• Speaker: Anna Weigandt, Massachusetts Institute of Technology
• Title: Combinatorial Aspects of Determinantal Varieties
• Date: 01/09/2023
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall (Virtual Meeting Link)
• Contact: 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.

• Speaker: 
• Title: G&T Seminar Organizational Meeting
• Date: 01/10/2023
• Time: 2:30 PM - 3:00 PM
• Place: C304 Wells Hall
• Contact: Peter Kilgore Johnson (john8251@msu.edu)

No abstract available.

• Speaker: Nathaniel Bottman, Max Planck Institute
• Title: What analysis, combinatorics, and quilted spheres can tell us about symplectic geometry
• Date: 01/10/2023
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall (Virtual Meeting Link)
• Contact: 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.

• Speaker: Aver St. Dizier, University of Illinois
• Title: A Polytopal View of Schubert Polynomials
• Date: 01/11/2023
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall (Virtual Meeting Link)
• Contact: 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.

• Speaker: Simon Foucart, Texas A&M University
• Title: ZOOM TALK (Passcode: the smallest prime > 100 ): Three uses of semidefinite programming in approximation theory
• Date: 01/12/2023
• Time: 2:30 PM - 3:30 PM
• Place: C304 Wells Hall (Virtual Meeting Link)
• Contact: 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.

• Speaker: Demetre Kazaras, Duke University
• Title: The geometry of scalar curvature and mass in general relativity
• Date: 01/12/2023
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall (Virtual Meeting Link)
• Contact: 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?

• Speaker: Alexander Watson, University of Minnesota
• Title: Mathematics of novel materials from atomic to macroscopic scales
• Date: 01/13/2023
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall (Virtual Meeting Link)
• Contact: 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.