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Colloquium
- Anna Weigandt, Massachusetts Institute of Technology
- Combinatorial Aspects of Determinantal Varieties
- 01/09/2023
- 4:10 PM - 5:00 PM
- C304 Wells Hall
(Virtual Meeting Link)
- Sabrina M Walton (waltons3@msu.edu)
Schubert calculus has its origins in enumerative questions asked by the geometers of the 19th century, such as “how many lines meet four fixed lines in three-space?” These problems can be recast as questions about the structure of cohomology rings of geometric spaces such as flag varieties. Borel’s isomorphism identifies the cohomology of the complete flag variety with a simple quotient of a polynomial ring. Lascoux and Schützenberger (1982) defined Schubert polynomials, which are coset representatives for the Schubert basis of this ring. However, it was not clear if this choice was geometrically natural. Knutson and Miller (2005) provided a justification for the naturality of Schubert polynomials via antidiagonal Gröbner degenerations of matrix Schubert varieties, which are generalized determinantal varieties. Furthermore, they showed that pre-existing combinatorial objects called pipe dreams govern this degeneration. In this talk, we study the dual setting of diagonal Gröbner degenerations of matrix Schubert varieties, interpreting these limits in terms of the “bumpless pipe dreams” of Lam, Lee, and Shimozono (2021). We then use the combinatorics of K-theory representatives for Schubert classes to compute the Castelnuovo-Mumford regularity of matrix Schubert varieties, which gives a bound on the complexity of their coordinate rings.
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Geometry and Topology
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- G&T Seminar Organizational Meeting
- 01/10/2023
- 2:30 PM - 3:00 PM
- C304 Wells Hall
- Peter Kilgore Johnson (john8251@msu.edu)
No abstract available.
Colloquium
- Nathaniel Bottman, Max Planck Institute
- What analysis, combinatorics, and quilted spheres can tell us about symplectic geometry
- 01/10/2023
- 4:10 PM - 5:00 PM
- C304 Wells Hall
(Virtual Meeting Link)
- Sabrina M Walton (waltons3@msu.edu)
A central tool for studying symplectic manifolds is the Fukaya category. In this talk, I will describe my program to relate the Fukaya categories of different symplectic manifolds. The key objects are "witch balls", which are coupled systems of PDEs whose domain is the Riemann sphere decorated with circles and points, and "2-associahedra", the configuration spaces of these domains. I will describe applications to symplectic geometry and algebraic geometry, and highlight the role of degenerating families of elliptic PDEs.
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Colloquium
- Aver St. Dizier, University of Illinois
- A Polytopal View of Schubert Polynomials
- 01/11/2023
- 4:10 PM - 5:00 PM
- C304 Wells Hall
(Virtual Meeting Link)
- Sabrina M Walton (waltons3@msu.edu)
Schubert polynomials are a family of multivariable polynomials whose product can be used to solve problems in enumerative geometry. Despite their many known combinatorial formulas, there remain mysteries surrounding these polynomials. I will describe Schubert (and the special case of Schur) polynomials with a focus on polytopes. From this perspective, I will address questions such as vanishing of Schubert coefficients, relative size of coefficients, and interesting properties of their support. Time permitting, I'll talk about my current work on generalizing the Gelfand–Tsetlin polytope, and its connections with representation theory and Bott–Samelson varieties.
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Applied Mathematics
- Simon Foucart, Texas A&M University
- ZOOM TALK (Passcode: the smallest prime > 100 ): Three uses of semidefinite programming in approximation theory
- 01/12/2023
- 2:30 PM - 3:30 PM
- C304 Wells Hall
(Virtual Meeting Link)
- Mark A Iwen (iwenmark@msu.edu)
In this talk, modern optimization techniques are publicized as fitting computational tools to attack several extremal problems from Approximation Theory which had reached their limitations based on purely analytical approaches. Three such problems are showcased: the first problem---minimal projections---involves minimization over measures and exploits the moment method; the second problem---constrained approximation---involves minimization over polynomials and exploits the sum-of-squares method; and the third problem---optimal recovery from inaccurate observations---is highly relevant in Data Science and exploits the S-procedure. In each of these problems, one ends up having to solve semidefinite programs.
Colloquium
- Demetre Kazaras, Duke University
- The geometry of scalar curvature and mass in general relativity
- 01/12/2023
- 4:10 PM - 5:00 PM
- C304 Wells Hall
(Virtual Meeting Link)
- Sabrina M Walton (waltons3@msu.edu)
In general relativity, the space we inhabit is modeled by a Riemannian manifold. The fundamental restriction this theory places upon spatial geometry is a lower bound on this manifold's scalar curvature. It is an important problem in pure geometry to understand the geometric and topological features of this condition. For instance, if a manifold has positive scalar curvature, what may we conclude about the lengths of its curves, the areas of its surfaces, and the topology of the underlying manifold? I will explain many results (originally proven by Schoen-Yau and Gromov-Lawson) in this direction, and sketch proofs by analyzing objects I call 'spacetime harmonic functions.' Leveraging these new ideas, I will also describe progress on geometric versions of the following questions: How flat is a gravitational system with little total mass? How can we tell when matter will coalesce to form a black hole?
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Colloquium
- Alexander Watson, University of Minnesota
- Mathematics of novel materials from atomic to macroscopic scales
- 01/13/2023
- 4:10 PM - 5:00 PM
- C304 Wells Hall
(Virtual Meeting Link)
- Sabrina M Walton (waltons3@msu.edu)
Materials' electronic properties arise from the complex dynamics of electrons flowing through the material. These dynamics are quantum mechanical and present many surprising phenomena without classical analogues. I will present analytical and numerical work clarifying these dynamics in three novel materials which have attracted intense theoretical and experimental attention in recent years: graphene, the first ``2D'' material, whose electronic properties can be captured by an effective Dirac equation, topological insulators, whose edges host surprising one-way edge currents, and twisted bilayer graphene, an aperiodic material whose properties can be captured by an effective system of Dirac equations with periodic coefficients. I will then present ongoing and future work focused on further clarifying the properties of twisted bilayer graphene, which was recently shown to superconduct when twisted to the ``magic'' twist angle 1 degree.
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Colloquium
- Cesar Cuenca, Harvard University
- Random matrices and random partitions at varying temperatures
- 01/17/2023
- 4:10 PM - 5:00 PM
- C304 Wells Hall
(Virtual Meeting Link)
- Sabrina M Walton (waltons3@msu.edu)
I will discuss the global-scale behavior of ensembles of random matrix eigenvalues and random partitions which depend on the "inverse temperature" parameter beta. The goal is to convince the audience of the effectiveness of the moment method via Fourier-like transforms in characterizing the Law of Large Numbers and Central Limit Theorems in various settings. We focus on the regimes of high and low temperatures, that is, when the parameter beta converges to zero and infinity, respectively. Part of this talk is based on joint projects with F. Benaych-Georges -- V. Gorin, and M. Dolega -- A. Moll.
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Colloquium
- Charles Ouyang, UMass Amherst
- Compactifications of Hitchin components
- 01/18/2023
- 4:10 PM - 5:00 PM
- C304 Wells Hall
(Virtual Meeting Link)
- Sabrina M Walton (waltons3@msu.edu)
Hitchin components are natural generalizations of the classical Teichmüller space. In the setting of SL(3,R), the Hitchin component parameterizes the holonomies of convex real projective structures, which are related to hyperbolic affine spheres. By studying Blaschke metrics, which are Riemannian metrics associated to hyperbolic affine spheres, along with their limits, we obtain a compactification of the SL(3,R)-Hitchin component. We show the boundary objects are hybrid structures, which are in part flat metric and in part laminar. These hybrid objects are natural generalizations of measured laminations, which are the boundary objects in Thurston's compactification of Teichmüller space.
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Applied Mathematics
- Madeleine Udell, Stanford University
- ZOOM TALK (Passcode: the smallest prime > 100 ): Low rank approximation for faster optimization
- 01/19/2023
- 2:30 PM - 3:30 PM
- C304 Wells Hall
(Virtual Meeting Link)
- Mark A Iwen ()
Low rank structure is pervasive in real-world datasets. This talk shows how to accelerate the solution of fundamental computational problems, including eigenvalue decomposition, linear system solves, composite convex optimization, and stochastic optimization (including deep learning), by exploiting this low rank structure. We present a simple method based on randomized numerical linear algebra for efficiently computing approximate top eigendecompositions, which can be used to replace large matrices (such as Hessians and constraint matrices) with low rank surrogates that are faster to apply and invert. The resulting solvers for linear systems (NystromPCG), composite convex optimization (NysADMM), and deep learning (SketchySGD) demonstrate strong theoretical and numerical support, outperforming state-of-the-art methods in terms of speed and robustness to hyperparameters.
Colloquium
- March Tian Boedihardjo, ETH Zurich
- Freeness and matrices
- 01/19/2023
- 4:10 PM - 5:00 PM
- C304 Wells Hall
(Virtual Meeting Link)
- Sabrina M Walton (waltons3@msu.edu)
I will begin by giving some background on Free Probability motivated by the freeness in free groups. I will then demonstrate how Free Probability can be used to obtain a sharp non-asymptotic random matrix estimate for general use. This talk will be concluded by a recent application of our result to the Matrix Spencer Conjecture. Joint work with Afonso Bandeira and Ramon van Handel.
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Dynamical Systems
- Fan Yang, Michigan State University
- Lorenz attractor and singular flows: expansivity, entropy, and equilibrium states
- 01/20/2023
- 3:00 PM - 5:00 PM
- C304 Wells Hall
- Fan Yang (yangfa31@msu.edu)
No abstract available.
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Colloquium
- Zhongshan An, University of Michigan
- Geometric boundary conditions for the Einstein equations and quasi-local mass
- 01/23/2023
- 4:10 PM - 5:00 PM
- C304 Wells Hall
(Virtual Meeting Link)
- Sabrina M Walton (waltons3@msu.edu)
The Einstein equations are the most fundamental equations for spacetimes in general relativity. They relate the geometry (curvatures) of a spacetime with its physical property. When a spacetime has nonempty boundary, it is natural to ask what geometric boundary conditions are well-posed for the Einstein equations. The investigation of geometric boundary conditions both gives rise to interesting geometric PDE problems in differential geometry, and also plays an important role in the study of quasi-local mass for compact spacetimes in general relativity. In this talk, we will discuss geometric boundary conditions for the vacuum Einstein equations, from both the hyperbolic and elliptic aspects. Furthermore, we will talk about applications of these geometric boundary value problems in the construction of quasi-local mass.
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CoIntegrate Mathematics
- Vince Melfi, MSU; Jenny Green, MSU; John Keane, MSU
- Fostering a Culture of Instructional Development in the Department of Statistics and Probability: Our Journey with First-Year Graduate Teaching Assistants
- 01/24/2023
- 1:00 PM - 2:30 PM
- 115 Erickson Hall
- Lisa Keller (kellerl@msu.edu)
How do we support graduate students to teach introductory statistics classes, which themselves are undergoing dramatic transformation? In this talk, we will get to engage with
lessons learned and questions still unanswered as we embarked on the journey of developing an instructional mentoring program for the Department of Statistics and Probability.
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Combinatorics and Graph Theory
- Yibo Gao, University of Michigan
- CANCELLED: Symmetric structures in the strong Bruhat order
- 01/25/2023
- 3:00 PM - 3:50 PM
- C304 Wells Hall
- Bruce E Sagan (bsagan@msu.edu)
The Bruhat order encodes algebraic and topological information of Schubert varieties in the flag manifold and possesses rich combinatorial properties. In this talk, we discuss three interrelated stories regarding the Bruhat order: self-dual Bruhat intervals, Billey-Postnikov decompositions and automorphisms of the Bruhat graph. This is joint work with Christian Gaetz.
Probability
- Wlodzimierz Bryc, University of Cincinnati
- Stationary measures of the Kardar-Parisi-Zhang equation and their limits
- 01/25/2023
- 3:00 PM - 3:50 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Konstantin Matetski (matetski@msu.edu)
I will overview recent results of [Corwin and Knizel, 2021] on the existence of stationary measures for the KPZ equation on an interval and [Barraquand and Le Doussal, 2022], [B.-Kuznetsov-Wang-Wesolowski, 2022] who found two different probabilistic descriptions of the stationary measures as a Markov process and as a measure with explicit Radon-Nikodym derivative with respect to the Brownian motion. The Markovian description leads to rigorous proofs of some of the limiting results claimed in [Barraquand and Le Doussal, 2022]. I shall discuss how the stationary measures of the KPZ equation on [0,L] behave at large scale as L goes to infinity which according to [Barraquand and Le Doussal, 2022] depending on the normalization, should correspond to stationary measures of a hypothetical KPZ fixed point on [0,1], to the stationary measure for the KPZ equation on the half-line, and to the stationary measure of a hypothetical KPZ fixed point on the half-line.
The talk is based mostly on a joint work with Alexey Kuznetsov (ALEA 2022).
Mathematics Education Colloquium Series
- Katie Lewis, University of Washington
- Disability Equity in Mathematics Education: Accessibility, Re-mediation, and CompensationAbstract
- 01/25/2023
- 3:30 PM - 5:00 PM
- 252 EH
(Virtual Meeting Link)
- Lisa Keller (kellerl@msu.edu)
Equity in mathematics education research has only recently begun to consider students with disabilities. In this talk, I focus specifically on students with mathematics disabilities – students who have a neurological difference in how their brains process numerical information. Prior research on mathematics disabilities (i.e., dyscalculia) has predominantly taken up a deficit frame, documenting the ways in which students with dyscalculia are deficient in terms of speed and accuracy. In my work, I argue that this deficit orientation is problematic, and I offer an alternative. I take up an explicitly anti-deficit framing and draw upon sociocultural learning theories and Disability Studies to orient my work. In this talk I use multiple case studies to explore ideas about accessibility, re-mediation, and compensation across a range of mathematical topics. This anti-deficit work provides an alternative vantage point to understand disability in mathematics education and suggests avenues to work towards equity. I close by considering ways that mathematics education equity research can be in service of and in partnership with the populations that we study. Zoom option: https://msu.zoom.us/j/95059549382 Passcode: PRIME
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Operator Algebras Reading
- Lucas Hall, MSU
- Approximately Finite Dimensional C*-algebras
- 01/30/2023
- 4:00 PM - 5:30 PM
- C517 Wells Hall
- Brent Nelson (banelson@msu.edu)
I’ll tour through the study of finite dimensional C*-algebras and homomorphisms between them, and use this as a basis to define and study approximately finite dimensional (AF) algebras.
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Algebra
- Theodore Voronov, University of Manchester
- From homotopy Lie brackets to thick morphisms of supermanifolds and non-linear functional-algebraic duality (NOTE UNUSUAL DAY)
- 01/31/2023
- 3:00 PM - 4:00 PM
- C204A Wells Hall
- Michael Shapiro (mshapiro@msu.edu)
I will give a motivation for homotopy Lie brackets and the corresponding morphisms preserving brackets "up to homotopy" (more precisely, for L-infinity morphisms and L-infinity algebras), and show how to describe them using supergeometry. So, instead of a single Poisson or Lie bracket, there is a whole sequence of operations with n arguments, n=1,2,3,..., satisfying a linked infinite sequence of identities replacing the familiar Jacobi identity for a Lie bracket; and, instead of a morphism as a linear map mapping a bracket to a bracket, there is a sequence of multi-linear mappings mixing brackets with different numbers of arguments, and, in particular, the binary bracket is preserved only up to an (algebraic) homotopy. Geometrically, such a sequence of multi-linear mappings assembles into one non-linear map of supermanifolds.
For the case of homotopy brackets of functions ("higher Poisson" or "homotopy Poisson" structure), this leads us to the question about a natural construction of non-linear mappings between algebras of smooth functions generalizing the usual pull-backs. I discovered such a construction some years ago. These are "thick morphisms" of (super)manifolds generalizing ordinary smooth maps. From a more general perspective, we arrive in this way at a non-linear analog of the classical functional-algebraic duality between spaces and algebras.
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