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Series: School of Mathematics Colloquium

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.

Series: School of Mathematics Colloquium

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.

Series: School of Mathematics Colloquium

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. .

Series: School of Mathematics Colloquium

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.

Series: School of Mathematics Colloquium

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.

Series: School of Mathematics Colloquium

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.

Series: School of Mathematics Colloquium

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.

Series: School of Mathematics Colloquium

Kick-off of the <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.

Series: School of Mathematics Colloquium

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.

Series: School of Mathematics Colloquium

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.