We study N-player repeated Cournot competitions that model the determination of price in an oligopoly where firms choose quantities. These are nonzero-sum (ordinary and stochastic) differential games, whose value functions may be characterized by systems of nonlinear Hamilton-Jacobi-Bellman partial differential equations. When the quantity being produced is in finite supply, such as oil, exhaustibility enters as boundary conditions for the PDEs. We analyze the problem when there is an alternative, but expensive, resource (for example solar technology for energy production), and give an asymptotic approximation in the limit of small exhaustibility. We illustrate the two-player problem by numerical solutions, and discuss the impact of limited oil reserves on production and oil prices in the dupoly case. Joint work with Chris Harris (Cambridge University) and Sam Howison (Oxford University).

Cardiac cells, like toilets, are excitable: Giving a sufficiently strong push to the handle of a quiescent toilet elicits a dramatic response (flush) followed by a gradual return to the resting state. Likewise, supplying a sufficiently strong electrical stimulus to a quiescent cardiac cell elicits a prolonged elevation of the membrane potential (an action potential).

Suppose that one end of an excitable fiber of cardiac cells is paced periodically. If the period is large, the generic response is a stable periodic wave train of the sort associated with normal, coordinated contraction of heart muscle tissue. Reducing the period (think "speeding up the heart rate") can cause the onset of an instability which can have devastating physiological consequences. Echebarria and Karma (Chaos, 2002) argued that if one attempts to stabilize the periodic wave train by using feedback control to perturb the pacing period, success can be achieved only within some small radius of the stimulus site. Those authors used a special case of the ETDAS control method that Dan Gauthier and Josh Socolar devised.

Here, I will offer an explanation as to WHY algorithms like ETDAS, applied locally, cannot achieve global results in this context. Then, I'll argue that it actually IS possible to stabilize the periodic wave train if the perturbations are chosen more carefully. While these findings may seem encouraging from an experimental or clinical standpoint, I will close by describing some recent work of Flavio Fenton which I believe is even more promising.

On a compact 4-manifold, every self-dual connection and every anti-self-dual connection minimizes the Yang-Mills energy. In this talk, I will answer the converse question for compact homogeneous 4-manifolds. I will also survey related stability results in other dimensions.

We revisit the singular IIB supergravity solution describing M fractional 3-branes on the conifold (hep-th/0002159). Its 5-form flux decreases, which we explain by showing that the relevant N=1 SUSY SU(N+M) x SU(N) gauge theory undergoes repeated Seiberg-duality transformations in which N -> N-M. Far in the IR the gauge theory confines; its chiral symmetry breaking removed the singularity of hep-th/0002159 by deforming the conifold. We propose a non-singular pure-supergravity background dual to the field theory on all scales, with small curvature everywhere if the 't Hooft coupling (g_s M) is large. In the UV it approaches that of hep-th/0002159, incorporating the logarithmic flow of couplings. In the IR the deformation of the conifold gives a geometrical realization of chiral symmetry breaking and confinement. We suggest that pure N=1 Yang-Mills may be dual to strings propagating at small (g_s M) on a warped deformed conifold. We note also that the standard model itself may lie at the base of a duality cascade.

I will start with the deterministic dynamics generated by a vector field that has several unstable critical points connected by heteroclinic orbits. A perturbation of this system by white noise will be considered. I will study the limit of the resulting stochastic system in distribution (under appropriate time rescaling) as the noise intensity vanishes. It is possible to describe the limiting process in detail, and, in particular, interesting non-Markov effects arise. There are situations where this result provides more precise exit asymptotics than the classical Wentzell-Freidlin theory.

Colmez conjectured a formula relating periods of abelian varieties with complex multiplication to derivatives of Artin L-functions. I’ll explain how to prove an averaged version of Colmez’s conjectural formula, using the arithmetic of integral models of orthogonal Shimura varieties. This is joint work with F. Andreatta, E. Goren, and K. Madapusi Pera.

RNA-Seq is a new technology for measuring the content of a transcriptome using high-throughput sequencing technology. I will provide a self-contained introduction to the technology, and proceed to discuss some interesting mathematical questions we have had to address in order to realize the potential of "comparative transcriptomics" for comparing and contrasting transcriptomes. We will start with the "freshman's dream", and proceed to examine issues related to maximum matching, the (phylogenetic) space of trees and Simpson's paradox. This is joint work with my current and former students Natth Bejraburnin, Nicolas Bray, Adam Roberts, Cole Trapnell and Meromit Singer.

We consider a coupled system of Schroedinger equations, arising in quantum mechanics via the so-called time-dependent self-consistent field method. Using Wigner transformation techniques we study the corresponding classical limit dynamics in two cases. In the first case, the classical limit is only taken in one of the two equations, leading to a mixed quantum-classical model which is closely connected to the well-known Ehrenfest method in molecular dynamics. In the second case, the classical limit of the full system is rigorously established, resulting in a system of coupled Vlasov-type equations. In the second part of our work, we provide a numerical study of the coupled semiclassically scaled Schroedinger equations and of the mixed quantum-classical model obtained via Ehrenfest's method. A second order (in time) method is introduced for each case. We show that the proposed methods allow time steps independent of the semi-classical parameter(s) while still capturing the correct behavior of physical observables. It also becomes clear that the order of accuracy of our methods can be improved in a straightforward way.

This is joint work with Y. Pan that applies previous joint work with M. Sullivan. Let $\Lambda \subset \mathbb{R}^{3}$ be a Legendrian knot with respect to the standard contact structure. The Legendrian contact homology (LCH) DG-algebra, $\mathcal{A}(\Lambda)$, of $\Lambda$ is functorial for exact Lagrangian cobordisms in the symplectization of $\mathbb{R}^3$, i.e. a cobordism $L \subset \mathit{Symp}(\mathbb{R}^3)$ from $\Lambda_-$ to $\Lambda_+$ induces a DG-algebra map, $f_L:\mathcal{A}(\Lambda_+) \rightarrow \mathcal{A}(\Lambda_-).$ In particular, if $L$ is an exact Lagrangian filling ($\Lambda_-= \emptyset$) the induced map is an augmentation $\epsilon_L: \mathcal{A}(\Lambda_+) \rightarrow \mathbb{Z}/2.$ In this talk, I will discuss an extension of this construction to the case of immersed, exact Lagrangian cobordisms based on considering the Legendrian lift $\Sigma$ of $L$. When $L$ is an immersed, exact Lagrangian filling a choice of augmentation $\alpha$ for $\Sigma$ produces an induced augmentation $\epsilon_{(L, \alpha)}$ for $\Lambda_+$. Using the cellular formulation of LCH, we are able to show that any augmentation of $\Lambda$ may be induced by such a filling.

The need to deduce reduced computational models from discrete observations of complex systems arises in many climate and engineering applications. The challenges come mainly from memory effects due to the unresolved scales and nonlinear interactions between resolved and unresolved scales, and from the difficulty in inference from discrete data.

We address these challenges by introducing a discrete-time stochastic parametrization framework, through which we construct discrete-time stochastic models that can take memory into account. We show by examples that the resulting stochastic reduced models that can capture the long-time statistics and can make accurate short-term predictions. The examples include the Lorenz 96 system (which is a simplified model of the atmosphere) and the Kuramoto-Sivashinsky equation of spatiotemporally chaotic dynamics.

The AdS/CFT duality buys us a consistent non-perturbative description of gravity (in some class of backgrounds) at the price of manifest spacetime causal structure and locality. In this talk I will describe the bare beginnings of an attempt to understand these issues by studying the duality in Lorentzian signature and showing how states, operators, and classical fields are mapped between the dual theories. I will then describe two examples of classical D-brane and string probes of AdS backgrounds and show that locality in AdS shows up as scale locality for the dual boundary configurations. I will close with some suggestions for what the horizon and singularity of the AdS black hole might mean in the boundary CFT.

Braverman and Kazhdan, L. Lafforgue, Ngo, and Sakellaridis have pursued a set of conjectures asserting that analogues of the Poisson summation formula are valid for all spherical varieties. If proven, these conjectures imply the analytic continuation and functional equations of quite general Langlands L-functions (and thus, by converse theory, much of Langlands functoriality). I will explain techniques for proving the conjectures in special cases that include the first known case where the underlying spherical variety is not a generalized flag variety.

A well-known theorem of Lickorish and Wallace states that any closed orientable 3-manifold can be obtained by surgery on a link in the 3-sphere. For a given 3-manifold one can ask how "simple" a link can be used to obtain it, e.g., whether a manifold satisfying certain obvious necessary conditions on its fundamental group always arises by surgery on a knot. This question turns out to be rather subtle, and progress has been limited, but in general the answer is known to be "no." Here I’ll summarize some recent results including joint work with Matt Hedden, Min Hoon Kim, and Kyungbae Park that give the first examples of 3-manifolds with the homology of S^1 x S^2 and having fundamental group of weight 1 that do not arise by surgery on a knot in the 3-sphere.

Categorification can be viewed as the process of lifting scalar and polynomial invariants to homology theories having those invariants as (graded) Euler characteristics. In this talk, we will discuss categorification in general and as manifested in specific examples (ie Khovanov homology and knot Floer homology). Examples will be given showing how the categorified invariants are stronger and often more useful than the original invariants. I will motivate categorification using familiar constructions from (very basic) topology. It is my hope that this will make the discussion accessible to a wide audience. No prior knowledge of knot theory or category theory needed!

I will discuss a version of stochastic Loewner evolution with branching introduced in my student Vivian Olsiewski Healey's 2017 thesis. Our main motivation was to find natural conformal processes that embed Aldous' continuum random tree in the upper half plane. Unlike previous attempts that rely on lattice models or conformal welding, our model relies on a careful choice of driving measure in the Loewner evolution and the theory of continuous state branching processes. The most important feature of our model is that it has a very nice scaling limit, where the driving measure is a superprocess.

Introductory presentations for DOmath 2022.

D-branes on Calabi-Yau threefolds are interesting both as alternative probes of quantum geometry, and as nontrivial realizations of gauge theories with four supercharges. Of course, these issues are not independent. In this talk I will briefly discuss the former by describing and characterizing D-brane states at the Gepner point in the moduli space of the quintic. I will then use such D-branes to realize d=4 N=1 SUSY gauge theories, and discuss the computation of the superpotential of these theories, particularly for those realized by D6-branes wrapped around special Lagrangian submanifolds of the threefold. I will close with some speculations on the implications of mirror symmetry for these superpotentials.

Mumford-Tate groups arise as the natural symmetry groups of Hodge structures and their variations. I describe recent work with Matt Kerr on computing the Mumford-Tate group of the Kato-Usui boundary components of a degeneration of Hodge structure.

In this talk, I will discuss the geometry of finite volume complex hyperbolic surfaces and their compactifications. Finally, applications at the common edge between Riemannian and complex algebraic geometry are given.

The k-dilation of a map measures how much the map stretches k-dimensional volumes. The 1-dilation is the usual Lipschitz constant. We consider the problem of finding the smallest k-dilation among all degree 1 maps from one rectangle to another rectangle. (These are n-dimensional rectangles.) In general the linear map is far from optimal.

Two metrics are geodesically equivalent if they have the same (unparameterized) geodesics. During my talk I describe geodesically equivalent metrics on closed manifolds (which is an answer to Beltrami's question) and explain the proof of Lichnerowicz-Obata conjecture (which is an answer on the infinitesimal version of the Beltrami question known as Schouten problem).

In this talk, I will discuss a generalized Moran process from the evolutionary game theory. The generalization incorporates arrangement of by graphs and games among individuals. For these additional features, there has been consistent interest in using general spatial structure as a way to explain the ubiquitous game behavior in biological evolutions; the introduction of games leads to technical complications as basic as nonlinearity and asymmetry in the model. The talk will be centered around a seminal finding in the evolutionary game theory that was obtained more than a decade ago. By an advanced mean-field method, it reduces the infinite-dimensional problem of solving for the game fixation probabilities to a one-dimensional diffusion problem in the limit of a large population. The recent mathematical results and some related mathematical methods will be explained.

David Aldous, Probability Seminar Sept 30, 2021 TITLE: Can one prove existence of an infectiousness threshold (for a pandemic) in very general models of disease spread? ABSTRACT: Intuitively, in any kind of disease transmission model with an infectiousness parameter, there should exist a critical value of the parameter separating a very likely from a very unlikely resulting pandemic. But even formulating a general conjecture is challenging. In the most simplistic model (SI) of transmission, one can prove this for an essentially arbitrary large weighted contact network. The proof for SI depends on a simple lemma concerning hitting times for increasing set-valued Markov processes. Can one extend to SIR or SIS models over similarly general networks, where the lemma is no longer applicable?

A large class of string compactifications are warped products of the compactification manifold and our four-dimensional spacetime, where the spacetime metric depends on the position in the compactification manifold. New, calculable regimes will arise when warp factors are large and when several different regions with interesting physics are spatially separated in the compactification manifold. We discuss a variant of the Randall-Sundrum scenario which, at low energies, captures the essential features of this regime. We discuss the low-energy effective theory of such a scenario, including the important scales and leading irrelevant operators. We comment critically on the conjectured relationship between the Randall-Sundrum proposal and the AdS/CFT correspondence. Finally, we discuss some phenomenological consequences of these models: namely, a new mediation mechanism for supersymmetry breaking, and a dark matter candidate which might explain some puzzles in the structure of galactic dark matter halos.

The inviscid Burgers' equation has the remarkable property that its dynamics preserve the class of spectrally negative L\'{e}vy initial data, as observed by Carraro and Duchon (statistical solutions) and Bertoin (entropy solutions). Further, the evolution of the L\'{e}vy measure admits a mean-field description, given by the Smoluchowski coagulation equation with additive kernel. In this talk we discuss ongoing efforts to generalize this result to scalar conservation laws, a special case where this is done, and a connection with integrable systems. Includes work with F. Rezakhanlou.