Abstract: We define and compute ``analytic'' intersection numbers of quadratic CM-cycles on Lubin-Tate (LT) space at infinite level. This is based on the formalism of tropical (p,q)-forms by Gubler-KÃ¼nnemann and the description of the infinitel level LT-space by Scholze-Weinstein.
The intersection problem itself plays a role in the linear Arithmetic Fundamental Lemma conjecture of W. Zhang. Our approach is motivated by a recent result of Q. Li who gave a formula for the corresponding intersection numbers on formal models. A posteriori, we see that our analytically defined numbers coincide the ones from formal models.
Abstract: By elementary linear algebra, any complex matrix in the special linear group can be factored into a product of elementary matrixes, i.e. matrixes with ones on the diagonal and no more than one non-zero element outside the diagonal. The corresponding factorisation problem for SLn valued holomorphic functions on Stein manifolds is called the Gromov-Wasserstein problem and was solved by Ivarsson and Kutzschebauch in 2008. In this talk I will adress a 'vector bundle analog' of this problem. In particular, I will provide a theorem ruling out topological obstructions. This is joint work with Erlend F Wold at University of Oslo.
Abstract: Optimization problems with partial differential equations (PDEs) as
constraints is known as PDE constrained optimization. In this talk, we
will discuss an abstract formulation of the problem as well as methods
for solving such problems. We then present two specific problems. One
application involves elastic waves propagating through a piezoelectric
solid where the PDE constraints take the form of a coupled PDE
system.The other application involves fractional (nonlocal) PDE
constraints, which have various applications including image denoising.
Abstract: Generally, Vlasov equation and Vlasov-Maxwell equations are considered as kinetic equations and are addressed as Cauchy problems. Though, when one listens to the Physics community, they often describe their results as 'the response' of a plasma, stressing on the uniqueness of a solution without prescribing the Cauchy data.
This is only an apparent contradiction, and in this talk (joint work with Omar Maj, NMPP, IPP, Max Planck Institute Garching) describes the obtention of this response for the linearized Vlasov-Maxwell equations around a Maxwellian-type density function using distribution theory.
Abstract: In this talk, I shall consider products of i.i.d. matrices $g_j(t),\ j\ge 1,$ where $t$ is a parameter, $\ t\in T,$ and $T$ is a compact metric space. Matrices $g(\cdot)$ are continuous functions of $t$. I shall discuss necessary and sufficient conditions under which with probability 1
\[
\frac1n \ln\| g_n(t)\ldots g_1(t)\| \lambda(t)\ \ \text{ uniformly in $t\in T$,}
\]
where $\lambda(t)$ is the corresponding Lyapunov exponent.
I shall then explain what happens when $g_j$ are matrices corresponding to the Anderson model and the parameter is the energy $E$ and, time permitting, shall discuss some open problems.
Abstract: Minimal discrete energy problems arise in a variety of scientific contexts---such as crystallography, nanotechnology, information theory, and viral morphology, to name but a few. Our goal is to analyze the structure of configurations generated by optimal (and near optimal) N-point configurations that minimize the Riesz s-energy over a bounded surface in Euclidean space. The Riesz s-energy potential is simply given by 1/r^s, where r denotes the distance between a pair of points; it is a generalization of the familiar Coulomb potential. We show how such potentials and their minimizing point configurations are ideal for use in sampling surfaces (and even generating a "near perfect" poppy-seed bagel). Connections to the recent breakthrough results by M. Viazovska et al on best-packing and universal optimality in 8 and 24 dimensions will be discussed.
Abstract: In this talk, I will describe experiments on a chaotic electronic circuit that can be used as a high speed true random number generator. This circuit can be modified to act as a Physically Unclonable Function, which are novel cybersecurity devices used for device authentication, tamper-proofing, and key generation.
Abstract: In the first part of the talk I will review several limit theorems for stationary processes such as hyperbolic and expanding dynamical systems.
In the second part I will discuss recent results for random, uniform and non uniform, distance expanding dynamical systems. The main focus will be the local central limit theorem, and if time permits I will also discuss additional results such as the Berry-Esseen theorem (optimal convergence rate in the central limit theorem) and an almost sure central limit theorem. This part is partially based on joint work with Yuri Kifer.
Abstract: I will present a paper by Bin Guo and Jian Song in which they derive interior Schauder estimates for linear elliptic and parabolic equations with background Kaehler metric of conical singularities along a divisor of simple normal crossings.
Abstract: If we want the solution to the Schrodinger equation to converge to its initial data pointwise, what's the minimal regularity condition for the initial data should be? I will present recent progress on this classic question of Carleson. This pointwise convergence problem is closely related to other problems in PDE and geometric measure theory, including spherical average Fourier decay rates of fractal measures, Falconer's distance set conjecture, etc. All these problems essentially ask how to control Schrodinger solutions on sparse and spread-out sets, which can be partially answered by several recent results derived from induction on scales and Bourgain-Demeter's decoupling theorem.
Abstract: In this talk, I present computational methodologies for extracting dynamic neural functional networks that underlie behavior. These methods aim at capturing the sparsity, dynamicity and stochasticity of these networks, by integrating techniques from high-dimensional statistics, point processes, state-space modeling, and adaptive filtering. I demonstrate their utility using several case studies involving auditory processing, including 1) functional auditory-prefrontal interactions during attentive behavior in the ferret brain, 2) network-level signatures of decision-making in the mouse primary auditory cortex, and 3) cortical dynamics of speech processing in the human brain.