Seminars and Colloquia by Series

Saturating the Jacobian ideal of a line arrangement via rigidity theory

Series
Algebra Seminar
Time
Monday, March 6, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Michael DiPasqualeUniversity of South Alabama

A line arrangement is a collection of lines in the projective plane.  The intersection lattice of the line arrangement is the set of all lines and their intersections, ordered with respect to reverse inclusion.  A line arrangement is called free if the Jacobian ideal of the line arrangement is saturated.  The underlying motivation for this talk is a conjecture of Terao which says that whether a line arrangement is free can be detected from its intersection lattice.  This raises a question - in what ways does the saturation of the Jacobian ideal depend on the geometry of the lines and not just the intersection lattice?  A main objective of the talk is to introduce planar rigidity theory and show that 'infinitesimal rigidity' is a property of line arrangements which is not detected by the intersection lattice, but contributes in a very precise way to the saturation of the Jacobian ideal.  This connection builds a theory around a well-known example of Ziegler.  This is joint work with Jessica Sidman (Mt. Holyoke College) and Will Traves (Naval Academy).

TBA by Abdon Moutinho

Series
PDE Seminar
Time
Tuesday, February 28, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Online: https://gatech.zoom.us/j/95574359880?pwd=cGpCa3J1MFRkY0RUeU1xVFJRV0x3dz09
Speaker
Abdon MoutinhoLAGA, Université Sorbonne Paris Nord

TBA

Crossing the transcendental divide: from translation surfaces to algebraic curves

Series
Algebra Seminar
Time
Monday, February 27, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Yelena MandelshtamUC Berkeley

A translation surface is obtained by identifying edges of polygons in the plane to create a compact Riemann surface equipped with a nonzero holomorphic one-form. Every Riemann surface can be given as an algebraic curve via its Jacobian variety. We aim to construct explicitly the underlying algebraic curves from their translation surfaces, given as polygons in the plane. The key tools in our approach are discrete Riemann surfaces, which allow us to approximate the Riemann matrices, and then, via theta functions, the equations of the curves. In this talk, I will present our algorithm and numerical experiments. From the newly found Riemann matrices and equations of curves, we can then make several conjectures about the curves underlying the Jenkins-Strebel representatives, a family of examples that until now, lived squarely on the analytic side of the transcendental divide between Riemann surfaces and algebraic curves.

Regularity of Hele-Shaw flow with source and drift: Flat free boundaries are Lipschitz

Series
PDE Seminar
Time
Tuesday, February 14, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yuming Paul ZhangAuburn University

The classical Hele-Shaw flow describes the motion of incompressible viscous fluid, which occupies part of the space between two parallel, nearby plates. With source and drift, the equation is used in models of tumor growth where cells evolve with contact inhibition, and congested population dynamics. We consider the flow with Hölder continuous source and Lipschitz continuous drift. We show that if the free boundary of the solution is locally close to a Lipschitz graph, then it is indeed Lipschitz, given that the Lipschitz constant is small. This is joint work with Inwon Kim.

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