Talk_id | Date | Speaker | Title |
31514
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Monday 1/9 4:10 PM
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Anna Weigandt, Massachusetts Institute of Technology
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Combinatorial Aspects of Determinantal Varieties
- 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.
|
31521
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Tuesday 1/10 4:10 PM
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Nathaniel Bottman, Max Planck Institute
|
What analysis, combinatorics, and quilted spheres can tell us about symplectic geometry
- 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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31519
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Wednesday 1/11 4:10 PM
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Aver St. Dizier, University of Illinois
|
A Polytopal View of Schubert Polynomials
- 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.
|
31511
|
Thursday 1/12 4:10 PM
|
Demetre Kazaras, Duke University
|
The geometry of scalar curvature and mass in general relativity
- 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?
|
31528
|
Friday 1/13 4:10 PM
|
Alexander Watson, University of Minnesota
|
Mathematics of novel materials from atomic to macroscopic scales
- 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.
|
31510
|
Tuesday 1/17 4:10 PM
|
Cesar Cuenca, Harvard University
|
Random matrices and random partitions at varying temperatures
- 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.
|
31524
|
Wednesday 1/18 4:10 PM
|
Charles Ouyang, UMass Amherst
|
Compactifications of Hitchin components
- 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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31518
|
Thursday 1/19 4:10 PM
|
March Tian Boedihardjo, ETH Zurich
|
Freeness and matrices
- 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.
|
31512
|
Monday 1/23 4:10 PM
|
Zhongshan An, University of Michigan
|
Geometric boundary conditions for the Einstein equations and quasi-local mass
- 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.
|
30460
|
Thursday 2/23 4:10 PM
|
Brendan Hassett, Brown University
|
TBA
- Brendan Hassett, Brown University
- TBA
- 02/23/2023
- 4:10 PM - 5:00 PM
- Online (virtual meeting)
- Joseph Waldron (waldro51@msu.edu)
No abstract available.
|
29375
|
Thursday 3/2 4:10 PM
|
Katy Craig, UCSB
|
TBA
- Katy Craig, UCSB
- TBA
- 03/02/2023
- 4:10 PM - 5:00 PM
- C304 Wells Hall
- Olga Turanova (turanova@msu.edu)
TBA
|
29404
|
Thursday 3/16 4:10 PM
|
John Baldwin, Boston College
|
TBA
- John Baldwin, Boston College
- TBA
- 03/16/2023
- 4:10 PM - 5:00 PM
- C304 Wells Hall
- Joseph Waldron (waldro51@msu.edu)
No abstract available.
|
29388
|
Sunday 3/26 4:10 PM
|
Robin Walters, Northeastern University
|
TBA
- Robin Walters, Northeastern University
- TBA
- 03/26/2023
- 4:10 PM - 5:00 PM
- C304 Wells Hall
- Olga Turanova (turanova@msu.edu)
TBA
|
29381
|
Thursday 3/30 4:10 PM
|
Tim Hoheisel, McGill University
|
TBA
- Tim Hoheisel, McGill University
- TBA
- 03/30/2023
- 4:10 PM - 5:00 PM
- C304 Wells Hall
- Olga Turanova (turanova@msu.edu)
TBA
|
29382
|
Thursday 4/6 4:10 PM
|
Michael Brannan, University of Waterloo
|
TBA
- Michael Brannan, University of Waterloo
- TBA
- 04/06/2023
- 4:10 PM - 5:00 PM
- C304 Wells Hall
- Olga Turanova (turanova@msu.edu)
TBA
|
30453
|
Thursday 4/13 4:10 PM
|
David Fisher, Indiana University Bloomington
|
TBA
- David Fisher, Indiana University Bloomington
- TBA
- 04/13/2023
- 4:10 PM - 5:00 PM
- C304 Wells Hall
- Olga Turanova (turanova@msu.edu)
TBA
|
29405
|
Thursday 4/20 4:10 PM
|
Robert Pollack, Boston University
|
TBA
- Robert Pollack, Boston University
- TBA
- 04/20/2023
- 4:10 PM - 5:00 PM
- C304 Wells Hall
- Joseph Waldron (waldro51@msu.edu)
No abstract available.
|
29406
|
Thursday 4/27 4:10 PM
|
Tony Feng, University of California, Berkeley
|
TBA
- Tony Feng, University of California, Berkeley
- TBA
- 04/27/2023
- 4:10 PM - 5:00 PM
- C304 Wells Hall
- Joseph Waldron (waldro51@msu.edu)
No abstract available.
|