I will present several results (that have been obtained jointly with Stefanos Aretakis and Dejan Gajic) from the analysis of solutions of linear and nonlinear wave equations on extremal Reissner-Nordstrom spacetimes, including sharp asymptotics on the horizon and at infinity for linear waves, and instability phenomena for nonlinear waves. These results can be seen as stepping stones to the fully nonlinear problem of stability/instability of extremal black holes.

There is a rich connection between homogeneous dynamics and number theory, especially when dynamical results are effective (i.e. when rates of convergence for dynamical phenomena are known). In this final defense, I describe my research on the asymptotic distribution of almost-prime times in horospherical flows on the space of lattices, as well as on compact quotients of SL(n,R). In the compact setting, I obtain a result that implies density for almost-primes in horospherical flows, where the number of prime factors is independent of the basepoint, and in the space of lattices I show the density of almost-primes in abelian horospherical orbits of points satisfying a certain Diophantine condition. To prove this, I first give an effective equidistribution result for arbitrary horospherical flows on the space of lattices, which I then use to prove an effective rate for the equidistribution of arithmetic progressions in abelian horospherical flows, to which I then apply a combinatorial sieve.

Motivated by the study of mirror symmetry, Strominger-Yau-Zaslow (SYZ) conjectured that Calabi-Yau manifolds admit certain minimal Lagrangian fibrations. These minimal Lagrangians are the special Lagrangian submanifolds studied earlier by Harvey-Lawson. Many of the implication of the SYZ conjecture is proved and it has been the guiding principle for studying mirror symmetry for a long time. However, not many special Lagrangians are known in the literature. In this talk, I will prove the existence of special Lagrangian fibration on the complement of a smooth anti-canonical divisor in a (weak) Del Pezzo surface. If the time allows, I will explain its impact to mirror symmetry. This is joint work with Tristan Collins and Adam Jacob.

A challenging problem in the brain imaging research is a principled incorporation of information from different imaging modalities in regression models. Frequently, data from each modality is analyzed separately using, for instance, dimensionality reduction techniques, which result in a loss of information. We propose a novel regularization method, griPEER (generalized ridgified Partially Empirical Eigenvectors for Regression) to estimate the association between the brain structure features and a scalar outcome within the generalized linear regression framework. griPEER provides a principled approach to use external information from the structural brain connectivity to improve the regression coefficient estimation. Our proposal incorporates a penalty term, derived from the structural connectivity Laplacian matrix, in the penalized generalized linear regression. We address both theoretical and computational issues and show that our method is robust to the incomplete structural brain connectivity information. griPEER is evaluated via extensive simulation studies and it is applied in classification of the HIV+ and HIV- individuals.

The talk concerns inequalities on functions of matrix variables, in particular a convex set C of matrices defined by them. The functions are typically (noncommutative) polynomials or rational functions and the sets include matrices of all sizes, hence are dimension free
convex sets.

Extreme points are getting to be understood. But optimizing a linear functional leads in our experiments to surprising properties which are unexplained. The talk describes the sets, the extreme points and the experimental findings.

Ratner's celebrated equidistribution theorem states that the trajectory of any point in a homogeneous space under a unipotent flow is getting equidistributed with respect to some algebraic measure. In the case where the action is horospherical, one can deduce an effective equidistribution result by mixing methods, an idea that goes back to Margulis' thesis. When the homogeneous space is non-compact, one needs to impose further
diophantine conditions'' over the base point, quantifying some recurrence rates, in order to get a quantified equidistribution result. In the talk I will discuss certain diophantine conditions, and in particular I will show how a new Margulis' type inequality for translates of horospherical orbits helps verify such conditions, leading to a quantified equidistribution result for a large class of points, akin to the results of A. Strombergsson regarding the SL2 case. In particular we deduce a fully effective quantitative equidistribution statement for horospherical trajectories of lattices defined over number fields.

The generalized Dickman distribution ${\cal D}_\theta$ with parameter $\theta>0$ is the unique solution to the distributional equality
$W=_d W^*$, where
\begin{align*}
W^*=_d U^{1/\theta}(W+1),
\end{align*}
with $W$ non-negative with probability one, $U \sim {\cal U}[0,1]$ independent of $W$, and $=_d$ denoting equality in distribution. Members of this family appear
in the study of algorithms, number theory, stochastic geometry, and perpetuities.

The Wasserstein distance $d(\cdot,\cdot)$ between such a $W$ with finite mean, and $D \sim {\cal D}_\theta$ obeys
\begin{align*} d(W,D) \le (1+\theta)d(W^*,W).
\end{align*}
The specialization of this bound to the case $\theta=1$ and coupling constructions yield for $n \ge 1$ that
\begin{align*}
d_1(W_n,D) \le \frac{8\log (n/2)+10}{n} \quad \mbox{where } \quad W_n=\frac{1}{n}C_n-1,
\end{align*}
and $C_n$ is the number of comparisons made by the Quickselect algorithm to find the smallest element of a list of $n$ distinct numbers.

Joint with Bhattacharjee, using Stein's method, bounds for Wasserstein type distances can also be computed between ${\cal D}_\theta$ and weighted sums arising
in probabilistic number theory of the form
\begin{align*}
S_n=\frac{1}{\log(p_n)} \sum_{k=1}^n X_k \log(p_k)
\end{align*}
where $(p_k)_{k \ge 1}$ is an enumeration of the prime numbers in increasing order and $X_k$ is, for instance, Geometric with parameter $1-1/p_k$.

Queueing systems operating under the processor sharing discipline are relevant for studying time-sharing in computer and communication systems. Measure-valued processes, which track the residual service times of all jobs in the system, have been used to describe the dynamics of such systems. However, exact analysis of these infinite-dimensional stochastic processes is rarely possible. As a tool for approximate analysis of such systems, it has been proved that a fluid model arises as a functional law of large numbers limit of a multi-class processor sharing queue. This talk will focus on the asymptotic behavior of such a fluid model in the interesting regime of critical loading, where the average inflow of work to the system is equal to the capacity of the system to process that load.

Using an approach involving a certain relative entropy functional, we show that critical fluid model solutions converge to a set of invariant states as time goes to infinity, uniformly for all initial conditions lying in certain relatively compact sets. This generalizes an earlier single-class result of Puha and Williams to the more complex multiclass setting. In particular, several new challenges are overcome, including formulation of a suitable relative entropy functional and identifying a convenient form of the time derivative of the relative entropy applied to trajectories of critical fluid model solutions.

This is joint work with Justin A. Mulvany (USC) and Ruth J. Williams (UCSD)

This talk is concerned with the inverse problem of recovering a discrete measure on the torus consisting of S atoms, given M consecutive noisy Fourier coefficients. Super-resolution is sensitive to noise when the distance between two atoms is less than 1/M. We connect this problem to the minimum singular value of non-harmonic Fourier matrices. New results for the latter are presented, and as consequences, we derive results regarding the information theoretic limit of super-resolution and the resolution limit of subspace methods (namely, MUSIC and ESPRIT). These results rigorously establish the super-resolution phenomena of these algorithms that were empirically discovered long ago, and numerical results indicate that our bounds are sharp or nearly sharp. We also discuss how to take advantage of redundant measurements for the purpose of reducing quantization error. Interesting connections to trigonometric interpolation and uncertainty principles are also presented. Joint work with John Benedetto, Albert Fannjiang, Sinan Gunturk, and Wenjing Liao.

I will describe a new set of estimates for the 2d water waves problem, in which the free surface has an angled crest (or corner) with a time-dependent angle that changes with the evolution of the water wave, and with a corner vertex that can move in all directions. There are no symmetry constraints on the crest, and the fluid can have bulk vorticity. This is joint work with D. Coutand.

Lattices of minimal covolume have been studied fairly
intensively in real Lie groups, particularly in the hyperbolic
isometry groups. On the other hand, their $p$-adic analogues only
have been determined in (some) lower rank groups. In this talk,
we will discuss the higher rank behavior of lattices of minimal
covolume in $\mathrm{SL}_n(\mathbb{Q}_p)$. We will briefly introduce
their general structure, then use Prasad's volume formula and
Borel-Prasad techniques to compute their covolume. This quantity involves
a variety of number-theoretical objets, and its understanding gives rise
to some number-theoretical questions. As this is work in progress, joint
with Alireza Salehi Golsefidy, the scope of the talk will be to give a
general overview of the techniques and problems involved, rather than
stating precise results.

For environmental microbial communities, environment is destiny in the sense that, frequently, microbial community form and function are strongly linked to chemical and physical conditions. Moreover, most environments outside of the lab are physically and chemically heterogeneous, further shaping and complicating the metabolisms of their resident microbial communities: spatial variation introduce physics such as diffusive and advective transport of nutrients and byproducts for example. Conversely, microbial metabolic activity can strongly effect the environment in which the community must function. Hence it is important to link metabolism at the cellular level to physics and chemistry at the community level.

In order to introduce metabolism to community-scale population dynamics, many modeling methods rely on large numbers of reaction kinetics parameters that are unmeasured and likely effectively unmeasurable (because they are themselves coupled to environmental conditions), also making detailed metabolic information mostly unusable. The bioengineering community has, in response to these difficulties, moved to kinetics-free formulations at the cellular level, termed flux balance analysis. These cellular level models should respond to system level environmental conditions. To combine and connect the two scales, we propose to replace classical kinetics functions (almost) entirely in community scale models and instead use cell-level metabolic models to predict metabolism and how it is influenced and influenced by the environment. Further, our methodology permits assimilation of many types of measurement data.

In his influential disjointness paper, H. Furstenberg proved that weakly-mixing systems are disjoint from irrational rotations (and in general, Kronecker systems), a result that inspired much of the modern research in dynamics. Recently, A. Venkatesh managed to prove a quantitative version of this disjointness theorem for the case of the horocyclic flow on a compact Riemann surface. I will discuss Venkatesh's disjointness result and present a generalization of this result to more general actions of nilpotent groups, utilizing structural results about nilflows proven by Green-Tao-Ziegler. If time permits, I will discuss applications of such theorems in sparse equidistribution problems and number theory.

This talk will include an introduction to the topic of V(D)J rearrangements of particular subsets of T cells and B cells of the adaptive human immune system, in particular of IgG heavy chains. There are many statistical problems that arise in understanding these cells. This presentation will be my attempt to provide some mathematical and computational details that arise in trying to understand the data.

Reaction-diffusion equations have been widely used to describe biological pattern formation. Nonuniform steady states of reaction-diffusion models correspond to stationary spatial patterns supported by these models. Frequently these steady states are not unique, which correspond to various spatial patterns observed in biology. Traditionally, time-marching methods or steady state solvers based on Newton’s method were used to compute such solutions. However, the solution that any of these methods leads to highly depends on the initial condition/guess. In this talk, I present a systematic method to compute multiple nonuniform steady states for reaction-diffusion models and determine the dependence on model parameters. The method is based on homotopy continuation techniques and multigrid methods. We apply the method to two classic reaction-diffusion models and compare our results with available theoretical analysis in the literature. The first is the Schnakenberg model that has been used to describe biological pattern formation due to diffusion-driven instability. The second is the Gray-Scott model which was proposed in 1980’s to describe autocatalytic glycolysis reactions. In each case, our method uncovers many, if not all, nonuniform steady states and their stabilities. We also compared our computational results with analytical results in the literature and the comparison suggests some errors in prior results obtained using asymptotic analysis.