Seminars and Colloquia by Series

Monday, February 22, 2016 - 16:00 , Location: Skiles 005 , Michael Loss , School of Mathematics, Georgia Tech , Organizer:
The Caffarelli-Kohn-Nirenberg inequalities form a two parameter family of inequalities that interpolate between Sobolev's inequality and Hardy's inequality. The functional whose minimization yields the sharp constant is invariant under rotations. It has been known for some time that there is a region in parameter space where the optimizers for the sharp constant are {\it not} radial. In this talk I explain this and related problems andindicate a proof that, in the remaining parameter region, the optimizers are in fact radial. The novelty is the use of a flow that decreases the functional unless the function is a radial optimizer. This is joint work with Jean Dolbeault and Maria Esteban.
Thursday, February 11, 2016 - 16:05 , Location: Skiles 005 , Rachel Kuske , University of British Columbia , Organizer: Christine Heitsch
There have been many recent advances for analyzing the complex deterministic behavior of systems with discontinuous dynamics. With the identification of new types of nonlinear phenomena exploding in this realm, one gets the feeling that almost anything can happen. There are many open questions about noise-driven and noise-sensitive phenomena in the non-smooth context, including the observation that noise can facilitate or select "regular" dynamics, thus clarifying the picture within the seemingly endless sea of possibilities. Familiar concepts from smooth systems such as escapes, resonances, and bifurcations appear in unexpected forms, and we gain intuition from seemingly unrelated canonical models of biophysics, mechanics, finance, and climate dynamics. The appropriate strategy is often not immediately obvious from the area of application or model type, requiring an integration of multiple scales techniques, probabilistic models, and nonlinear methods.
Tuesday, February 9, 2016 - 15:30 , Location: Skiles 005 , Patrick Gerard , Université Paris-Sud , patrick.gerard@math.u-psud.fr , Organizer: Michael Damron
In the world of Hamiltonian partial differential equations, complete integrability is often associated to rare and peaceful dynamics, while wave turbulence rather refers to more chaotic dynamics. In this talk I will first try to give an idea of these different notions. Then I will  discuss the example of the cubic Szegö equation, a nonlinear wave toy model which surprisingly displays both properties. The key is a Lax pair structure involving Hankel operators from classical analysis, leading to the inversion of large ill-conditioned matrices. .
Monday, February 8, 2016 - 16:00 , Location: Skiles 005 , Jesus De Loera , University of California, Davis , Organizer: Greg Blekherman
Convex analysis and geometry are tools fundamental to the foundations of several applied areas (e.g., optimization, control theory, probability and statistics),  but at the same time convexity intersects in lovely ways with topics considered pure (e.g., algebraic geometry, representation theory and of course number theory). For several years I have been interested interested on how convexity relates to lattices and discrete subsets of Euclidean space. This is part of mathematics  H. Minkowski named in 1910 "Geometrie der Zahlen''.  In this talk I will use two well-known results, Caratheodory's & Helly's theorems, to explain my most recent work on lattice points on convex sets. The talk is for everyone! It is designed for non-experts and grad students should understand the key ideas. All new theorems are joint work with subsets of the following mathematicians I. Aliev, C. O'Neill, R. La Haye, D. Rolnick, and P. Soberon.
Thursday, February 4, 2016 - 11:00 , Location: Skiles 006 , Prof. Dr. Adilson E. Motter , Northwestern University , motter@northwestern.edu , Organizer: Molei Tao
The recent interest in network modeling has been largely driven by the prospect that network optimization will help us understand the workings of evolution in natural systems and the principles of efficient design in engineered systems. In this presentation, I will reflect on unanticipated properties observed in three classes of network optimization problems. First, I will discuss implications of optimization for the metabolic activity of living cells and its role in giving rise to the recently discovered phenomenon of synthetic rescues. I will then comment on the problem of controlling network dynamics and show that theoretical results on optimizing the number of driver nodes often only offer a conservative lower bound to the number actually needed in practice. Finally, I will discuss the sensitive dependence of network dynamics on network structure that emerges in the optimization of network topology for dynamical processes governed by eigenvalue spectra, such as synchronization and consensus processes. It follows that optimization is a double-edged sword for which desired and adverse effects can be exacerbated in network systems due to the high dimensionality of their phase spaces.
Thursday, January 28, 2016 - 11:05 , Location: Skiles 006 , Leonid Bunimovich , Georgia Institute of Technology , leonid.bunimovich@math.gatech.edu , Organizer: Michael Damron
Hepatitis C virus (HCV) has the propensity to cause chronic infection. HCV affects an estimated 170 million people worldwide. Immune escape by continuous genetic diversification is commonly described using a metaphor of "arm race" between virus and host. We developed a mathematical model that explained all clinical observations which could not be explained by the "arm race theory". The model applied to network of cross-immunoreactivity suggests antigenic cooperation as a mechanism of mitigating the immune pressure on HCV variants. Cross-immunoreactivity was observed for dengue, influenza, etc.   Therefore antigenic cooperation is a new target for therapeutic- and vaccine- development strategies.  Joint work with P.Skums and Yu. Khudyakov (CDC).  Our model is in a sense simpler than old one. In the speaker's opinion it is a good example to discuss what Math./Theor. Biology is and what it should be. Such (short) discussion is expected. NO KNOWLEDGE of Biology is expected to understand this talk.
Thursday, January 14, 2016 - 11:00 , Location: Skiles 006 , Eric Katz , University of Waterloo , eekatz@uwaterloo.ca , Organizer: Michael Damron
We discuss applications of Hodge theory which is a part of algebraic geometry to problems in combinatorics, in particular to Rota's Log-concavity Conjecture.  The conjecture was motivated by a question in enumerating proper colorings of a graph which are counted by the chromatic polynomial.  This polynomial's coefficients were conjectured to form a unimodal sequence by Read in 1968.  This conjecture was extended by Rota in his 1970 ICM address to assert the log-concavity of the characteristic polynomial of matroids which are the common combinatorial generalizations of graphs and linear subspaces.  We discuss the resolution of this conjecture which is joint work with Karim Adiprasito and June Huh. The solution draws on ideas from the theory of algebraic varieties, specifically Hodge theory, showing how a question about graph theory leads to a solution involving Grothendieck's standard conjectures.  This talk is a preview for the upcoming workshop at Georgia Tech.
Friday, December 4, 2015 - 16:00 , Location: Skiles 006 , Karen Vogtmann , University of Warwick , Organizer: John Etnyre

Kick-off of the&nbsp;<a href="http://ttc.gatech.edu/">Tech Topology Conference</a>, December 4-6, 2015

Finite metric graphs are used to describe many phenomena in mathematics and science, so we would like to understand the space of all such graphs, which is called the moduli space of graphs.  This space is stratified by subspaces consisting of graphs with a fixed number of loops and leaves.  These strata generally have complicated structure that is not at all well understood.  For example, Euler characteristic calculations indicate a huge number of nontrivial homology classes, but only a very few have actually been found.   I will discuss the structure of these moduli spaces, including recent progress on the hunt for homology based on  joint work with Jim Conant, Allen Hatcher and Martin Kassabov.
Thursday, November 19, 2015 - 11:00 , Location: Skiles 006 , Jesse Johnson , Google , Organizer: John Etnyre
Topological data analysis is the study of Machine Learning/Data Mining problems using techniques from geometry and topology. In this talk, I will discuss how the scale of modern data analysis has made the geometric/topological perspective particularly relevant for these subjects. I'll then introduce an approach to the clustering problem inspired by a tool from knot theory called thin position.
Thursday, November 12, 2015 - 11:00 , Location: Skiles 006 , Prof. Dr. Matthew Kahle , Ohio State University , mkahle@math.osu.edu , Organizer: Molei Tao
The study of random topological spaces: manifolds, simplicial complexes, knots, and groups, has received a lot of attention in recent years. This talk will focus on random simplicial complexes, and especially on a certain kind of topological phase transition, where the probability that that a certain homology group is trivial passes from 0 to 1 within a narrow window. The archetypal result in this area is the Erdős–Rényi theorem, which characterizes the threshold edge probability where the random graph becomes connected. One recent breakthrough has been in the application of Garland’s method, which allows one to prove homology-vanishing theorems by showing that certain Laplacians have large spectral gaps. This reduces problems in random topology to understanding eigenvalues of certain random matrices, and the method has been surprisingly successful. This is joint work with Christopher Hoffman and Elliot Paquette.

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