Certain imaging applications such as x-ray crystallography require the recovery
of an underlying signal from intensity (or magnitude) measurements - a problem
commonly referred to as Phase Retrieval. In this talk, we discuss a framework for
solving the discrete phase retrieval problem using block circulant measurement
constructions and angular synchronization. We develop an algorithm which is
near-linear time, making it computationally feasible for large dimensional
signals. Theoretical and experimental results demonstrating the method's
speed, accuracy and robustness will be presented. We also present an extension
of the framework to sparse phase retrieval, including the first known
sublinear-time compressive phase retrieval algorithm.

Random matrix theory is filled with laws of large numbers and central limit theorems, all specialized to different statistics and scaling regimes. The two most famous laws of large numbers are for Gaussian Wigner (Hermitian) matrices: the bulk (empirical distribution) of eigenvalues converge a.s. to the semicircle law, and the largest eigenvalue converges to twice the common variance of the entries. The corresponding central limit theorems give Gaussian fluctuations in the bulk, but a completely different distribution (the Tracy-Widom Law) for the largest eigenvalue.

One can view a Gaussian Wigner matrix as (a marginal of) Brownian motion on the Lie algebra of Hermitian matrices. It is then natural to study the analogous questions for the eigenvalues of the Brownian motion on the associated Lie group: the unitary group. For the bulk, the a.s. limit empirical eigenvalue distribution was discovered by Biane and Rains independently in the late 1990s; the corresponding bulk central limit result was largely found by L\'evy and Ma\"ida in 2010

I will give an overview of Kodaira's vanishing theorem and its applications in higher-dimensional algebraic geometry. I will then proceed to give an introduction to formal schemes and their applications. Then I will state a version of the vanishing theorem for formal schemes, along with some possible further results to be investigated.

Elliptic curves appear prominently in Wiles's resolution of Fermat's Last Theorem, the Birch and Swinnerton-Dyer Conjecture (a Millennium Problem), and Lenstra's factoring algorithm, and they are also important in modern cryptography. In this talk, we will give a gentle introduction to elliptic curves and then explain some of these applications without assuming any background in algebraic geometry or number theory.

Sponsored by the GSA.

We investigate timelike and null vector flows on closed Lorentzian manifolds and their relationship to Ricci curvature. The guiding observation, first observed for closed Riemannian 3-manifolds by Harris & Paternain '13, is that positive Ricci curvature tends to yield contact forms, namely, 1-forms metrically equivalent to unit vector fields with geodesic flow. We carry this line of thought over to the Lorentzian setting. First, we observe that the same is true on a closed Lorentzian 3-manifold: if X is a global timelike unit vector field with geodesic flow satisfying $Ric(X,X) > 0$, then $g(X,•)$ is a contact form with Reeb vector field X, at least one of whose integral curves is closed. Second, we show that on a closed Lorentzian 4-manifold, if X is a global null vector field satisfying $\nabla_XX = X$ and $Ric(X) > divX - 1$, then $dg(X,•)$ is a symplectic form and X is a Liouville vector field.

Electrostatic interactions play an important role in many complex charged systems, such as biological molecules, soft matter material, nanofluids, and electrochemical devices. In this work, we develop mathematical theories and computational methods to understand such interactions, particularly in charged biological molecular systems. Our main contributions include: 1. Theoretical studies of mean-field variational models of ionic solution and that of molecular surfaces with the Poisson-Boltzmann electrostatics; 2. Design and implementation of the corresponding computational algorithms, and conduct extensive Monte Carlo simulations and numerical solutions of partial differential equations for charge-charge interactions; 3. Discovery of various interesting properties of charged molecules, validate some experimental results, and clarify some confusion in literature. A common theme of this work is the variational approach. Many physical effects such as ionic size effects, solvent entropy, concentration dependent dielectric response can be incorporated into a mean-field free-energy functional of ionic concentrations coupled with the Poisson equation for electrostatics. The techniques of analysis developed in this work may help improve the understanding of the underlying physical properties of charged systems and provide new ways of studying analytically and numerically other problems in the calculus of variations.

A celebrated 1942 result of Schoenberg characterizes all entry-wise functions which preserve positivity of matrices of any size. I will present a characterization of polynomials which preserve positivity when applied entry-wise on matrices of a fixed dimension. All put in historical context and motivated by recent demands of statistics of large data and optimization theory. A sketch of the proof will take a detour through the representation theory of the symmetric group.

An algebraic cycle homologous to zero on a variety leads to an
extension of Hodge-theoretic data, and in a variational context to a
family of extensions called a normal function. These may be viewed as
"horizontal" sections of a bundle of complex tori, and are used to detect
cycles modulo algebraic (or rational) equivalence. Conversely, the
existence of normal functions can be used to predict that interesting
cycles are present... or absent: a famous theorem of Green and Voisin
states that for projective hypersurfaces of large enough degree, there are
no normal functions (into the intermediate Jacobian bundle associated to
these hypersurfaces) over any etale neighborhood of the coarse moduli
space. Inspired by recent work of Friedman-Laza on Hermitian variations of
Hodge structure and Oort's conjecture on special (i.e. Shimura)
subvarieties in the Torelli locus, R. Keast and I wondered about the
existence of normal functions over etale neighborhoods of Shimura
varieties. Here the function is supposed to take values in a family of
intermediate Jacobians associated to a representation of a reductive
group. In this talk I will explain our classification of the cases where a
Green-Voisin analogue does *not* hold and where one therefore expects
interesting cycles to occur, and give some evidence that these predictions
might be "sharp".