Seminars and Colloquia by Series

Mixing and Explosions for the Generalized Recurrent Set

Series
CDSNS Colloquium
Time
Monday, October 21, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skyles 006
Speaker
Jim WisemanAgnes Scott

We consider the strong chain recurrent set and the generalized recurrent set for continuous maps of compact metric spaces.  Recent work by Fathi and Pageault has shown a connection between the two sets, and has led to new results on them.  We discuss a structure theorem for transitive/mixing maps, and classify maps that permit explosions in the size of the recurrent sets.

Surfaces: BIG and small

Series
Time
Monday, October 21, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Dr. Marissa LovingGeorgia Tech

As a geometric group theorist, my favorite type of manifold is a surface and my favorite way to study surfaces is by considering the mapping class group, which is the collection of symmetries of a surface. In this talk, we will think bigger than your average low-dimensional topologist and consider surfaces of infinite type and their associated “big” mapping class groups.

The geometry of subgroup combination theorems

Series
Geometry Topology Seminar
Time
Monday, October 21, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker

While producing subgroups of a group by specifying generators is easy, understanding  the structure of such a subgroup is notoriously difficult problem.  In the case of hyperbolic groups, Gitik utilized a local-to-global property for geodesics to produce an elegant condition that ensures a subgroup generated by two elements (or more generally generated by two subgroups) will split as an amalgamated free product over the intersection of the generators. We show that the mapping class group of a surface and many other important groups have a similar local-to-global property from which an analogy of Gitik's result can be obtained.   In the case of the mapping class group, this produces a combination theorem for the dynamically and topologically important convex cocompact subgroups.  Joint work with Davide Spriano and Hung C. Tran.

Tropical convex hulls of convex sets

Series
Student Algebraic Geometry Seminar
Time
Monday, October 21, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Cvetelina HillGeorgia Tech

This talk is based on work in progress with Sara Lamboglia and Faye Simon. We study the tropical convex hull of convex sets and of tropical curves. Basic definitions of tropical convexity and tropical curves will be presented, followed by an overview of our results on the interaction between tropical and classical convexity. Lastly, we study a tropical analogue of an inequality bounding the degree of a projective variety in classical algebraic geometry; we show a tropical proof of this result for a special class of tropical curves.

Groups as geometric objects

Series
Geometry Topology Seminar Pre-talk
Time
Monday, October 21, 2019 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker

Gromov revolutionized the study of finitely generated groups by showing that an intrinsic metric on a group is intimately connected with the algebra of the group. This point of view has produced deep applications not only in group theory, but also topology, geometry, logic, and dynamical systems. We will start at the beginning of this story with the definitions of these metrics on groups and how notions from classical geometry can be generalized to this context.  The focus will be on how the "hyperbolic groups" exhibit geometric and dynamical feature reminiscent of the hyperbolic plane and its isometries.

New mechanisms of instability in Hamiltonian systems

Series
CDSNS Colloquium
Time
Monday, October 21, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 06
Speaker
Tere M. SearaUniv. Politec. de Catalunya

In this talk we present some recent results which allow to prove
instability in near integrable Hamiltonian systems. We will show how
these mechanisms are suitable to apply to concrete systems but also are
useful to obtain Arnold diffusion in a large set  of Hamiltonian systems.

Twisted Schubert polynomials

Series
Combinatorics Seminar
Time
Friday, October 18, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ricky LiuNorth Carolina State University

We will describe a twisted action of the symmetric group on the polynomial ring in n variables and use it to define a twisted version of Schubert polynomials. These twisted Schubert polynomials are known to be related to the Chern-Schwartz-MacPherson classes of Schubert cells in the flag variety. Using properties of skew divided difference operators, we will show that these twisted Schubert polynomials are monomial positive and give a combinatorial formula for their coefficients.

Oral Exam-Bounds on regularity of quadratic monomial ideals and Pythagoras numbers on projections of Rational Normal Curves

Series
Other Talks
Time
Friday, October 18, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jaewoo JungGeorgia Tech

In this talk, I will introduce my old(1.) and current works(2.).

1. Bounds on regularity of quadratic monomial ideals

We can understand invariants of monomial ideals by invariants of clique (or flag) complex of  corresponding graphs. In particular, we can bound the Castelnuovo-Mumford regularity (which is a measure of algebraic complexity) of the ideals by bounding homol0gy of corresponding (simplicial) complex. The construction and proof of our main theorem are simple, but it provides (and improves) many new bounds of regularities of quadratic monomial ideals.

2. Pythagoras numbers on projections of Rational Normal Curves

Observe that forms of degree $2d$ are quadratic forms of degree $d$. Therefore, to study the cone of  sums of squares of degree $2d$, we may study quadratic forms on Veronese embedding of degree $d$.  In particular,  the rank of sums of squares (of degree $2d$) can be studied via Pythagoras number  (which is a classical notion) on the Veronese embedding of degree $d$. In this part, I will compute the Pythagoras number on rational normal curve (which is a veronese embedding of $\mathbb{P}^1$) and discuss about how Pythagoras numbers are changed when we take some projections away from some points.

On the breakdown of small amplitude breathers for the reversible Klein-Gordon equation

Series
Time
Friday, October 18, 2019 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 06
Speaker
Marcel GuardiaUniv. Politec. de Catalunya

Breathers are periodic in time spatially localized solutions of evolutionary PDEs. They are known to exist for the sine-Gordon equation but are believed to be rare in other Klein-Gordon equations. Exchanging the roles of time and position, breathers can be interpreted as homoclinic solutions to a steady solution. In this talk, I will explain how to obtain an asymptotic formula for the distance between the stable and unstable manifold of the steady solution when the steady solution has weakly hyperbolic one dimensional stable and unstable manifolds. Their distance is exponentially small with respect to the amplitude of the breather and therefore classical perturbative techniques cannot be applied. This is a joint work with O. Gomide, T. Seara and C. Zeng.

On the circumference of essentially 4-connected planar graphs

Series
Time
Thursday, October 17, 2019 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael WigalGeorgia Tech
Carsten Thomassen showed for planar graphs $G$ that there exists a cycle $C$ such that every component of $G - C$ has at most three neighbors on C. This implies that 4-connected planar graphs are hamiltonian. A natural weakening is to find the circumference of essentially 4-connected planar graphs. We will cover an outline of Thomassen's proof and what is currently known on circumference bounds for essentially 4-connected planar graphs.