CMSA Q&A Seminar: Noam Elkies
2024-09-18 12:00 - 13:00 (America/New_York) | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Noam Elkies, Harvard Math
Topic: How to show E8 and Leech lattices have optimal sphere packings?

Feynman graph integrals from topological holomorphic theories
2024-09-19 10:00 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Minghao Wang (Boston University)
Title: Feynman graph integrals from topological holomorphic theories
Abstract: Feynman graph integrals from topological theories were developed by M. Kontsevich, S. Axelrod and I. M. Singer in 1990s. These integrals have many mathematical applications, such as knot invariants, operad theory and formality theorems. In this talk, I will talk about Feynman graph integrals from topological-holomorphic theories. In particular, I will prove the finiteness of Feynman graph integrals when spacetime is flat spaces and a vanishing result of graph integrals. Combining the vanishing result with Batalin-Vilkovisky(BV) formalism, we can show the absence of anomalies of topological-holomorphic theories on flat spaces with at least two topological dimensions. As a consequence, we can construct factorization algebras of quantum observables. This is a joint with Brian Williams.

Topics in Deep Learning Theory
2024-09-19 14:30 - 15:45, 2024-09-24 14:30 - 15:45, 2024-09-26 14:30 - 15:45, 2024-10-01 14:30 - 15:45 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topics in Deep Learning Theory
Eli Grigsby

Math and Machine Learning Program Discussion
2024-09-20 10:30 - 12:00, 2024-09-23 10:30 - 12:00, 2024-09-25 10:30 - 12:00, 2024-09-27 10:30 - 12:00, 2024-09-30 10:30 - 12:00, 2024-10-02 10:30 - 12:00, 2024-10-04 10:30 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States

Communication Complexity of Combinatorial Auctions
2024-09-20 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Tomer Ezra (CMSA)
Title: Communication Complexity of Combinatorial Auctions
Abstract: We study the communication complexity of welfare maximization in combinatorial auctions with m items and two subadditive bidders. A 2-approximation can be guaranteed by a trivial randomized protocol with zero communication, or a trivial deterministic protocol with O(1) communication. We show that outperforming these trivial protocols requires exponential communication, settling an open question of [DobzinskiNS10, Feige09].
Specifically, we show that any (randomized) protocol guaranteeing a o(logm)-approximation requires communication exponential in m. We complement it by presenting an O(logm)-approximation in poly(m) communication.

Classification and Construction of crystalline topological superconductors and insulators in interacting fermion systems
2024-09-20 14:00 - 15:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter Seminar
Speaker: Zhengcheng Gu, Chinese University of Hong Kong
Title: Classification and construction of crystalline topological superconductors and insulators in interacting fermion systems
Abstract: The construction and classification of crystalline symmetry protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Crystalline SPT phases are not only of conceptual importance, but also provide us great opportunities towards experimental realization since space group symmetries naturally exist for any realistic material. In this talk, I will discuss how to construct and classify crystalline topological superconductors (TSC) and topological insulators (TI) in interacting fermion systems. I will also discuss the relationship between internal symmetry protected SPT phases and crystalline symmetry protected SPT Phases.

Symmetry groups in infinite dimensions
2024-09-23 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Lisa Carbone, Rutgers University
Title: Symmetry groups in infinite dimensions
Abstract: The study of many physical theories requires an understanding of symmetries of infinite dimensional Lie algebras. The construction of groups of automorphisms for infinite dimensional Lie algebras is challenging, but there is well established theory for the class of Kac-Moody algebras. A generalization of Kac-Moody algebras known as Borcherds algebras arise in string theory models, but the methods for constructing Kac-Moody groups break down for this more general class. We discuss the challenges that arise and describe several approaches to constructing groups for Borcherds algebras. Our main example is the Monster Lie algebra which plays an important role in the solution of Monstrous Moonshine and which is a symmetry algebra of a model of the compactified Heterotic String.

New Energy Inequality in AdS
2024-09-24 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Diandian Wang, Harvard University
Title: New Energy Inequality in AdS
Abstract: I will describe evidence for a new energy inequality in asymptotically AdS spacetimes whose conformal boundary contains a spatial circle. It is in some sense analogous but crucially different to the Penrose inequality. In the AdS4 case, this generalizes the Horowitz-Myers conjecture. I will show how static solutions play an interesting role in determining the shape of the function that bounds the gravitational energy.

Math and Machine Learning Program Open Discussion/Tea
2024-09-24 16:00 - 17:00, 2024-10-01 16:00 - 17:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Open Discussion/Tea

Symplectic duality in examples
2024-09-24 16:15 - 18:15 | Science Center Hall E, 1 Oxford Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Vasily Krylov, Harvard CMSA & Math
Title: Symplectic duality in examples
Abstract: Over the past twenty years, mathematicians and physicists have shown increasing interest in studying certain Poisson varieties, known as “symplectic singularities.” Many of these objects naturally arise as Higgs or Coulomb branches of certain TQFTs and, therefore, fall within the framework of 3D mirror symmetry, also known as symplectic duality. The first part of the talk will provide a gentle introduction to the theory of symplectic singularities, with an emphasis on various examples. In the second part, we will discuss how the symplectic duality works in examples, beginning with the simplest cases. We will then discuss a particular phenomenon called the Hikita-Nakajima conjecture, which predicts a deep and nontrivial relationship between dual varieties. It is particularly intriguing that this conjecture was formulated by mathematicians and still requires further understanding from a physical perspective.

Infinite Limits and Scaling Laws for Deep Neural Networks
2024-09-25 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Blake Bordelon
Title: Infinite Limits and Scaling Laws for Deep Neural Networks
Abstract: Scaling up the size and training horizon of deep learning models has enabled breakthroughs in computer vision and natural language processing. Empirical evidence suggests that these neural network models are described by regular scaling laws where performance of finite parameter models improves as model size increases, eventually approaching a limit described by the performance of an infinite parameter model. In this talk, we will first examine certain infinite parameter limits of deep neural networks which preserve representation learning and then describe how quickly finite models converge to these limits. Using dynamical mean field theory methods, we provide an asymptotic description of the learning dynamics of randomly initialized infinite width and depth networks. Next, we will empirically investigate how close the training dynamics of finite networks are to these idealized limits. Lastly, we will provide a theoretical model of neural scaling laws which describes how generalization depends on three computational resources: training time, model size and data quantity. This theory allows analysis of compute optimal scaling strategies and predicts how model size and training time should be scaled together in terms of spectral properties of the limiting kernel. The theory also predicts how representation learning can improve neural scaling laws in certain regimes. For very hard tasks, the theory predicts that representation learning can approximately double the training-time exponent compared to the static kernel limit.

Witten deformation for non-Morse functions and gluing formulas
2024-09-26 10:00 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Physics and Algebraic Geometry
Speaker: Junrong Yan (Northeastern University)
Title: Witten deformation for non-Morse functions and gluing formulas
Abstract: Witten deformation is a versatile tool with numerous applications in mathematical physics and geometry. In this talk, we will focus on the analysis of Witten deformation for a family of non-Morse functions, which leads to a new technique for studying the gluing formulas of global spectral invariants (such as eta invariants, analytic torsions, and some invariants related to Feynman diagrams, etc.). We will then discuss some applications of this new method.

Going to the other side .... in algebra, topology, and maybe physics
2024-09-27 09:00 - 10:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics
Speaker: Sergei Gukov (Caltech)

Title: Going to the other side …. in algebra, topology, and maybe physics

Abstract: Inspired by Eugene Wigner’s reflections on the ‘unreasonable effectiveness of mathematics in the natural sciences,’ this talk is about the surprising and pervasive role of a peculiar phenomenon that, a priori, seemed to have no reason to exist. Yet, it emerges across many different areas of mathematics and theoretical physics, including:

the Kazhdan-Lusztig correspondence
quantum invariants of 3-manifolds
the study of 2d (0,2) boundary conditions in 3d N=2 theories
resurgent analysis

Although each of these fields approaches the phenomenon from a different perspective, the results align in striking and unexpected ways.

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

Organizers:

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

Quasinormal Corrections to Near-Extremal Black Hole Thermodynamics
2024-10-01 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Daniel Kapec, Harvard
Title: Quasinormal Corrections to Near-Extremal Black Hole Thermodynamics
Abstract: Recent work on the quantum mechanics of near-extremal non-supersymmetric black holes has identified a characteristic scaling of the low temperature black hole partition function. This result has only been derived using the path integral in the near-horizon region and relies on many assumptions. We discuss how to derive the scaling for the near-extremal rotating BTZ black hole from a calculation in the full black hole background using the Denef-Hartnoll-Sachdev (DHS) formula, which expresses the 1-loop determinant of a thermal geometry in terms of a product over the quasinormal mode spectrum. We also derive the spectral measure for fields of any spin in Euclidean BTZ and use it to provide a new proof of the DHS formula and a new, direct derivation of the BTZ heat kernel. The computations suggest a path to proving the scaling for the asymptotically flat 4d Kerr black hole.

Topological Invariants of gapped states through cosheaves
2024-10-01 16:15 - 18:15 | Science Center Hall E, 1 Oxford Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Bowen Yang, Harvard CMSA
Title: Topological Invariants of gapped states through cosheaves
Abstract: We provide a proper mathematical framework for the constructions of topological invariants of gapped quantum states and interpret topological invariants of gapped states as lattice analogs of ’t Hooft anomalies in Quantum Field Theory. Our secondary goal is to generalize this construction in various directions. In particular, we show how to define topological invariants of lattice spin systems living on well-behaved subsets of the lattice.

CMSA Q&A Seminar: Cliff Taubes
2024-10-02 12:00 - 13:00 | MA
CMSA Q&A Seminar
Speaker: Cliff Taubes, Harvard Mathematics
Topic: What are Z/2 harmonic 1-forms?

Hierarchical data structures through the lenses of diffusion models
2024-10-02 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Antonio Sclocchi, EPFL
Title: Hierarchical data structures through the lenses of diffusion models
Abstract: The success of deep learning with high-dimensional data relies on the fact that natural data are highly structured. A key aspect of this structure is hierarchical compositionality, yet quantifying it remains a challenge.
In this talk, we explore how diffusion models can serve as a tool to probe the hierarchical structure of data. We consider a context-free generative model of hierarchical data and show the distinct behaviors of high- and low-level features during a noising-denoising process. Specifically, we find that high-level features undergo a sharp transition in reconstruction probability at a specific noise level, while low-level features recombine into new data from different classes. This behavior of latent features leads to correlated changes in real-space variables, resulting in a diverging correlation length at the transition.
We validate these predictions in experiments with real data, using state-of-the-art diffusion models for both images and texts. Remarkably, both modalities exhibit a growing correlation length in changing features at the transition of the noising-denoising process.
Overall, these results highlight the potential of hierarchical models in capturing non-trivial data structures and offer new theoretical insights for understanding generative AI.

Enumerative geometry and modularity in two-modulus K3-fibered Calabi-Yau threefolds
2024-10-03 10:00 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Chuck Doran, Harvard CMSA
Title: Enumerative geometry and modularity in two-modulus K3-fibered Calabi-Yau threefolds
Abstract: Smooth $M_m$-polarized K3-fibered Calabi-Yau (CY) 3-folds have been classified in terms of the choice of a generalized functional invariant and, in the case $m=1$, a generalized homological invariant. The resulting geometries generally exhibit a small number of complex structure moduli greater or equal to two. A concrete choice of these invariants realizes (almost all of) the known Calabi-Yau geometries with exactly two moduli and allows us to describe completely the structure of the corresponding moduli spaces. The corresponding variations of Hodge structure are entirely determined by the regular periods, for which we obtain a generic expression in terms of $m$ and three integers $i,j,s$. Using the form of this period and Batyrev-Borisov mirror symmetry we explicitly construct the corresponding mirror CY 3-folds with two Kaehler moduli and show consistency with the DHT conjecture. In the cases with $s=0$, the mirror CY 3-folds are again K3-fibered but with the mirror $<2m>$-polarization. The generic form of the periods allows us to derive generic modular expressions for the A-model topological string free energies and we argue that those are a consequence of a Tyurin degeneration of the generalized functional invariant with the central fiber being an $M_m$-polarized K3. This is joint work with Boris Pioline and Thorsten Schimannek.

Holography and Regge Phases at Large U(1) Charge
2024-10-04 09:00 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Giulia Fardelli, Boston University
Title: Holography and Regge Phases at Large U(1) Charge
Abstract: A single Conformal Field Theory (CFT) can have a rich phase diagram with qualitatively different emergent behaviors in a range of different regimes parameterized by the conserved charges of the theory. In this talk, I will consider a CFT with a global U(1) current and explore the phase diagram as a function of the U(1) charge Q and angular momentum J, particularly at large J and Q. By taking the large J limit first, we are able to employ a dual holographic interpretation in AdS_{d+1} to predict the energy spectrum of Q-particle states. This limit has been studied in detail for Q=2, yielding very general results applicable to unitary CFTs in d>2. When Q is also taken to be large, the description is more complicated; nevertheless, we can draw interesting conclusions about the energy spectrum under certain assumptions. I will conclude with a concrete example, the O(2) model in 3d, highlighting interesting connections with recent (and less recent) results in this context.
