Tesler matrices are certain integral matrices counted by the Kostant partition function and have appeared recently in Haglund's study of diagonal harmonics. In 2014, Drew Armstrong defined a poset on such matrices and conjectured that the characteristic polynomial of this poset is a power of $(q-1)$. We will use a method of Bruce Sagan and Joshua Hallam to prove Armstrong's conjecture and explore how this result can improve the bounds on the number of Tesler matrices.

We continue the previous studies in two papers of Huang-Yin on the flattening problem of a CR singular point of real codimension two sitting in a submanifold in $C^{n+1}$ with $n + 1 ≥ 3$, whose CR points are non-minimal. We give a very general flattening theorem for a non-degenerate CR singular point. This is joint work with Xiaojun Huang.

The theory supporting conforming finite elements on meshes of squares and cubes requires some care when extended to more general meshes of quadrilaterals and hexahedra in order to preserve desired rates of convergence. In this talk, I'll present two areas of research related to these issues. First, I'll describe a new family of methods called ``trimmed serendipity elements'' that fit within the same framework described by the Periodic Table of the Finite Elements (see https://femtable.org). The computational effort required to employ a trimmed serendipity element method is significantly less that what is required for comparable alternatives from the table, thereby presenting a host of potential benefits to the speed and accuracy of square/cube finite element methods in practice. Second, I'll present the loss of convergence issues that arise when square/cube elements are mapped non-affinely as well as some recent techniques that restore convergence order. In addition, I'll show why general quad/hex meshes are of increasing interest in application contexts, including an cardiac electrophysiology example carried out in collaboration with Andrew McCulloch's research group and NBCR.

I will discuss recent joint results with B. Lamel addressing the $C^\infty$ regularity problem for CR mappings between smooth CR submanifolds in complex spaces of possibly different dimension. We essentially show that a nowhere smooth CR map must have its image contained in the set of D’Angelo infinite type points of the target manifold, from which we derive a number of new regularity results, even in the hypersurface case. Applications to the boundary regularity of proper holomorphic mappings will also be mentioned.

An important task of olfactory sensing is the discrimination of different odors. An odor captures the chemical state of the environment in a mixture of smell molecules, called odorants. Olfactory sensing is realized by the selective binding of odorants to a set of olfactory receptors, which in turn activates the corresponding olfactory sensory neurons, constructing the brain's first representation of the odor. Despite the high-dimensional nature of olfactory sensing, recent measurements with human olfactory receptors suggest that the odorant-receptor interaction is sparse; only a small fraction of all available pairs interact. What are the optimal interaction structures for effective olfactory discrimination, and are these optimal solutions employed by the real system? We investigate these questions by combining studies of model systems and analyses of experimental data. We show that optimization depends on the statistical properties of the olfactory environment, and furthermore suggest that the human olfactory receptors are adapted to the natural odor statistics.

Holomorphic classification of real submanifolds in complex space is one of the central goals in complex analysis in several variables. This classification is well understood and approaches are well developed for submanifolds satisfying certain bracket generating conditions of Hormander type, while very little is known in more degenerate setting. In particular, somewhat surprizingly, the classification problem is still completely open for hypersurfaces in complex 2-space. The class of hypersurfaces bringing conceptual difficulties here is the class of (Levi-nonflat) infinite type hypersurfaces.
In our joint work with Ebenfelt and Lamel, we develop the equivalence theory for infinite type hypersurfaces in $C^2$. We do so by providing a normal for for such hypersurfaces. We extensively use the newly developed approach of Associated Differential Equations. The normal form construction is performed in two steps: (i) we provide a normal for for associated ODES; (ii) we use the normal form of ODEs for solving the equivalence problem for hypersurfaces.
Somewhat similarly to the Poincare-Dulac theory in Dynamical System, our classification theory exhibits resonances, convergence and divergence phenomena, Stokes phenomenon and sectorial regularity phenomena.

We will discuss a few interesting open problems related to the analysis of PDEs which model fluid motion. While some of these models were derived by Euler in the 1750s, understanding the dynamics of solutions continues to be a major challenge despite many advancements in recent years.

Consider the scaling invariant, sharp log entropy (functional) introduced by Weissler on noncompact manifolds with nonnegative Ricci curvature. It can also be regarded as a sharpened version of Perelman's W entropy in the stationary case. We prove that it has a minimizer if and only if the manifold is isometric to the Euclidean space. Using this result, it is proven that a class of noncompact manifolds with nonnegative Ricci curvature is isometric to $R^n$. Comparing with earlier well known flatness results on asymptotically flat manifolds and asymptotically locally Euclidean (ALE) manifolds, their decay or integral condition on the curvature tensor is replaced by the condition that the metric converges to the Euclidean one in $C^1$ sense at infinity. No second order condition on the metric is needed.

Solving an unconstrained quadratic program means solving a linear equation where the matrix is symmetric and positive definite. This is a fundamental subproblem in nonlinear optimization. We discuss the behavior of the method of conjugate gradients and quasi-Newton methods on a quadratic problem. We first derive the method of conjugate gradients and then give necessary and sufficient conditions for an exact line search quasi-Newton method to generate a search direction which is parallel to that of the method of conjugate gradients. We analyze update matrices and show how the secant condition fits the discussion of giving parallel search directions.

Our interest is limited-memory quasi-Newton methods tailored for interior methods. The talk describes the fundamental properties for the exact quadratic case, which is the foundation for the work.

The talk is based on joint work with David Ek and Tove Odland.

The hydrodynamic limit of the Kawasaki dynamics states that a certain stochastic evolution of a lattice system converges macroscopically to a deterministic non-linear heat equation. We will discuss how the statement of the hydrodynamic limit can be made quantitative. The key step is to introduce an additional evolution on a mesoscopic scale that emerges from projecting the microscopic observables onto splines. The hydrodynamic limit is then deduced in two steps. In the first step one shows the convergence of the microscopic to the mesoscopic evolution and in the second step one deduces the convergence of the mesoscopic to the macroscopic evolution.

The talk is about a joint work with Deniz Dizdar, Felix Otto and Tianqi Wu.

A certain infinite determinant arises as the constant in the block case of the Szego-Widom Limit Theorem for Toeplitz matrices. These are matrices whose block entries are constant on the diagonals. While the constant has a very nice form, often a more explicit form, analogous to the one found in the scalar case, is needed in applications. This talk will survey some of the ways the constant can be more explicitly computed (often using functional analysis techniques) and also some applications.

In a 1987 letter to Tate, Serre showed that the Hecke
eigensystems appearing in mod p modular forms are the same as those
appearing in mod p functions on a finite double coset constructed from
the quaternion algebra ramified at p and infinity. At the end of the
letter, he asked whether there might be a similar relation between
p-adic modular forms and p-adic functions on the quaternion algebra. We
show the answer is yes: the completed Hecke algebra of p-adic modular
forms is the same as the completed Hecke algebra of naive p-adic
automorphic functions on the quaternion algebra. The resulting p-adic
Jacquet-Langlands correspondence is richer than the classical
Jacquet-Langlands correspondence -- for example, Ramanujan's delta
function, which is invisible to the classical correspondence, appears.
The proof is a lifting of Serre's geometric argument from characteristic
p to characteristic zero; the quaternionic double coset is realized as a
fiber of the Hodge-Tate period map, and eigensystems are extended off of
the fiber using a variant of Scholze's fake Hasse invariants.

Nonconvex optimization problems are the bottleneck in many applications in science and technology. Often these problems are NP-hard and they are approached with ad hoc methods that frequently fail to yield satisfactory results. This issue becomes even more prevalent in the ``Big Data Regime''.
In my talk I will report on recent breakthroughs in solving some important nonconvex optimization problems. In particular, I will discuss the problems of phase retrieval, blind deconvolution, and blind source separation. The most notorious among these three is arguably phase retrieval, which is the century-old problem of reconstructing a function from intensity measurements, typically from the modulus of a diffracted wave. Phase retrieval problems arise in numerous areas including X-ray crystallography, differential geometry, astronomy, diffraction imaging, and quantum physics and are very difficult to solve numerically.

Combining tools from optimization, random matrix theory and harmonic analysis, we will derive rigorous mathematical methods that can solve the aforementioned problems under meaningful practical conditions. The proposed methods come with rigorous theoretical guarantees, are numerically efficient and stable in the presence of noise, and require little or no parameter tuning, thus making them useful for massive datasets. I will also discuss connections to the emerging field of self-calibration, which is based on the idea of equipping a sensor with a smart algorithm that can compensate automatically for the sensor's imperfections. The effectiveness of our methods will be illustrated in applications such as astronomy, X-ray crystallography, terahertz imaging, and the Internet-of-Things.

The gonality of a smooth projective curve is the smallest degree of a map from the curve to the projective line. If a curve is embedded in projective space, it is natural to ask whether the gonality is related to the embedding. In my talk, I will discuss recent work with James Hotchkiss. Our main result is that, under mild degree hypotheses, the gonality of a complete intersection curve in projective space is computed by projection from a codimension 2 linear space, and any minimal degree branched covering of $P^1$ arises in this way.