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 BuildingGromov 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 BuildingTwo 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.
