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Series: Job Candidate Talk

Dynamical billiards constitute a very natural class of Hamiltonian systems: in
1927 George Birkhoff conjectured that, among all billiards inside smooth
planar convex domains, only billiards in ellipses are integrable. In this talk
we will prove a version of this conjecture for convex domains that are
sufficiently close to an ellipse of small eccentricity.
We will also describe some remarkable relation with inverse spectral theory
and spectral rigidity of planar convex domains. Our techniques can in fact be
fruitfully adapted to prove spectral rigidity among generic (finitely) smooth
axially symmetric domains which are sufficiently close to a circle. This gives
a partial answer to a question by P. Sarnak.

Series: Job Candidate Talk

Hybrid particle/grid numerical methods have been around for a long time, andtheir usage is common in some fields, from plasma physics to artist-directedfluids. I will explore the use of hybrid methods to simulate many differentcomplex phenomena occurring all around you, from wine to shaving foam and fromsand to the snow in Disney's Frozen. I will also talk about some of thepractical advantages and disadvantages of hybrid methods and how one of theweaknesses that has long plagued them can now be fixed.

Series: Job Candidate Talk

We consider the problem of how to control the measures of false scientific discoveries in high-dimensional models. Towards this goal, we focus on the uncertainty assessment for low dimensional components in high-dimensional models. Specifically, we propose a novel decorrelated likelihood based framework to obtain valid p-values for generic penalized M-estimators. Unlike most existing inferential methods which are tailored for individual models, our method provides a general framework for high-dimensional inference and is applicable to a wide variety of applications, including generalized linear models, graphical models, classifications and survival analysis. The proposed method provides optimal tests and confidence intervals. The extensions to general estimating equations are discussed. Finally, we show that the p-values can be combined to control the false discovery rate in multiple hypothesis testing.

Series: Job Candidate Talk

The eigendecomposition of an adjacency matrix provides a way to embed a graph as points in finite dimensional Euclidean space. This embedding allows the full arsenal of statistical and machine learning methodology for multivariate Euclidean data to be deployed for graph inference. Our work analyzes this embedding, a graph version of principal component analysis, in the context of various random graph models with a focus on the impact for subsequent inference. We show that for a particular model this embedding yields a consistent estimate of its parameters and that these estimates can be used to accurately perform a variety of inference tasks including vertex clustering, vertex classification as well as estimation and hypothesis testing about the parameters.

Series: Job Candidate Talk

Bootstrap is one of the most powerful and common tools in statistical
inference. In this talk a multiplier bootstrap procedure is
considered for construction of likelihood-based confidence sets.
Theoretical results justify the bootstrap validity for a small or
moderate sample size and allow to control the impact of the parameter
dimension p: the bootstrap approximation works if p^3/n is small,
where n is a sample size. The main result about bootstrap validity
continues to apply even if the underlying parametric model is
misspecified under a so-called small modelling bias condition. In the
case when the true model deviates significantly from the considered
parametric family, the bootstrap procedure is still applicable but it
becomes conservative: the size of the constructed confidence sets is
increased by the modelling bias. The approach is also extended to the
problem of simultaneous confidence estimation. A simultaneous
multiplier bootstrap procedure is justified for the case of
exponentially large number of models. Numerical experiments for
misspecified regression models nicely confirm our theoretical
results.

Series: Job Candidate Talk

Random graphs are the basic mathematical models for large-scale
disordered networks in many different fields (e.g., physics, biology,
sociology).
Their systematic study was pioneered by Erdoes and Renyi around 1960, and
one key feature of many classical models is that the edges appear
independently.
While this makes them amenable to a rigorous analysis, it is desirable
(both mathematically and in terms of applications) to understand more
complicated situations.
In this talk I will discuss some of my work on so-called Achlioptas
processes, which (i) are evolving random graph models with dependencies
between the edges and (ii) give rise to
more interesting percolation phase transition phenomena than the classical
Erdoes-Renyi model.

Series: Job Candidate Talk

Polynomials defined either by some type of orthogonality or
satisfying differential equations are pervasive in approximation theory,
random matrix theory, special functions, harmonic analysis, scientific
computing and applications. Numerical simulations show that their zeros
exhibit a common feature: they align themselves along certain curves on the
plane. What are these curves? In some cases we can answer this question, at
least asymptotically. The answer connects fascinating mathematical objects,
such as extremal problems in electrostatics, Riemann surfaces, trajectories
of quadratic differentials, algebraic functions; this list is not complete.
This talk is a brief survey of some ideas related to this problem, from the
breakthrough developments in the 1980-ies to nowadays, finishing with some
recent results and open problems.

Series: Job Candidate Talk

The goal of this lecture is to explain and motivate the connection between Aubry-Mather theory (Dynamical Systems), and viscosity solutions of the Hamilton-Jacobi equations (PDE). The connection is the content of weak KAM Theory. The talk should be accessible to the ''generic" mathematician. No a priori knowledge of any of the two subjects is assumed.

Series: Job Candidate Talk

We discuss asymptotic-in-time behavior of time-like constant meancurvature hypersurfaces in Minkowski space. These objects model extended relativistic test objects subject to constant normal forces, and appear in the classical field theory foundations of the theory of vibrating strings and membranes. From the point of view of their Cauchy problem, these hypersurfaces evolve according to a geometric system of quasilinear hyperbolic partial differential equations. Inthis talk we will focus on three explicit solutions to the equations:the Minkowski hyperplane, the static catenoid, and the expanding de Sitter space. Their stability properties in the context of the Cauchy problem will be discussed, with emphasis on the geometric origins of the various mechanisms and obstacles that come into play.

Series: Job Candidate Talk

Low-rank structures are common in modern data analysis and signal processing, and they usually
play essential roles in various estimation and detection problems. It is challenging to recover the underlying
low-rank structures reliably from corrupted or undersampled measurements. In this talk, we will introduce
convex and nonconvex optimization methods for low-rank recovery by two examples.
The first example is community detection in network data analysis. In the literature, it has been formulated
as a low-rank recovery problem, and then SDP relaxation methods can be naturally applied. However,
the statistical advantages of convex optimization approaches over other competitive methods, such as spectral
clustering, were not clear. We show in this talk that the methodology of SDP is robust against arbitrary
outlier nodes with strong theoretical guarantees, while standard spectral clustering may fail due to a small
fraction of outliers. We also demonstrate that a degree-corrected version of SDP works well for a real-world
network dataset with a heterogeneous distribution of degrees.
Although SDP methods are provably effective and robust, the computational complexity is usually high
and there is an issue of storage. For the problem of phase retrieval, which has various applications and
can be formulated as a low-rank matrix recovery problem, we introduce an iterative algorithm induced by
nonconvex optimization. We prove that our method converges reliably to the original signal. It requires far
less storage and has much higher rate of convergence compared to convex methods.