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Series: School of Mathematics Colloquium

Unitary representations of Lie groups appear in many guises in
mathematics: in harmonic analysis (as generalizations of classical
Fourier analysis); in number theory (as spaces of modular and
automorphic forms); in quantum mechanics (as "quantizations" of
classical mechanical systems); and in many other places. They have
been the subject of intense study for decades, but their
classification has only recently emerged. Perhaps
surprisingly, the classification has inspired connections with
interesting geometric objects (equivariant mixed Hodge modules on
flag varieties). These connections have made it possible to extend
the classification scheme to other related settings.
The purpose of this talk is to
explain a little bit about the history
and motivation behind the study of unitary representations and offer
a few hints about the algebraic and geometric ideas which enter into
their study. This is based on joint work with Adams, van Leeuwen,
and Vogan.

Series: School of Mathematics Colloquium

While the fields named in the title seem unrelated, there is a strong
link between them. This amazing connection came to life during a meeting
between Freeman Dyson and Hugh Montgomery at the Institute for Advanced
Study. Random matrices are now known to predict many number theoretical
statistics, such as moments, low-lying zeros and correlations between
zeros. The goal of this talk is to discuss this connection, focusing on
number theory. We will cover both basic facts about the zeta functions
and recent developments in this active area of research.

Series: School of Mathematics Colloquium

One of the simplest and most natural ways of representing geometry and information in three and higher dimensions is using point clouds, such as scanned 3D points for shape modeling and feature vectors viewed as points embedded in high dimensions for general data analysis. Geometric understanding and analysis of point cloud data poses many challenges since they are unstructured, for which a global mesh or parametrization is difficult if not impossible to obtain in practice. Moreover, the embedding is highly non-unique due to rigid and non-rigid transformations.
In this talk, I will present some of our recent work on geometric understanding and analysis of point cloud data. I will first discuss a multi-scale method for non-rigid point cloud registration based on the Laplace-Beltrami eigenmap and optimal transport. The registration is defined in distribution sense which provides both generality and flexibility. If time permits I will also discuss solving geometric partial differential equations directly on point clouds and show how it can be used to “connect the dots” to extract intrinsic geometric information for the underlying manifold.

Series: School of Mathematics Colloquium

From theoretical to applied, we present curiosity driven research which
goes beyond classical dynamical systems theory and (i) extend the notion
of chaos to actions of topological semigroups, (ii) model how the human
bone renews, (iii) study transient dynamics as it occurs e.g. in
oceanography, (iv) understand how to protect houses from hurricane
damage. The talk introduces concepts from topological dynamics,
mathematical biology, entropy theory and mechanics.

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.