Past events

Stochastic optimization for sequential decision problems under uncertainty arises in many settings, and as a result as evolved under several canonical frameworks with names such as dynamic programming, stochastic programming, optimal control, robust optimization, and simulation optimization (to name a few).  This is in sharp contrast with the universally accepted canonical frameworks for deterministic math programming (or deterministic optimal control).  We have found that these competing frameworks are actually hiding different classes of policies to solve a single problem which encompasses all of these fields.  In this talk, I provide a canonical framework which, while familiar to some, is not universally used, but should be.  The framework involves solving an objective function which requires searching over a class of policies, a step that can seem like mathematical hand waving.  We then identify four fundamental classes of policies, called policy function approximations (PFAs), cost function approximations (CFAs), policies based on value function approximations (VFAs), and lookahead policies (which themselves come in different flavors).  With the exception of CFAs, these policies have been widely studied under names that make it seem as if they are fundamentally different approaches (policy search, approximate dynamic programming or reinforcement learning, model predictive control, stochastic programming and robust optimization).  We use a simple energy storage problem to demonstrate that minor changes in the nature of the data can produce problems where each of the four classes might work best, or a hybrid.  This exercise supports our claim that any formulation of a sequential decision problem should start with a recognition that we need to search over a space of policies.

I will discuss some recent results on the distribution of the real-analytic Eisenstein series on thin sets, such as a geodesic segment. These investigations are related to mean values of the Riemann zeta function, and have connections to quantum chaos.

Despite many years of intensive research, the modeling of contact lines moving by spreading and/or evaporation still remains a subject of debate nowadays, even for the simplest case of a pure liquid on a smooth and homogeneous horizontal substrate. In addition to the inherent complexity of the topic (singularities, micro-macro matching, intricate coupling of many physical effects, …), this also stems from the relatively limited number of studies directly comparing theoretical and experimental results, with as few fitting parameters as possible. In this presentation, I will address various related questions, focusing on the physics invoked to regularize singularities at the microscale, and discussing the impact this has at the macroscale. Two opposite “minimalist” theories will be detailed: i) a classical paradigm, based on the disjoining pressure in combination with the spreading coefficient; ii) a new approach, invoking evaporation/condensation in combination with the Kelvin effect (dependence of saturation conditions upon interfacial curvature). Most notably, the latter effect enables resolving both viscous and thermal singularities altogether, without needing any other regularizing effects such as disjoining pressure, precursor films or slip length. Experimental results are also presented about evaporation-induced contact angles, to partly validate the first approach, although it is argued that reality might often lie in between these two extreme cases.

We present a trust region algorithm for solving nonconvex optimization problems that, in the worst-case, is able to drive the norm of the gradient of the objective below a prescribed threshold $\epsilon > 0$ after at most ${\cal O}(\epsilon^{-3/2})$ function evaluations, gradient evaluations, or iterations.  Our work has been inspired by the recently proposed Adaptive Regularisation framework using Cubics (i.e., the ARC algorithm), which attains the same worst-case complexity bound.  Our algorithm is modeled after a traditional trust region algorithm, but employs modified step acceptance criteria and a novel trust region updating mechanism that allows it to achieve this desirable property.  Importantly, our method also maintains standard global and fast local convergence guarantees.

Inspired by a question posed by Lax, in recent years it has received  

an increasing attention the study of quantitative compactness  

estimates for the solution operator $S_t$, $t>0$ that associates to  

every given initial data $u_0$ the corresponding solution $S_t u_0$ of  

a conservation law or of a first order Hamilton-Jacobi equation.



Estimates of this type play a central roles in various areas of  

information theory and statistics as well as of ergodic and learning  

theory. In the present setting, this concept could provide a measure  

of the order of ``resolution'' of a numerical method for the  

corresponding equation.



In this talk we shall first review the results obtained in  

collaboration with O. Glass and K.T. Nguyen, concerning the  

compactness estimates for solutions to conservation laws. Next, we  

shall turn to the  analysis of the Hamilton-Jacobi equation pursued in  

collaboration with P. Cannarsa and K.T.~Nguyen.

We say a group is accessible if the process of iteratively decomposing G as an amalgamated free product or HNN extension over a finite group terminates in a finite number of steps. We will see Dunwoody's proof that FP2 groups are accessible, but that finitely generated groups need not be. If time permits, we will examine generalizations by Bestvina-Feighn, Sela and Louder.

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The notion of prime decomposition will be defined and illustrated for
manifolds. Two proofs of existence will be given, including Kneser's
classical proof using normal surface theory.

Let G be a transitive permutation group. If G is finite, then a classical theorem of Jordan implies the existence of fixed-point-free elements, which we call derangements. This result has some interesting and unexpected applications, and it leads to several natural problems on the abundance and order of derangements that have been the focus of recent research. In this talk, I will discuss some of these related problems, and I will report on recent joint work with Hung Tong-Viet on primitive permutation groups with extremal derangement properties.

I present a new method for optimizing quadratic functions subject to simple bound constraints.  If the problem happens to be strictly convex, the algorithm reduces to a highly efficient method by Dostal and Schoberl.  Our algorithm, however, is also able to efficiently solve nonconcex problems. During this talk I will present the algorithm, a sketch of the convergence theory, and numerical results for convex and nonconvex problems.

I will present a joint work with Leonid Kovalev and Jani Onninen. The proofs are  based on topological arguments (degree theory)  and the properties  of planar  quasiconformal mappings. These new ideas  apply well to inner variational equations of conformally invariant energy integrals; in particular, to the Hopf-Laplace equation for the Dirichlet integral.

We work in the context of Markovian rough paths associated to a class of uniformly subelliptic Dirichlet forms and prove an almost-Gaussian tail-estimate for the accumulated local p-variation functional, which has been introduced and studied by Cass, Litterer and Lyons. We comment on the significance of these estimates to a range of currently-studied problems, including the recent results of Ni Hao, and Chevyrev and Lyons.

The triangulation conjecture stated that any n-dimensional topological manifold is homeomorphic to a simplicial complex. It is true in dimensions at most 3, but false in dimension 4 by the work of Casson and Freedman. In this talk I will explain the proof that the conjecture is also false in higher dimensions. This result is based on previous work of Galewski-Stern and Matumoto, who reduced the problem to a question in low dimensions (the existence of elements of order 2 and Rokhlin invariant one in the 3-dimensional homology cobordism group). The low-dimensional question can be answered in the negative using a variant of Floer homology, Pin(2)-equivariant Seiberg-Witten Floer homology. At the end I will also discuss a related version of Heegaard Floer homology, which is more computable.

The main goal of the talk is to set up a framework for constructing the likelihood for discretely observed RDEs. The main idea is to contract a function mapping the discretely observed data to the corresponding increments of the driving noise. Once this is known, the likelihood of the observations can be written as the likelihood of the increments of the corresponding noise times the Jacobian correction.

Constructing a function mapping data to noise is equivalent to solving the inverse problem of looking for the input given the output of the Ito map corresponding to the RDE. First, I simplify the problem by assuming that the driving noise is linear between observations. Then, I will introduce an iterative process and show that it converges in p-variation to the piecewise linear path X corresponding to the observations. Finally, I will show that the total error in the likelihood construction is bounded in p-variation.

A long-standing problem in almost complex geometry has been the question of existence of (complete) inhomogeneous nearly Kahler 6-manifolds. One of the main motivations for this question comes from $G_2$ geometry: the Riemannian cone over a nearly Kahler 6-manifold is a singular space with holonomy $G_2$.

Viewing Euclidean 7-space as the cone over the round 6-sphere, the induced nearly Kahler structure is the standard $G_2$-invariant almost complex structure on the 6-sphere induced by octonionic multiplication. We resolve this problem by proving the existence of exotic (inhomogeneous) nearly Kahler metrics on the 6-sphere and also on the product of two 3-spheres. This is joint work with Lorenzo Foscolo, Stony Brook.

This talk will describe methods for computing sharp upper bounds on the probability of a random vector falling outside of a convex set, or on the expected value of a convex loss function, for situations in which limited information is available about the probability distribution. Such bounds are of interest across many application areas in control theory, mathematical finance, machine learning and signal processing. If only the first two moments of the distribution are available, then Chebyshev-like worst-case bounds can be computed via solution of a single semidefinite program. However, the results can be very conservative since they are typically achieved by a discrete worst-case distribution. The talk will show that considerable improvement is possible if the probability distribution can be assumed unimodal, in which case less pessimistic Gauss-like bounds can be computed instead. Additionally, both the Chebyshev- and Gauss-like bounds for such problems can be derived as special cases of a bound based on a generalised definition of unmodality.

In joint work with Todor Tsankov, we show that the automorphism groups of countable, omega-categorical structures have Kazhdan's property (T). The proof uses Tsankov's work on the unitary representations of these groups, together with a construction of a particular free subgroup of the automorphism group.