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Math & Stats Algebra & Geometry Seminar
Monday, March 10, 2025

Time: 4:30 p.m.  Place: Jeffery Hall, Room 116

Speaker: Gregory G. Smith (Queen's University)

Title: Smooth Quot Schemes

Abstract: How can we understand all coherent sheaves on a projective scheme? Quot schemes offer a geometric answer by parameterizing the sheaves, and the Hilbert polynomial gives a natural stratification into disjoint pieces. We will survey the key features of these parameter spaces for the quotients of a finite direct sum of the structure sheaf on projective space. Moreover, we will completely classify the smooth Quot schemes of this form. This talk is based on joint work with Roy Skjelnes and Mike Stillman.

Math & Stats Department Colloquium
Friday, March 14, 2025

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Giuseppe Maria Coclite (Politecnico di Bari)

Title: Measure valued solutions for an optimal harvesting problem

Abstract: In this lecture we consider a model for the harvesting of marine resources. Since the cost functionals have linear growth with respect to the pointwise intensity of fishing effort, optimal solutions are in general measure-valued. For the control problem, we prove the existence of optimal strategies. Those results were obtained in collaboration with A. Bressan, G. Devillanova, M. Garavello, W. Shen, S. F. Solimini, L. V. Spinolo.

PDEs & Applications Seminar

Friday, March 7, 2025

Time: 10:30 a.m.  Place: Jeffery Hall, Room 319

Speaker: Grigalius Taujanskas (University of Cambridge)

Title: Mathematical Theory of Carrollian Fluids in 1+1 Dimensions

Abstract: Due to connections to flat space holography, Carrollian geometry, physics and fluid dynamics have received an explosion of interest over the last two decades. In the "Carrollian limit" of vanishing speed of light c, relativistic fluids reduce to a set of PDEs called the Carrollian fluid equations. Although in general these equations are not well understood analytically, and their PDE theory does not appear to have been studied, in dimensions 1+1 it turns out that there is a duality with the Galilean compressible Euler equations in 1+1 dimensions inherited from the isomorphism of the Carrollian (c to 0) and Galilean (c to infinity) contractions of the Poincaré algebra. Under this duality time and space are interchanged, leading to different dynamics in evolution. I will discuss recent work with N. Athanasiou (Thessaloniki), M. Petropoulos (Paris) and S. Schulz (Pisa) in which we establish the first rigorous PDE results for these equations by introducing a notion of Carrollian isentropy and studying the equations using Lax’s method and compensated compactness. In particular, I will explain that there is global existence in rough norms but finite-time blow-up in smoother norms.

Math & Stats Algebra & Geometry Seminar
Monday, March 3, 2025

Time: 4:30 p.m.  Place: Jeffery Hall, Room 116

Speaker: Yaolong Shen (University of Ottawa)

Title: Quantum supersymmetric pairs and ıSchur duality

Abstract: Let g be a semisimple Lie algebra and θ be an involution of g. The quantization (U_q(g),U^ı) of the symmetric pair (g,g^θ) was systematically developed by Letzter, where U_q(g) is the Drinfeld-Jimbo quantum group, and U^ı is a coideal subalgebra of it. We usually refer U^ı as the ıquantum group. Over the last decade, many fundamental constructions in quantum groups have been generalized to ıquantum groups by Wang and his collaborators. In this talk, we will discuss the super analog of U^ı and introduce one specific family which unites ıquantum groups (non-super) of type AI and AII. We will also demonstrate an ıSchur duality between this specific family and the q Brauer algebra. This duality can be viewed as a quantization of the classical duality between the orthosymplectic Lie superalgebra and the Brauer algebra. This is joint work with Weiqiang Wang

PDEs & Applications Seminar

Friday, February 28, 2025

Time: 10:30 a.m.  Place: Jeffery Hall, Room 319

Speaker: Massimiliano Rosini (Uniwersytet Marii Curie-Sklodowskiej & Universita' “G. d’Annunzio” di Chieti-Pescara)

Title: Micro and macro descriptions of traffic

Abstract: We will explore the interplay between deterministic particle systems and one-dimensional conservation laws, with applications to vehicular traffic models and pedestrian flows.

We begin by discussing follow-the-leader dynamics, described by systems of ordinary differential equations, and show how they can approximate solutions of conservation laws. The talk focuses on the Cauchy problem for the Lighthill-Whitham-Richards (LWR) traffic model and extends to initial-boundary value problems, highlighting its adaptability to diverse traffic scenarios.The seminar also examines the Aw-Rascle-Zhang (ARZ) traffic model, which generalizes the LWR framework to account for multi-population dynamics, and the Hughes model for pedestrian flow, which incorporates decision-making processes such as exit selection. This presentation demonstrates how deterministic particle systems enhance our understanding of macroscopic models in traffic and pedestrian dynamics, offering valuable insights into complex real-world phenomena.

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Wednesday, February 12, 2025

Time: 9:30 a.m.  Place: Jeffery Hall, Room 110

Speaker: Eamon Quinlan-Gallego (The University of Utah)

Title: The b-function in positive characteristic

Abstract: The b-function (or Bernstein-Sato polynomial) of a holomorphic function is a classical invariant that is defined using differential operators and is now a foundational tool in D-module theory and the theory of singularities in algebraic geometry over the complex numbers. After explaining the definition, and going through some classical results, I will explain what happens when one replaces the complex numbers with a field of positive characteristic: I will explain how one can think of differential operators over such fields, and I will survey some result on the b-function for polynomials over them.

Monday, February 10, 2025

Time: 3:30 p.m.  Place: Jeffery Hall, Room 110

Speaker: Michael Brown (Auburn)

Title: The BGG correspondence for toric varieties

Abstract: The classical Bernstein-Gel’fand-Gel’fand (BGG) correspondence exhibits a close relationship between the homological properties of a polynomial ring and an exterior algebra. More precisely, it gives an equivalence of the derived categories of these two rings. I will discuss a geometric version of the BGG correspondence involving toric varieties.