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Math & Stats Algebra & Geometry Seminar
Monday, March 17, 2025

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

Speaker: Maya Banks (University of Illinois Chicago)

Title: Minimal Degree Curves and Scrolls in Weighted Projective Space

Abstract: A nondegenerate variety in projective space satisfies the condition that its degree is bounded below by 1 more than its codimension. Varieties attaining this lower bound—so-called "varieties of minimal degree"—were classified by Bertini and Del Pezzo and exhibit several nice properties, for instance, they are generated by quadrics and have linear syzygies. This theory of minimal degree varieties may be generalized to study varieties in weighted projective spaces. We will discuss bounds on the degree of a nondegenerate variety in certain weighted projective spaces, as well as curves and 'weighted scrolls' which attain the minimal degree bound, and some of their syzygetic properties.

Math & Stats Department Colloquium
Friday, March 14, 2025

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

Speaker: Giuseppe Maria Coclite (Politecnico di Bari)

Title: Vanishing viscosity versus Rosenau approximation for scalar conservation laws: the fractional case

Abstract: In this talk, we consider approximations of scalar conservation laws by adding nonlocal diffusive operators. In particular, we consider solutions associated with fractional Laplacian and fractional Rosenau perturbations and show that, for any $t>0$, the mutual $L^1$ distance of their profiles is negligible as compared to their common distance to the underlying inviscid entropy solution.We provide explicit examples showing that our rates are optimal in the supercritical and critical cases, in one space dimension and for strictly convex fluxes. For subcritical equations, our rates are not optimal but they remain explicit. Those results were obtained in collaboration with N. Alibaud, M. Dalery, and C. Donadello.

PDEs & Applications Seminar

Friday, March 21, 2025

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

Speaker: Fabian Bleitner (McMaster University)

Title: Buoyancy-Driven Flows With Navier-Slip Boundary Conditions

Abstract: In this talk two-dimensional buoyancy-driven flows are investigated. While usually the Navier-Stokes equations are equipped with no-slip boundary conditions, here we focus on the Navier-slip conditions that, depending on the system at hand, better reflect the physical behavior. In particular, we study two systems, Rayleigh-Bénard convection and a closely related problem without thermal diffusion. In the former, bounds on the vertical heat transfer, given by the Nusselt number, with respect to the strength of the buoyancy force, characterized by the Rayleigh number, are derived. These bounds hold for a broad range of applications, allowing for non-flat boundaries, any sufficiently smooth positive slip coefficient, and are valid over all ranges of the Prandtl number, a system parameter determined by the fluid. For the thermally non-diffusive system, regularity estimates are proven. Up to a certain order, these bounds hold uniformly in time, which, combined with estimates for their growth, provide insight into the long-time behavior. In particular, solutions converge to the hydrostatic equilibrium, where the fluid's velocity vanishes and the buoyancy force is balanced by the pressure gradient.

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.