CMSA

Combinatorics and geometry of the amplituhedron
2024-09-09 16:30 - 17:30 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Lauren Williams, Harvard University
Title: Combinatorics and geometry of the amplituhedron
Abstract: The amplituhedron is a geometric object introduced by Arkani-Hamed and Trnka to compute scattering amplitudes in N=4 super Yang Mills theory. It generalizes interesting objects such as cyclic polytopes and the positive Grassmannian. It has connections to tropical geometry, cluster algebras, and combinatorics (plane partitions, Catalan numbers). I’ll give a gentle introduction to the amplituhedron, then survey some recent progress on some of the main conjectures about the amplituhedron: the Magic Number Conjecture, the BCFW tiling conjecture, and the Cluster Adjacency conjecture. Based on joint works withEvan-Zohar, Lakrec, Parisi, Sherman-Bennett, and Tessler.

BPS Algebras in Landau-Ginzburg Models
2024-09-10 16:00 - 18:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Ahsan Khan (CMSA)
Title: BPS Algebras in Landau-Ginzburg Models
Abstract: The study of BPS states in supersymmetric quantum field theory has been a fruitful source of both mathematical and physical insights. In particular their study often leads to rich algebraic structures – from the “Algebra of the Infrared” of Gaiotto-Moore-Witten to the “Cohomological Hall Algebras” of Kontsevich-Soibelman. In this talk, I will provide an overview of some of these algebraic constructions, with a particular emphasis on BPS states in two-dimensional Landau-Ginzburg models. In the second half of the talk, I will discuss how these algebraic structures can be extended to more general Landau-Ginzburg models defined by closed holomorphic one-forms.

Abundance for mixed characteristic threefolds
2024-09-13 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Iacopo Brivio (CMSA)
Title: Abundance for mixed characteristic threefolds
Abstract: The Minimal Model Program (MMP) predicts that every algebraic variety X is birational to either a fibration in Fano varieties, or it admits a “minimal model” X’, that is a birational model with nef canonical bundle K_X’. The Abundance conjecture predicts then that K_X’ is actually semiample, in particular it endows X’ with the structure of a Calabi-Yau fibration. These conjectures were initially phrased for complex varieties, but more recently there has been a lot of interest in working over positive characteristic fields, or even mixed characteristic rings. In this talk I will give a broad overview of the subject, starting from the case of complex surfaces. In the last part I will outline a proof of the Abundance conjecture for mixed characteristic threefolds (based on joint work with F. Bernasconi and L. Stigant).

Freedman CMSA Seminar
2024-09-13 14:30 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Freedman CMSA Seminar

2:00-3:30 pm ET
Speaker: Mike Freedman, Harvard CMSA
Title: Detecting hidden structures in linear maps
Abstract: I’ll consider the problem of detecting spectral features and tensor structures within linear maps both in a quantum and classical contexts. In the quantum context there is the question of whether a Hamiltonian is local, and if so, local in distinct coordinate systems (a “duality”). Also, in the case of a unitary described by a quantum circuit, does it possess unusual spectral features or tensor structure? In ML one optimizes many linear maps. How would we know – and would we care – if the resulting maps (approximately) tensor factored?

3:30-4:00 pm ET
Break/Discussion

4:00-5:30 pm ET
Speaker: Ryan O’Donnell, Carnegie Mellon University
Title: Quartic quantum speedups for planted inference
Abstract: Consider the following task (“noisy 4XOR”), arising in CSPs, optimization, and cryptography. There is a ‘secret’ Boolean vector x in {-1,+1}^n. One gets m randomly chosen pairs (S, b), where S is a set of 4 coordinates from [n] and b is x^S := prod_{i in S} x_i with probability 1-eps, and -x^S with probability eps. Can you tell the difference between the cases eps = 0.1 and eps = 0.5?
It depends on m. The best known algorithms use the “Kikuchi method” and run in time ~n^L when m ~ n^2/L. We will review this method, and also show that the running time can be improved to roughly n^{L/4} with a quantum algorithm.
Joint work with Alexander Schmidhuber (MIT), Robin Kothari (Google), and Ryan Babbush (Google).

Ringdown in the SYK model
2024-09-16 09:30 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Joint BHI/CMSA Foundation Seminar
Speaker: Matthew Dodelson (Harvard)
Title: Ringdown in the SYK model
Abstract: Thermal correlators in large N systems equilibrate at late times, but the precise late-time behavior is unknown away from holographic and free field limits. In this talk I will analyze this problem in the case of the SYK model away from the low-temperature limit. The basic technique is a resummation of perturbation theory which is reminiscent of the double cone construction. We will also discuss the interpretation of the result in terms of a dual stringy black hole.

Periodic pencils of flat connections and their p-curvature
2024-09-16 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Pavel Etingof (MIT)
Title: Periodic pencils of flat connections and their p-curvature
A periodic pencil of flat connections on a smooth algebraic variety is a linear family of flat connections , where are local coordinates on and are matrix-valued regular functions. A pencil is periodic if it is generically invariant under the shifts up to isomorphism. I will explain that periodic pencils have many remarkable properties, and there are many interesting examples of them, e.g. Knizhnik-Zamolodchikov, Dunkl, Casimir connections and equivariant quantum connections for conical symplectic resolutions with finitely many torus fixed points. I will also explain that in characteristic , the -curvature operators of a periodic pencil are isospectral to the commuting endomorphisms , where is the Frobenius twist of . This allows us to compute the eigenvalues of the -curvature for the above examples, and also to show that a periodic pencil of connections always has regular singularites. This is joint work with Alexander Varchenko.
(Abstract link (pdf)

Mathematics around Twisted Holography
2024-09-17 16:00 - 18:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Keyou Zeng (CMSA)
Title: Mathematics around Twisted Holography
Abstract: The holography principle is an important idea in physics and has been widely studied since the 90s. Twisted holography offers a way to simplify physical holography models through the procedure called twisting. In the first part of the talk, I’ll introduce some of the mathematical structures underlying this twisted version of holography, such as Koszul duality.
In the second part, I’ll discuss the concept of vertex algebras in symmetric monoidal categories, specifically in Deligne category. This framework will serve as a tool to rigorously define the “large N” algebra that emerges from twisted holography.

CMSA Q&A Seminar: Noam Elkies
2024-09-18 12:00 - 13:00 | 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?

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.

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.

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.

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.

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.

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.

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.

CMSA Q&A Seminar: Nazim Bouatta
2024-10-16 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Nazim Bouatta (HMS)
Topic: What are AlphaFold2 and OpenFold

From Word Prediction to Complex Skills: Data Flywheels for Mathematical Reasoning
2024-10-16 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Anirudh Goyal (University of Montreal)
Title: From Word Prediction to Complex Skills: Data Flywheels for Mathematical Reasoning
Abstract: This talk examines how large language models (LLMs) evolve from simple word prediction to complex skills, with a focus on mathematical problem solving. A major driver of AI products today is the fact that new skills emerge in language models when their parameter set and training corpora are scaled up. This phenomenon is poorly understood, and a mechanistic explanation via mathematical analysis of gradient-based training seems difficult. The first part of the talk focuses on analysing emergence using the famous (and empirical) Scaling Laws of LLMs. Then I talk about howc LLMs can verbalize these skills by assigning labels to problems and clustering them into interpretable categories. This metacognitive ability allows us to leverage skill-based prompting, significantly improving performance on mathematical reasoning. I then present a framework that combines LLMs with human oversight to generate challenging, out-of-distribution math questions. This process led to the creation of the MATH^2 dataset, which enhances both model and human performance, driving further advances in mathematical reasoning capabilities.

Bosonic and fermionic 1-form symmetries and anomaly matching
2024-10-18 09:00 - 10:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
*via Zoom only*
Speaker: Rajath Radhakrishnan (ICTP, Trieste)
Title: Bosonic and fermionic 1-form symmetries and anomaly matching
Abstract: In this talk, I will consider bosonic and fermionic (non-invertible) 1-form symmetries in 2+1d QFTs. These are 1-form symmetries implemented by topological line operators with real spins. I will present a classification of topological quantum field theories in which all line operators have real topological spins, and use this framework to classify the anomalies associated with these 1-form symmetries. Additionally, I will discuss the anomaly matching condition for these symmetries under an RG flow. I will illustrate this condition in concrete examples.

Positive mass and rigidity theorems in Riemannian geometry
2024-10-18 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Puskar Mondal
Title: Positive mass and rigidity theorems in Riemannian geometry
Abstract: Positive mass theorem proved by Schoen-Yau, Witten, Taubes-Parker is one of the most important results in scalar curvature geometry in asymptotically flat settings. Since then several versions have been proven and generalized to other geometries such as asymptotically hyperbolic manifolds. The analogous theorem for strictly positive curvature geometries is absent. There have been counterexamples but a precise quantification does not exist.I prove a scalar curvature rigidity theorem for spheres. In particular, I prove that $n+1~(ngeq 2)$ dimensional spherical caps with constant positive mean curvature totally umbilic boundaries are rigid under smooth perturbations, and such rigidity results fail for the hemisphere. The assertion of this result is based on the notion of a real Killing connection and solution of the boundary value problem associated with its Dirac operator. Additionally, an improved eigenvalue estimate for the Dirac operator on hypersurfaces in positively curved manifolds is obtained.

Foundation Seminar: Singularity Theorems, Part I
2024-10-21 09:30 - 10:30 | Black Hole Initiative, 20 Garden Street, Cambridge MA, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Foundation Seminar (Joint Seminar with BHI)
Location: BHI
Title: Singularity Theorems, Part I
Journal Club Discussion
