(FAU) 
Copula function: a formula that killed Wall Street  Copula function is a classical probabilistic formula designed to capture the dependency structure between two random variables. Recently this approach has gained some momentum, primarily because it addresses the nonGaussian settings which in turn cannot be described by correlation coefficients. The talk will have a large, nontechnical portion, where I plan to introduce some basic facts and motivations. I also plan to describe my contribution to this field as well as some new ideas and projects in progress.  
(Universitat de Barcelona) 
Zeros of optimal polynomial approximants: Jacobi matrices and Jentzschtype theorems  I will present a recent work with B\´en\´eteau, Khavinson, Liaw and Simanek where we study the structure of the zeros of polynomials appearing in the study of cyclicity in Hilbert spaces of analytic functions. We find the minimum possible modulus of occurring zeros via a nonlinear extremal problem associated with norms of Jacobi matrices. We examine global properties of these zeros and prove Jentzschtype theorems describing where they accumulate.  
(FAU) 
A short introduction to constructive analysis  Constructive analysis may be characterized by the fact that it only allows the construction of solutions to problems that are 'well posed' as described by Hadamard in his "Lectures on Cauchy's problem in linear partial differential equations". For this characterization to work we have to weaken Hadamard's notion, but the key idea remains that the solution must depend continuously on the data. As an illustration, we will look at Brouwer's fixed point theorem and the fundamental theorem of algebra, both of which can be thought of as problems: we must construct a fixed point or a root.  
(FAU) 
Compressed Sensing and Machine Learning  In this talk we will see how sparse vectors can be randomly undersampled and then perfectly recovered w.h.p. Connections to highdimensional geometry and applications in cryptography and machine learning will be discussed.  

The Solution of Uhlenbeck’s Unsolved Problem B  Boltzmann (transport) equations arise in semiconductors, plasmas, nuclear reactors, and any other situations in which one needs to keep track of the velocity (momentum) distribution of particles at each point in space, instead of just the spatial density. In these problems, one can prescribe boundary conditions only for velocities pointing into the region; obtaining the distribution of particles for outgoing velocities is part of the solution. Analyzing these problems requires constructing halfrange expansions: unlike standard eigenfunction expansions, in which one seeks to reconstruct a given function over its entire domain using all the eigenfunctions, in halfrange expansions one needs to reconstruct a given function over just half the domain, but one can only use half the eigenfunctions. We use complex variable theory, namely, a variant of the WeinerHopf technique, to create a method for constructing such halfrange expansions explicitly. This method is then used to solve Uhlenbeck’s Unsolved Problem B, where it is found that the dimensionless Milne length is given by the Riemann zeta function at 1/2, namely 1.46035…  


(VU Amsterdam) 
Validated computations for connecting orbits of ODEs.  In this talk we present a computerassisted procedure for proving the existence of transverse heteroclinic orbits connecting hyperbolic equilibria of polynomial vectorfields. The idea is to compute highorder Taylor approximations of local charts on the (un)stable manifolds by using the Parameterization Method and to use Chebyshev series to parameterize the orbit in between. The existence of a heteroclinic orbit can then be established by setting up an appropriate fixedpoint problem amenable to computerassisted analysis. In addition, we explain how this method can be used to perform validated continuation. This is joint work with Jan Bouwe van den Berg and Christian Reinhardt.  
(FAU) 
Validated computation of transport barriers using rigorous integration techniques  
(Oklahoma State) 
Equidistribution of Zeros of Random Polynomials  pdf of abstract  
(VU Amsterdam) 
Rigorous computation of invariant manifolds of periodic orbits for vector fields  Invariant sets as attractors, periodic orbits, (un)stable manifolds are fundamental objects in the analysis of dynamical systems, because they provide a skeleton for long time dynamics and can be used to prove existence of chaotic motion. In this talk it is presented a technique to compute within rigorous bounds the local invariant manifolds associated to hyperbolic periodic orbits. Following the Parametrisation Method and the Floquet theory, a FourierTaylor power series is introduced to parameterise the invariant manifold. Combining rigorous computations and analytical estimates, the coefficients of the series are computed and the convergence is proven within explicit bounds. Some examples are given to illustrate the methodology.  
(Nova Southeastern University) 
Fixed Point Theorems and Applications to Differential Equations  A common method of showing the existence of (positive) solutions to various types of integral, difference, differential, and dynamic equations is by applying a fixed point theorem to an operator. In this talk, a brief introduction into fixed point theory is provided including wellknown fixed point theorems and applications. Afterward, we will discuss a new class of fixed point theorems dubbed Avery type fixed point theorems which are extensions of the LeggettWilliams fixed point theorem. An application of one such theorem will be provided.  
(FAU) 
Parametrization method for periodic manifolds using Chebyshev polynomials.  Computation of manifolds is a key element to understand the dynamics of a system. They provide a better understanding of the behavior of orbits starting near a given equilibrium. In this talk I will introduce Chebyshev polynomials and how one can use them to compute (un)stable manifolds for periodic orbits. The method will be applied to the Lorenz system and the circular restricted three body problem to illustrate results.  
(FAU) 
Parameterization of invariant vector bundles with applications to stability of nonlinear waves  I'll discuss a parameterization method for invariant vector/frame bundles associated with stable/unstable manifolds of a hyperbolic fixed point, and illustrate some applications.  
(HEC Montreal) 
Price Dynamics in a General Markovian Limit Order Book  We propose a simple stochastic model for the dynamics of a limit order book, extending the recent work of Cont and de Larrard, where the price dynamics are endogenous, resulting from market transactions. We also show that the diffusion limit of the price process is the socalled Brownian meander.  

