A negative control outcome (NCO) is an outcome that is associated with unobserved confounders of the effect of a treatment on an outcome in view, and is a priori known not to be causally impacted by the treatment. In the first half of the talk, we discuss the single proxy control (SPC) framework, a formal NCO method to detect and correct for residual confounding bias. We establish nonparametric identification of the average causal effect for the treated (ATT) by treating the NCO as an error-prone proxy of the treatment-free potential outcome, a key assumption of the SPC framework. We characterize the efficient influence function for the ATT under a semiparametric model in which nuisance functions are a priori unrestricted. Moreover, we develop a consistent, asymptotically linear, and locally semiparametric efficient estimator of the ATT using modern machine learning theory. Shifting to the second half of the talk, we introduce the single proxy synthetic control (SPSC) framework, an extension of the SPC framework designed for a synthetic control setting, where a single unit is treated and pre- and post-treatment time series data are available on the treated unit and a heterogeneous pool of untreated control units. Similar to SPC, the SPSC framework views the outcomes of untreated control units as proxies of the treatment-free potential outcome of the treated unit, a perspective we formally leverage to construct a valid synthetic control. Under this framework, we establish alternative identification and estimation methodology for synthetic controls and, in turn, for the ATT. Additionally, we adapt a conformal inference approach to perform inference on the treatment effect, obviating the need for a large number of post-treatment data. We illustrate the SPC and SPSC approaches with real-world applications from the Zika virus outbreak in Brazil and the 1907 financial crisis.

In this paper, we address the problem of continuous-time reinforcement learning in scenarios where the dynamics follow a stochastic differential equation. When the underlying dynamics remain unknown and we have access only to discrete-time information, how can we effectively perform policy evaluation? We first demonstrate that the commonly used Bellman equation is a first-order approximation to the true value function. We then introduce a higher order PDE-based Bellman equation called PhiBE. We show that the solution to the i-th order PhiBE is an i-th order approximation to the true value function. Additionally, even the first-order PhiBE outperforms the Bellman equation in approximating the true value function when the system dynamics change slowly. We develop a numerical algorithm based on Galerkin method to solve PhiBE when we possess only discrete-time trajectory data. Numerical experiments are provided to validate the theoretical guarantees we propose.

We prove that $G_{n,p=c/n}$ whp has a $k$-regular subgraph if $c$ is at least  $e^{-\Theta(k)}$ above the threshold for the appearance of a subgraph with minimum degree at least $k$; i.e. an non-empty $k$-core. In particular, this pins down the threshold for the appearance of a $k$-regular subgraph to a window of size $e^{-\Theta(k)}$.

This is a joint work with Dieter Mitsche and Pawel Pralat; see arXiv:1804.04173

In the first half of the talk, I will give a brief introduction to quantum computing from the perspective of a computer scientist/mathematician. While it may seem obvious that quantum computers should be better than classical computers, this can be surprisingly hard to rigorously prove, especially using the types of quantum computers that are available today. In the second half of the talk, I will describe this quest for provable quantum advantage and some of the research directions I find most interesting.

I will present a line of work on the computational complexity of several algorithmic tasks on random inputs, including hypothesis testing, sampling, and "certification" for optimization problems (where an algorithm must output a bound on a problem's optimum rather than just a high-quality solution). Surprisingly, these diverse tasks admit a unified analysis involving the same two main ingredients. The first is the study of algorithms that output low-degree polynomial functions of their inputs. Such algorithms are believed to be optimal for many statistical tasks and can be understood with the theory of orthogonal polynomials, leading to strong evidence for the hardness of certain hypothesis testing problems. The second is a strategy of "planting" unusual structures in problem instances, which gives reductions from hypothesis testing to tasks like sampling and certification. I will focus on examples of the latter motivated by statistical physics: (1) sampling from Ising models, and (2) certifying bounds on the Hamiltonian of the Sherrington-Kirkpatrick spin glass model.

Next, by examining the sum-of-squares hierarchy of semidefinite programs, I will demonstrate how reasoning with planted solutions can show computational hardness of certification problems not only in random settings under strong distributional assumptions, but also for more generic problem instances. As an extreme example, I will show how some of the above ideas may be completely derandomized and applied in a deterministic setting. Using as a testbed the long-standing open problem in number theory and Ramsey theory of bounding the clique number of the Paley graph, I will give an analysis of semidefinite programming that suggests both new theoretical approaches to proving stronger bounds on the clique number and refined notions of pseudorandomness capturing deterministic versions of phenomena from random matrix theory.

In the setting of zero-entropy transformations, the class of subshifts--closed shift-invariant subsets $X$ of $\mathcal{A}^{\mathbb{Z}}$ for a finite alphabet $\mathcal{A}$--possesses a quantitative measure of complexity: the number of distinct `words' of a given length $p(q) = |\{ w \in \mathcal{A}^{q} : \exists x \in X \text{ s.t. w is a substring of x}\}|$.

I will discuss my work, some joint with R. Pavlov, pinning down the relationship between this quantitative notion of complexity with the qualitative dynamical complexity properties of probability-preserving systems known as strong and weak mixing.

Specifically, I will present results that strong mixing can occur with word complexity arbitrarily close to linear but cannot occur when $\liminf p(q)/q < \infty$ and that weak mixing can occur when $\limsup p(q)/q = 1.5$ but cannot occur when $\limsup p(q)/q < 1/5$.

The condition that $\limsup p(q)/q < 1.5$ is a (much) stronger version of zero entropy. A corollary of our work is that the celebrated Sarnak conjecture holds for all such systems.

Given any closed Riemannian manifold $M$, we construct a reversible diffusion process on the space $\mathcal{P}(M)$ of probability measures on $M$ that is
 

• reversible w.r.t. the entropic measure $\mathbb{P}^\beta$ on $\mathcal{P}(M)$, heuristically given as 

$$d\mathbb{P}^\beta(\mu) =\frac{1}{Z} e^{-\beta \, \text{Ent}(\mu | m)}\ d\mathbb{P}^0(\mu);$$

• associated with a regular Dirichlet form with carré du champ derived from the Wasserstein gradient in the sense of Otto calculus

$$\mathcal{E}_W(f)=\liminf_{\tilde f\to f}\ \frac12\int_{\mathcal{P}(M)} \big\|\nabla_W \tilde f\big\|^2(\mu)\ d\mathbb{P}^\beta(\mu);$$

• non-degenerate, at least in the case of the $n$-sphere and the $n$-torus.

Since Vaughan Jones introduced the theory of subfactors in 1983, it has been an open problem to determine the set of Jones indices of irreducible, hyperfinite subfactors. Not much is known about this set.

My student Julio Caceres and I could recently show that certain indices between 4 and 5 are realized by new hyperfinite subfactors with Temperley-Lieb-Jones standard invariant. This leads to a conjecture regarding Jones' problem. Our construction involves commuting squares, a graph planar algebra embedding theorem, and a few tricks that allow us to avoid solving large systems of linear equations to compute invariants of our subfactors. If there is time, I will mention a few connections to quantum Fourier analysis and quantum information theory.

 I will discuss mathematical models of stem cell dynamics in tissues at homeostasis, focusing on the ability of negative feedback loops within cell lineages to contribute to homeostatic control. These dynamics will be examined both in non-spatial and spatially explicit computational models, highlighting how spatial interactions can change dynamics and conclusions. The talk will further discuss the evolution of cells towards escape from homeostatic control, which gives rise to cancerous growth of cells. In the cancer cell growth dynamics, the models will be used to examine factors that determine the fraction of cancer stem cells in tumors, which in turn can determine the degree to which tumors respond to chemotherapies. Higher stem cell fractions correlate with increased resistance to therapy. This theory will be applied to data from bladder cancer, with the aim to better understand the heterogeneity that is observed in the responses among different patients to treatments.   

The $S$-unit equation has vast applications in number theory. We will discuss an implementation of an algorithm to solve the $S$-unit equation in the mathematical software Sage.  The mathematical foundation for this implementation and some applications will be outlined, including an asymptotic version of Fermat's Last Theorem for totally real cubic number fields with bounded discriminant in which 2 is totally ramified. We will conclude with a discussion on current and future work toward improving the existing Sage functionality.

[pre-talk at 1:20PM, in person only]

Representations of groups over $p$-adic fields arise naturally in Number Theory. Stable lattices serve as integral models for such representations. I will provide an example of these representations. I will  discuss the connection between the set of lattices and a combinatorial object called the Bruhat-Tits building. If time permits, I will discuss open problems.

Have you ever thrown a rock into a still pond, stared at the concentric waves for a while and realized that the farther the waves travel, the smaller their crests seem to get? In this talk, we'll discuss a class of equations that, like the water waves in a pond, disperse. We'll discuss common techniques to study existence and qualitative properties of solutions to nonlinear dispersive PDEs using equations that model the strong nuclear force: the Dirac-Klein-Gordon system.