Seminars and Colloquia by Series

Joint UGA-GT Topology Seminar at UGA: Stein domains in complex two space with prescribed boundary

Series
Geometry Topology Seminar
Time
Monday, March 2, 2020 - 14:30 for 1 hour (actually 50 minutes)
Location
Boyd
Speaker
Bolent Tosun – University of Alabama

A Stein manifold is a complex manifold with particularly nice convexity properties. In real dimensions above 4, the existence of a Stein structure is essentially a homotopical question, but for 4-manifolds the situation is more subtle. In this talk we will consider the existence of such structures in the ambient settings, that is, manifolds/domains with various degree of convexity as open/compact subsets of a complex manifold, e.g. complex 2-space C^2. In particular, I will discuss the following question: Which homology spheres embed in C^2 as the boundary of a Stein domain? This question was first considered and explored in detail by Gompf. At that time, he made a fascinating conjecture that no non-trivial Brieskorn homology sphere, with either orientation, embeds in C^2 as a Stein boundary. In this talk, I will survey what we know about this conjecture, and report on some closely related recent work in progress that ties to an interesting symplectic rigidity phenomena in low dimensions.

Lower Deviations and Convexity

Series
Stochastics Seminar
Time
Thursday, February 27, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Petros Valettas – University of Missouri, Columbia

While deviation estimates above the mean is a very well studied subject in high-dimensional probability, for their lower analogues far less are known. However, it has been observed, in several key situations, that lower deviation inequalities exhibit very different and stronger behavior. In this talk I will discuss how convexity can serve as a key feature to (a) explain this distinction, (b) obtain improved lower tail bounds, and (c) characterize the tightness of Gaussian concentration. 

Introduction to algebraic graph theory

Series
Time
Wednesday, February 26, 2020 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
James Anderson – Georgia Tech

Continuing from Biggs’s Algebraic Graph Theory, we discuss the properties of the Laplacian Matrix of a graph and how it relates to the tree number.

Gaussian methods in randomized Dvoretzky theorem

Series
High Dimensional Seminar
Time
Wednesday, February 26, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Petros Valettas – University of Missouri, Columbia

The cornerstone in local theory of Banach spaces is Dvoretzky’s theorem, which asserts that almost euclidean structure is locally present in any high-dimensional normed space. The random version of this remarkable phenomenon was put forth by V. Milman in 70’s, who employed the concentration of measure on the sphere. Purpose of the talk is to present how Gaussian tools from high-dimensional probability (e.g., Gaussian convexity, hypercontractivity, superconcentration) can be exploited for obtaining optimal results in random forms of Dvoretzky’s theorem. Based on joint work(s) with Grigoris Paouris and Konstantin Tikhomirov.

Cosmetic surgeries on knots in S^3

Series
Geometry Topology Student Seminar
Time
Wednesday, February 26, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hugo Zhou – Georgia Tech

Cosmetic surgeries (purely cosmetic surgeries) are two distinct surgeries on a knot that produce homeomorphic 3-manifolds (as oriented manifolds). It is one of the ways Dehn surgeries on knots could fail to be unique. Gordon conjectured that there are no nontrivial purely cosmetic surgeries on nontrivial knots in S^3. We will recap the progress of the problem over time, and mainly discuss Ni and Wu's results in their paper "Cosmetic surgeries on knots in S^3".

Geometric averaging operators and points configurations

Series
Analysis Seminar
Time
Wednesday, February 26, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Eyvindur Ari Palsson – Virginia Tech –

Two classic questions -- the Erdos distinct distance problem, which asks about the least number of distinct distances determined by N points in the plane, and its continuous analog, the Falconer distance problem -- both focus on the distance, which is a simple two point configuration. When studying the Falconer distance problem, a geometric averaging operator, namely the spherical averaging operator, arises naturally. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as 3-point configurations. In this talk I will give a brief introduction to the motivating point configuration questions and then report on some novel geometric averaging operators and their mapping properties.

Modeling malaria development in mosquitoes: How fast can mosquitoes pass on infection?

Series
Mathematical Biology Seminar
Time
Wednesday, February 26, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lauren Childs – Virginia Tech

The malaria parasite Plasmodium falciparum requires a vertebrate host, such as a human, and a vector host, the Anopheles mosquito, to complete a full life cycle. The portion of the life cycle in the mosquito harbors both the only time of sexual reproduction, expanding genetic complexity, and the most severe bottlenecks experienced, restricting genetic diversity, across the entire parasite life cycle. In previous work, we developed a two-stage stochastic model of parasite diversity within a mosquito, and demonstrated the importance of heterogeneity amongst parasite dynamics across a population of mosquitoes. Here, we focus on the parasite dynamics component to evaluate the first appearance of sporozoites, which is key for determining the time at which mosquitoes first become infectious. We use Bayesian inference techniques with simple models of within-mosquito parasite dynamics coupled with experimental data to estimate a posterior distribution of parameters. We determine that growth rate and the bursting function are key to the timing of first infectiousness, a key epidemiological parameter.

Existence and uniqueness to a fully non-linear version of the Loewner-Nirenberg problem

Series
PDE Seminar
Time
Tuesday, February 25, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yanyan Li – Rutgers University –

We consider the problem of finding on a given bounded and smooth
Euclidean domain \Omega of dimension n ≥ 3 a complete conformally flat metric whose Schouten
curvature A satisfies some equation of the form  f(\lambda(-A)) =1. This generalizes a problem
considered by Loewner and Nirenberg for the scalar curvature. We prove the existence and uniqueness of
locally Lipschitz solutions. We also show that the Lipschitz regularity is in general optimal.

Hankel index of a projection of rational normal curve.

Series
Student Algebraic Geometry Seminar
Time
Monday, February 24, 2020 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jaewoo Jung – Georgia Tech –

The dual of the cone of non-negative quadratics (on a variety) is included in the dual of the cone of sums of squares. Moreover, all (points which span) extreme rays of the dual cone of non-negative quadratics is point evaluations on real points of the variety. Therefore, we are interested in extreme rays of the dual cone of sums of squares which do not come from point evaluations. The dual cone of sums of squares on a variety is called the Hankel spectrahetron and the smallest rank of extreme rays which do not come from point evaluations is called Hankel index of the variety. In this talk, I will introduce some algebraic (or geometric) properties which control the Hankel index of varieties and compute the Hankel index of rational curves obtained by projecting rational normal curve away from a point (which has almost minimal degree).

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