For more information: https://scgp.stonybrook.edu/archives/39876

Title:   Rhombic staircase tableaux and Koornwinder polynomials
Speaker:   Lauren Williams
Abstract:   In earlier work with Corteel and Mandelshtam, we introduced rhombic staircase tableaux and used them to give a combinatorial formula for the stationary distribution of the two-species ASEP on a line with open boundaries. I will discuss work-in-progress in which we use these tableaux to give formulas for some special Koornwinder polynomials.

Title:   Moduli spaces of pseudo-holomorphic curves
Speaker:   John Pardon [Stony Brook University]

Abstract:   We will discuss various analytical, topological, and categorical aspects of moduli spaces of pseudo-holomorphic curves. Topics may include elliptic regularity, infinity-categories, topological stacks, derived smooth manifolds, log smooth manifolds, gluing, and Gromov compactness.
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Title:   Weighted random Motzkin paths and the stationary distribution for an open Asymmetric Simple Exclusion Process
Speaker:   Woldek Bryc
Abstract:   This expository talk explains how one can use non-uniformly weighted random Motzkin paths to describe the stationary distribution for an open Asymmetric Simple Exclusion Process (ASEP) in the fan region. This relationship motivated the study of asymptotics for random Motzkin paths in several recent papers with Yizao Wang, Jacek Wesolowski, and Alexey Kuznetsov. The connections between Motzkin paths and ASEP have been known in the literature for a while, but the relation to be presented has not been widely used. It is implicit in the matrix model of Enaud and Derrida (2004) and appears explicitly in Derrida, Enaud, and Lebowitz (2004), and in a somewhat different form in Barraquand and Le Doussal (2023). The stationary distribution of ASEP was also expressed in terms of weighted Motzkin path in Brak, Corteel, Essam, Parviainen, and Rechnitzer (2006) and in Corteel, Josuat-Vergès, and Williams (2011), which were based on the well-known matrix model of Uchiyama, Sasamoto, and Wadati (2004). The latter representation of the stationary distribution of ASEP is also discussed in a recent paper Nestoridi and Schmid (2023). In this talk, I will first review the Motzkin paths and the stationary distribution of ASEP. I will then discuss the relationship between the them. If time permits, I will discuss some of the results on asymptotics for random Motzkin paths.

Speaker:   Viktor Eisler

Title:   Entanglement Hamiltonians in quantum many-body systems

Abstract:   Understanding the structure of entanglement in quantum many-body systems, both in and out of equilibrium, has been a long-standing goal of the last two decades. It has been routinely characterized via the Renyi entanglement entropies, which provide information about the moments of the reduced density matrix. Despite their success as a measure of overall entanglement, they leave many questions about the precise structure of the reduced state unanswered. To fill this gap, one could study the entanglement Hamiltonian, which is nothing but the logarithm of the reduced density matrix. It turns out that, under simple assumptions about the underlying state, this operator has a remarkably universal structure. In particular, it is local and in various situations can be written as a simple deformation of the physical Hamiltonian. In this talk I will give a review of this topic, highlighting exact theoretical results as well as recent breakthroughs in experiments.

Special Talk for High School Students 

TITLE: Tigers’ Stripes and Leopard’s Spots

Abstract:   A leopard cannot change its spots” — but how did it get the spots in the first place? The animal kingdom is full of creatures with beautiful markings, from the stripes of tigers and zebras to the apparently random triangles on some seashells. The mathematician Alan Turing is best known for his work on codebreaking during World War II and for early research on computing and artificial intelligence, but he was also interested in biology. In 1952 he proposed a mathematical explanation of how the patterns of animal markings form, based on chemicals called ‘morphogens’ that react and diffuse during key stages of the animal’s development.
The talk will explain the underlying mathematics in simple terms, mainly through pictures. It will be illustrated by numerous examples in the animal kingdom, and mention recent research that confirms some aspects of Turing’s theory.

Title:   Current fluctuations for symmetric processes
Speaker:   Tomohiro Sasamoto
Abstract:   In the field of integrable probability, it has been common to study “asymmetric models” in which particles hop with asymmetric rates in left and right and then often show KPZ behaviors. But their symmetric versions already show various non-trivial behaviors and there remain various interesting questions. In this presentation we will explain how one can study current fluctuations of symmetric models. The main focus would be on the case of one-dimensional symmetric exclusion but we may also discuss other processes. The talk is mainly based on a joint work with Takashi Imamura and Kirone Mallick [1], and discussions with Cristian Giardina. Also the talk is somewhat complimentary to the talk by Kirone Mallick. References: [1] T. Imamura, K. Mallick, T. Sasamoto, Distribution of a tagged particle position in the one-dimensional symmetric simple exclusion process with two-sided Bernoulli initial condition, Commun. Math. Phys. 384: 1409-1444, 2021.

Title:   Graduate Student theorems
Speaker:   Jiahao Hu [Stony Brook University]

Abstract:   I will discuss some theorems proved by Stony Brook Graduate students through collaboration.
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Title:   Decorrelation of the KPZ fixed point from the flat initial condition
Speaker:   Ofer Busani
Abstract:   The KPZ class is a set of 1+1 random growth interface models that are believed to model the statistics of an interface of a growing two dimensional surface. Upon a time-space scaling , it is believed (and was proven for a handful of models) that all such models should converge to a universal scaling limit called the KPZ-fixed point. We shall discuss spatial decorrelation for the KPZ-fixed point. In particular, when the KPZ-fixed point starts from a specific initial condition called 'flat', we obtain the first order term in the exponent of the decay. Joint work with Riddhipratim Basu and Patrik Ferrari.

Title:   Probing Supersymmetric Black Holes with Surface Defects

Abstract:   Supersymmetric black holes in Anti de-Sitter space have recently been shown to have a large number of exactly degenerate microstates, and it is an open problem to probe and ultimately distinguish these microstates in terms of the bulk degrees of freedom. In the first part of the talk, we will review how AdS5 black hole microstates may be reliably counted in the dual N=4 SYM theory using the 1/16 BPS superconformal index and how this result is reproduced by the bulk gravitational path integral in the near BPS / N=2 super JT gravity limit. This perspective also suggests a ""mass gap"" between BPS and near-BPS quantum black holes.  With the goal of understanding more detailed properties involving the BPS black holes using the bulk description, we will turn to the question of whether there are supersymmetric probes of the black hole which have an exact field theory dual. We find one such candidate is the superconformal index with the insertion of a Gukov-Witten surface operator of N=4 SYM, dual to a D3 brane which wraps the AdS5 black hole horizon. We find a saddle with a large N growth which can be exactly matched to the probe brane action, and further conjecture a generalized Cardy formula for surface defects in holographic 4d SCFT's. In addition to detecting the familiar deconfinement transition associated to the dominance of the bulk black hole saddle, this provides an example of a system in which a black hole interacts with other degrees of freedom which has a microscopic description.

Title:   Infinitesimal generators of quadratic harnesses - an algebraic approach
Speaker:   Jacek Wesolowski
Abstract:   Quadratic harnesses (equivalently, Askey-Wilson processes) appeared to be useful in studying properties of the ASEP with open boundaries under the stationary measure. Quadratic harnesses (QH processes) are typically determined by 5 numerical constants and thus denoted by QH(\eta,\theta,\sigma,\tau,\gamma). In particular, in BW(2017) we used infinitesimal generators of QH(\eta,\theta,0,0,\gamma) (the bi-Poisson process) to derive formulas for difference of average occupancy of neighbouring sites. In this talk we will show that infinitesimal generators of QH processes are special integro-differential operators parametrized by a measure of integration, which often can be associated to some Askey-Wilson measure (or special conditional distribution of some QH process). In AW(2015) we introduced some algebraic methodology which appeares to be quite useful in studying infinitesimal generators of QH processes. In particular, this methodology lead to the form of the operator for "free" QH process, (\gamma=-\sigma\tau). Recently, with my PhD student, A. Zięba, this algebraic methodology was considerably extended. Firstly, the method was extended to cover the case of QH(\eta,\theta,0,\tau,\gamma) (see WZ(2023)) and then the full range of parameters was covered in a clever, though quite complicated, algebraic proof given in Z(2023). References: BW(2015): W. Bryc, J. Wesołowski, Infinitesimal generators for a class of polynomialprocesses. Studia Math., 229(1) (2015):73–93. BW(2017): W. Bryc, J. Wesołowski, Asymmetric simple exclusion process with open boundaries and quadratic harnesses. J. Stat. Phys., 167(2) (2017): 383–415.
WZ(2023): J. Wesołowski, A. Zieba,Infinitesimal generators for a family of polynomial
processes – an algebraic approach. arXiv: 2305.00198 (2023): 1-21. Z(2023): A. Zięba, Infinitesimal generators of quadratic harnesses, PhD Thesis, Warsaw Uni. Tech., 2023: 1-133.

Hashtag: #workshop

Title:   Universality for multicomponent stochastic systems
Speaker:   Alessandra Occelli
Abstract:   We study the equilibrium fluctuations of an interacting particle system evolv- ing on the discrete ring TN with three species of particles that we name A,B and C, subject to the exclusion rule: at each site there is only one particle. The interaction rates depend on the type of particles involved via three constants EA, EB and EC, and on the size of the system. This model can be seen as a multi-species generalisation of the weakly asymmetric simple exclusion process. We analyse proper choices of the density fluctuation fields associated to the conserved quantities (the densities of particles for each species), that are given by linear combinations of the fields that match those from nonlinear fluctuating hydrodynamics theory [1]: we show that they converge, in the limit N → ∞, to a system of stochastic partial differential equations, that, according to the asymmetry of the jumps, can either be the Ornstein–Uhlenbeck equation or the Stochastic Burgers’ equation. Based on a joint work with G. Cannizzaro, P. Gon ̧calves and R. Misturini. [1] Spohn, H. Nonlinear Fluctuating Hydrodynamics for Anharmonic Chains, Journal of Statistical Physics, 154.5, 1191–1227 (2014).

Title:   Exact solutions of the macroscopic fluctuation theory for the symmetric exclusion process
Speaker:   Kirone Mallick
Abstract:   At hydrodynamic scale, large deviations if symmetric exclusion process can can studied from a variational principle – due to Kipnis, Olla and Varadhan and generalized to diffusive processes by G. Jona-Lasinio and his collaborators. This framework, known as the Macroscopic Fluctuation Theory (MFT), expresses the optimal rare fluctuations as the solutions of two coupled non-linear PDEs with mixed and non-local boundary conditions conditions. In this talk, we shall show that, for the exclusion process, the MFT equations are classically integrable in the sense of Liouville and Lax, by using the inverse scattering method. By solving exactly the associated Riemann- Hilbert problem, we shall calculate the large deviation function of the cur- rent (that embodies its statistics) and the optimal evolution that generates a required fluctuation, both at initial and final times [1]. This work, at macro- scopic level, complements the microscopic integrability approach, presented by Tomohiro Sasamoto.
[1] K. Mallick, H. Moriya and T. Sasamoto, Exact Solution of the Macroscopic Fluctuation Theory for the Symmetric Exclusion Process, Phys. Rev. Lett. 129, 040601 (2022).

Title:   Exact Solution of a Lifted Asymmetric Exclusion Process
Speaker:   Fabian Essler
Abstract:   Markov-chain Monte Carlo has countless applications in science and technology. The underlying Markov chains are typically taken to be time-reversible and satisfy the detailed-balance condition, which encodes the absence of flows. The resulting exploration of sample space is diffusive and hence slow. In recent years the use of non-reversible Markov chains has been explored. In particular expanding the sample space through a procedure called lifting has been shown to lead to substantial speedups in converging to steady states a range of problems.
I introduce the “lifted” totally asymmetric simple exclusion process (TASEP) as a solvable paradigm for lifted non-reversible Markov chains. I then show that it can be solved by a coordinate Bethe Ansatz with some unusual features. By working out the leading eigenvalues of the transition matrix from the Bethe Ansatz and comparing them to numerical simulations I show that the lifted TASEP indeed reaches the steady state faster than the TASEP.

Title:   Brill-Noether loci and degree of irrationality of surfaces
Speaker:   Federico Moretti [Stony Brook University]

Abstract:   Given a variety of dimension n, I will study rational maps to a
projective space of the same dimension using the associated kernel
(syzygy) sheaf. I will mainly focus on the case of K3 surfaces with Picard
rank 1. In this context, I will explain how this perspective allows us to
show that maps of degree at most d, induced by the primitive linear
system, move in families. I will then study and characterize, in some
cases, projections of minimal degree for primitively polarized K3 surfaces
up to genus 14. This can be seen as a first step towards computing the
degree of irrationality of these surfaces. This is a joint work with
Andrès Rojas.
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Title:   TASEP with a moving wall
Speaker:   Patrik Ferrari
Abstract:   We study TASEP on Z with the step initial condition, under the additional restriction that the first particle cannot cross a deterministically moving wall. We prove that such a wall may induce asymptotic fluctuation distributions of particle positions equal to the probability that the Airy2 process is below a barrier function g. This is the same class of distributions that arises as one-point asymptotic fluctuations of TASEPs with arbitrary initial conditions.

Title:   On the computation of limits of stationary measures of open ASEP
Speaker:   Yizao Wang
Abstract:   Recently, several advances have been made on limits of stationary measures of open ASEP. This talk will present an overview on the method underlying many of these developments. The starting point of the method is a new representation of the probability generating function of stationary measures of open ASEP in terms of the so-called Askey-Wilson Markov processes introduced by Bryc and Wesolowski (2010, 2017). Compared to the well-known Derrida’s matrix ansatz from the 1990s, the advantage of this new representation is that for the Laplace transform of stationary measures of open ASEP it becomes much easier to compute its asymptotics. Computing the limit of Laplace transform then leads to the limit of stationary measures. This conceptually straightforward computation, however, consists of two further delicate steps. One is the computation of the so-called tangent process of Askey-Wilson process, and the other is the derivation of a duality formula of Laplace transforms of certain Markov processes. The talk shall present how the method has been applied and adapted, particularly regarding these two steps, in various setups of open ASEP, and comment on other potential applications. Based on a series of joint works with Wlodek Bryc, Alexey Kuznetsov, Jacek Wesolowski and Zongrui Yang.

Speaker:   Berislav Buca

Title:   Eigenoperator thermalization theory

Abstract:   I will provide a framework for time-averaged dynamics in locally interacting systems in any dimension. It is based on pseudolocal dynamical symmetries generalising pseudolocal charges and unifies seemingly disparate manifestations of quantum non-ergodic dynamics including quantum many-body scars, continuous, discrete and dissipative time crystals, Hilbert space fragmentation, lattice gauge theories, and disorder-free localization. In the process novel pseudo-local classes of operators are introduced: "projected local", which are local only for some states, and "crypto-local", whose locality is not manifest in terms of any finite number of local densities. Using the theory two novel types of phase transitions are introduced: 1) The "scarring phase transition" where the order parameter is the locality of the projected local quantities - for certain initial states persistent oscillations are present. 2) The "fragmentation phase transition" for which long-range order is established in an entire phase due to presence of certain non-local strings. Two prototypical, but otherwise mostly intractable, models are solved using the theory: 1) a spin 1 scarred model and 2) the t-J_z model with fragmentation.

Title:   Tagged particles, voter models and fractional Gaussian noise
Speaker:   Alan Hammond
Abstract:   Let 2N+1 independent one-dimensional Brownian motions begin at the origin. By considering the order statistics of this system, we may locate a bulk curve that has N motions above it, and N below. As Dürr, Goldstein and Lebowitz showed, the law of the bulk curve asymptotically in high N is a Gaussian process with t^{1/4}-order fluctuations. In this talk, we will discuss fractional Gaussian fields and how they arise from models such as the tagged particle in the symmetric simple exclusion process and power-law Polya urns and voter models.

Title:   Large deviations for diffusion in random media: integrable crossover from macroscopic fluctuation theory to weak noise KPZ equation
Speaker:   Pierre Le Doussal
Abstract:   The large deviations for the diffusion of a tracer in a 1D time dependent medium can be described, on diffusive scales, by the macroscopic fluctuation theory (MFT). The corresponding MFT variational equations are mapped to the integrable derivative non-linear Schrodinger equation. We provide a solution using inverse scattering methods, and obtain the large deviation rate function for the sample to sample fluctuation of the probability of the tracer position. Furthermore by varying the position of the tracer, i.e. the asymmetry, we uncover the full integrable crossover from the MFT to the weak noise theory of the KPZ equation, matching our previous results for the latter problem.
Based on
Krajenbrink, A., & Le Doussal, P. (2023). Crossover from the macroscopic fluctuation theory to the Kardar-Parisi-Zhang equation controls the large deviations beyond Einstein's diffusion. Physical Review E, 107(1), 014137.

Title:   The second class particle shock process in TASEP
Speaker:   Peter Nejjar
Abstract:   We consider TASEP with initial data such that a shock is created, and a second class particle following the shock. We show that the particle's limit process is the difference of two independent limit processes, coming from the left resp. the right of the shock. We directly relate the particle's position to the difference of height functions and show that the latter decouple, avoiding any usage of LPP. Joint work with Patrik Ferrari.

Title:   Surgery Exact Triangles in Instanton Theory
Speaker:   Deeparaj Bhat [MIT]

Abstract:   We prove an exact triangle relating the knot instanton homology to the instanton homology of surgeries along the knot. As the knot instanton homology is computable in many instances, this sheds some light on the instanton homology of closed 3-manifolds. We illustrate this with computations in the case of some surgeries on the trefoil which are different from the analogous groups in other Floer theories such as Heegaard Floer and monopole Floer. Finally, we sketch the proof of the triangle.
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Title:   Ergodic stationary measures of the Kardar-Parisi-Zhang equation
Speaker:   Timo Seppäläinen
Abstract:   This talk describes multifunction invariant distributions of the KPZ equation on the line. These measures are finite-dimensional distributions of the function-valued Busemann process, indexed by the asymptotic spatial slope which is a conserved quantity. These functions are eternal solutions of the equation and attractors of the evolution. The Busemann process has a countable dense set of discontinuities in the slope variable. These are values of the conserved slope at which there are multiple eternal solutions, thereby breaking the one force-one solution principle. Based on joint projects with Sean Groathouse (Utah), Chris Janjigian (Purdue), Firas Rassoul-Agha (Utah), Evan Sorensen (Columbia).

TITLE: Hidden symmetries in lattice networks

ABSTRACT: Networks of coupled dynamical systems are a current topic of research in many areas of science. The network topology influences the generic dynamic phenomena, often in curious ways. A fundamental example is synchrony, when two or more nodes have the same time series. Patterns of synchrony in networks of coupled dynamical systems can be represented as colorings, in which nodes with the same color are synchronous. Balanced colorings, where nodes of the same color have color-isomorphic input sets, determine robust synchrony patterns, and define a ‘quotient network’ whose dynamics gives that of the synchronous clusters. Subgroups of the symmetry group of the network lead to balanced colorings, but the converse is false.
We consider doubly periodic balanced colorings of the square and hexagonal lattices with nearest-neighbor coupling, and classify the `exotic’ cases where the  quotient network has extra automorphisms  not induced from automorphisms of the lattice. These comprise some simple infinite families and a few isolated exceptions. The results have unexpected implications for bifurcations to doubly periodic patterns in lattice models.

Title:   The environment seen from a geodesic in last-passage percolation, and the TASEP seen from a second-class particle.
Speaker:   James Martin
Abstract:   We study directed last-passage percolation in Z^2 with i.i.d. exponential weights. What does a geodesic path look like locally, and how do the weights on and nearby the geodesic behave? We show convergence of the distribution of the "environment" as seen from a typical point along the geodesic in a given direction, as its length goes to infinity. We describe the limiting distribution, and can calculate various quantities such as the density function of a typical weight, or the proportion of "corners" along the path. The analysis involves a link with the TASEP seen from an isolated second-class particle, and we obtain some convergence and ergodicity results for that process. The talk is based on joint work with Allan Sly and Lingfu Zhang.

Title:   Q&A with Jae Ho Cho
Speaker:   Jae Ho Cho [Morgan Stanley]

Abstract:   In this presentation, I will provide a brief introduction to my work experience at Morgan Stanley, key points to know when applying, and how mathematics is used. The primary focus of the presentation would be Q&A session following the short introduction.
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Title:   Askey-Wilson signed measures and open ASEP in the shock region
Speaker:   Zongrui Yang
Abstract:   In this talk, we will present a new method for studying the stationary measure of the open asymmetric simple exclusion process (ASEP) in the shock region. We introduce a family of multi-dimensional Askey-Wilson signed measures and then describe the joint generating function of the open ASEP stationary measure in terms of integrations with respect to these Askey-Wilson signed measures. As an application, we offer a rigorous derivation of the density profile and limit fluctuations of the open ASEP in the shock region, confirming the existing physics postulations. This is a joint work with Yizao Wang and Jacek Wesolowski.

Title:   The influence of edges in first-passage percolation on Z^d
Speaker:   Dor Elboim
Abstract:   We study the probability that a geodesic passes through a prescribed edge in first-passage percolation on. Benjamini, Kalai and Schramm famously conjectured that this probability tends to zero as the length of the geodesic tends to infinity, as long as the edge is not too close to the endpoints of the geodesic. I will present a short proof that this probability is arbitrarily small for all edges except for constantly many of them. This is a joint work with Ron Peled and Barbara Dembin.

Title:   Colored Interacting Particle Systems on the Ring: Stationarity from Yang-Baxter
Speaker:   Leonid Petrov
Abstract:   Recently, there has been much progress in understanding stationary measures for colored (also called multi-species or multi-type) interacting particle systems motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials). I will describe a unified approach to constructing stationary measures for colored ASEP, q-Boson, and q-PushTASEP systems based on integrable stochastic vertex models and the Yang-Baxter equation. Stationary measures become partition functions of new "queue vertex models" on the cylinder, and stationarity is a direct consequence of the Yang-Baxter equation. Our construction recovers and generalizes known stationary measures constructed using multiline queues and the Matrix Product Ansatz. In the quadrant, Yang-Baxter implies a colored version of Burke's theorem, which produces stationary measures for particle systems on the line. We also compute the colored particle currents in stationarity. Joint work with Amol Aggarwal and Matthew Nicoletti.

Title:   Alterations: Three-points lemma
Speaker:   TBA [Stony Brook University]

Abstract:   TBA
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Public holiday

Observance
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Mini-course by Gregory Falkovich

September 26 - November 28, 2023 on Tuesdays at 2:30pm in room 313

Title:   Introduction to Information Theory

Selected Subjects: 

Measuring Uncertainty

Mutual Information and Entanglement Entropy, Redundancy of Language and Genetic Code

Information is Life and Money

Theory of Mind – Active Inference

Information in Quantum Mechanics and Statistics

Title:   No Seminar: Fall Break


Abstract:  
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Title:   Moduli spaces of pseudo-holomorphic curves
Speaker:   John Pardon [Stony Brook University]

Abstract:   We will discuss various analytical, topological, and categorical aspects of moduli spaces of pseudo-holomorphic curves. Topics may include elliptic regularity, infinity-categories, topological stacks, derived smooth manifolds, log smooth manifolds, gluing, and Gromov compactness.
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Title:   TBA
Speaker:   Georgina Spence [Stony Brook University]

Abstract:   TBA
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Title:   Compact surface automorphisms
Speaker:   Eric Bedford [Stony Brook University]

Abstract:  
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Title:   Moduli of curves and K-stability
Speaker:   Junyan Zhao [University of Illinois at Chicago]

Abstract:   The K-moduli theory provides us with an approach to study moduli of curves. In this talk, I will introduce the K-moduli of certain log Fano pairs and how it relates to moduli of curves. We will see that the K-moduli spaces interpolate between different compactifications of moduli of curves. In particular, the K-moduli gives the last several Hassett-Keel models of moduli of curves of genus six.
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Title:   TBA
Speaker:   Marcelo Atallah [Université de Montréal ]

Abstract:  
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Speaker:   Vladimir Matveev (Friedrich-Schiller-Universität Jena, Germany)

Title:   Applications of Nijenhuis Geometry: finite-dimensional reductions and integration in quadratures of certain non-diagonalizable systems of hydrodynamic type.

Abstract:   Nijenhuis Geometry is a recently initiated research program, I will recall its philosophic motivation and fundamental results.  New part of my talk is related to applications of these results in the theory of infinite-dimensional integrable systems and includes the following topics(1)  Construction of a large (the freedom is a number of functions of one variable)   family of integrable systems of hydrodynamic type.   Different from most previously known examples, the corresponding generators are not diagonalizable.  
(2)  Finite-dimensional reductions of such systems. The commuting functions of the corresponding finite-dimensional integrable systems are quadratic in momenta and can be viewed as a metric and its (commuting) Killing tensors.
(3) Integration of such systems in quadratures.
This is a work in progress in collaboration with Alexey Bolsinov and Andrey Konyaev.

Title:   Homology cobordism and the geometry of hyperbolic three-manifolds
Speaker:   Francesco Lin [Columbia University]

Abstract:   The three-dimensional homology cobordism group is a fundamental object of study in low-dimensional topology. A major challenge in the study of its structure is to understand the interaction between hyperbolic geometry and homology cobordism. In this talk, after introducing the main protagonists and explaining the role they play in topology, I will discuss how monopole Floer homology (a subtle invariant of three-manifolds obtained by suitably counting solutions to the Seiberg-Witten equations) can be used to study some basic properties of certain subgroups of the homology cobordism group generated by hyperbolic homology spheres satisfying natural geometric constraints of Riemannian and spectral nature.
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Last day to submit an approved adjustment form for selected AMS, MAT, and MAP and PHY courses to the Office of Registrar. Changes must be processed by 4:00 PM