Math and Machine Learning Program Discussion
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, 2024-10-07 10:30 - 12:00, 2024-10-09 10:30 - 12:00, 2024-10-11 10:30 - 12:00, 2024-10-14 10:30 - 12:00 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States

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.

Topics in Deep Learning Theory
2024-10-01 14:30 - 15:45, 2024-10-08 14:30 - 15:45, 2024-10-10 14:30 - 15:45, 2024-10-15 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 Open Discussion/Tea
2024-10-01 16:00 - 17:00, 2024-10-08 16:00 - 17:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Open Discussion/Tea

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
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.

High-dimensional learning of narrow neural networks
2024-10-04 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Hugo Cui, CMSA
Title: High-dimensional learning of narrow neural networks
Abstract: This talk explores the interplay between neural network architectures and data structure through the lens of high-dimensional asymptotics. We focus on a class of narrow neural networks, namely networks possessing a finite number of hidden units, while operating in high dimensions. In the limit of large data dimension and comparably large number of samples, we derive a tight asymptotic characterization of the learning of these architectures. As an illustration, we discuss how this characterization enables the analysis of a solvable model of dot-product attention. We show how the latter can learn to implement either a positional attention mechanism (with tokens attending to each other based on their respective positions), or a semantic attention mechanism (with tokens attending to each other based on their meaning), and evidence a phase transition with sample complexity from positional to semantic learning.

Local complexity measures in modern parameterized function classes for supervised learning
2024-10-07 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Elisenda Grigsby, Boston College
Title: Local complexity measures in modern parameterized function classes for supervised learning
Abstract: The parameter space for any fixed architecture of neural networks serves as a proxy during training for the associated class of functions – but how faithful is this representation? For any fixed feedforward ReLU network architecture, it is well-known that many different parameter settings can determine the same function. It is less well-known that the degree of this redundancy is inhomogeneous across parameter space. I’ll discuss two locally-applicable complexity measures for ReLU network classes and what we know about the relationship between them: (1) the local functional dimension, and (2) a local version of VC dimension called persistent pseudodimension. The former is easy to compute on finite batches of points, the latter should give local bounds on the generalization gap. I’ll speculate about how this circle of ideas might help guide our understanding of the double descent phenomenon. All of the work described in this talk is joint with Kathryn Lindsey. Some portions are also joint with Rob Meyerhoff, David Rolnick, and Chenxi Wu.

Continuation of solutions of Einstein's equations
2024-10-08 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Oswaldo Vazquez, Northeastern University
Title: Continuation of solutions of Einstein’s equations
Abstract: Klainerman-Rodnianski improved the continuation criterion for the solutions of Einstein’s equations proved by Michael Anderson using Kirchoff-Sobolev type parametrix and geometric Littlewood-Paley theory. Using their technique but a new parametrix we prove a continuation condition in the context of 3+1 dimensional vacuum Einstein gravity in Constant Mean extrinsic Curvature (CMC) gauge. More precisely, we obtain quantitative criteria under which the physical spacetime can be extended indefinitely in the future as a solution to the Cauchy problem of the Einstein equations given regular initial data. In particular, we show that a gauge-invariant H^2 Sobolev norm of the spacetime Riemann curvature remains bounded in the future time direction provided the so-called deformation tensor of the unit timelike vector field normal to the chosen CMC hypersurfaces verifies a spacetime L^{infty} bound.

Skein traces and curve counting
2024-10-08 16:15 - 18:15 | Science Center Hall E, 1 Oxford Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Sunghyuk Park, Harvard CMSA
Title: Skein traces and curve counting
Abstract: Skein modules are vector space-valued invariants of 3-manifolds describing the space of line defects modulo skein relations (determined by a choice of a ribbon category). When the 3-manifold is S x I for some surface S, the skein module has a natural algebra structure and is called the skein algebra of S.
In 2010, Bonahon and Wong constructed an algebra embedding (named “quantum trace”) of the sl_2 skein algebra into a quantum cluster variety called the “quantum Teichmuller space” for punctured surfaces, which has applications to the representation theory of skein algebras.
In the first half of this talk, I will give an overview of these concepts and explain how the quantum trace map can be generalized to the 3-dimensional setup.
In the second half, I will discuss how everything above can be generalized to HOMFLYPT skeins and has natural interpretation in terms of counts of holomorphic curves.

CMSA Q&A Seminar: Laura DeMarco
2024-10-09 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Laura DeMarco, Harvard University
Topic: What is Teichmuller geometry and why is it important?

Dolbeault Virasoro algebra and M5 branes
2024-10-11 09:00 - 10:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Brian Williams, Boston University
Title: Dolbeault Virasoro algebra and M5 branes
Abstract: The worldvolume theory on a stack of M5 branes in M-theory is superconformal. We propose a conjecture that in the holomorphic twist of the theory on a stack of M5 branes an infinite-dimensional enhancement of the (twisted) superconformal algebra is a symmetry. This algebra is closely related to the exceptional infinite-dimensional Lie superalgebra called E(3|6). We show that under the usual AGT correspondence this enhanced algebra degenerates to the Virasoro algebra at a particular central charge.

Scattering Amplitude from a Twistor Point of View
2024-10-11 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Keyou Zeng
Title: Scattering Amplitude from a Twistor Point of View
Abstract: Scattering amplitude is a key quantity in quantum field theory. Although challenging to compute at higher loops and for large particle numbers, physicists have developed various tools to gain a deeper understanding of amplitudes. In this seminar, I will introduce a novel approach, initiated by K. Costello and N. Paquette, which makes use of twistor correspondence. This approach enables the computation of certain amplitudes using chiral algebra. I will also present an upcoming work that constructs a top-down holography model for computing tree and loop amplitudes of certain non-SUSY theories in various self-dual backgrounds.

Gravitational collapse to extremal Reissner-Nordström and the third law of black hole thermodynamics
2024-10-15 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Christoph Kehle, MIT
Title: Gravitational collapse to extremal Reissner-Nordström and the third law of black hole thermodynamics
Abstract: In this talk, I will present a proof that extremal Reissner-Nordström black holes can form in finite time in gravitational collapse of charged matter. In particular, this construction provides a definitive disproof of the “third law” of black hole thermodynamics. I will also discuss recent works showing that extremal black holes take on a central role in gravitational collapse, giving rise to a new conjectural picture of “extremal critical collapse.” This is joint work with Ryan Unger (Stanford).

Topological Modular Forms, its equivariant refinements and relation with supersymmetric quantum field theories
2024-10-15 16:15 - 18:15 | Science Center Hall E, 1 Oxford Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Mayuko Yamashita, Kyoto University
Title: Topological Modular Forms, its equivariant refinements and relation with supersymmetric quantum field theories
Abstract: This talk is about the Segal-Stolz-Teichner program, which is one of the most deep and interesting topics relating homotopy theory and physics. Mathematically, they propose a geometric model of TMF, the spectrum (in homotopy theory) of Topological Modular Forms, in terms of supersymmetric quantum field theories. Their proposal, although far from solid formulation or a proof, has been a guiding principle leading us to many new interesting ideas and discoveries in both mathematics and physics. In this talk, I will give an overview of this topic, as well as my current works using equivariant twisted TMF.
