The process of converting analog signals, or continuous functions, to digital signals is a classical problem in signal processing. Many analog to digital converters follow the paradigm of taking lots of samples and then compressing afterwards. One might wonder if you could be more prudent and only take as many samples as you'd need to guarantee that you can reconstruct a given signal. That is, can you compress while simultaneously measuring a signal? Compressed sensing's hallmark achievement is proving that for a large class of structured signals this is indeed possible.
In this talk, I will introduce compressed sensing by defining a mathematical model for signal acquisition and outline procedures that guarantee signal reconstruction. There will be lots of pictures, and lots of solving $Ax=b$

We will discuss a 1995 paper of Koll\'ar, which showed via degeneration to characteristic 2 the nonrationality of many low degree hypersurfaces in projective space.

Let $G$ be a finite group. The faithful dimension of $G$ is defined to be the smallest possible dimension for a faithful complex representation of $G$. Aside from its intrinsic interest, the problem of determining the faithful dimension of finite groups is intimately related to the notion of essential dimension, introduced by Buhler and Reichstein.

In this paper, we will address this problem for groups parameterized by a prime parameter $p$ (e.g., Heisenberg groups over finite fields with $p$) and study the question of the dependence of the essential dimension on $p$. As it will be shown, in general, this is always a piecewise polynomial function along certain ``number-theoretically defined'' sets, while, in some specific cases, it is given by a uniform polynomial in $p$.

This talk is based on a joint work with Mohammad Bardestani and Hadi Salmasian.

I will discuss a recent line of research that uses properties of real rooted polynomials to get quantitative estimates in combinatorial linear algebra problems. I will start by discussing the main result that bridges the two areas (the ``method of interlacing polynomials'') and show some examples of where it has been used successfully (e.g. Ramanujan families and the Kadison Singer problem). I will then discuss some more recent work that attempts to make the method more accessible by providing generic tools and also attempts to explain the accuracy of the method by linking it to random matrix theory and (in particular) free probability. I will end by mentioning some current research initiatives as well as possible future directions.

We study the 2D Euler equations linearized around smooth, radially symmetric vortices with strictly decreasing vorticity profiles. Under a trivial orthogonality condition, we prove that the perturbation vorticity winds up around the vortex and weakly converges to a radially symmetric configuration, as time goes to infinity. This process is known as ``vortex axisymmetrization'' in the physics literature and is thought to stabilize vortex structures such as hurricanes and cyclones. Additionally, the velocity field converges strongly in L2 to the corresponding equilibrium (as time goes to infinity) and we give optimal decay rates in weighted L2 spaces. Interestingly, the rate of decay is faster for the linearized 2D Euler equations than for the passive scalar equation. The passive scalar rate is degraded by the slow mixing at the vortex core, but the linearized 2D Euler equations expel vorticity from the origin leading to a faster decay rate.

What is a random graph and what do random graphs look like? We will discuss some basic properties, mathematical methods and applications of random graphs.

In this talk, we introduce the LRHB algorithm, which is an extension of the reduced-Hessian method of Gill and Leonard for unconstrained problems to problems with simple bound constraints. Numerical results for LRHB will be presented. We will also consider computational and practical issues with methods for nonlinear optimization and present results on a large test collection of problems indicating the reliability and efficiency of sequential quadratic programming methods and interior-point methods on certain classes of problems. This is joint work with Michael Ferry and Philip E. Gill.

Deninger and Werner developed an analogue for $p$-adic curves of the
classical correspondence of Narasimhan and Seshadri between stable
bundles of degree zero and unitary representations of the topological
fundamental group for a complex smooth proper curve. Using parallel
transport, they associated functorially to every vector bundle on a
$p$-adic curve whose reduction is strongly semi-stable of degree 0 a
$p$-adic representation of the \'etale fundamental group of the curve. They
asked several questions: whether their functor is fully faithful and
what is its essential image; whether the cohomology of the local systems
produced by this functor admits a Hodge-Tate filtration; and whether
their construction is compatible with the $p$-adic Simpson correspondence
developed by Faltings. We will answer these questions in this talk.

Gene conversion is a ubiquitous phenomenon that leads to the exchange of genetic information between homologous DNA regions and maintains co-evolving multi-gene families in most pro- and eukaryotic organisms. In this talk, I will consider its implications for the evolution of a single functional gene with a silenced duplicate, using two different mathematical models of evolution on rugged fitness landscapes. Our analytical and numerical results show that, by helping to circumvent valleys of low fitness, gene conversion with an inactive duplicate gene can cause a significant speedup of adaptation which depends non-trivially on the frequency of gene conversion and the structure of the landscape. Stochastic effects due to finite population sizes further increase the likeliness of exploiting this evolutionary pathway. Our results reveal the potential for duplicate genes to act as a ``scratch paper'' that frees evolution from being limited to strictly beneficial mutations in strongly selective environments.