Seminars and Colloquia by Series

Isodiametry, variance, and regular simplices from particle interactions

Series
PDE Seminar
Time
Tuesday, October 1, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tongseok Lim – ShanghaiTech University –

We study the geometry of minimizers of the interaction energy functional. When the interaction potential is mildly repulsive, it is known to be hard to characterize those minimizers due to the fact that they break the rotational symmetry, suggesting that the problem is unlikely to be resolved by the usual convexity or symmetrization techniques from the calculus of variations. We prove that, if the repulsion is mild and the attraction is sufficiently strong, the minimizer is unique up to rotation and exhibits a remarkable simplex-shape rigid structure. As the first crucial step we consider the maximum variance problem of probability measures under the constraint of bounded diameter, whose answer in one dimension was given by Popoviciu in 1935.

Mason's Conjecture

Series
Lorentzian Polynomials Seminar
Time
Tuesday, October 1, 2019 - 14:50 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Trevor Gunn – Georgia Tech

Using what we have studied in the Brändén-Huh paper, we will go over the proof of the ultra-log-concavity version of Mason's conjecture.

Existence of a family of solutions in state-dependent delay equations

Series
Dynamical Systems Working Seminar
Time
Tuesday, October 1, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jiaqi Yang – Georgia Tech –
Given an analytic two-dimensional ordinary differential equation which admits a limit cycle, we consider the singular perturbation of it by adding a state-dependent delay. We show that for small enough perturbation, there exist a limit cycle and a two-dimensional family of solutions to the perturbed state-dependent delay equation (SDDE), which resemble the solutions of the original ODE. 
More precisely, for the original ODE, there is a parameterization of the limit cycle and its stable manifold. We show that, there is a very similar parameterization that gives a 2-dimensional family of solutions of the SDDE. 
In our work, we analyze the parameterization, and find functional equations to be satisfied (invariance equations). We prove a theorem in \emph{``a posteriori''} format, that is, if there are approximate solutions of the invariance equations, then there are true solutions of the invariance equations nearby (with appropriate choices of norms). An algorithm which follows from the constructive proof of above theorem has been implemented. 
 
This is a joint work with Joan Gimeno and Rafael de la Llave.

Certifying solutions to a square analytic system

Series
Algebra Seminar
Time
Tuesday, October 1, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kisun Lee – Georgia Tech –

In this talk, we discuss about methods for proving existence and uniqueness of a root of a square analytic system in a given region. For a regular root, Krawczyk method and Smale's $\alpha$-theory are used. On the other hand, when a system has a multiple root, there is a separation bound isolating the multiple root from other roots. We define a simple multiple root, a multiple root whose deflation process is terminated by one iteration, and establish its separation bound. We give a general framework to certify a root of a system using these concepts.

Sharp diameter bound on the spectral gap for quantum graphs

Series
Math Physics Seminar
Time
Monday, September 30, 2019 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kenny Jones – Emory –

We establish an upper bound on the spectral gap for compact quantum graphs which depends only on the diameter and total number of vertices. This bound is asymptotically sharp for pumpkin chains with number of edges tending to infinity. This is a joint work with D. Borthwick and L. Corsi.

Variational Problems in Capillarity

Series
Undergraduate Seminar
Time
Monday, September 30, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
John McCuan – Georgia Tech

I will describe a few classical problems in capillarity and the associated classical variational framework.  These problems include the well-known shape and rise height problems for the meniscus in a tube as well as the problems associated with sessile and pendent drops. I will briefly discuss elements of recent modifications of the variational theory allowing floating objects.  Finally, I will describe a few open problems. 

Geometry Topology Seminar : Surface bundles and complex projective varieties by Corey Bregman

Series
Geometry Topology Seminar
Time
Monday, September 30, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Corey Bregman – Brandeis University –

Kodaira, and independently Atiyah, gave the first examples of surface bundles over surfaces whose signature does not vanish, demonstrating that signature need not be multiplicative.  These examples, called Kodaira fibrations, are in fact complex projective surfaces admitting a holomorphic submersion onto a complex curve, whose fibers have nonconstant moduli. After reviewing the Atiyah-Kodaira construction, we consider Kodaira fibrations with nontrivial holomorphic invariants in degree one. When the dimension of the invariants is at most two, we show that the total space admits a branched covering over a product of curves.

The essential variety and degrees of minimal problems

Series
Student Algebraic Geometry Seminar
Time
Monday, September 30, 2019 - 13:15 for 1 hour (actually 50 minutes)
Location
Skiles
Speaker
Tim Duff – GA Tech

It is a fundamental problem in computer vision to describe the geometric relations between two or more cameras that view the same scene -- state of the art methods for 3D reconstruction incorporate these geometric relations in a nontrivial way. At the center of the action is the essential variety: an irreducible subvariety of P^8 of dimension 5 and degree 10 whose homogeneous ideal is minimal generated by 10 cubic equations. Taking a linear slice of complementary dimension corresponds to solving the "minimal problem" of 5 point relative pose estimation. Viewed algebraically, this problem has 20 solutions for generic data: these solutions are elements of the special Euclidean group SE(3) which double cover a generic slice of the essential variety. The structure of these 20 solutions is governed by a somewhat mysterious Galois group (ongoing work with Regan et. al.)

We may ask: what other minimal problems are out there? I'll give an overview of work with Kohn, Pajdla, and Leykin on this question. We have computed the degrees of many minimal problems via computer algebra and numerical methods. I am inviting algebraic geometers at large to attack these problems with "pen and paper" methods: there is still a wide class of problems to be considered, and the more tools we have, the better.

Geometry Topology Seminar Pre-talk: Fundamental groups of projective varieties by Corey Bregman

Series
Geometry Topology Seminar Pre-talk
Time
Monday, September 30, 2019 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Corey Bregman – Brandeis University –

A question going back to Serre asks which groups arise as fundamental groups of smooth, complex projective varieties, or more generally, compact Kaehler manifolds.  The most basic examples of these are surface groups, arising as fundamental groups of 1-dimensional projective varieties.  We will survey known examples and restrictions on such groups and explain the special role surface groups play in their classification. Finally, we connect this circle of ideas to more general questions about surface bundles and mapping class groups. 

Finite element approximation of invariant manifolds by the parameterization method

Series
CDSNS Colloquium
Time
Monday, September 30, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jorge Gonzalez – Florida Atlantic University

We consider the problem of computing unstable manifolds for equilibrium solutions of parabolic PDEs posed on irregular spatial domains. This new approach is based on the parameterization method, a general functional analytic framework for studying invariant manifolds of dynamical systems. The method leads to an infinitesimal invariance equation describing the unstable manifold. A recursive scheme leads to linear homological equations for the jets of the manifold which are solved using the finite element method. One feature of the method is that we recover the dynamics on the manifold in addition to its embedding.  We implement the method for some example problems with polynomial and non-polynomial nonlinearities posed on various non-convex two dimensional domains. We provide numerical support for the accuracy of the computed manifolds using the natural notion of a-posteriori error admitted by the parameterization method. This is joint work with J.D. Mireles-James and Necibe Tuncer. 

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