Contract Design in Combinatorial Settings
2024-03-01 12:00 - 13:00 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Tomer Ezra (Harvard CMSA)
Title: Contract Design in Combinatorial Settings
Abstract: We study two combinatorial settings of the contract design problem, in which a principal wants to delegate the execution of a costly task. In the first setting, the principal delegates the task to an agent that can take any subset of a given set of unobservable actions, each of which has an associated cost. The principal receives a reward which is a combinatorial function of the actions taken by the agent. In the second setting, we study the single-principal multi-agent contract problem, in which the principal motivates a team of agents to exert effort toward a given task. We design (approximately) optimal algorithms for both settings along with impossibility results for various classes of combinatorial functions.
This talk is based on joint works with Paul Duetting, Michal Feldman, Thomas Kesselheim and Maya Schlesinger.

Strong bounds for arithmetic progressions
2024-03-04 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Raghu Meka (UCLA)
Title: Strong bounds for arithmetic progressions
Abstract: Suppose you have a set S of integers from {1,2,…,N} that contains at least N / C elements. Then for large enough N, must S contain three equally spaced numbers (i.e., a 3-term arithmetic progression)?
In 1953, Roth showed this is the case when C is roughly (log log N). Behrend in 1946 showed that C can be at most exp(sqrt(log N)). Since then, the problem has been a cornerstone of the area of additive combinatorics. Following a series of remarkable results, a celebrated paper from 2020 due to Bloom and Sisask improved the lower bound on C to C = (log N)^(1+c) for some constant c > 0.
This talk will describe a new work showing that C can be much closer to Behrend’s construction. Based on joint work with Zander Kelley.

CMSA Q&A Seminar 3/5/2024
2024-03-05 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q and A Seminar

Speaker: Cumrun Vafa

Question: What is swampland?

CMSA Q&A Seminar 3/5/2024
2024-03-05 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar

Speaker: Yannai Gonczarowski, Harvard University

Question: What do people mean when they say ‘the intersection between theoretical computer science and economic theory’?

LILO: Learning Interpretable Libraries by Compressing and Documenting Code
2024-03-06 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Gabe Grand, MIT CSAIL and Dept. of EE&CS
Title: LILO: Learning Interpretable Libraries by Compressing and Documenting Code
Abstract: While large language models (LLMs) now excel at code generation, a key aspect of software development is the art of refactoring: consolidating code into libraries of reusable and readable programs. In this paper, we introduce LILO, a neurosymbolic framework that iteratively synthesizes, compresses, and documents code to build libraries tailored to particular problem domains. LILO combines LLM-guided program synthesis with recent algorithmic advances in automated refactoring from Stitch: a symbolic compression system that efficiently identifies optimal lambda abstractions across large code corpora. To make these abstractions interpretable, we introduce an auto-documentation (AutoDoc) procedure that infers natural language names and docstrings based on contextual examples of usage. In addition to improving human readability, we find that AutoDoc boosts performance by helping LILO’s synthesizer to interpret and deploy learned abstractions. We evaluate LILO on three inductive program synthesis benchmarks for string editing, scene reasoning, and graphics composition. Compared to existing neural and symbolic methods – including the state-of-the-art library learning algorithm DreamCoder – LILO solves more complex tasks and learns richer libraries that are grounded in linguistic knowledge.

AQFT Lecture Series
2024-03-07 13:30 - 14:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AQFT Lecture Series
Speaker: An Huang (Brandeis)
Title: Tate’s thesis and p-adic strings
Abstract: I shall explain the relation between a family of deformations of genus zero p-adic string worldsheet action and Tate’s thesis, which in particular, gives rise to an attempt of physically deriving quadratic reciprocity. I shall then propose a genus one p-adic string worldsheet action, where the key is an explicit definition of a p-adic Laplacian operator on a Tate curve, and the computation of its Green’s function. I shall also mention a very recent attempt of developing a Lorentzian version of the p-adic worldsheet action.

AQFT Lecture Series
2024-03-07 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AQFT Lecture Series
Speaker: John Francis (Northwestern)
Title: Integrating braided categories over 3-manifolds
Abstract: I’ll describe a form of factorization homology, which gives an invariant of framed 3-manifolds given as input a rigid braided-monoidal category with duals. More generally, the construction gives an invariant of framed n-manifolds from an E_{n-1}-monoidal category with duals. This construction relies on a form of the tangle hypothesis, that a dualizable object in an E_{n-1}-monoidal category uniquely determines a functor from the category of tangles in n-space. This is joint work with David Ayala.

Symmetry in quantum field theory
2024-03-08 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Dan Freed (Harvard Mathematics and CMSA)
Title: Symmetry in quantum field theory
Abstract: In joint work with Greg Moore and Constantin Teleman we show how ideas and techniques in topological field theory apply to the study of symmetry in quantum field theory. I will discuss how this came about, beginning with some discussion of symmetry in mathematics more generally, and give some examples.

AQFT Lecture Series
2024-03-08 13:00 - 14:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AQFT Lecture Series
Speaker: Dihua Jiang (U Minnesota)
Title: Shalika Periods: Functoriality and Arithmetic
Abstract: Shalika periods of automorphic forms were first used by H. Jacquet and J. Shalika (1990) in their construction of global zeta integrals for exterior square L-functions of GL(2n). They were also used by S. Friedberg and H. Jacquet (1993) in their construction of global zeta integrals for the standard L-functions with connections to the linear periods. A. Ash and D. Ginzburg (1994) used them in their study of p-adic L-functions of GL(2n). In this talk, we will first review the implication of the Shalika periods to the Langlands functoriality and the connections with automorphic descents and theta correspondence. Then we turn to my recent work joint with B. Sun and F. Tian on an application of Shalika periods to algebraicity of critical L-values of cuspidal automorphic of GL(2n) of symplectic type.

AQFT Lecture Series
2024-03-15 11:45 - 12:45 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AQFT Lecture Series
Speaker: Baiying Liu (Purdue)
Title: Recent progress on certain problems related to local Arthur packets of classical groups
Abstract: In this talk, I will introduce recent progress on certain problems related to local Arthur packets of classical groups. First, I will introduce a joint work with Freydoon Shahidi towards Jiang’s conjecture on the wave front sets of representations in local Arthur packets of classical groups, which is a natural generalization of Shahidi’s conjecture, confirming the relation between the structure of wave front sets and the local Arthur parameters. Then, I will introduce a joint work with Alexander Hazeltine and Chi-Heng Lo on the intersection problem of local Arthur packets for symplectic and split odd special orthogonal groups, with applications to the Enhanced Shahidi’s conjecture, the closure relation conjecture, and the conjectures of Clozel on unramified representations and on unramified components of automorphic representations. This intersection problem also has been worked out independently at the same time by Hiraku Atobe. In a recent joint work with Alexander Hazeltine, Chi-Heng Lo, and Freydoon Shahidi, we made an upper bound conjecture on wavefront sets of admissible representations of connected reductive groups. In the last part of my talk, I will introduce our recent progress towards this conjecture.

AQFT Lecture Series
2024-03-15 14:15 - 15:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AQFT Lecture Series
Speaker: Tasho Kaletha (University of Michigan)
Title: Covers of reductive groups and functoriality
Abstract: When studying problems arising from Langlands’ functoriality principle, one often encounters groups that are extensions of complex reductive groups by Galois groups, but that do not necessarily satisfy all properties to be L-groups of reductive groups. We will show, in the case of a local base field F, that such group can be understood as L-groups of covers of reductive groups. This generalizes to the case of arbitrary local fields work of Adams–Vogan for real groups. These covers, for a fixed connected reductive group G, can be understood either as arising from a certain “universal” cover of the topological group G(F) by a certain “fundamental” group tildepi_1(G). We will present two concrete applications of this, one that gives a characterization of the local Langlands correspondence for supercuspidal L-parameters when p is sufficiently large, and one to the construction of transfer factors in the theory of endoscopy.

AQFT Lecture Series
2024-03-18 13:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AQFT Lecture Series
Speaker: Xinwen Zhu (Stanford)
Title: The tame categorical local Langlands correspondence.
Abstract: I will discuss a conjectural categorical form of the local Langlands correspondence for p-adic groups and establish the tame part of such correspondence (currently for unramified groups with connected center, and for Q_l-coefficients). I will also explain how to extract a (n enhanced) discrete Langlands parameter for depth zero supercuspidal representations from the categorical equivalence.

Koszul duality & twisted holography for asymptotically flat spacetimes
2024-03-18 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Natalie Paquette, University of Washington
Title: Koszul duality & twisted holography for asymptotically flat spacetimes
Abstract: Koszul duality has been understood in recent years to characterize order-type defects in twists of supersymmetric field theories. This notion has been generalized, from a physical point of view, by studying couplings between D-branes and closed string theories in the topological string. Computing the D-brane backreaction, and studying the resulting open/closed string duality, is the purview of the twisted holography program. Twisted holography seeks to study supersymmetric sectors of the AdS/CFT correspondence using these methods, and leverage the appropriate generalization of Koszul duality to elucidate the bulk/boundary map. When applying these methods to a topological string configuration on twistor space, one can construct an instance of twisted holography in which a 2d chiral algebra, supported on the “celestial sphere”, is dual to a 4d theory in an asymptotically flat spacetime. This is the first such top-down example of holography in a 4d asymptotically flat spacetime. This talk describes joint work done, variously, with Kevin Costello, Brian Williams, and Atul Sharma.

CMSA Q&A Seminar 3/19/2024
2024-03-19 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q and A Seminar

Speaker: Arthur Jaffe, Harvard University

Question: What is mathematical picture language?

AQFT Lecture Series
2024-03-20 11:00 - 12:15 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AQFT Lecture Series
Speaker: Stephen D. Miller (Rutgers University)
Title: What 4-graviton scattering amplitudes had to say about the unitary dual
Abstract: I’ll give an update on the problem of describing all unitary representations of a Lie group, including joint work with Michael Green and Pierre Vanhove that used intuition from string theory to show the unitarity of the “next to minimal” representation of E8, and more recent work with Joe Hundley and Jeff Adams, Marc van Leeuwen, and David Vogan.

Solving olympiad geometry without human demonstrations
2024-03-20 14:00 - 15:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Trieu H. Trinh, Google Deepmind and NYU Dept. of Computer Science
Title: Solving olympiad geometry without human demonstrations
Abstract: Proving mathematical theorems at the olympiad level represents a notable milestone in human-level automated reasoning, owing to their reputed difficulty among the world’s best talents in pre-university mathematics. Current machine-learning approaches, however, are not applicable to most mathematical domains owing to the high cost of translating human proofs into machine-verifiable format. The problem is even worse for geometry because of its unique translation challenges, resulting in severe scarcity of training data. We propose AlphaGeometry, a theorem prover for Euclidean plane geometry that sidesteps the need for human demonstrations by synthesizing millions of theorems and proofs across different levels of complexity. AlphaGeometry is a neuro-symbolic system that uses a neural language model, trained from scratch on our large-scale synthetic data, to guide a symbolic deduction engine through infinite branching points in challenging problems. On a test set of 30 latest olympiad-level problems, AlphaGeometry solves 25, outperforming the previous best method that only solves ten problems and approaching the performance of an average International Mathematical Olympiad (IMO) gold medallist. Notably, AlphaGeometry produces human-readable proofs, solves all geometry problems in the IMO 2000 and 2015 under human expert evaluation and discovers a generalized version of a translated IMO theorem in 2004.

AQFT Lecture Series
2024-03-21 11:00 - 12:15 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AQFT Lecture Series
Speaker: Ezra Getzler (Northwestern)
Title: Flat connections on derived stacks and the Gauss-Manin connection in derived algebraic geometry

Modularity and Fibrations in Mirror Symmetry
2024-03-22 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Chuck Doran (Harvard CMSA)
Title: Modularity and Fibrations in Mirror Symmetry
Abstract: We will introduce appearances of modularity in the study both of families of Calabi-Yau threefolds and of their enumerative invariants. An important role is played by the structure of fibrations and the DHT fibration-degeneration mirror correspondence, which clarifies how these notions of modularity are (and are not) related. This is joint work with Boris Pioline and Thorsten Schimannek.

AQFT Lecture Series
2024-03-22 13:45 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
AQFT Lecture Series
Speaker: Jayce Getz (Duke University)
Title: The Poisson summation conjecture and the fiber bundle method
Abstract: The Poisson summation conjecture of Braverman-Kazhdan, L. Lafforgue, Ngo, and Sakellaridis predicts that spherical varieties over a global field admit Schwartz spaces, Fourier transforms, and a generalized Poisson summation formula. In this talk I will state a rough form of the conjecture and explain how to deduce new cases of it from known cases using the fiber bundle method. Time permitting I will sketch an approach to proving the functional equation of triple product L-functions using this method. The approach is joint work with Pam Gu, Chun-Hsien Hsu, and Spencer Leslie.

Phase diagram and confining strings in a minimal model of nematopolar matter
2024-04-05 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Farzan Vafa
Title: Phase diagram and confining strings in a minimal model of nematopolar matter
Abstract: We investigate a minimal model of a nematopolar system. We analytically uncover a phase diagram consisting of a locked phase where the polar order and nematic order are locked, and unlocked phases which could be ordered or disordered. In particular, we develop two complementary perspectives on the locked phase: (i) the nematic order induces polar order, (ii) in the locked phase, all 1/2 integral nematic topological charges are confined. In particular, a polar +1 defect fattens from a point along a string with constant tension and confines a pair of nematic +1/2 defects at its ends.

Global weak solutions of 3+1 dimensional vacuum Einstein equations
2024-04-12 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Puskar Mondal
Title: Global weak solutions of 3+1 dimensional vacuum Einstein equations
Abstract: It is important to understand if the `solutions’ of non-linear evolutionary PDEs persist for all time or become extinct in finite time through the blow-up of invariant entities. Now the question of this global existence or finite time blow up in the PDE settings is well defined if the regularity of the solution is specified. Most physically interesting scenarios demand control of the point-wise behavior of the solution. Unfortunately, most times this level of regularity is notoriously difficult to obtain for non-linear equations. In this talk, I will discuss very low regularity solutions namely distributional (or weak) solutions of vacuum Einsten’s equations in 3+1 dimensions. I prove that on a globally hyperbolic spacetime foliated by closed connected oriented negative Yamabe slices, weak solutions of the Einstein equations exist for all time. The monotonicity of a Coercive Entity called reduced Hamiltonian that controls the minimum regularity required for the weak solution is employed. This is in the same spirit as Leray’s global weak solutions of Navier-Stokes in 3+1 dimensions and the first result in the context of Einstein equations.

CMSA Q&A Seminar 4/16/2024
2024-04-16 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q and A Seminar

Speaker: Cengiz Pehlevan, Harvard

Question: What is feature learning?

Member Seminar
2024-04-19 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Sunghyuk Park, Harvard CMSA
Title: 3D quantum trace map
Abstract: I will speak about my recent work (joint with Sam Panitch) constructing the 3d quantum trace map, a homomorphism from the Kauffman bracket skein module of an ideally triangulated 3-manifold to its (square root) quantum gluing module, thereby giving a precise relationship between the two quantizations of the character variety of ideally triangulated 3-manifolds. Our construction is based on the study of stated skein modules and their behavior under splitting, especially into face suspensions.

CMSA Q&A Seminar 4/23/2024
2024-04-23 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q and A Seminar

Speaker: Melanie Weber, Harvard

Question: What is the Ricci curvature of a graph?

Member Seminar
2024-04-26 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Matteo Parisi, Harvard CMSA

The DNA of Particle Scattering
2024-04-29 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Lance Dixon (SLAC, Stanford University)
Title: The DNA of Particle Scattering
Abstract: At the Large Hadron Collider, the copious scattering of quarks and gluons in quantum chromodynamics (QCD) produces Higgs bosons and many backgrounds to searches for new physics. At short distances, scattering in QCD can be evaluated in perturbation theory and leads to highly intricate, multivariate mathematical functions such as generalized polylogarithms. To gain further insight, one can study a cousin of QCD called planar N=4 super-Yang-Mills theory. Some processes in this theory can be computed to eighth order in perturbation theory, versus second or third order in QCD. The computation and analysis of these results rely on a Hopf algebra coaction on polylogarithms. Its maximal iteration is called the ‘symbol’, which serves as a ‘genetic code’ for amplitudes. The symbol is a linear combination of words, sequences of letters analogous to sequences of DNA base pairs. Understanding the alphabet, and then reading the code, exposes the physics and mathematics of quantum scattering, including bizarre new symmetries. For example, the two scattering amplitudes that are known to the highest orders in perturbation theory (8 loops) are related to each other by an ‘antipodal duality’, which involves reading the code backwards as well as forwards. A third scattering amplitude, which contains the other two as limits, has an antipodal self-duality which ‘explains’ the other duality. However, we still don’t know ‘who ordered’ this property, or what it really means.

CMSA Q&A Seminar 4/30/2024
2024-04-30 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Lakshminarayanan Mahadevan

Question: What is morphogenesis? (Morphogenesis: geometry and biology)

Liouville Theory and Weil-Petersson Geometry
2024-05-06 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Sarah Harrison (Northeastern University)
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.

On using ML for Economics
2024-05-07 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Sergiy Verstyuk
Title: On using ML for Economics
Abstract: I will introduce some tools from the field of machine learning and discuss how they can be leveraged to get a fresh perspective on economics.

On the landscape of 4d N=2 SCFTs
2024-05-10 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Robert Moscrop, Harvard CMSA
Title: On the landscape of 4d N=2 SCFTs
Abstract: Four-dimensional conformal field theories with sufficient (N = 2) supersymmetry are highly constrained. So much so, there has been an ongoing effort to classify them using only information about their moduli space of vacua. In this talk, I will review recent progress in this classification before detailing a subclass of theories for which the classification problem is particularly tractable.
