Fri, Feb 3

Geometry and Topology Seminar
Morgan Weiler (Cornell University)
ECH cobordism maps and infinite staircases of 4D symplectic embeddings
4:00PM, 122 Mathematics Building

The ellipsoid embedding function of a symplectic manifold measures the amount by which the symplectic form must be scaled in order to fit an ellipsoid of a given eccentricity. It generalizes the Gromov width and ball packing numbers. In 2012 McDuff and Schlenk computed the ellipsoid embedding function of the ball, showing that it exhibits a delicate piecewise linear pattern known as an infinite staircase. Since then, the embedding function of many other symplectic four-manifolds have been studied, and not all have infinite staircases. We classify those symplectic Hirzebruch surfaces whose embedding functions have an infinite staircase. We will emphasize the relationship between the geometric motivation for cobordism maps coming from embedded contact homology (ECH) and McDuff's methods for obstructing ellipsoid embeddings using Taubes' Gromov-Seiberg-Witten invariants of symplectic 4-manifolds. Based on work with Magill and McDuff and work in progress with Magill and Pires.

Mon, Feb 13

Applied Math Seminar
Jiyoung Kang, Pukyong National University
Brain Dynamics and its Control: Computational Approaches

2:00PM, Math 250 and on Zoom - contact mbichuch@buffalo.edu for link

The brain is a complex nonlinear system, and it is challenging to understand its dynamics. In addition, the brain possesses plasticity, which frequently results in a temporary improvement followed by a decline after brain disease treatment. We have been developing computational modeling techniques for brain dynamics analysis to address these issues. In this presentation, several computational modeling studies utilizing electrophysiology data, such as calcium imaging and voltage-sensitive dye imaging data, along with energy landscape analysis studies using fMRI data, will be discussed. I will conclude by discussing our recent computational framework for brain control, which can account for both brain dynamics and plasticity.

Mon, Feb 13

Algebra Seminar
Yiqiang Li, University at Buffalo
Quantum groups and edge contraction

4:00PM, 250 Math Bldg

Edge contraction is a simple operation on graphs that produces a new graph by merging two vertices on a given graph along an edge. In this talk, I will report recent studies on the behaviors of representation-theoretic objects attached to graphs under an edge contraction operation.

Thu, Feb 16

Special Event
Solitons and the inverse scattering transform: an overview
 Solitons and the inverse scattering  transform: an overview.Abstract: An exciting and extremely active area of research investigation is the study of solitons and the nonlinear partial differential equations that describe them. In this talk, we will discuss what solitons are, and what makes them so special. We will see when the first solitons were observed, and when the first math that describe them appeared. We will introduce ourselves to integrable systems, and we will describe how the technique of the inverse scattering transform is applied in soliton theory.  If time permits, we will give some examples of integrable systems and we will discuss their applications.  
4:00PM

Title: Solitons and the inverse scattering  transform: an overview.

Abstract: An exciting and extremely active area of research investigation is the study of solitons and the nonlinear partial differential equations that describe them. In this talk, we will discuss what solitons are, and what makes them so special. We will see when the first solitons were observed, and when the first math that describe them appeared. We will introduce ourselves to integrable systems, and we will describe how the technique of the inverse scattering transform is applied in soliton theory.  If time permits, we will give some examples of integrable systems and we will discuss their applications. 

Fri, Feb 17

Geometry and Topology Seminar
Yvon Verberne (University of Toronto)
Automorphisms of the fine curve graph

4:00PM, 122 Mathematics Building

The fine curve graph of a surface was introduced by Bowden, Hensel and Webb. It is defined as the simplicial complex where vertices are essential simple closed curves in the surface and the edges are pairs of disjoint curves. We show that the group of automorphisms of the fine curve graph is isomorphic to the group of homeomorphisms of the surface, which shows that the fine curve graph is a combinatorial tool for studying the group of homeomorphisms of a surface. This work is joint with Adele Long, Dan Margalit, Anna Pham, and Claudia Yao.

Mon, Mar 6

Algebra Seminar
Bangming Deng, Tsinghua U
Fourier transforms on Ringel-Hall algebras
4:00PM, Zoom - contact achirvas@buffalo.edu for link

We study Fourier transforms on the double Ringel-Hall algebra of a quiver and make a comparison between Lusztig's symmetries and the isomorphisms defined by Sevenhant and Van den Bergh via combining BGP-reflection isomorphisms and Fourier transforms on the double Ringel-Hall algebra.

Mon, Mar 13

Applied Math Seminar
Weinan Wang, University of Arizona
Recent progress on the well-posedness theory for some kinetic models

2:00PM, Math 250 and on Zoom - contact mbichuch@buffalo.edu for link

The Boltzmann and Landau equations are two fundamental models in kinetic theory. They are nonlocal and nonlinear equations for which (large data) global well-posedness is an extremely difficult problem that is nearly completely open. In this talk, I will discuss two more tractable and related questions: (1) Local well-posedness for the Boltzmann equation and (2) Schauder estimates and their application to uniqueness of solutions to the Landau equation. At the end of the talk, I will discuss some open problems and future work. This is based on joint work with Christopher Henderson.

Fri, Mar 17

Geometry and Topology Seminar
Assaf Bar-Natan (Brandeis University)
How the Thurston metric on Teichmuller space is (not) like L^(infty)

4:00PM, 122 Mathematics Building

The Thurston Metric, introduced by Thurston in 1986, is an asymmetric metric on Teichmuller space, which measures distance between surfaces using the Lipschitz constant of maps between them. In this talk, I will tell you what I know about geodesics in this metric. Specifically, I will tell you about the geodesic envelope, its shape (and how the Thurston metric is similar to L^(infty)), and its width (and how the Thurston metric is not similar to L^(infty)). We'll finish up with a theorem which gives sufficient conditions for geodesics between two points to be "essentially unique" (ie, uniformly bounded diameter from each other) for low complexity surfaces.

Mon, Mar 27

Algebra Seminar
Guanglian Zhang, Shanghai Jiao Tong University
Every type-A quiver locus is a Kazhdan-Lusztig variety

9:00AM, Note unusual time. On Zoom (please email achirvas@buffalo.edu)

The Zariski orbit closures of the representations of type-A Dynkin quivers under the action of general linear groups are related in deep ways to Schubert varieties. In this paper, we construct a scheme-theoretic isomorphism from a type-A quiver locus to the intersection of some opposite Schubert cell and Schubert variety, also known as a Kazhdan-Lusztig variety in geometric representation theory. This isomorphism is a generalization, and also an unification, of the Zelevinsky maps on equioriented type-A quiver loci and bipartite type-A quiver loci which are respectively presented by A. V. Zelevinsky in 1985 and by R. Kinser and J. Rajchgot in 2015. This result provides a more direct and natural connection between type-A quiver loci and Schubert varieties than prior similar work.

Mon, Mar 27

Applied Math Seminar
Qingguo Hong, Penn State
A priori error analysis and greedy training algorithms for neural networks solving PDEs.

2:00PM, Math 250 and on Zoom - contact mbichuch@buffalo.edu for link

We provide an a priori error analysis for methods solving PDEs using neural networks. We show that the resulting constrained optimization problem can be efficiently solved using greedy algorithms, which replaces stochastic gradient descent. Following this, we show that the error arising from discretizing the energy integrals is bounded both in the deterministic case, i.e. when using numerical quadrature, and also in the stochastic case, i.e. when sampling points to approximate the integrals. This innovative greedy  algorithm is tested on several benchmark examples to confirm its efficiency and robustness.

Fri, Mar 31

Geometry and Topology Seminar
Yuan Yao (UC Berkeley)
Computing embedded contact homology in Morse-Bott settings

4:00PM, 122 Mathematics Building

Embedded contact homology (ECH) is a Floer theory defined for contact 3-manifolds with generators periodic Reeb orbits and differential defined by counts of J-holomorphic curves. It has been shown to be isomorphic to a versions of monopole Floer homology and Heegard Floer homology. It has many applications to symplectic and contact geometry (e.g. symplectic embedding problems, dynamics of Reeb vector fields). In this talk we will first review the definition of ECH; then we will discuss how to define ECH in the Morse-Bott setting. Our main tools will be  1) the intersection theory of J-holomorphic curves, 2) understanding how J-holomorphic curves degenerate into geometric objects called cascades when the background contact form degenerates to a Morse-Bott contact form, as well as 3) a gluing theorem that tells us how to glue cascades back into J-holomorphic curves.

Mon, Apr 3

Algebra Seminar
Mariusz Tobolski, University of Wroclaw
Cohomology of free unitary quantum groups 
4:00PM, Mathematics Building room 250

In this talk, I will present the Hochschild and bialgebra cohomology with 1-dimensional coefficients of the \(*\)-algebras associated with free universal unitary quantum groups. The result is based on the free resolution of the counit of the free orthogonal quantum groups found by Collins, Härtel, and Thom which was then generalized by Bichon to the case of quantum groups associated with a nondegenerate bilinear form. In fact, we compute cohomology groups of the universal cosovereign Hopf algebras, which generalize free unitary quantum groups and are connected to quantum groups of non-degenerate bilinear forms. This is a joint work with U. Franz, M. Gerhold,A. Wysocza\'nska-Kula, and I. Baraquin.

Mon, Apr 10

Algebra Seminar
Robert Corless, Western University
Bohemian Matrix Geometry 
4:00PM, Mathematics Building room 250

A Bohemian matrix family is a set of matrices all of whose entries are drawn from a fixed, usually discrete and hence bounded, subset of a field of characteristic zero. Originally these were  integers---hence the name, from the acronym BOunded HEight Matrix of Integers(BOHEMI)---but other kinds of entries are also interesting. Some kinds of questions about Bohemian matrices can be answered by numerical computation, but sometimes exact computation is better. In this paper we explore some Bohemianfamilies                  (symmetric, upper Hessenberg, or Toeplitz) computationally, and answer some (formerly) open questions posed about the distributions of eigenvalue densities.

This work connects with several disparate areas of mathematics, including dynamical systems, combinatorics, probability and statistics, and number theory.  Because the thinking about the topic is so recent, most of the material is still quite exploratory, and this talk will be accessible to students as well as to faculty.  Several open problems remain open, and I would welcome your thoughts on them.

This is joint work with several people, including EuniceY.S. Chan, Leili Rafiee Sevyeri, Neil J. Calkin, Piers W. Lawrence, Laureano Gonzalez-Vega, Dan Piponi, Juana Sendra, and Rafael Sendra.

Fri, Apr 14

Geometry and Topology Seminar
Nima Hoda (Cornell University)
Normed polyhedral complexes and nonpositive curvature
4:00PM, 122 Mathematics Building

Much recent work in geometric group theory has involved the study of groups acting on metric spaces of nonpositive curvature modeled on \(\ell^1\) (median graphs and metric median spaces), \(\ell^2\) (CAT(0) spaces) and \(\ell^{\infty}\) (Helly graphs and injective spaces).  All three of these cases appear in CAT(0) cube complexes depending on the choice of metric (\(\ell^1\), \(\ell^2\) or \(\ell^{\infty}\)) placed on the cubes.  It is natural to ask if there are metric nonpositive curvature conditions that can be modeled on more general normed spaces.  In this talk, I will discuss recent work with Thomas Haettel and Harry Petyt in which we prove Busemann-convexity of CAT(0) cube complexes whose cubes are given an \(\ell^p\) metric, with \(1 < p < \infty\).  I will also discuss conditions we proved for normed polyhedral complexes to be Busemann-convex and strongly bolic.

Mon, Apr 17

Algebra Seminar
Jacopo Zanchettin, SISSA
Hopf algebroids and twists for quantum projectivespaces 
4:00PM, Mathematics Building room 250

The Ehresmann-Schauenburg (E-S) bialgebroid associatedwith a Hopf-Galois extension is the noncommutative analog of the gauge groupoidassociated with a principal bundle. As for a Hopf algebra, a Hopf algebroid isa bialgebroid with an invertible antipode. In this talk, after recalling somebasic notions about rings, coring, and bialgebroids, we first show how twists(a sub-group of characters) of a bialgebroid are related to antipodes in thegeneral case. Eventually, after a short introduction to Hopf-Galois extensions,we characterize them for the E-S bialgebroid. Finally, we work out the exampleof a family of \(O(U(1))\)-extensions over quantum projective spaces. This talkis based on joint work with L. Dabrowski and G. Landi arXiv:2302.12073

Fri, Apr 21

Geometry and Topology Seminar
Vasudevan Srinivas (Tata Institute)
What is the Hodge Conjecture?

4:00PM, 122 Mathematics Building

The Hodge Conjecture is one of the famous unsolved problems in algebraic geometry over the complex number field. This talk will give an accessible introduction to this problem, meant for the non-expert. At the end, I will briefly discuss some related work of mine with A. Rosenschon.

Mon, Apr 24

Algebra Seminar
Ana Agore, Max Planck Institut and Simion Stoilow Institute of Mathematics
Universal constructions for Poisson algebras. Applications. 
9:00AM, Zoom (please email achirvas@buffalo.edu)

We introduce the universal algebra of two Poisson algebras \(P\) and \(Q\) as a commutative algebra \(A := \mathcal{P}(P, Q)\) satisfying a certain universal property. The universal algebra is shown to exist for any finite-dimensional Poisson algebra \(P\) and several of its applications are highlighted. For any Poisson \(P\)-module \(U\), we construct a functor \(U\otimes-: {}_A\mathcal{M} \to {}_Q\mathcal{PM}\) from the category of \(A\)-modules to the category of Poisson \(Q\)-modules which has a left adjoint whenever \(U\) is finite-dimensional. Similarly, if \(V\) is an \(A\)-module, then there exists another functor \(-\otimes V:{}_P\mathcal{PM}\to {}_Q\mathcal{QM}\) connecting the categories of Poisson representations of \(P\) and \(Q\) and the latter functor also admits a left adjoint if \(V\) is finite-dimensional. If \(P\) is\(n\)-dimensional, then \(\mathcal{P}(P) := \mathcal{P}(P, P)\) is the initial object in the category of all commutative bialgebras coacting on \(P\). As an algebra,\(\mathcal{P}(P)\) can be described as the quotient of the polynomial algebra\(k[X_{ij} | i, j = 1, · · · , n]\) through an ideal generated by \(2n^3\)non-homogeneous polynomials of degree \(\le 2\). Two applications are provided. The first one describes the automorphisms group\(\mathrm{Aut}_{\mathrm{Poiss}}(P)\) as the group of all invertible group-like elements of the finite dual \(\mathcal{P}(P)^{\circ}\).  Secondly, we show that for an abelian group\(G\), all \(G\)-gradings on \(P\) can be explicitly described and classified in terms of the universal coacting bialgebra \(\mathcal{P}(P)\). Joint work with G.Militaru.

Mon, Apr 24

Applied Math Seminar
Boaz Ilan, UC Merced
NLS equations: solitons, dispersive shocks and singularity formation.

2:00PM, Zoom - contact mbichuch@buffalo.edu for link

The Nonlinear Schrödinger (NLS) equation is a universal model for nonlinear dispersive waves. It describes the mean field superfluidic dynamics of a dilute gas of bosons near the absolute zero temperature, called a Bose-Einsten condensate, intense laser propagation through matter, and many other nonlinear systems. NLS equations and their generalizations possess a rich variety of special solutions, such as solitons, dispersive shocks, and singularity formation. I will discuss recent analytical and computational results and open problems with application to BECs.

Wed, Apr 26

Analysis Seminar
Min Woong Ahn, SUNY at Buffalo
The error-sum function of Pierce expansions
4:00PM, 250 Math Building

The notion of the error-sum function was first studied by Ridley and Petruska in the context of the regular continued fraction expansion. The Pierce expansion is another classical representation of a real number. In this talk, I will introduce the error-sum function of Pierce expansions and discuss the basic properties of the function and the fractal property of the graph of the function.

Thu, Apr 27

Colloquium
Bena Tshishiku (Brown University)
Mapping class groups and Nielsen realization problems

4:00PM, 250 Mathematics Building

The mapping class group \(\mathit{Mod}(M)\) of a smooth manifold \(M\) is the group of diffeomorphisms of \(M\), modulo isotopy. The study of mapping class groups interacts with many areas, including geometric topology, group theory, dynamics, and algebraic geometry. We explain some of these connections from the point-of-view of the Nielsen realization question. This problem, versions of which were posed by Nielsen (1932) and Thurston (1977), asks when a subgroup \(G<\mathit{Mod}(M)\) can be lifted to \(\mathrm{Diff}(M)\) under the natural projection \(\mathit{Diff}(M)\to \mathit{Mod}(M)\). 

Fri, Apr 28

Geometry and Topology Seminar
Bena Tshishiku (Brown University)
Pseudo-Anosov theory in the Goeritz group

4:00PM, 122 Mathematics Building

For a surface \(S_g\) of genus \(g\), the Goeritz group is the subgroup of the mapping class group \(\mathit{Mod}(S_g)\) consisting of isotopy classes that extend to the handlebodies in the genus-g Heegaard splitting of the 3-sphere. There are many open questions about the algebra of this group, including whether or not it's finitely generated when \(g\geq 4\). This talk will focus on geometric aspects of the genus-2 Goeritz group. I will explain a refinement of the Nielsen-Thurston classification for this group and will show that its purely pseudo-Anosov subgroups are convex cocompact, which answers a question of Farb-Mosher in a special case. 

Mon, May 1

Algebra Seminar
Michael Brannan, University of Waterloo
Ulam stability for quantum groups 
4:00PM, 250 Mathematics Building

In recent years, there has been a growing interest in the study of approximate representations of various algebraic structures. This is due to some very deep connections with (1) approximation properties for groups and (2) questions about robustness in quantum information theory. The basic question that we are interested in is the following: If we are given a linear map from an algebra (or group) into the bounded operators on a Hilbert space that is “almost” multiplicative, under what conditions can we guarantee that this map is a small perturbation of an actual representation of the algebra? I will describe some of the history around this problem as well as some on going work with Junichiro Matsuda (Kyoto) and Jennifer Zhu (Waterloo), where we investigate the Ulam (=operator norm) stability of approximate representations for compact and discrete quantum groups.

Wed, May 3

Analysis Seminar
Daxun Wang. SUNY at Buffalo
Boundary actions of groups and their C*-algebras
4:00PM, 250 Math Building

Pure infiniteness of C*-algebras plays an important role in the classification of C*-algebras. In this talk, we will talk about boundary actions of some popular groups that arise in geometric group theory such as right angled Artin groups, right angled Coxeter groups and graphs of groups, and show that the reduced crossed product C*-algebras of these boundary actions are purely infinite. This is a joint work with Xin Ma.

Fri, May 12

Geometry and Topology Seminar
Adam Sikora (University at Buffalo)
On skein modules of rational homology spheres

4:00PM, 122 Mathematics Building

The Kauffman bracket skein module S(M) of a 3-manifold M classifies polynomial invariants of links in M satisfying Kauffman bracket skein relations. Witten conjectured that the skein module is finite dimensional for any closed M. This conjecture was proved by Gunningham, Jordan, and Safronov, however their work does not lead to an explicit computation of S(M). In fact, S(M) has been computed for a few specific families of closed 3-manifolds so far only. We introduce a novel method of computing these skein modules for certain rational homology spheres. (This is joint work with R. Detcherry and E. Kalfagianni.)

Wed, Sep 6

Analysis Seminar
Jintao Deng, SUNY at Buffalo
The equivariant coarse Baum-Connes conjecture
4:00PM, 250 Mth Building

The equivariant coarse Baum-Connes conjecture claims that a certain assembly map from the equivariant K-homology of a metric space with a group action to the K-theory of the Roe algebras is an isomorphism. It has important applications in the study of the existence of Riemannian metric with positive scalar curvature. In this talk, I will talk about the concept of Roe algebras which encode the large-scale geometry of a metric space and group actions. The higher index of an elliptic operator is an element of the K-theory of this algebra. The equivariant coarse Baum-Connes conjecture provides an algorithm to compute its K-theory. I will talk about our recent result that the equivariant coarse Baum-Connes conjecture holds for a metric space with a group action under the conditions that the group is amenable and the associated quotient space is coarsely embeddable into Hilbert space. This is a joint work with Qin Wang and Benyin Fu.

Thu, Sep 7

Colloquium
Demonstration of new GRADER app
4:00PM, Room 250

Mon, Sep 11

Algebra Seminar
Peter Koroteev, University at Buffalo
Opers and integrability
4:00PM, University at Buffalo, Buffalo, NY 14260, USA

I will introduce (\(q\)-)opers on a projective line in the presence of twists and singularities and will discuss the space of such opers. We will see how Bethe Ansatz equations for quantum spin chains and energy level equations of classical soluble models of Calogero-Ruijsenaars type naturally appear from the oper construction. Both can also be described in terms of so-called \(QQ\)-systems, which have their origins in algebra and representation theory. Our construction is universal and works for any simple, simply-connected complex Lie group \(G\).

Wed, Sep 13

Analysis Seminar
Jintao Deng, SUNY at Buffalo
The equivariant coarse Baum-Connes conjecture Part II
4:00PM, 250 Math Building

Title: The equivariant coarse Baum-Connes conjecture Part II

Mon, Sep 18

Algebra Seminar
Tomasz Maszczyk, University of Warsaw
Quantum symmetries of Frobenius algebras
4:00PM, University at Buffalo, Buffalo, NY 14260, USA

We introduce the notions of "quantum support" and "quantum fundamental cycle" for a Frobenius algebra. We realize the Pareigis Hopf algebra, which encodes the monoidal structure of the category of complexes (via the Pareigis transform which is the identity on objects), as a universal quantum symmetry of the dual numbers algebra. We show that under the Pareigis transform the category of corresponding equivariant quasicoherent sheaves on the double point is equivalent to the category of complexes with square zero homotopies. In particular, the Pareigis transform of the algebra of dual numbers is the terminal object of the extended Hinich category of local pseudo-compact algebras. We prove that the Pareigis transform of the Frobenius support of the algebra of dual numbers is a closed graded trace of dimension\(-1\) on the terminal Hinich algebra, being a boundary of the Pareigis transform of the augmentation of dual numbers. This can be understood as a DGA model of the empty set with a homologically trivial \((-1)\)-dimensional fundamental cycle. We also study symmetries of other truncated polynomial algebras and relate them to the representation theory of \(SL(2)\), Hamiltonian moment maps, Fourier transforms, and the Springer resolution of the singularity of the nilpotent cone of \(SL(2)\).

Wed, Sep 20

Analysis Seminar
Min Woong Ahn, SUNY at Buffalo
Hausdorff dimensions in Pierce expansions
4:00PM, 250 Math Building

The Pierce expansion is one of many real number representation systems. Shallit (1986) established the law of large numbers, the central limit theorem, and the law of the iterated logarithm of the digits of the Pierce expansions. Additionally, it was shown that the series of iterates under a mapping that yields the Pierce expansion converges Lebesgue-almost everywhere. In this talk, I will discuss the Hausdorff dimensions of such sets with Lebesgue measure zero.

Fri, Sep 22

Applied Math Seminar
Jiuhua Hu, University of Wisconsin at Madison
Wavelet-based Edge Multiscale Parareal Algorithm for Parabolic Equations with Heterogeneous Coefficients.

3:00PM, Math 122 and on Zoom - contact mbichuch@buffalo.edu for link

In this talk, I will start with an introduction of some multiscale methods to solve multiscale problems. Then I will talk about the Wavelet-based Edge Multiscale Parareal Algorithm to solve parabolic equations with heterogeneous coefficients. This algorithm combines the merits of multiscale methods that can deal with heterogeneity in the spatial domain effectively, with the robust capabilities of parareal algorithms for speeding up time evolution problems. The convergence rate of this algorithm can also be derived.  To illustrate the algorithm's performance, I will present some numerical tests. This is a joint work with Guanglian Li (The University of Hong Kong).

Tue, Feb 22

Applied Math Seminar
Applied Math Seminar: Alexander Korotkevich (UNM)
Numerical Verification of the 6-Wave 1D Kinetic Equation.Speaker: Alexander Korotkevich (University of New Mexico, Department of Math&Stat)
4:00PM, Zoom

We consider wave kinetic equation (WKE) for quintic nonlinear Schroedinger equation (qNLSE),corresponding to 6-waves 3-to-3 interaction. WKE was derived for periodic boundary conditions. Wepropose conditions of applicability of WKE for description of qNLSE. These conditions were confirmedduring comparison of simulation of dynamical equation (qNLSE) and WKE. We observed convergence ofsolution of qNLSE to solution of WKE with increase of the period of the system. If we stay inproposed range of applicability, good correspondence between WKE and qNLSE is observed.Simulations for different boundary conditions (Dirichlet and Neumann) were performed. Wedemonstrate, that conditions of applicability derived for periodic boundary conditions are notdirectly applicable and have to be corrected. It should be noted, that in laboratory wave tankexperiments boundary conditions are usually different from periodic ones. At the same time,researchers expect to observe good correspondence with WKE derived for infinite domain.6-waves qNLSE appears in some of the applications, e.g. if one considers one dimensional opticalfiber communication line and would like to consider turbulence of the waves propagating in it. Nextorder correction on intensity, with respect to the classical NLSE, would result in qNLSE, aftercanonical transformation eliminating 4-waves nonlinear term. Also 6-waves WKE appears in descriptionof waves on Kelvin vortices in superfluid Helium.

Fri, Feb 25

Special Event
Makoto Ozawa (Komazawa University) via Zoom only Friday

4:00PM

Geometry Topology Seminar


VIA ZOOM

Handlebody decompositions of 3-manifolds and polycontinuous patterns

Tue, Mar 1

Applied Math Seminar
Applied Math Seminar
Denis Silantyev (UCCS)
Generalized Constantin-Lax-Majda Equation: Collapse vs. Blow Up and Global ExistenceSpeaker: Denis Silantyev (UC Colorado Springs, Department of Mathematics)
4:00PM, Zoom

We investigate the behavior of the generalized Constantin-Lax-Majda (CLM) equation which is a 1D model for the advection and stretching of vorticity in a 3D incompressible Euler fluid. Similar to Euler equations the vortex stretching term is quadratic in vorticity, and therefore is destabilizing and has the potential to generate singular behavior, while the advection term does not cause any growth of vorticity and provides a stabilizing effect. We study the influence of a parameter a which controls the strength of advection, distinguishing a finite time singularity formation (collapse or blow-up) vs. global existence of solutions. We find a new critical value ac=0.6890665337007457... below which there is finite time singularity formation that has a form of self-similar collapse, with the spatial extent of blow-up shrinking to zero, and above which up to a=1 we have an expanding blow up solutions. We identify the leading order complex singularity for general values of a which controls the leading order behavior of the collapsing solution. We also rederive a known exact collapsing solution for a=0 and we find a new exact analytical collapsing solution at a=1/2. For ac<a≤1, we find a blow-up solution on the real line in which the spatial extent of the blow-up region expands infinitely fast at the singularity time. For a>1, we find that the solution exists globally with exponential-like growth of the solution amplitude in time.

Tue, Mar 8

Applied Math Seminar
Dr. Kai Yang, Florida International University
Numerical methods for the KdV-type equations
4:00PM, Zoom: for link see email announcement or contact sergeyd at buffalo dot edu

We review several numerical approaches for KdV-type equations, including the generalized KdV and Benjamin-Ono equations as well as the KdV equation with fractional Laplacian. The spatial discretization is achieved by using the Fourier spectral method for fast decay solutions (e.g., in KdV equation), or the spectral method from the Wiener rational basis functions for both fast and slow decay solution cases. Both of these two spatial discretizations preserve the Hamiltonian in the spatial discrete sense. We also discuss the arbitrarily high order Hamiltonian conservative schemes that are constructed by applying the Scalar Auxiliary Variable (SAV) reformulation with the symplectic Runge-Kutta method in the time evolution.

Thu, Mar 10

Colloquium
Cary Malkiewich, Binghamton University
Brave new fixed-point theory
4:00PM, Zoom: for link see email announcement or contact badzioch at buffalo dot edu

The development of algebraic topology in the 20th century could be characterized as a gradual passage from combinatorial, numerical invariants such as the Betti numbers to group-valued invariants such as homology, and then from homological algebra to spectral or "brave new" algebra in the latter half of the century. In brave new algebra, integers are replaced by maps of spheres. What is amazing is that just about every concept in algebra can be transformed to accommodate this new paradigm.
The effects of this development continue to ripple through topology and nearby subjects, including the very classical topic of Nielsen fixed-point theory. I'll explain how work of Dold in the 1970s and of Ponto in the 2000s and 2010s infused the "brave new" perspective into fixed point theory. The result is that, amazingly, we can now work with the Lefschetz number and its generalizations without ever triangulating or mentioning homology groups, and the definitions now generalize easily to parametrized families of fixed-point problems.

Mon, Mar 14

Algebra Seminar
Benjamin Passe, United States Naval Academy
Boundary representations and isolated points
4:00PM, Zoom. Contact achirvas AT buffalo DOT edu for link.

Operator systems provide a way to examine convexity (and a generalization called matrix convexity) in operator algebraic terms. Extreme points in particular match up with a special class of irreducible representations, called boundary representations, defined by Arveson. While a single operator system \(S\) may have many different concrete forms, the boundary representations from each form nonetheless generate the same \(C^*\)-envelope of \(S\). Similar analogues of exposed points and isolated extreme points exist within the boundary representations, though these subsets are not as commonly used. In joint work with Ken Davidson, we determined exactly when a smallest concrete presentation of a separable operator system exists, as well as how it can be identified up to unitary equivalence using a restricted class of boundary representations.

Tue, Mar 15

Applied Math Seminar
Pavel Lushnikov, University of New Mexico
Conformal mappings and integrability of surface dynamics
4:00PM, Zoom: for link contact sergeyd@buffalo.edu

A Hamiltonian formulation of the time dependent potential flow of ideal incompressible fluid with a free surface is considered in two-dimensional geometry. It is well known that the dynamics of small to moderate amplitudes of surface perturbations can be reformulated in terms of the canonical Hamiltonian structure for the surface elevation and Dirichlet boundary condition of the velocity potential. Arbitrary large perturbations can be efficiently characterized through a time-dependent conformal mapping of a fluid domain into the lower complex half-plane. We reformulate the exact Eulerian dynamics through a non-canonical nonlocal Hamiltonian system for the pair of new conformal variables. The corresponding non-canonical Poisson bracket is non-degenerate, i.e., it does not have any Casimir invariant. Any two functionals of the conformal mapping commute with respect to Poisson bracket. We also consider a generalized hydrodynamics for two components of superfluid Helium which has the same non-canonical Hamiltonian structure. In both cases the fluid dynamics is fully characterized by the complex singularities in the upper complex half-plane of the conformal map and the complex velocity. Analytical continuation through the branch cuts generically results in the Riemann surface with infinite number of sheets. An infinite family of solutions with moving poles are found on the Riemann surface.Residues of poles are the constants of motion. There constants commute with each other in the sense of underlying non-canonical Hamiltonian dynamics which provides an argument in support of the conjecture of complete Hamiltonian integrability of surface dynamics.

Fri, Mar 18

Geometry and Topology Seminar
Subhankar Dey, University of Alabama
Detection results in link Floer homology

4:00PM, 122 Mathematics Building

 In this talk I will briefly describe link Floer homology toolbox and its usefulness. Then I will show how link Floer homology can detect links with small ranks, using a rank bound for fibered links by generalizing an existing result for knots, and some very useful spectral sequences. I will also show that stronger detection results can be obtained in a sense that the knot Floer homology can be shown to detect \(T(2,8)\) and \(T(2,10)\), and that link Floer homology detects \((2,2n)\)-cables of trefoil and the figure eight knot. This talk is based on joint work with Fraser Binns.

Thu, Mar 31

Colloquium
Colloquium: Michael Brannan (University of Waterloo)Via Zoom
4:00PM

Fri, Apr 1

Geometry and Topology Seminar
Hong Chang, University at Buffalo
 Efficient geodesics in the curve complex and their dot graphs

4:00PM, 122 Mathematics Building

 The notion of {\em efficient geodesics} in \(\mathcal{C}(S_{g>1})\), the complex of curves of a closed orientable surface of genus \(g\),  was first introduced in "Efficient geodesics and an effective algorithm for distance in the complex of curves".  There it was established that there exists (finitely many) efficient geodesics between any two vertices, \( v_{\alpha} , v_{\beta} \in \mathcal{C}(S_g)\), representing homotopy classes of simple closed curves, \(\alpha , \beta \subset S_g\). The main tool for used in establishing the existence of efficient geodesic was a {\em dot graph}, a booking scheme for recording the intersection pattern of a {\em reference arc}, \(\gamma \subset S_g\), with the simple closed curves associated with the vertices of geodesic path in the zero skeleton, \(\mathcal{C}^0(S_g)\).  In particular, it was shown that any curve corresponding to the vertex that is distance one from \(v_\alpha\) in an efficient geodesic intersects any \(\gamma\) at most \(d -2\) times, when the distance between \(v_\alpha\) and \(v_\beta\) is \(d \geq 3\).  In this note we make a more expansive study of the characterizing ``shape'' of the dot graphs over the entire set of vertices in an efficient geodesic edge-path.  The key take away of this study is that the shape of a dot graph for any efficient geodesic is contained within a {\em spindle shape} region.

Thu, Apr 7

Special Event
Colloquium: Gino Biondini, University at Buffalo
Two adventures in integrable systems: thenonlinear Schrodinger equation with non-trivial boundary conditions 
4:00PM, Room 250 Math Building, North Campus

A significant advance in mathematical physics in thesecond half of the twentieth century was the development of the theory ofmodern integrable systems. These systems are nonlinear evolution equations ofphysical significance that provide the nonlinear counterpart to the classicalPDEs of mathematical physics.

One such equation, and in some respects the mostimportant one, is the nonlinear Schrödinger (NLS) equation. The NLS equation isa universal model for weakly nonlinear dispersive wave packets, and arises in avariety of physical settings, including deep water, optics, acoustics, plasmas,condensed matter, etc. In addition, the NLS equation is a completelyintegrable, infinite-dimensional Hamiltonian system, and as a result itpossesses a remarkably deep and beautiful mathematical structure. At the rootof many of these properties is the existence of Lax pair, namely the fact thatthe NLS equation can be written as the compatibility condition of anoverdetermined pair of linear ODEs. The first half of the Lax pair for the NLSequation is the Zakharov-Shabat scattering problem, which is equivalent to aneigenvalue problem for a one-dimensional Dirac operator.

Even though the NLS equation has been extensively studiedthroughout the last sixty years, it continues to reveal new phenomena and offermany surprises. In particular, the focusing NLS equation with nontrivialboundary conditions has received renewed attention in recent years. This talkis devoted to presenting two recent results in this regard. Specifically, Iwill discuss: (i) A characterization of the universal nonlinear stage ofmodulational instability, achieved by studying the long-time asymptotics ofsolutions of the NLS equation with non-zero background; (ii) A characterizationof a two-parameter family of elliptic finite-band potentials of thenon-self-adjoint ZS operator, which are associated with purely real spectrum ofHill’s equation (i.e., the time-independent Schrodinger equation with periodiccoefficients) with a suitable complex potential.

Fri, Apr 8

Geometry and Topology Seminar
Sahana Hassan Balasubramanya, University of Münster
Actions of solvable groups on hyperbolic spaces
4:00PM, Zoom

(joint work with A.Rasmussen and C.Abbott) Recent papers of Balasubramanya and Abbott-Rasmussen have classified the hyperbolic actions of several families of classically studied solvable groups. A key tool for these investigations is the machinery of confining subsets of Caprace-Cornulier-Monod-Tessera. This machinery applies in particular to solvable groups with virtually cyclic abelianizations.

Mon, Apr 11

Algebra Seminar
Xiuping Su, University of Bath
Kac's Theorem for a class of string algebras of affine type \(\mathbf {C}\).

4:00PM, Contact achirvas@buffalo.edu for zoom link

Quiver representation theory and Lie theory are closely related via Gabriel's Theorem, Kac's Theorem and Dlab-Ringel's Theorem.  When a quiver (i.e. an oriented simply laced graph) \(Q\) is of finite representation type, Gabriel proved that a quiver is of finite representation type if and only if \(Q\) is Dykin and the map sending a representation \(X\) to its dimension vector provides a one-to-one correspondence between the isomorphism classes of indecomposable representations of \(Q\) and the positive roots of \(Q\) . The latter was generalized by Kac to any quiver. For non-simply laced graphs of finite or affine types, Dlab-Ringel generalized Gabriel's Theorem, using modulated (or valued) quivers.
Recently, Geiss-Leclerc-Schroer introduced a class of Iwanaga-Gorenstein algebras \(H\) via quivers \(Q\) with relations associated with symmetrizable Cartan matrices and studied \(\tau\)-locally free\(H\)-modules. Among other things, they proved that when the Cartan matrix is of finite type, there is a one-to-one correspondence between the dimension vectors of indecomposable \(\tau\)-locally free \(H\)-modules and the positive roots of the associated Lie algebra and conjectured that Kac's Theorem holds for any \(H\).
In this talk, I will describe the Auslander-Reiten quivers of some string algebras of affine type \(\mathbf{C}\), which are Iwanaga-Gorenstein algebras \(H\) associated to Cartan matrices of affine type \(\mathbf{C}\) and confirm GLS-conjecture for this case.

Tue, Apr 12

Applied Math Seminar
Dmitry Zakharov, Central Michigan U
Lump chains in the KP-I equation


4:00PM, Zoom - contact sergeydy@buffalo.edu for link

The Kadomstev--Petviashvili equation is one of the fundamental equations in the theory of integrable systems. The KP equation comes in two physically distinct forms: KP-I and KP-II. The KP-I equation has a large family of rational solutions known as lumps. A single lump is a spatially localized soliton, and lumps can scatter on one another or form bound states. The KP-II equation does not have any spatially localized solutions, but has a rich family of line soliton solutions that form evolving polygonal patterns.
I will discuss two new families of solutions of the KP-I equation, obtained using the Grammian form of the tau-function. The first is the family of lump chain solutions. A single lump chain consists of a linear arrangement of lumps, similar to a line soliton of KP-II. More generally, lump chains can form evolving polygonal arrangements whose structure closely resembles that of the line soliton solutions of KP-II. I will also show how lump chains and line solitons may absorb, emit, and reabsorb individual lumps.
Joint work with Andrey Gelash, Charles Lester, Yury Stepanyants, and Vladimir Zakharov.

Thu, Apr 14

Colloquium
Peter Thomas (Case Western U)
Phase and phase-amplitude reduction for stochastic oscillators
4:00PM, 250 Math Bldg, also accessible via Zoom - contact badzioch@buffalo for link

Phase reduction is a powerful and widely used tool for studying synchronization,
entrainment, and parametric sensitivity of limit cycle oscillators. The
classical phase reduction framework goes back at least 50 years for
deterministic ODE systems. In both natural and engineered systems, however,
stochastic dynamics are ubiquitous. For stochastic systems, the appropriate
analog of phase reduction remains a matter of debate. In 2013 Schwabedal and
Pikovsky introduced a notion of phase reduction for stochastic oscillators based
on a first-passage-time analysis. In 2014 Thomas and Lindner introduced an alternative
asymptotic phase for Markovian stochastic oscillators based on a spectral
decomposition of the Koopman operator (a.k.a. the generator of the Markov process,
or the adjoint Kolmogovor operator). I will report on recent advances in
understanding and expanding these ideas, including (i) reformulation of the
first-passage-time (FPT) phase in terms of the solution of a partial
differential equation with nonstandard boundary conditions, (ii) quantitative
comparison of the FPT and spectral phase for planar stochastic systems, and
(iii) extension of the spectral phase to a novel "phase-amplitude" reduction for
stochastic oscillators.
This is joint work with Benjamin Lindner (Humboldt University, Dept. of Physics
and Bernstein Center for Computational Neuroscience) and Alberto Perez-Cervera
(Complutense University of Madrid, Dept. of Applied Mathematics).

Fri, Apr 15

Geometry and Topology Seminar
Daxun Wang, University at Buffalo
Boundary action of CAT(0) groups and their \(C^\ast\)-algebras.

4:00PM, 122 Mathematics Building

(joint with Xin Ma) Boundaries of certain CAT(0) spaces and group actions on them play important roles in geometric group theory. In this talk, we will talk about boundary actions of CAT(0) spaces from a point of view of topological dynamics and \(C^\ast\)-algebras. In particular, we will describe the actions of right angled Coxeter groups and right angled Artin groups on certain boundaries. This provides some pure infiniteness results for reduced crossed product \(C^\ast\)-algebra of these actions. Next, we will talk about the action of fundamental groups of graph of groups on the visual boundaries of their Bass-Serre trees. This provides a new method in identifying \(C^\ast\)-simple generalized Baumslag-Solitar groups.

Tue, Apr 19

Applied Math Seminar
Bernard Deconinck, U of Washington
The water wave pressure problem

4:00PM, Zoom - contact sergeyd@buffalo.edu for link

I will discuss a new method to recover the water-wave surface elevation from pressure data obtained at the bottom of a fluid. The new method requires he numerical solution of a nonlocal nonlinear equation relating the pressure and the surface elevation which is obtained from the Euler formulation of the water-wave problem without approximation. From this new equation, a variety of different asymptotic formulas are derived. The nonlocal equation and the asymptotic formulas are compared with both numerical data and physical experiments, demonstrating excellent agreement, significantly beyond what is obtained using Archimedes' p=rho g h. 

Fri, Apr 22

Geometry and Topology Seminar
Matt Durham, UC Riverside/Cornell University
Local quasicubicality and sublinear Morse geodesics in mapping class groups and Teichmuller space

4:00PM, 122 Mathematics Building

Random walks on spaces with hyperbolic properties tend to sublinearly track geodesic rays which point in certain hyperbolic-like directions. Qing-Rafi-Tiozzo recently introduced the sublinear-Morse boundary to more broadly capture these generic directions.In joint work with Abdul Zalloum, we develop the geometric foundations of sublinear-Morseness in the mapping class group and Teichmuller space. We prove that their sublinearly-Morse boundaries are visibility spaces and admit continuous equivariant injections into the boundary of the curve graph. Moreover, we completely characterize sublinear-Morseness in terms of the hierarchical structure on these spaces.Our techniques include developing tools for modeling sublinearly-Morse rays via CAT(0) cube complexes. Part of this analysis involves establishing a direct connection between the geometry of the curve graph and the combinatorics of hyperplanes in these cubical models.

Mon, Apr 25

Algebra Seminar
Daniel Sage, LSU
The Deligne–Simpson problem for connections on \(\mathbb{G}_m\) with a maximally ramified singularity 
4:00PM, Mathematics Building Room 250

A natural question in the theory of systems of meromorphic differential equations (or equivalently, meromorphic connections) on \(\mathbb{P}^1\) is whether there exists a global connection with specified local  behavior at a collection of singular points.  The Deligne-Simpson problem is concerned with a variant of this question.  The classical additive Deligne–Simpson problem is the existence problem for Fuchsian connections whose residues at the singular points are contained in specified adjoint orbits.  Crawley-Boevey solved this problem in 2003 by reinterpreting it in terms of quiver varieties. A more general version of the problem, solved by Hiroe in 2017, allows additional unramified irregular singularities. We apply the theory of fundamental and regular strata due to Bremer and Sage to formulate a version of the Deligne–Simpson problem in which certain ramified singularities are allowed. These ramified singular points are called toral singularities; they are singularities whose leading term with respect to a Moy–Prasad filtration is regular semisimple. We solve this problem in a special case that plays an important role in recent work on the geometric Langlands program: connections on \(\mathbb{G}_m\) with a maximally ramified singularity at 0 and possibly an additional regular singular point at infinity. We also give a complete characterization of all such connections that are rigid, under the additional hypothesis of unipotent monodromy at infinity.

Tue, Apr 26

Applied Math Seminar
Svetlana Roudenko, Florida International University
The gKdV world thru the NLS lens
4:00PM, Zoom, contact sergeyd@buffalo.edu for link

In this talk we discuss the family of generalized KdV equations borrowing tools and approaches from the NLS equation. We address the wellposedness (for any power of nonlinearity), show formation and behavior of solitons (and thus, soliton resolution) as well as solutions behavior in the \(L^2\)-critical and supercritical settings including the finite time blow-up and the description of its dynamics.

Thu, Apr 28

Colloquium
Juanita Pinzón Caicedo, University of Notre Dame
Four-manifolds and knot concordance

4:00PM, 250 Math Bldg. Also via Zoom - contact badzioch@buffalo.edu for link.

 The study of 4-dimensional objects is special: a manifold can admit infinitely many non-equivalent smooth structures,  and manifolds can be homeomorphic but not diffeomorphic. This difference between topological and smooth structures,  can be addressed in terms of the study of knots as boundaries of surfaces embedded in 4D space. In this talk I will focus  on some knot operators known as satellites and will show that satellites can bound very different surfaces in the smooth  and topological category.
This talk is organized jointly by the Math Department and the UB chapter of AWM. 

Fri, Apr 29

Geometry and Topology Seminar
Juanita Pinzon Caicedo, University of Notre Dame
Satellite Operations that are not homomorphisms.

4:00PM, 122 Mathematics Building

Two knots \(K_0\) and \(K_1\) are said to be smoothly concordant if the connected sum \(K_0\#m({K_1}^r)\) bounds a disk smoothly embedded in the 4-ball. Smooth concordance is an equivalence relation, and the set \(\mathcal{C}\) of smooth concordance classes of knots is an abelian group with connected sum as the binary operation. Satellite operations, or the process of tying a given knot P along another knot K to produce a third knot P(K), are powerful tools for studying the algebraic structure of the concordance group. In this talk I will describe conditions on the pattern P that suffice to conclude that the function \(P:\mathcal{C}\to \mathcal{C}\) is not a homomorphism. This is joint work with Tye Lidman and Allison Miller.

Mon, May 2

Special Event
Nicolle González, UCLA
A skein theoretic \(A_{q,t}\) algebra 
4:00PM, Mathematics Building room 250

The \(A_{q,t}\) algebra first arose in connection to the celebrated proof of the shuffle theorem given by Carlsson and Mellit. This algebra is given as an extension of two copies of the affine Hecke algebra by certain raising and lowering operators. Its polynomial representation, which played a critical role in the proof given by Carlsson and Mellit, was later realized geometrically by Carlsson-Mellit and Gorsky in the context of parabolic flag Hilbert schemes. In this talk I will present a skein theoretic formulation of this representation given by certain skein-Heisenberg diagrams on a punctured annulus. This formulation recovers the original algebraic description of Carlsson and Mellit, but given the simplicity of the diagrams allows many computations to be more straightforward and intuitive. More interestingly, this diagrammatic presentation is primed for a direct categorification via the category of Soergel bimodules. This is joint work with Matt Hogancamp.

Fri, May 6

Geometry and Topology Seminar
Ciprian Manolescu, Stanford University
A knot Floer stable homotopy type
Knot Floer homology (introduced by Ozsváth–Szabó and Rasmussen) is an invariant whose definition is based on symplectic geometry, and whose applications have transformed knot theory over the last two decades. Starting from a grid diagram of a knot, I will explain how to construct a spectrum whose homology is knot Floer homology. Conjecturally, the homotopy type of the spectrum is an invariant of the knot. The construction does not use symplectic geometry, but rather builds on a combinatorial definition of knot Floer homology. We inductively define models for the moduli spaces of pseudo-holomorphic strips and disk bubbles, and patch them together into a framed flow category. The inductive step relies on the vanishing of an obstruction class that takes values in a complex of positive domains with partitions. (This is joint work with Sucharit Sarkar.)
4:00PM, Zoom

Title: A knot Floer stable homotopy type
Knot Floer homology (introduced by Ozsváth–Szabó and Rasmussen) is an invariant whose definition is based on symplectic geometry, and whose applications have transformed knot theory over the last two decades. Starting from a grid diagram of a knot, I will explain how to construct a spectrum whose homology is knot Floer homology. Conjecturally, the homotopy type of the spectrum is an invariant of the knot. The construction does not use symplectic geometry, but rather builds on a combinatorial definition of knot Floer homology. We inductively define models for the moduli spaces of pseudo-holomorphic strips and disk bubbles, and patch them together into a framed flow category. The inductive step relies on the vanishing of an obstruction class that takes values in a complex of positive domains with partitions. (This is joint work with Sucharit Sarkar.)

Mon, May 9

Algebra Seminar
Jie Ren, UB
Quivers and 2-Calabi-Yau categories
 
4:00PM

The framework of Calabi-Yau categories is appropriate forthe theory of motivic Donaldson-Thomas invariants. I will give an introductionto the Calabi-Yau categories associated to quivers, and the analyticity oftheir stability structures.

Mon, May 9

Special Event
Jie Ren, UB
Quivers and 2-Calabi-Yau categories
 
4:00PM

The framework of Calabi-Yau categories is appropriate forthe theory of motivic Donaldson-Thomas invariants. I will give an introductionto the Calabi-Yau categories associated to quivers, and the analyticity oftheir stability structures.

Tue, May 10

Applied Math Seminar
Panayotis Kevrekidis, U Mass
Some Vignettes of Nonlinear Waves in Granular Crystals: From Modeling and Analysis to Computations and Experiments
4:00PM, Zoom - contact sergeyd@buffalo.edu for link

In this talk, we will provide an overview of some results in the setting of granular crystals, consisting of beads interacting through Hertzian contacts. In 1d we show that there exist three prototypical types of coherent nonlinear waveforms: shock waves, traveling solitary waves and discrete breathers. The latter are time-periodic, spatially localized structures. For each one, we will discuss the existence theory, presenting connections to prototypical models of nonlinear wave theory, such as the Burgers equation, the Korteweg-de Vries equation and the nonlinear Schrodinger (NLS) equation, respectively. We will also explore the stability of such structures, presenting some explicit stability criteria analogous to the famous Vakhitov-Kolokolov criterion in the NLS model. Finally, for each one of these structures, we will complement the mathematical theory and numerical computations with recent experiments, allowing their quantitative identification and visualization. Finally, time permitting, ongoing extensions of these themes will be briefly touched upon, most notably in higher dimensions, in heterogeneous or disordered chains and in the presence of damping and driving; associated open questions will also be outlined.

Thu, May 26

Special Event
Tara Hudson
12:30PM

Tue, Aug 16

Special Event
103 Desk needs to be moved
12:30PM

Wed, Aug 31

Analysis Seminar
Yi Wang, Chongqing University
 Helton-Howe trace, Connes-Chern character and quantization

8:00PM, On Zoom - contact hfli@math.buffalo.edu for link

We study the Helton-Howe trace and the Connes-Chern character for Toeplitz operators on weighted Bergman spaces via the idea of quantization. We prove a local formula for the large t-limit of the Connes-Chern character as the weight goes to infinity. And we show that the Helton-Howe trace of Toeplitz operators is independent of the weight and obtain a local formula for the Helton-Howe trace for all weighted Bergman spaces. The proofs are based on an integration by parts formula and some harmonic analysis. This talk is based on joint work with Xiang Tang and Dechao Zheng. 

Tue, Sep 6

Applied Math Seminar
Maxim Bichuch, SUNY Buffalo
Introduction to Decentralized Finance

4:00PM, Math 250 and on Zoom - contact mbichuch@buffalo.edu for link

Decentralized Finance (DeFi) is an addition/alternative to the traditional financial system to handle financial transactions. The key difference between the traditional financial system and DeFi is the lack of central authority that establishes trust and regulates the financial system. In this talk we will explain the main concept of DeFi and consider three examples: Decentralized Payment Clearing using Blockchain and Optimal Bidding; Pricing by Staking; Axiomatic Automated Market Making.

Wed, Sep 7

Analysis Seminar
Mariusz Tobolski, University of Wroclaw
The Stone-von Neumann theorem for locally compact quantum groups
4:00PM, On Zoom - contact hfli@mah.buffalo.edu for link

The Stone-von Neumann theorem is a mathematical result that rigorously proves the equivalence between the two fundamental approaches to quantum mechanics, i.e. the matrix mechanics of Heisenberg and the wave mechanics of Schrodinger. It was then formulated by Mackey as a theorem about certain unitary representations of locally compact abelian groups. In my talk, based on yet another formulation due to Rieffel, I will present a Stone-von Neumann-type theorem in the setting of locally compact quantum groups introduced by Kustermans and Vaes and independently by Woronowicz.

Fri, Sep 9

Geometry and Topology Seminar
Bill Menasco (UB)
Surface Embeddings in \(\mathbb{R}^2 \times \mathbb{R}\)
4:00PM, 122 Mathematics Building

In this joint work with Margaret Nichols, we consider \(\mathbb{R}^3\) as having the product structure \(\mathbb{R}^2 \times \mathbb{R}\) and let \(\pi : \mathbb{R}^2 \times \mathbb{R} \rightarrow \mathbb{R}^2\) be the natural projection map onto the Euclidean plane. Let \( \epsilon : S_g \rightarrow \mathbb{R}^2 \times \mathbb{R}\) be a smooth embedding of a closed oriented genus \(g\) surface such that the set of critical points for the map \(\pi \circ \epsilon\) is a piece-wise smooth (possibly multi-component) \(1\)-manifold, \(\mathcal{C} \subset S_g\). We say \(\mathcal{C}\) is the {\em crease set of \(\epsilon\)} and two embeddings are in the same {\em isotopy class} if there exists an isotopy between them that has \(\mathcal{C}\) being an invariant set. The case where \(\pi \circ \epsilon |_\mathcal{C}\) restricts to an immersion is readily accessible, since the turning number function of a smooth curve in \(\mathbb{R}^2\) supplies us with a natural map of components of \(\mathcal{C}\) into \(\mathbb{Z}\). The Gauss-Bonnet Theorem beautifully governs the behavior of \(\pi \circ \epsilon (\mathcal{C})\), as it implies \(\chi(S_g) = 2 \sum_{\gamma \in \mathcal{C}} t( \pi \circ \epsilon (\gamma))\), where \(t\) is the turning number function. Focusing on when \(S_g \cong S^2\), we give a necessary and sufficient condition for when a disjoint collection of curves \(\mathcal{C} \subset S^2\) can be realized as the crease set of an embedding \(\epsilon: S^2 \rightarrow \mathbb{R}^2 \times \mathbb{R}\).

Mon, Sep 12

Algebra Seminar
Shichen Tang
Arithmetic stability of higher rank Artin-Schreier-Witt towers
4:00PM, 250 Mathematics Building

Let \(q=p^a\), \(K\) the rational function field over the field of \(q\) elements, and \(G\) be the absolute Galois group of \(K\).For any continuous p-adic representation of \(G\), one can construct a tower of finite Galois extensions of \(K\). A conjecture of Daqing Wan states that if this representation "comes from algebraic geometry", then the slopes of the zeta functions of the fields in this tower have a stable behavior. In general, Wan's conjecture is wide open and already very hard when this tower is an Artin-Schreier-Witt tower. In this talk, we will discuss some recent progress related to Wan's conjecture for higher rank Artin-Schreier-Witt towers.

Wed, Sep 14

Analysis Seminar
Hanfeng Li, SUNY at Buffalo
Entropy and asymptotic pairs
4:00PM, 250 Math Building and on Zoom - contact hfli@buffalo.edu for Zoom link

Positive entropy and the existence of nontrivial asymptotic pairs are both kind of chaotic properties in topological dynamics. I will discuss the relation between these two properties for algebraic actions of amenable groups, and how this is related to the strong Atiyah conjecture in L2-invariants theory. This is joint work with Sebastian Barbieri and Felipe Garcia-Ramos.

Fri, Sep 16

Geometry and Topology Seminar
Yulan Qing (Fudan University/ University of Toronto)
Gromov boundary extended


4:00PM, 122 Mathematics Building

Gromov boundary provides a useful compactification for all infinite-diameter Gromov hyperbolic spaces. It consists of all geodesic rays starting at a given base-point and it has been an essential tool in the study of the coarse geometry of hyperbolic groups. In this study we introduce two topological spaces that are natural analogs of the Gromov boundary for a larger class of metric spaces. First we construct the sublinearly Morse boundaries and show that it is a QI-invariant topological space that can be associated to all finitely generated groups. Furthermore, for many groups, the sublinear boundary can be identified with the Poisson boundaries of the associated group, thus providing a QI-invariant model for Poisson boundaries. This result answers the open problems regarding QI-invariant models of CAT(0) groups and the mapping class group. Lastly,  for a subset of the metric spaces we define a compactification of the sublinearly Morse boundary and show that in these cases they are naturally identified with the Bowditch boundary. This is a series of joint work with Kasra Rafi and Giulio Tiozzo.

Mon, Sep 19

Algebra Seminar
Mariusz Tobolski, University of Wrocław
4:00PM, 250 Mathematics Building

Tue, Sep 20

Applied Math Seminar
Yangwen Zhang, CMU
A new reduced order model of linear parabolic PDEs.

4:00PM, Math 250 and on Zoom - contact mbichuch@buffalo.edu for link

How to build an accurate reduced order model (ROM) for multidimensional time dependent partial differ- ential equations (PDEs) is quite open. In this paper, we propose a new ROM for linear parabolic PDEs. We prove that our new method can be orders of magnitude faster than standard solvers, and is also much less memory intensive. Under some assumptions on the problem data, we prove that the convergence rates of the new method is the same with standard solvers. Numerical experiments are presented to confirm our theoretical result.

Tue, Sep 20

Special Event
250 Sexual Harassment Prevention Training
5:15PM, 250 Mathematics Building

Thu, Sep 22

Colloquium
Abdul Zalloum (University of Toronto)
 
4:00PM, 250 Mathematics Building

The field of geometric group theory investigates theconnections between the algebraic structure of a group and the geometriesof the metric spaces on which that group acts. This modern approach to grouptheory has revolutionized the study of finitely generated groups andproduced deep applications in logic, topology, geometry, and dynamical systems.

Group actions on hyperbolic spaces tend to be particularlyinformative of the algebraic structure of the acting group.One very broad class of groups that can be studied using actions onhyperbolic spaces is the class of hierarchically hyperbolicgroups which includes free groups, surface groups, mapping class groups,fundamental groups of 3–manifolds without Nil or Sol components, and manyothers.

I will show how the introduction, formalism and study of such a class haveled to the resolution of many long-standing open problems in the fieldincluding Farb's quasi-flats conjecture and the semi-hyperbolicity conjectureof the mapping class group. Finally, I will discuss some of my recentwork with Sisto where we show that the geometric model introduced byHaettel, Hoda and Petyt is the first geometric model for mapping classgroups where pseudo-Anosov elements have stronglycontracting axes confirming Thurston's conjecture on genericity of theseelements in such a geometric model.

Fri, Sep 23

Geometry and Topology Seminar
Abdul Zalloum (University of Toronto)
Hyperbolic models for CAT(0) spaces

4:00PM, 122 Mathematics Building

Two of the most well-studied topics in geometric group theory are CAT(0) cube complexes and mapping class groups. This is in part because they both admit powerful combinatorial-like structures that encode their (coarse) geometry: hyperplanes for the former and curve graphs for the latter. In recent years, analogies between the two theories have become more apparent. For instance: there are counterparts of curve graphs for CAT(0) cube complexes and rigidity theorems for such counterparts that mirror the surface setting, and both can be studied using the machinery of hierarchical hyperbolicity. However, the considerably larger class of CAT(0) spaces is left out of this analogy, as the lack of a combinatorial-like structure presents a difficulty in importing techniques from those areas. In this talk, I will speak about recent work with Petyt and Spriano where we bring CAT(0) spaces into the picture by developing analogues of hyperplanes and curve graphs for them. The talk will be accessible to everyone, and all the aforementioned terms will be defined.

Mon, Oct 3

Algebra Seminar
Li Li, Oakland University
Cluster algebras and Nakajima's graded quivervarieties 
4:00PM, 250 Mathematics Building

Nakajima's graded quiver varieties are complex algebraicvarieties associated with quivers. They are introduced by Nakajima in the studyof representations of universal enveloping algebras of Kac-Moody Lie algebras,and can be used to study cluster algebras. In the talk, I will explain how toprecisely locate the supports of the triangular basis of skew-symmetric rank-2quantum cluster algebras by applying the decomposition theorem to variousmorphisms related to quiver varieties, thus prove a conjecture proposed byLee-Li-Rupel-Zelevinsky in 2014.

Wed, Oct 5

Special Event
2022 Myhill lecture Series: Gigliola Staffilani October 5-7
The study of wave interactions: where beautiful mathematical ideas come together.


4:00PM, 250 Mathematics Building,

  Phenomena involving interactions of waves happen at different scales and in different media: from gravitational waves to the waves on the surface of the ocean, from our milk and coffee in the morning to infinitesimal particles that behave like wave packets in quantum physics. These phenomena are difficult to study in a rigorous mathematical manner, but maybe because of this challenge mathematicians have developed interdisciplinary approaches that are powerful and beautiful. I will describe some of these approaches and show for example how the need to understand certain multilinear and periodic interactions gave also the tools to prove a famous conjecture in number theory or how classical tools in probability gave the right framework to still have viable theories behind certain deterministic counterexamples.

Tue, Oct 11

Applied Math Seminar
Zechuan Zhang, SUNY Buffalo
Soliton resolution and asymptotic stability of N-soliton solutions for the defocusing mKdV equation with finite density type initial data

4:00PM, Math 250 and on Zoom - contact mbichuch@buffalo.edu for link

We consider the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with finite density type initial data. According to the \(\bar\partial\) steepest descent method, we introduce a series of transformation to the original Riemann-Hilbert problem \(m(z)\) to extrapolate the leading order approximation to the solution of mKdV for large time in the solitonic space-time region \(|x/t + 4| < 2\), and we give bounds for the error which decay as \(t\to\infty\). Our results also provide a verification of the soliton resolution conjecture and asymptotic stability of N-soliton solutions for mKdV equation with finite density type initial data.

Wed, Oct 12

Analysis Seminar
Yuqing (Frank) Lin, Texas A&M University
Entropy for actions of free groups under bounded orbit equivalence

4:00PM, 250 math Building and on Zoom - contact hfli@math.buffalo.edu for link

Joint work with Lewis Bowen. The f-invariant is a notion of entropy for probability measure preserving (pmp) actions of free groups. It is invariant under measure conjugacy and is an extension of Kolmogorov-Sinai entropy for actions of the integers. Two pmp actions are orbit equivalent if their orbits can be matched almost everywhere in a measurable fashion. Although entropy is not invariant under orbit equivalence in general, work of Austin and Kerr-Li has shown in various settings that entropy is invariant under certain stronger notions of quantitative orbit equivalence. We add to these results by showing that the f-invariant is invariant under the assumption of bounded orbit equivalence. 

Fri, Oct 14

Geometry and Topology Seminar
Bojun Zhao (UB)
Left orderability and taut foliations with one-sided branching
4:00PM, 122 Mathematics Building

For a closed orientable irreducible 3-manifold \(M\) that admits a co-orientable taut foliation with one-sided branching, we show that \(\pi_1(M)\) is left orderable.

Mon, Oct 17

Algebra Seminar
Mihai Fulger, U of Connecticut
Positivity vs. semi-stability for bundles with vanishing discriminant
4:00PM, Zoom - contact achirvas@buffalo.edu for link

Positivity properties of vector bundles like ampleness or nefness are rich research topics, mostly studied in the rank 1 case where they have important applications to birational geometry. Semi-stability is another important property due to its application to the construction of moduli spaces. On curves there are connections between the two first observed in work of Hartshorne. In higher dimension a connection exists under the additional assumption that the discriminant of the bundle vanishes. We give algebraic proofs of this valid in arbitrary characteristic. This is in joint work with Adrian Langer.

Tue, Oct 18

Colloquium
Jie Shen, Purdue University
Efficient positivity/bound preserving schemes for complex nonlinear systems
4:00PM, Math Bldg Room 250

Solutions of a large class of partial differential equations (PDEs) arising from sciences and engineering
applications are required to be positive or within a specified bound, and also energy dissipative.
It is of critical importance that their numerical approximations preserve these structures at the discrete level, as violation of these structures may render the discrete problems ill posed or inaccurate.
I will review the existing approaches for constructing positivity/bound preserving schemes, and then present
several efficient and accurate approaches:
(i) through reformulation as Wasserstein gradient flows;
(ii) through a suitable functional transform
(iii) through a Lagrange multiplier.
These approaches have different advantages and limitations, are all relatively easy to implement and can be combined with most spatial discretizations.

Fri, Oct 21

Geometry and Topology Seminar
José Román Aranda Cuevas (Binghamton University)

4:00PM, 122 Mathematics Building

Take two 3-dimensional handlebodies with the same boundary surface. One can tell them apart by studying the curves on the boundary surface bounding disks on each handlebody. Hempel studied Heegaard splittings of closed 3-manifolds by comparing these disk sets in the curve complex. For trisections of 4-manifolds, one can measure the length of loops in some complex passing through the disk set of each 3-dimensional handlebody. Kirby and Thompson used cut systems this way to define the L-invariant of a trisection of a closed 4-manifold. Other authors extended this definition for relative trisections and bridge trisections. Naturally, L is hard to compute. We will discuss lower bounds for (b,c)-bridge trisections of closed surfaces. This is joint work with Taylor, Pongtanapaisan, and Zhang.

Mon, Oct 24

Algebra Seminar
Doyon Kim, Rutgers University
The existence and uniqueness of Whittaker functionals for \(GL(n,R)\): an algebraic-geometric proof
4:00PM, Zoom; please email achirvas@buffalo.edu for meeting info

The "multiplicity one theorem," proved by Piatetski-Shapiro and Shalika, asserts that the space of Whittaker functionals on unitary irreducible representations of \(GL(n,R)\) is at most one-dimensional. In this talk, we discuss a new, algebraic-geometric proof that the space of Whittaker functionals on principal series representations of \(GL(n,R)\) is exactly one-dimensional. Additionally, we discuss its possible applications on Jacquet integrals.

Tue, Oct 25

Applied Math Seminar
Naoki Masuda, SUNY Buffalo
Core-periphery structure in networks.

4:00PM, Math 250 and on Zoom - contact mbichuch@buffalo.edu for link

Core-periphery structure is a mesoscale structure of networks, with which nodes in the core set are densely interconnected, peripheral nodes are connected to core nodes to different extents, and peripheral nodes are sparsely interconnected. I will introduce new scalable and principled algorithms to find core-periphery structure in networks. Core-periphery structure composed of a single core and a single periphery has been observed for various networks. In contrast, our algorithm aims to find multiple non-overlapping groups of core-periphery structure in a network. We also argue that, relative to a standard random graph model, core-periphery structure in the given network is mathematically possible only when we allow at least three blocks of nodes, thereby excluding conventional core-periphery structure composed of two blocks, i.e., a core block and a peripheral block. We illustrate our algorithms with empirical networks with applications.

Mon, Oct 31

Special Event
Jesse Huang, University of Alberta
Some attempts to build NCCRs for higher dimensional toric Gorenstein rings 
4:00PM, Zoom; please email achirvas@buffalo.edu for meeting info

A noncommutative crepant resolution (NCCR) is a nice endomorphism algebra of a sum of modules that ``resolves'' a normal Gorenstein ring. In the toric context, mirror symmetry suggests that questions surrounding the existence of NCCRs and derived equivalences among them could have geometric answers. In this talk, I will discuss some speculations on a geometric method to construct NCCRs as a quiver algebra for certain toric Calabi-Yau singularities ,potentially generalizing results of Mozgovoy and Bocklandt in dimension 3.

Tue, Nov 1

Applied Math Seminar
Scott Rich, Krembil Brain Institute
Resilience through diversity: Reduced heterogeneity in human epilepsy destabilizes neuronal circuits and promotes seizure-like transitions.

4:00PM, Math 250 and on Zoom - contact mbichuch@buffalo.edu for link

 
Epilepsy is the most common serious neurological disorder in the world, characterized by sudden transitions between sparse and asynchronous neuronal activity into hyper-synchronous and hyper-active oscillatory dynamics that envelop the entire brain. This phenomenon echoes effects of bifurcations in dynamical systems, supporting a robust subfield of computational neuroscience focused on modeling seizure dynamics. This pursuit is made more challenging considering the multitude of etiologies that can yield recurrent seizure, and in turn be classified under the broad banner of epilepsy. However, many of these etiologies can be understood as reduced variability in the properties of neuronal circuits, yielding the hypothesis that epileptogenesis can be recontextualized as a progressive loss of biophysical heterogeneity.
In this talk, I will present interdisciplinary results from my recently published work supporting this novel perspective on epilepsy. Neurons from human cortical tissue exhibit decreased heterogeneity when taken from brain regions that initiate seizure. When implemented in computational neuronal circuits, these reduced heterogeneities yield sudden transitions into synchronous oscillatory dynamics that are absent with physiological levels of heterogeneity. Perhaps most interestingly, this phenomenon is explained at a fundamental level by the differing stability structures of these systems with low and high heterogeneities. Alongside further research in preparation, these results showcase a fundamental role for biophysical heterogeneity in the brain in imparting resilience to pathological dynamica such as seizure.

Wed, Nov 9

Analysis Seminar
Hongming Nie, SUNY at Stony Brook
A metric on hyperbolic components

4:00PM, 250 Math Building and on Zoom - contact hfli@math.buffalo.edu for Zoom link

In this talk, under a mild condition, I will introduce a metric on hyperbolic components of rational maps. This metric is constructed by considering the measure-theoretic entropy with respect to some equilibrium state. Moreover, this metric is conformal equivalent to the pressure from the thermodynamics. It is a joint work with Y.M. He.

Tue, Nov 15

Applied Math Seminar
Anita Layton, Waterloo
TBA

4:00PM, Zoom - contact mbichuch@buffalo.edu for link

TBA

Wed, Nov 16

Analysis Seminar
Sagun Chanillo, Rutgers University
Local Version of Courant's Nodal Domain Theorem
4:00PM, On Zoom - conact hfli@math.buffalo.edu for link

Let \((M^n, g)\) denote a smooth, compact Riemannian manifold with no boundary. A fundamental object on this manifold is the Laplace-Beltrami operator which has a discrete spectrum. If we arrange the eigenvalues of the Laplacian in increasing order (for the negative of the Laplacian) with multiplicity, Courant's theorem states, that the number of nodal components for the k-th eigenfunction is at most k. A nodal component of an eigenfunction u is a connected component of the set where u does not vanish. In this talk we study a local version of the global result of Courant. The local question was raised by Donnelley and C.Fefferman in the late1980s. Our theorems are joint work with A. Logunov, E. Mallinikova and D. Mangoubi.

Thu, Nov 17

Colloquium
Colloquium Hossein Shahmohamad
Graphs & Their potent Energy Drinks 
4:00PM

Speaker: Hossein Shahmohamad, RIT

Title: Graphs & Their potent Energy Drinks

Fri, Nov 18

Geometry and Topology Seminar
Yvon Verberne (University of Toronto)
Postponed to Spring 2023 due to Storm
Automorphisms of the fine curve graph

4:00PM, 122 Mathematics Building

The fine curve graph of a surface was introduced by Bowden, Hensel and Webb. It is defined as the simplicial complex where vertices are essential simple closed curves in the surface and the edges are pairs of disjoint curves. We show that the group of automorphisms of the fine curve graph is isomorphic to the group of homeomorphisms of the surface, which shows that the fine curve graph is a combinatorial tool for studying the group of homeomorphisms of a surface. This work is joint with Adele Long, Dan Margalit, Anna Pham, and Claudia Yao.

Wed, Nov 30

Analysis Seminar
Joseph Hundley, SUNY at Buffalo
Functorial Descent in the Exceptional Groups
4:00PM, Zoom - contact hfli@math.buffalo.edu for link

In this talk, I will discuss the method of functorial descent. This method, which was discovered by Ginzburg, Rallis and Soudry, uses Fourier coefficients of residues of Eisenstein series to attack the problem of characterizing the image of Langlands functorial liftings. We'll discuss the general structure, the results of Ginzburg, Rallis and Soudry in the classical groups, some recent attempts to extend the same ideas to the exceptional groups, and challenges and new phenomena which merge in these attempts. The new work discussed will mainly be joint with Baiying Liu. Time permitting I may also comment on unpublished work of Ginzburg and joint work with Ginzburg.

Tue, Dec 6

Applied Math Seminar
Weiqi Chu, UCLA
Non-Markovian opinion models inspired by random processes on networks

4:00PM, Math 250 and on Zoom - contact mbichuch@buffalo.edu for link

The study of opinion dynamics models opinion evolution as dynamical processes on social networks. For social networks, nodes encode social entities (such as people and twitter accounts), while edges encode relationship or events between entities. Traditional models of opinion dynamics consider how opinions evolve either on time-independent networks or on temporal networks with edges that follow Poisson statistics. However, in many real-life networks, interactions between individuals (and hence the edges in a network) follow non-Poisson processes, which leads to dynamics on networks with memory-dependent effects (such as stereotypes). In this talk, we model social interactions as random processes on temporal networks and derive the opinion model that is governed by an arbitrary waiting-time distribution (WTD). When random processes have non-Poisson interevent statistics, the corresponding opinion models yield non-Markovian dynamics naturally. We analyze the convergence to consensus of these models and illustrate a variety of induced opinion models from common WTDs (including Dirac delta, exponential, and heavy-tailed distributions). When the opinion model does not have an explicit form (such as models induced by heavy-tailed WTDs), we provide a discrete-time approximation method and derive an associate set of discrete-time opinion-dynamics models.

Sat, Dec 17

Special Event
150 KIm Javor
10:00AM

Mon, Dec 19

Special Event
Liviu Paunescu, Simion Stoilow Institute of Mathematics

4:00PM, : Zoom; please email achirvas@buffalo.edu for meeting info

Two permutations that almost commute are close to two commuting permutations. The same question can be asked for other relations, not only the commutant. Moreover, the answer to this question depends only on the group that the equations describe. We then survey some recent results where this question is answered affirmatively or negatively, depending on the group, and study the connections to the theory of sofic groups.