Seminars and Colloquia by Series

Tropical covers with an abelian group action

Series
Algebra Seminar
Time
Tuesday, October 29, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dmitry Zakharov – Central Michigan University –

Given a graph X and a group G, a G-cover of X is a morphism of graphs X’ --> X together with an invariant G-action on X’ that acts freely and transitively on the fibers. G-covers are classified by their monodromy representations, and if G is a finite abelian group, then the set of G-covers of X is in natural bijection with the first simplicial cohomology group H1(X,G).

In tropical geometry, we are naturally led to consider more general objects: morphisms of graphs X’ --> X admitting an invariant G-action on X’, such that the induced action on the fibers is transitive, but not necessarily free. A natural question is to classify all such covers of a given graph X. I will show that when G is a finite abelian group, a G-cover of a graph X is naturally determined by two data: a stratification S of X by subgroups of G, and an element of a cohomology group H1(X,S) generalizing the simplicial cohomology group H1(X,G). This classification can be viewed as a tropical version of geometric class field theory, and as an abelianization of Bass--Serre theory.

I will discuss the realizability problem for tropical abelian covers, and the relationship between cyclic covers of a tropical curve C and the corresponding torsion subgroup of Jac(C). The realizability problem for cyclic covers of prime degree turns out to be related to the classical nowhere-zero flow problem in graph theory.

Joint work with Yoav Len and Martin Ulirsch.

Quantum fate of classical solitons

Series
Math Physics Seminar
Time
Monday, October 28, 2019 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael Pustilnik – School of Physics, Georgia Tech –
This talk will focus on one-dimensional interacting quantum systems near the classical limit described by the Korteweg–de Vries (KdV) equation. Classical excitations in this regime are the well-known solitons, i.e., localized disturbances with particle-like properties, and delocalized waves of density, or phonons. It turns out, however, that the semiclassical description inevitably breaks down at long wavelengths. In this limit, quantum effects become dominant, the system is best described in terms of weakly interacting fermions, and classical solitons and phonons reach their ultimate quantum fate of being demoted to fermionic particles and holes.
 
We will give simple heuristic arguments in support of this claim and present the exact solution for the spectra of elementary excitations. The results are universally applicable to all quantum one-dimensional systems with a well-defined classical limit described by the KdV equation. This includes identical bosons with a weak short-range repulsion and identical particles, either bosons or fermions, with a strong long-range repulsion.

Knots, Legendrian Knots, and Their Invariants

Series
Undergraduate Seminar
Time
Monday, October 28, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Dr. Caitlin Leverson – Georgia Tech
A knot can be thought of as a piece of string tied up, that then has its ends glued together. As long as we don’t cut the string, any way we move the string in space doesn’t change the knot we are considering. A surprisingly hard and interesting problem is, when handed two knots, how to determine if they are the same knot or not. We can further give structure to our knots and thus the problem, by adding geometric constraints to our knots, yielding what are called Legendrian knots. We can once again try to determine if two Legendrian knots are the same or not. In this talk I will introduce knots, Legendrian knots, and some ways we have to try to distinguish two knots or Legendrian knots, called knot invariants.

Connected Floer homology of covering involutions

Series
Geometry Topology Seminar
Time
Monday, October 28, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skile 006
Speaker
Sungkyung Kang – Chinese University of Hong Kong

Using the covering involution on the double branched cover of S3 branched along a knot, and adapting ideas of Hendricks-Manolescu and Hendricks-Hom-Lidman, we define new knot (concordance) invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots. This is a joint work with A. Alfieri and A. Stipsicz.

Analysis and Applications of Nonsmooth Bifurcations

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 28, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 05
Speaker
Oleg Makarenkov – Univ Texas at Dallas –
In this talk I will first give a brief overview of how nonsmooth bifurcations (border-splitting, grazing, and fold-fold bifurcations) help to rigorously explain the existence of nonsmooth limit cycles in the models of anti-lock braking systems, power converters, integrate-and-fire neurons, and climate dynamics. I will then focus on one particular application that deals with nonsmooth bifurcations in dispersing billiards. In [Nonlinearity 11 (1998)] Turaev and Rom-Kedar discovered that every periodic orbit that is tangent to the boundary of the billiard produces an island of stability upon smoothening the boundary of the billiard. The result to be presented in the talk (joint work with Turaev) proves that any dispersing billiard admits such an arbitrary small perturbation that ensures the occurrence of a tangent periodic orbit.

Nonnegative symmetric polynomials and sums of squares with many variables

Series
Student Algebraic Geometry Seminar
Time
Monday, October 28, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Jose Gabriel Acevedo Habeych – Georgia Tech –

Please Note: By using the representation theory of the symmetric group we try to compare, with respect to two different bases of the vector space of symmetric forms, the cones of symmetric nonnegative forms and symmetric sums of squares of a fixed even degree when the number of variables goes to infinity.

Heegaard Floer homology and Seifert manifolds

Series
Geometry Topology Seminar Pre-talk
Time
Monday, October 28, 2019 - 12:45 for 1 hour (actually 50 minutes)
Location
Skile 006
Speaker
Sungkyung Kang – Chinese University of Hong Kong

Heegaard Floer homology gives a powerful invariant of closed 3-manifolds. It is always computable in the purely combinatorial sense, but usually computing it needs a very hard work. However, for certain graph 3-manifolds, its minus-flavored Heegaard Floer homology can be easily computed in terms of lattice homology, due to Nemethi. I plan to give the basic definitions and constructions of Heegaard Floer theory and lattice homology, as well as the isomorphism between those two objects.

Effective bounds for the measure of rotations

Series
CDSNS Colloquium
Time
Monday, October 28, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 05
Speaker
Alex Haro – Univ. de Barcelona

A fundamental question in Dynamical Systems is to identify regions of
phase/parameter space satisfying a given property (stability,
linearization, etc).  In this talk, given a family of analytic circle
diffeomorphisms depending on a parameter, we obtain effective (almost
optimal) lower bounds of the Lebesgue measure of the set of parameters
for which that diffeomorphism is conjugate to a rigid rotation.
We estimate this measure using an a-posteriori KAM
scheme that relies on quantitative conditions that
are checkable using computer-assistance. We carefully describe
how the hypotheses in our theorems are reduced to a finite number of
computations, and apply our methodology to the case of the
Arnold family, in the far-from-integrable regime.

This is joint work with Jordi Lluis Figueras and Alejandro Luque.
 

Spin Dynamics: Algorithms and Spin of Planets

Series
SIAM Student Seminar
Time
Friday, October 25, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Renyi Chen – GT Math

In this talk, we will focus on the spin dynamics of rigid bodies.
Algorithm part: There are many algorithms designed for N body simulations. 
But, to study the climates of a planet, we need to extend the simulation from point mass bodies to rigid bodies.
In the N-rigid-body simulations, we will consider the orientation and angular momentum of the rigid body to understand the spin.
In terms of the algorithm, symplectic integrators are designed by splitting methods. 
Physical part: We studied the spin dynamics of an Earth-like planet in circumbinary systems.
Canonical Delaunay variables and Andoyer variables are applied to split the variables to be slow part and fast part.
Applying averaging method, we approximated the spin dynamics.
From the approximated dynamics, we may draw some interesting physical conclusions.
 

The proxy point method for rank-structured matrices

Series
Dissertation Defense
Time
Friday, October 25, 2019 - 13:30 for 1.5 hours (actually 80 minutes)
Location
Skiles 311
Speaker
Xin Xing – School of Mathematics, Georgia Tech –

Rank-structured matrix representations, e.g., H2 and HSS, are commonly used to reduce computation and storage cost for dense matrices defined by interactions between many bodies. The main bottleneck for their applications is the expensive computation required to represent a matrix in a rank-structured matrix format which involves compressing specific matrix blocks into low-rank form.
We focus on the study and application of a class of hybrid analytic-algebraic compression methods, called the proxy point method. We address several critical problems concerning this underutilized method which limit its applicability. A general form of the method is proposed, paving the way for its wider application in the construction of different rank-structured matrices with kernel functions that are more general than those usually used. Further, we extend the applicability of the proxy point method to compress matrices defined by electron repulsion integrals, which accelerates one of the main computational steps in quantum chemistry. 

Committee members: Prof. Edmond Chow (Advisor, School of CSE, Georgia Tech), Prof. David Sherrill (School of Chemistry and Biochemistry, Georgia Tech), Prof. Jianlin Xia (Department of Mathematics, Purdue University), Prof. Yuanzhe Xi (Department of Mathematics, Emory University), and Prof. Haomin Zhou (School of Mathematics, Georgia Tech).

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