CMSA

The string/black hole transition in anti de Sitter space
2023-03-02 16:00 - 17:00 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Erez Urbach, Weizmann Institute of Science
Title: The string/black hole transition in anti de Sitter space
Abstract: String stars, or Horowitz-Polchinski solutions, are string theory saddles with normalizable condensates of thermal-winding strings. In the past, string stars were offered as a possible description of stringy (Euclidean) black holes in asymptotically flat spacetime, close to the Hagedorn temperature. I will discuss the thermodynamic properties of string stars in asymptotically (thermal) anti-de Sitter background (including AdS3 with NS-NS flux), their possible connection to small black holes in AdS, and their implications for holography. I will also present new “winding-string gas” saddles for confining holographic backgrounds such as the Witten model, and their relation to the deconfined phase of 3+1 pure Yang-Mills.

Categorical Symmetry of the Standard Model from Gravitational Anomaly
2023-03-07 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Juven Wang
Title: Categorical Symmetry of the Standard Model from Gravitational Anomaly
Abstract: In the Standard Model, the total “sterile right-handed” neutrino number n_{νR} is not equal to the family number Nf. The anomaly index (-Nf+n_{νR}) had been advocated to play an important role in our previous work on Cobordism and Deformation Class of the Standard Model [2112.14765, 2204.08393] and Ultra Unification [2012.15860] in order to predict new highly entangled sectors beyond the Standard Model. Moreover, the invertible baryon minus lepton number B−L symmetry current conservation can be violated quantum mechanically by gravitational backgrounds such as gravitational instantons. In specific, we show that a noninvertible categorical counterpart of the B−L symmetry still survives in gravitational backgrounds. In general, we propose a construction of noninvertible symmetry charge operators as topological defects derived from invertible anomalous symmetries that suffer from mixed gravitational anomalies. Examples include the perturbative local and nonperturbative global anomalies classified by ℤ and ℤ16 respectively. For this construction, we utilize the anomaly inflow concept and the 3d Witten-Reshetikhin-Turaev-type topological quantum field theory corresponding to a 2d rational conformal field theory with an appropriate chiral central charge, or the 3d boundary topological order of 4d ℤTF4-time-reversal symmetric topological superconductor [2302.14862].

Conformal symmetry, Optimization algorithms and the Critical Phenomena
2023-03-08 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Ning Su, University of Pisa
Title: Conformal symmetry, Optimization algorithms and the Critical Phenomena
Abstract: In the phase diagram of many substances, the critical points have emergent conformal symmetry and are described by conformal field theories. Traditionally, physical quantities near the critical point can be computed by perturbative field theory method, where conformal symmetry is not fully utilized. In this talk, I will explain how conformal symmetry can be used to determine certain physical quantities, without even knowing the fine details of the microscopic structure. To compute the observables precisely, one needs to develop powerful numerical techniques. In the last few years, we have invented many computational tools and algorithms, and predicted critical exponents of Helium-4 superfluid phase transition and Heisenberg magnet to very high precision.

How to steer foundation models?
2023-03-08 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Jimmy Ba, University of Toronto
Title: How to steer foundation models?
Abstract: By conditioning on natural language instructions, foundation models and large language models (LLMs) have displayed impressive capabilities as general-purpose computers. However, task performance depends significantly on the quality of the prompt used to steer the model. Due to the lack of knowledge of how foundation models work, most effective prompts have been handcrafted by humans through a demanding trial-and-error process. To reduce the human effort in this alignment process, I will discuss a few approaches to steer these powerful models to excel in various downstream language and image tasks.

Randomized algorithms in combinatorics
2023-03-14 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Michael Simkin
Title: Randomized algorithms in combinatorics
Abstract: Randomized algorithms have been a computational workhorse for almost as long as there have been computers. Surprisingly, such algorithms can also be used to attack problems that are neither algorithmic nor probabilistic. Time permitting I will discuss the following combintorial examples:

Enumerative combinatorics and the n-queens problem: In how many ways can one place n queens on an n x n chessboard so that no queen threatens any other?
Constructions of combinatorial designs and the Erdos high-girth Steiner triple system problem: An order-n Steiner triple system (STS) is a collection of triples on n vertices such that every pair of vertices is contained in exactly one triple. In 1973 Erdos conjectured that there exist STSs with arbitrary large girth (informally, no small set of vertices spans many triples). I will discuss the use of randomized algorithms to prove this conjecture. Joint work with Kwan, Sah, and Sawhney.
Thresholds in random graphs and hypergraphs: I will discuss how randomized algorithms can be combined with the recent resolution of the Kahn–Kalai conjecture to determine thresholds in random (hyper) graph theory. Joint work with Pham, Sah, and Sawhney.

Quantum Gravity constraints beyond asymptotic regimes
2023-03-21 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Max Wiesner
Title: Quantum Gravity constraints beyond asymptotic regimes
Abstract: Not every effective field theory that is consistent in the absence of gravity can be completed to a consistent theory of quantum gravity. The goal of the Swampland program is to find general criteria that distinguish effective field theories, that can be obtained as a low-energy approximation of quantum gravity, from those that are inconsistent in the presence of gravity. These criteria are oftentimes motivated by patterns observed in explicit compactifications of perturbative string theory and have passed many non-trivial tests in asymptotic regions of the field space such as, e.g., weak coupling limits. Still, the Swampland criteria should equally apply to effective theories that do not arise in asymptotic regions of the field space of string theory compactifications. In this talk I will summarize some of my recent works that studies the interior of regions of the field space of string theory in the context of the Swampland program.

Synchronization in a Kuramoto Mean Field Game
2023-03-22 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Mete Soner (Princeton University)
Title: Synchronization in a Kuramoto Mean Field Game
Abstract: Originally motivated by systems of chemical and biological oscillators, the classical Kuramoto model has found an amazing range of applications from neuroscience to Josephson junctions in superconductors, and has become a key mathematical model to describe self organization in complex systems. These autonomous oscillators are coupled through a nonlinear interaction term which plays a central role in the long term behavior of the system. While the system is not synchronized when this term is not sufficiently strong, fascinatingly, they exhibit an abrupt transition to a full synchronization above a critical value of the interaction parameter. We explore this system in the mean field formalism. We treat the system of oscillators as an infinite particle system, but instead of positing the dynamics of the particles, we let the individual particles determine endogenously their behaviors by minimizing a cost functional and eventually, settling in a Nash equilibrium. The mean field game also exhibits a bifurcation from incoherence to self-organization. This approach has found interesting applications including circadian rhythms and jet-lag recovery. This is joint work with Rene Carmona of Princeton and Quentin Cormier of INRIA, Paris.

Monotonicity of quasilocal mass for asymptotically flat Riemannian manifolds
2023-03-28 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Jue Liu
Title: Monotonicity of quasilocal mass for asymptotically flat Riemannian manifolds
Abstract: The study of quasilocal mass in general relativity has a long history. In previous papers by many authors we have a deep understanding of the properties of quasilocal mass such as positivity, rigidity and asymptotics etc. In this talk I will focus on the monotonicity of quasilocal mass, which means the quasilocal mass of a finite region is smaller than that of a larger region containing the previous region. Motivated by the definition of Bartnik mass and Brown-York mass, we define a new quasilocal mass using the Hamilton method. With this definition, we derive the sufficient condition such that the quasilocal mass is monotonic. As an application I will also discuss the relationship with the Bartnik conjecture, that is under what conditions there will be a static extension to the given Bartnik data.

Scattering amplitudes in quantum field theory
2023-03-29 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Ruth Britto (Trinity College Dublin)
Title: Scattering amplitudes in quantum field theory
Abstract: Particle collider experiments require a detailed description of scattering events, traditionally computed through sums of Feynman diagrams. However, it is not practical to evaluate Feynman diagrams directly for all significant scattering processes. Moreover, adding all diagrams reveals many cancellations: scattering amplitudes in theories such as QCD take remarkably simple forms. This simplicity is a clue that the perturbative theory is perhaps best understood without reference to Feynman diagrams. In fact, it has recently become possible to explain some of this simplicity. I will show how to derive many amplitudes efficiently and elegantly, and propose taming the remaining complexity with ideas drawn from combinatorics and geometry.

Black hole microstate counting from the gravitational path integral
2023-04-03 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Luca Iliesiu, Stanford
Title: Black hole microstate counting from the gravitational path integral
Abstract: Reproducing the integer count of black hole micro-states from the gravitational path integral is an important problem in quantum gravity. In the first part of the talk, I will show that, by using supersymmetric localization, the gravitational path integral for 1/16-BPS black holes in supergravity can reproduce the index obtained in the string theory construction of such black holes. A more refined argument then shows that not only the black hole index but also the total number of black hole microstates within an energy window above extremality that is polynomially suppressed in the charges also matches this string theory index. In the second part of the talk, I will present a second perspective on this state count and show how the BPS Hilbert space can be obtained by directly preparing states using the gravitational path integral. While such a preparation naively gives rise to a Hilbert space of BPS states whose dimension is much larger than expected, I will explain how non-perturbative corrections in the overlap of such states are again responsible for reproducing the correct dimension of the Hilbert space.

Optimal Dynamic Allocation: Simplicity through Information Design
2023-04-04 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Faidra Monachou
Title: Optimal Dynamic Allocation: Simplicity through Information Design
Abstract: We study dynamic nonmonetary markets where objects are allocated to unit-demand agents with private types. An agent’s value for an object is supermodular in her type and the quality of the object, and her payoff is quasilinear in her waiting cost. The social planner’s objective is a linear combination of allocative efficiency (i.e., the sum of values) and welfare (i.e., the sum of payoffs). We identify the optimal mechanism in the class of direct-revelation mechanisms that elicit agents’ types and assign them to objects over time. We show that, when the social planner can design the information disclosed to the agents about the objects, the optimal mechanism has a simple implementation: a first-come first-served waitlist with deferrals. In this implementation, the information disclosed about each object is an interval containing the object quality, rather than the exact quality. These intervals partition the quality space. We also show that when the planner’s objective weight on welfare increases, these intervals become coarser, and optimal disclosure policies less informative. A direct corollary is that mechanisms that achieve higher welfare also induce lower distributional inequality, in terms of the Lorenz order.

Spectral gap and two-point functions in spin glasses
2023-04-11 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Changji Xu
Title: Spectral gap and two-point functions in spin glasses
Abstract: Many have worked on spin glass models over the past 50 years, including physicists, mathematicians, and computers. A question that arises is whether computers yield dependable simulation results. In this talk, I will discuss some recent mathematical progress on spectral gap and two-point functions in spin glasses models.

Unexpected Uses of Neural Networks: Field Theory and Metric Flows
2023-04-12 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: James Halverson (Northeastern University)

Title: Unexpected Uses of Neural Networks: Field Theory and Metric Flows
Abstract: We are now quite used to the idea that deep neural networks may be trained in a variety of ways to tackle cutting-edge problems in physics and mathematics, sometimes leading to rigorous results. In this talk, however, I will argue that breakthroughs in deep learning theory are also useful for making progress, focusing on applications to field theory and metric flows. Specifically, I will introduce a neural network approach to field theory with a different statistical origin, that exhibits generalized free field behavior at infinite width and interactions at finite width, and that allows for the study of symmetries via the study of correlation functions in a different duality frame. Then, I will review recent progress in approximating Calabi-Yau metrics as neural networks and cast that story into the language of neural tangent kernel theory, yielding a theoretical understanding of neural network metric flows induced by gradient descent and recovering famous metric flows, such as Perelman’s formulation of Ricci flow, in particular limits.

Recent progress on coupled Einstein-Yang-Mills dynamics
2023-04-18 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Puskar Mondal
Title: Recent progress on coupled Einstein-Yang-Mills dynamics
Abstract: I will discuss my recent work with prof. S.T. Yau on Einstein-Yang-Mills theory. I’ll discuss the main ideas behind the stability of the Minkowski and Milne space time under coupled gauge-gravity perturbations. Construction of the gauge invariant energies that control the suitable norms of the physical degrees of freedom in a double-null framework play an important role. I’ll sketch how the special structure of the Einstein and Yang-Mills nonlinearities play a crucial role in obtaining the stability result of non-expanding spacetimes in particular AF spacetimes.

Black hole collider physics
2023-04-20 13:30 - 14:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Julio Parra Martinez, Caltech
Title: Black hole collider physics
Abstract: Despite more than a century since the development of Einstein’s theory, the general relativistic two-body problem remains unsolved. A precise description of its solution is now essential, as it is necessary for understanding the strong-gravity dynamics of compact binaries observed at LIGO/VIRGO/KAGRA and in future gravitational wave observatories. In this talk, I will describe how considering the scattering of black holes and gravitons can shed new light on this problem. I will explain how using modern ideas from collider and particle physics we can calculate scattering observables in classical gravity, and extract the basic ingredients that describe the bound binary dynamics. Such calculations have produced state-of-art predictions for current and future gravitational wave observatories, which open the door for further discovery as we enter this new era of precision gravitational physics.

On complete Calabi-Yau metrics and some related Monge-Ampere equations
2023-04-25 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Freid Tong
Title: On complete Calabi-Yau metrics and some related Monge-Ampere equations
Abstract: We will give a basic introduction to constructions for complete Calabi-Yau metrics. A systematic approach to construct such metrics using PDE methods was proposed in the work of Tian-Yau in the 90s and have attracted a lot of attention in recent years. I will discuss some joint work with B. Guo and T. Collins on a singular version of such a construction, as well as some ongoing work with Prof. Yau on some related boundary value problems.

Boundary behavior at classical and quantum phase transitions
2023-04-26 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Max Metlitski (MIT)
Title: Boundary behavior at classical and quantum phase transitions
Abstract: There has been a lot of recent interest in the boundary behavior of materials. This interest is driven in part by the field of topological states of quantum matter, where exotic protected boundary states are ubiquitous. In this talk, I’ll ask: what happens at a boundary of a system, when the bulk goes through a phase transition. While this question was studied in the context of classical statistical mechanics in the 70s and 80s, basic aspects of the boundary phase diagram for the simplest classical phase transitions have been missed until recently. I’ll describe progress in this field, as well as some extensions to quantum phase transitions.

Toolformer: Language Models Can Teach Themselves to Use Tools
2023-04-26 14:00 - 15:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Timo Schick, Meta AI
Title: Toolformer: Language Models Can Teach Themselves to Use Tools
Abstract: Language models exhibit remarkable abilities to solve new tasks from just a few examples or textual instructions, especially at scale. They also, paradoxically, struggle with basic functionality, such as arithmetic or factual lookup, where much simpler and smaller models excel. In this talk, we show how these limitations can be overcome by letting language models teach themselves to use external tools via simple APIs. We discuss Toolformer, a model trained to independently decide which APIs to call, when to call them, what arguments to pass, and how to best incorporate the results into future token prediction. Through this, it achieves substantially improved zero-shot performance across a variety of downstream tasks without sacrificing its core language modeling abilities.

Toroidal Positive Mass Theorem
2023-05-02 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Aghil Alaee
Title: Toroidal Positive Mass Theorem
Abstract: In this talk, we review the positive mass conjecture in general relativity and prove a toroidal version of this conjecture in an asymptotically hyperbolic setting.

Generative Adversarial Networks (GANs): An Analytical Perspective
2023-05-03 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Xin Guo, UC Berkeley
Title: Generative Adversarial Networks (GANs): An Analytical Perspective
Abstract: Generative models have attracted intense interests recently. In this talk, I will discuss one class of generative models, Generative Adversarial Networks (GANs). I will first provide a gentle review of the mathematical framework behind GANs. I will then proceed to discuss a few challenges in GANs training from an analytical perspective. I will finally report some recent progress for GANs training in terms of its stability and convergence analysis.

Modern Hopfield Networks for Novel Transformer Architectures
2023-05-10 14:00 - 15:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Dmitry Krotov, IBM Research – Cambridge
Title: Modern Hopfield Networks for Novel Transformer Architectures
Abstract: Modern Hopfield Networks or Dense Associative Memories are recurrent neural networks with fixed point attractor states that are described by an energy function. In contrast to conventional Hopfield Networks, which were popular in the 1980s, their modern versions have a very large memory storage capacity, which makes them appealing tools for many problems in machine learning and cognitive and neurosciences. In this talk, I will introduce an intuition and a mathematical formulation of this class of models and will give examples of problems in AI that can be tackled using these new ideas. Particularly, I will introduce an architecture called Energy Transformer, which replaces the conventional attention mechanism with a recurrent Dense Associative Memory model. I will explain the theoretical principles behind this architectural choice and show promising empirical results on challenging computer vision and graph network tasks.

Quantum information and extended topological quantum field theory
2023-05-15 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Gabriel Wong
Title: Quantum information and extended topological quantum field theory
Abstract: Recently, ideas from quantum information theory have played an important role in condensed matter and quantum gravity research. Most of these applications focus on the entanglement structure of quantum states, and the computation of entanglement measures such as entanglement entropy has been an essential part of the story. In this talk, we will address some subtleties that arise when trying to define entanglement entropy in quantum field theory and quantum gravity. In particular, we will explain why extended topological field theory provides a useful framework to define and compute entanglement entropy in a continuous system. Time permitting, we will explain some recent applications of these ideas in low dimensional quantum gravity and to topological string theory.

Quantum UV-IR map and curve counts in skeins
2023-09-15 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Sunghyuk Park
Title: Quantum UV-IR map and curve counts in skeins
Abstract: Quantum UV-IR map (a.k.a. q-nonabelianization map), introduced by Neitzke and Yan, is a map from UV line defects in a 4d N=2 theory of class S to those of the IR. Mathematically, it can be described as a map between skein modules and is a close cousin of quantum trace map of Bonahon and Wong.
In this talk, I will discuss how quantum UV-IR map can be generalized to a map between HOMFLYPT skein modules, using skein-valued curve counts of Ekholm and Shende.

CMSA Q and A Seminar 9/19/2023
2023-09-19 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speakers: Dan Freed (Harvard Math and CMSA), Denis Auroux (Harvard Math)

The TinyStories Dataset: How Small Can Language Models Be And Still Speak Coherent
2023-09-20 14:00 - 15:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Ronen Eldan, Microsoft Research
Title: The TinyStories Dataset: How Small Can Language Models Be And Still Speak Coherent
Abstract: While generative language models exhibit powerful capabilities at large scale, when either the model or the number of training steps is too small, they struggle to produce coherent and fluent text: Existing models whose size is below a few billion parameters often do not generate coherent text beyond a few sentences. Hypothesizing that one of the main reasons for the strong reliance on size is the vast breadth and abundance of patterns in the datasets used to train those models, this motivates the following question: Can we design a dataset that preserves the essential elements of natural language, such as grammar, vocabulary, facts, and reasoning, but that is much smaller and more refined in terms of its breadth and diversity?
In this talk, we introduce TinyStories, a synthetic dataset of short stories that only contain words that 3 to 4-year-olds typically understand, generated by GPT-3.5/4. We show that TinyStories can be used to train and analyze language models that are much smaller than the state-of-the-art models (below 10 million parameters), or have much simpler architectures (with only one transformer block), yet still produce fluent and consistent stories with several paragraphs that are diverse and have almost perfect grammar, and demonstrate certain reasoning capabilities. We also show that the trained models are substantially more interpretable than larger ones, as we can visualize and analyze the attention and activation patterns of the models, and show how they relate to the generation process and the story content. We hope that TinyStories can facilitate the development, analysis and research of language models, especially for low-resource or specialized domains, and shed light on the emergence of language capabilities in LMs.

Modularity of Landau-Ginzburg Models
2023-09-22 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Chuck Doran
Title: Modularity of Landau-Ginzburg Models
Abstract: Fano varieties are the basic building blocks of algebraic varieties. Smooth Fano varieties have been classified in dimensions one (the projective line), two (del Pezzo surfaces), and three (Mori-Mukai classification). What does Mirror Symmetry have to say about such classifications? By studying the Landau-Ginzburg models mirror to smooth Fano threefolds we can transform the Mori-Mukai classification into an effective uniruledness result for moduli spaces of certain K3 and abelian surfaces. This is joint work with Andrew Harder, Ludmil Katzarkov, Mikhail Ovcharenko, and Victor Przjalkowski (arXiv:2307.15607).

CMSA Q and A Seminar 9/26/2023
2023-09-26 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q and A Seminar
Speakers: Michael Douglas (CMSA) and Mayuko Yamashita (Kyoto University)
Topics:
Michael Douglas: “What is non-commutative field theory?”

Mayuko Yamashita: “What is differential cohomology?”

Transformers for maths, and maths for transformers
2023-09-27 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: François Charton, Meta AI
Title: Transformers for maths, and maths for transformers
Abstract: Transformers can be trained to solve problems of mathematics. I present two recent applications, in mathematics and physics: predicting integer sequences, and discovering the properties of scattering amplitudes in a close relative of Quantum ChromoDynamics.
Problems of mathematics can also help understand transformers. Using two examples from linear algebra and integer arithmetic, I show that model predictions can be explained, that trained models do not confabulate, and that carefully choosing the training distributions can help achieve better, and more robust, performance.

Moduli of vector bundles on curve and semiorthogonal decomposition
2023-09-29 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Kai Xu (CMSA)
Title: Moduli of vector bundles on curve and semiorthogonal decomposition
Abstract: In this talk we construct semiorthogonal decompositions of moduli of vector bundles on a curve into its symmetric powers. The essential ingredients in the proof include Borel-Weil-Bott theory for loop groups, derived Schur-Weyl duality for current groups and derived Θ-stratification.

Gravitational Instantons
2023-10-02 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Yu-Shen Lin (Boston University)
Title: Gravitational Instantons
Abstract: Gravitational instantons were introduced by Hawking as building blocks of his Euclidean quantum gravity theory back in the 1970s. These are non-compact Calabi-Yau surfaces with L2 curvature and thus can be viewed as the non-compact analogue of K3 surfaces. K3 surfaces are 2-dimensional Calabi-Yau manifolds and are usually the testing stone before conquering the general Calabi-Yau problems. The moduli space of K3 surfaces and its compactification on their own form important problems in various branches in geometry. In this talk, we will discuss the Torelli theorem of gravitational instantons, how the cohomological invariants of a gravitational instanton determine them. As a consequence, this leads to a description of the moduli space of gravitational instantons.
