberg-Witten theory.

Adversarial KA
2025-04-25 03:00 - 16:00 | Virtual
Freedman CMSA Seminar
Speaker: Slava Dzhenzher, MIPT
Title: Adversarial KA
Abstract: Regarding the representation theorem of Kolmogorov and Arnold (KA) as an algorithm for representing or «expressing» functions, we test its robustness by analyzing its ability to withstand adversarial attacks. We find KA to be robust to countable collections of continuous adversaries, but unearth a question about the equi-continuity of the outer functions that, so far, obstructs taking limits and defeating continuous groups of adversaries. This question on the regularity of the outer functions is relevant to the debate over the applicability of KA to the general theory of NNs. Based on https://arxiv.org/abs/2504.05255

Bass-Note Spectra of locally uniform geometries
2025-04-28 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Peter Sarnak, IAS & Princeton University
Title: Bass-Note Spectra of locally uniform geometries
Abstract: We formulate and report on the problem of the Bass-Note Spectrum of an invariant operator as one varies over locally uniform geometries. In the Euclidean setting this recasts classical problems of Mahler from the geometry of numbers in a new light. For certain operators homogeneous dynamics can be used decisively. In the non-Euclidean setting of hyperbolic manifolds we review some recent developments using the conformal bootstrap method and of random covers to study the Bass-Note spectra. We highlight the theme and impact of rigidity.

Algebraic approach to the bow construction
2025-04-29 16:15 - 18:15 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speakers: Anadil Saeed Rao (Northeastern) and Vasily Krylov (CMSA/Harvard)
Title: Algebraic approach to the bow construction
Abstract: Anadil will finish his part of the talk. Vasily will then explain how to understand moduli spaces of instantons on R^4, monopoles on R^3, and instantons on Taub-NUT spaces algebraically as so-called bow varieties. We will see how affine type A Nakajima quiver varieties and Slodowy slices arise as special cases of bow varieties. Time permitting, we will also discuss applications to 3D mirror symmetry and Coulomb branches.

From superspace to twisted supergravity
2025-05-01 10:00 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Fabian Hahner, University of Washington
Title: From superspace to twisted supergravity
Abstract: In this talk, I will present a geometric perspective on the pure spinor superfield formalism, which proves fruitful for studying twisted supergravity. For eleven-dimensional supergravity, we use this technique to construct the full interacting theory together with all its twists in a uniform and geometric way as homotopy Poisson–Chern–Simons theories. In addition to simplifying the computation of twists immensely, this also provides fresh insights into the supergeometric origin of supergravity. Building on these ideas, we further construct local dg Lie algebras that recover conformal supergravity multiplets and their twists in terms of a geometric moduli problem on superspace.

Incentives for data sharing in federated learning
2025-05-02 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Han Shao, Harvard CMSA
Title: Incentives for data sharing in federated learning
Abstract: Federated learning has recently emerged as a powerful approach for enabling collaboration across large populations of learning agents. However, agents may have incentives to defect from the collaboration—that is, to withdraw or contribute less data than expected—due to the costs of data curation and privacy concerns. This raises several key questions: What happens when agents defect, and how can we prevent such defections?

Quantum Field Theory and Physical Mathematics
2025-05-05 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar

Thinking Outside the Ballot Box
2025-05-05 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Ariel Procaccia, Harvard University
Title: Thinking Outside the Ballot Box
Abstract: How should one design unprecedented democratic processes capable of handling enormous sets of alternatives like all possible policies, bills, or statements? I argue that this challenge can be addressed through a framework called generative social choice, which fuses the rigor of social choice theory with the flexibility and power of large language models. I then explore an application of generative social choice to the problem of identifying a proportionally representative slate of opinion statements. This includes a discussion of desired properties, an algorithm that provably achieves them, an implementation using GPT, and insights from an end-to-end pilot. By providing guarantees, generative social choice could alleviate concerns about AI-driven democratic innovation and help unlock its potential.

Geometry and Quantum Theory Seminar
2025-05-06 16:15 - 18:15 | Science Center 507, 1 Oxford Street, Cambridge, 02138

Residues and homotopy Lie algebras
2025-05-08 10:00 - 11:00 | Virtual
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Zhenping Gui, Shanghai Institute for Mathematics and Interdisciplinary Sciences
Title: Residues and homotopy Lie algebras
Abstract: I will introduce the notion of a chiral operad for any compact Riemann surface. This operad consists of compositions of residue operations, which give rise to the Chevalley-Cousin complex and lead to the definition of chiral homology (derived conformal blocks). I will explain how to use this machinery to rigorously define certain Feynman integrals in 2D chiral CFTs. Subsequently, I will present a polysimplicial construction of a series of chain models for the configuration space of points in an affine space and study residue operations. These residue operations can be described by a homotopy Lie algebra structure, and the latter defines a higher-dimensional analog of the Chevalley-Cousin complex. This is based on joint work in progress with Charles Young and Laura Felder.

Asset pricing with heterogeneous agents
2025-05-09 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Sergiy Verstyuk, Harvard CMSA
Title: Asset pricing with heterogeneous agents
Abstract: This talk will introduce the basics of continuous-time finance, discuss important existing theories and models, as well as present some new asset pricing results in a setting with many heterogeneous investors. (Joint work with Puskar Mondal.)

An index for 2d invertible phases of quantum many-body systems
2025-05-12 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Nikita Sopenko, IAS
Title: An index for 2d invertible phases of quantum many-body systems

Factorizations for data analysis
2025-05-12 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Anna Seigal, Harvard University
Title: Factorizations for data analysis
Abstract: We can find structure in data by factoring it into building blocks, which should be interpretable for the context at hand. A classical example is principal component analysis (PCA), which uses the eigendecomposition of the covariance matrix to find axes of variation in a dataset. Starting from PCA, I will discuss matrix and tensor factorizations for data analysis, and the linear and multilinear algebra that underpins their theoretical properties. We will see examples from causal inference, independent component analysis, and dimensionality reduction.

Report on the Perimeter Institute Theory+AI Workshop
2025-05-14 12:00 - 13:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Conference Reports
Speaker: Hugo Cui, Harvard CMSA
Title: Report on the Perimeter Institute Theory+AI Workshop
Abstract: I will give a survey and brief summary of some of the talks given at the Theory+AI Workshop: Theoretical Physics for AI event organized by Perimeter Institute in April, on approaches to machine learning theory inspired from physics.
Link : https://events.perimeterinstitute.ca/event/993/

Resurgence, number theory, and quantum mirror curves
2025-05-15 10:00 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Claudia Rella (IHES)
Title: Resurgence, number theory, and quantum mirror curves
Abstract: Resurgence provides a powerful toolbox to access the non-perturbative sectors hidden within the divergent asymptotic series of quantum theories. Under some special assumptions, the non-perturbative data extracted via resurgent methods possess intrinsic number-theoretic properties that are deeply rooted in the symmetries and arithmetic of the geometry underlying the quantum theory. The framework of modular resurgence aims to formalise this observation. In this talk, after introducing the basics of modular resurgence, I will consider the TS/ST correspondence for toric Calabi-Yau threefolds and focus on the fermionic spectral traces of quantum mirror curves. Here, a complete realisation of the modular resurgence paradigm is found in the spectral theory of local P^2—where the bridge between non-perturbative physics and the arithmetic properties of the geometry takes the form of an exact strong-weak symmetry—and is now being generalised to all local weighted projective spaces. This talk is based on arXiv:2212.10606, 2404.10695, 2404.11550, and work in progress.

Echo Chamber: RL Post-training Amplifies Behaviors Learned in Pretraining
2025-05-16 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Samy Jelassi, CMSA
Title: Echo Chamber: RL Post-training Amplifies Behaviors Learned in Pretraining
Abstract: Reinforcement Learning has become a crucial step in training state-of-the-art language models such as DeepSeek-R1 for solving mathematical problems. In this talk, I will first review the mechanisms of Reinforcement Learning fine-tuning. Then, I will present a systematic end-to-end study of RL fine-tuning for mathematical reasoning, training models entirely from scratch on different mixtures of fully open datasets and fine-tuning them with RL. Doing so allows us to investigate the effects of the pretraining data mixture on the behavior of RL, and its interaction with the model size and choices of the algorithm hyperparameters. Our study reveals that RL algorithms consistently converge towards a dominant output distribution, amplifying patterns in the pretraining data. We also find that models of different scales trained on the same data mixture will converge to distinct output distributions, suggesting that there are scale-dependent biases in model generalization.
The second part of the talk is based on a joint work with Rosie Zhao, Alex Meterez, Cengiz Pehlevan, Sham Kakade and Eran Malach: https://arxiv.org/abs/2504.07912

Higher Gauge Theory and Integrability
2025-05-22 10:00 - 11:00 | Virtual
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Joaquin Liniado, Instituto de Física La Plata
Title: Higher Gauge Theory and Integrability
Abstract: Integrable field theories are remarkable for possessing an infinite number of conserved quantities, which often allow for their exact solvability. In two dimensions, this structure is elegantly captured by the existence of a Lax connection, whose path ordered exponential allows for the systematic construction of an infinite number of conserved quantities. In 2019, Costello, Witten and Yamazaki introduced a four-dimensional holomorphic extension of Chern-Simons theory that provides the first attempt at explaining the appearance of the Lax connection, whose origin had remained somewhat mysterious until then.
In this talk, we present a generalization of these ideas to three-dimensional field theories, guided by the so-called “categorical ladder = dimensional ladder” principle. The central idea is that conserved quantities arise from surface-ordered exponentials of higher-rank tensors, defining a higher categorical notion of the Lax connection. We show that such a structure naturally emerges from a five-dimensional holomorphic extension of higher Chern-Simons theory. This work, carried out in collaboration with Hank Chen, provides a framework that enables the systematic construction of integrable field theories in three dimensions.

Summer School in Total Positivity and Quantum Field Theory
2025-06-02 09:00 - 2025-06-04 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Summer School in Total Positivity and Quantum Field Theory
Dates: June 2–4, 2025
Location: CMSA, 20 Garden Street, Cambridge MA

In the past decade, there has been a great deal of interest and progress in the study of algebro-combinatorial and geometric structures appearing across diverse areas of physics, from particle physics to cosmology. As these research programs expand, there is an ever-growing need for mathematicians and physicists to collaborate effectively and build a shared language. Join us at Harvard University’s Center of Mathematical Sciences and Applications for a week-long summer school dedicated to addressing these interdisciplinary connections. The school welcomes graduate students, postdocs, and early-career researchers drawn to the intersection of mathematics and physics. Whether you are an algebraic combinatorialist looking for a better grasp on the physics, a high energy theorist trying to figure out the math, or a newcomer to both fields, this summer school offers an ideal opportunity for you to learn.

Courses taught by both mathematicians and physicists will connect ideas from total positivity, matroid theory, discrete geometry, and real algebraic geometry with fundamental questions in quantum field theory. Topics will include amplituhedra, cluster algebras, and positive geometry as they relate to scattering amplitudes and cosmological correlators in high-energy physics. Our courses are designed to be accessible to a varied audience; speakers will be mindful of the diverse backgrounds of the participants from both fields.
Amid this exciting period of collaboration between mathematicians and physicists, we look forward to exploring these rich, cutting-edge topics with you.

Courses:

Positive Grassmannian and Cluster Algebras, Lara Bossinger (Instituto de Matemáticas Universidad Nacional Autónoma de México)
slides | exercises

Positive Geometry and Canonical Forms, Simon Telen (MPI Leipzig)
slides

Scattering Amplitudes and Amplituhedra, Marcus Spradlin (Brown)
exercises

Cosmology and Cosmological Polytopes, Nima Arkani-Hamed (IAS)

Organizers: Jonathan Boretsky (McGill University) | Matteo Parisi (Harvard CMSA and IAS Princeton) | Lauren Williams (Harvard University)

Youtube Playlist
Schedule
Monday, June 2, 2025

8:30–9:00 am
Morning Reception

9:00–10:00 am
Lara Bossinger: Positive Grassmannian and Cluster Algebras I

10:00–10:30 am
Coffee Break

10:30–11:10 am
Exercises

11:10 am–12:10 pm
Nima Arkani-Hamed: Cosmology and Cosmological Polytopes I

12:10–2:00 pm
Lunch Break

2:00–3:00 pm
Nima Arkani-Hamed: Cosmology and Cosmological Polytopes II

3:00–3:30 pm
Coffee Break

3:30–4:10 pm
Exercises

4:10–5:10 pm
Nima Arkani-Hamed: Cosmology and Cosmological Polytopes III

Tuesday, June 3, 2025

8:30–9:00 am
Morning Reception

9:00–10:00 am
Lara Bossinger: Positive Grassmannian and Cluster Algebras II

10:00–10:30 am
Coffee Break

10:30–11:30 am
Marcus Spradlin: Scattering Amplitudes and Amplituhedra I

11:30 am–12:10 pm
Exercises

12:10–2:00 pm
Lunch Break

2:00–3:00 pm
Simon Telen: Definitions and first examples of positive geometries

3:00–3:30 pm
Coffee Break

3:30–4:30 pm
Lightning Talks

4:30–5:30 pm
Simon Telen: Positive geometry of polytopes

Wednesday, June 4, 2025

8:30–9:00 am
Morning Reception

9:00–10:00 am
Lara Bossinger: Positive Grassmannian and Cluster Algebras III

10:00–10:30 am
Coffee Break

10:30–11:10 am
Exercises

11:10 am–12:10 pm
Marcus Spradlin: Scattering Amplitudes and Amplituhedra II

12:10–2:00 pm
Lunch Break

2:00–3:00 pm
Marcus Spradlin: Scattering Amplitudes and Amplituhedra III

3:00–3:30 pm
Coffee Break

3:30–4:30 pm
Simon Telen: Positive geometry of polypols

4:30–5:10 pm
Exercises

Image credit: Annabel Ma (Harvard College)
