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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.

Series: School of Mathematics Colloquium

Although the genetic information in each cell within an organism is
identical, gene expression varies widely between different cell types. The
quest to understand this phenomenon has led to many interesting mathematics
problems. First, I will present a new method for learning gene regulatory
networks. It overcomes the limitations of existing algorithms for learning
directed graphs and is based on algebraic, geometric and combinatorial
arguments. Second, I will analyze the hypothesis that the differential gene
expression is related to the spatial organization of chromosomes. I will
describe a bi-level optimization formulation to find minimal overlap
configurations of ellipsoids and model chromosome arrangements. Analyzing
the resulting ellipsoid configurations has important implications for the
reprogramming of cells during development. Any knowledge of biology which
is needed for the talk will be introduced during the lecture.

Series: School of Mathematics Colloquium

How well can a convex body be approximated by a polytope? This is a fundamental question in convex geometry, also in view of applications in many other areas of mathematics and related fields. It often involves side conditions like a prescribed number of vertices, or, more generally, k-dimensional faces and a requirement that the body contains the polytope or vice versa. Accuracy of approximation is often measured in the symmetric difference metric, but other metrics can and have been considered. We will present several results about these issues, mostly related to approximation by “random polytopes”.

Series: School of Mathematics Colloquium

Though the modern analytic celestial mechanics has been existing for more
than 300 years since Newton, there are still many basic questions
unanswered, for instance, there is still no rigorous mathematical proof
explaining why our solar system has been stable for such a long time (five
billion years) hence no guarantee that it would remain stable for the next
five billion years. Instead, it is known that there are various instability
behaviors in the Newtonian N-body problem.
In this talk, we mention three types instability behaviors in Newtonian
N-body problem. The first type we will talk about is simply chaotic
motions, which include for instance the oscillatory motions, in which case,
one body travels back and forth between neighborhoods of zero and infinity.
The second type is “organized” chaotic motions, also known as Arnold
diffusion or weak turbulence. Finally, we will talk about our work on the
existence of the most wild unstable behavior, non collision singularities,
also called finite time blow up solution.
The talk is mostly expository. Zero background on celestial mechanism or
dynamical systems is needed to follow the lecture.

Series: School of Mathematics Colloquium

Approximate separable representation of the Green’s functions for
differential operators is a fundamental question in the analysis of
differential equations and development of efficient numerical
algorithms. It can reveal intrinsic complexity, e.g., Kolmogorov n-width
or degrees of freedom of the corresponding differential
equation. Computationally, being able to approximate a Green’s function
as a sum with few separable terms is equivalent to the existence of low
rank approximation of the discretized system which can be explored for
matrix compression and fast solution techniques such as in fast multiple
method and direct matrix inverse solver. In this talk, we will mainly
focus on Helmholtz equation in the high frequency limit for which we
developed a new approach to study the approximate separability of
Green’s function based on an geometric characterization of the relation
between two Green's functions and a tight dimension estimate for the
best linear subspace approximating a set of almost orthogonal vectors.
We derive both lower bounds and upper bounds and show their sharpness
and implications for computation setups that are commonly used
in practice. We will also make comparisons with other types of
differential operators such as coercive elliptic differential operator
with rough coefficients in divergence form and hyperbolic differential
operator. This is a joint work with Bjorn Engquist.