Logistic regression is arguably the most widely used and studied non-linear model in statistics. It has found widespread applicability in varied domains, such as genetics, health care, e-commerce, etc. Classical maximum-likelihood theory for this model hinges on the fundamental results---(1) the maximum-likelihood-estimate (MLE) is asymptotically unbiased (2) its variability can be quantified via the inverse Fisher Information (3) the likelihood-ratio-test (LRT) is asymptotically a Chi-Square. These results are universally used for statistical inference. Our findings reveal, however, when the number of features p and the sample size n both diverge, with the ratio p/n converging to a positive constant, classical results are far from accurate. For a certain class of logistic models, we observe (1) the MLE is biased, (2) its variability is much higher than classically estimated, and (3) the LRT is not distributed as a Chi-Square. We develop a new theory that quantifies the asymptotic bias and variance of the MLE, and characterizes asymptotic distribution of the LRT under certain assumptions on the covariate distribution. Empirical findings demonstrate that our results provide extremely accurate inference in finite samples. These novel results depend on the underlying regression coefficients through a single scalar, the overall signal strength, and we discuss a procedure to estimate this parameter accurately. This is based on joint work with Emmanuel Candes and Yuxin Chen.

Regularity results for linear elliptic and parabolic systems
with measurable coefficients play an important role in the calculus of
variations. Morrey showed that in two dimensions, solutions to linear
elliptic systems are continuous. We will discuss some surprising recent
examples of discontinuity formation in the plane for the parabolic
problem.

Nowadays, data is exploding at a faster rate than computer architectures can handle. For that reason, mathematical techniques to analyze large-scale data need be developed. Stochastic iterative algorithms have gained interest due to their low memory footprint and adaptability for large-scale data. In this talk, we will present the Randomized Kaczmarz algorithm for solving extremely large linear systems of the form Ax=y. In the spirit of large-scale data, this talk will act under the assumption that the entire data matrix A cannot be loaded into memory in a single instance. We consider different settings including when a only factorization of A is available, when A is missing information, and a time-varying model. We will also present applications of these Kaczmarz variants to problems in data science.

In the theory of von Neumann algebras, fundamental unsolved problems going back to the 1930s have seen remarkable progress in the last two decades due to Sorin Popa's breakthrough deformation/rigidity theory. Popa's discovery hinges on the fact that, just as stirring a soup allows one to locate its most rigid (and desirable) hidden components, 'deformability' of an algebra $M$ allows one to precisely locate 'rigid' subalgebras known to exist only via a supposed isomorphism $M \cong N$.

This talk will focus on joint work with Rolando de Santiago, Ben Hayes, and Thomas Sinclair, which shows that any diffuse subalgebra which is rigid with respect to a mixing $s$-malleable deformation is in fact contained in subalgebra which is uniquely maximal with respect to that rigidity. In particular, an algebra generated by a family of rigid subalgebras which intersect diffusely must itself be rigid with respect to that deformation. The case where this family has two members answers a question of Jesse Peterson asked at the American Institute of Mathematics (AIM), but the result is most striking when the family is infinite.

It is not obvious why natural selection would've selected for animals that can compute integrals, yet we exist. In this talk we'll look at some explanations from cognitive science, examples from the wild, and perhaps a look at the future of how maths could be done.

Bernstein operators are vertex operators that create and annhilate Schur polynomials. These operators play a significant role in the mathematical formulation of the Boson-Fermion correspondence due to Kac and Frenkel. The role of this correspondence in mathematical physics has been widely studied as it bridges the actions of the infinite Heisenberg and Clifford algebras on Fock space. Cautis and Sussan conjectured a categorification of this correspondence within the framework of Khovanov's Heisenberg category. I will discuss how to categorify the Bernstein operators and settle the Cautis-Sussan conjecture, thus proving a categorical Boson-Fermion correspondence.

Li-Chung Chen and Mark Haiman studied a family of symmetric functions called Catalan (symmetric) functions which are indexed by pairs consisting of a partition contained in the staircase $(n-1, \dots, 1, 0)$ (of which there are Catalan many) and a composition weight of length $n$. They include the Schur functions, the Hall-Littlewood polynomials and their parabolic generalizations. They can be defined by a Demazure-operator formula, and are equal to the $GL$-equivariant Euler characteristics of vector bundles on the flag variety by the Borel-Weil-Bott Theorem. We have discovered various properties of Catalan functions, providing new insight on the existing theorems and conjectures inspired by the Macdonald Positivity Conjecture.

A key discovery in our work is an elegant set of ideals of roots whose associate Catalan functions are $k$-Schur functions, proving that graded $k$-Schur functions are $GL$-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We exposed a new shift invariance property of the graded $k$-Schur functions and resolved the Schur positivity and $k$-branching conjectures by providing direct combinatorial formulas using strong marked tableaux. We conjectured that Catalan functions with a partition weight are $k$-Schur positive which strengthens the Schur positivity of the Catalan function conjecture by Chen-Haiman and resolved the conjecture with positive combinatorial formulas in cases which capture and refine a variety of problems.

This is joint work with Jonah Blasiak, Jennifer Morse, and Daniel Summers.

Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three paradigmatic facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems.

Modern statistical analysis often requires an integration of statistical thinking and algorithmic thinking. In many problems, statistically sound estimation procedures (e.g., the MLE) may be difficult to compute, at least in the naive form. This challenge calls for a new look into simple statistical methods such as the spectral methods (including PCA), as well as an examination of optimization algorithms from the statistical lens.

In this talk, I will sample two typical modern statistical problems: one addresses network type data (community detection), and the other involves pairwise comparison data (phase synchronization). I will show that in high dimensions, spectral methods exhibit a very interesting new phenomenon in entrywise behavior, which leads to new theoretical insights and has practical relevance. Also, for a complex nonconvex problem, I will show how algorithmic analysis can benefit from classical statistical ideas.

This talk features joint work with (alphabetically) Emmanuel Abbe, Nicolas Boumal, Jianqing Fan, and Kaizheng Wang.

In this talk, we present some recent work on discontinuous Galerkin (DG) methods for waves and fluid flow.
Three topics will be covered, including (1) new energy-conserving DG methods for linear hyperbolic waves, and nonlinear dispersive waves, (2) globally divergence-free DG methods for incompressible flow, and (3) hybridizable DG methods for Darcy flow on polygonal meshes. The key idea of each DG scheme will be addressed.

Let k be a perfect field of characteristic p, and let Gal(k)
denote the absolute Galois group of k. By a classical result of Katz, the
category of
finite-dimensional $F_p-vector spaces$ with an action of Gal(k) is equivalent
to the category of finite-dimensional vector spaces over k with a
Frobenius-semilinear automorphism. In this talk, I'll discuss some joint
work with Bhargav Bhatt which generalizes Katz's result, replacing the
field k by an arbitrary $F_p-scheme X$. In this case, there is a
correspondence relating p-torsion etale sheaves on X to quasi-coherent
sheaves on X equipped with a Frobenius-semilinear automorphism, which can
be viewed as a ``mod p'' version of the Riemann-Hilbert correspondence for
complex algebraic varieties.