Analysis and Probability Seminar


The seminar is joint for the division of Analysis and Probability. Unless otherwise indicated, the seminar takes place Tuesdays at 1515-1615 in MVL:14.

Past Seminars:

18-01-23
Jérémie Joudioux (Radboud University Nijmegen)
The vector-field method for the relativistic transport equation with application to the Einstein-Vlasov system
The vector-field method, developed by Klainerman in the 80s was the key to understand the global existence and asymptotic behaviour of solutions to some nonlinear wave equation. This method is based on the construction of appropriate commutators with the wave equation. Hence, coercive estimates for a solution of a wave equation can be extended to derivatives of this solution, and pointwise weighted estimates can be derived. I will briefly present in this talk this vector-field method, and explain how it extends to relativistic transport equations. Then, after briefly introducing the stability problems in General Relativity, I will explain how this method can be used to prove the stability of the "ground state" of the Einstein equations in the presence of a matter field modelled by a transport equation (Einstein-Vlasov system). This is joint work with David Fajman (Vienna) and Jacques Smulevici (Orsay-ENS) (arXiv:1707.06141).

The first two parts of the talk will be kept as elementary as possible to an audience familiar with basic PDE estimates. The last part of the talk will give a quick and simplified overview on the proof of the stability of Minkowski space.


18-01-30
Hasse Carlsson
Estimates of the renewal measure
Let $\mu$ be a nonlattice probability meassure on the line, and $\nu=\sum_0^\infty \mu^{n*}$ its renewal measure. Blackwell's renewal theorem states that
$$
\lim_{x\to+\infty} \nu(x+I)=|I|/\mu_1,
$$
where $\mu_1=\int xd\mu(x)$ is the first moment of $\mu$ and $|I|$ the length of the interval $I$.
A lot of papers have studied the rate of this convergence. My talk is yet another attempt in this quest. In particular I am intersted in the case where $\mu$ has finite moments of order $\alpha$ (i.e. $\int|x|^\alpha d\mu(x)<\infty$) when $1>\akpha<2$.
The methods are Fourier analytic.

abstract_fornyelse.pdfDownload Abstract

The talk is aimed at the whole division.


18-02-06
Peter Sjögren
Estimates for some operators associated with the Laplacian with drift in Euclidean space
Let
$v\neq 0$ be a vector in $\mathbb{R}^n$. Consider the Laplacian on $\mathbb{R}^n$ with drift $\Delta_v=\Delta+2v\cdot\Nabla$ and the measure $d\mu(x)=e^{2\langle v,x\rangle}dx$, with respect to which
$\Delta_v$ is self-adjoint. This measure has exponential growth with respect to the Euclidean distance. We study weak type $(1,1)$ and other sharp endpoint estimates for the related Riesz transforms of any order, and also for the vertical and horizontal Littlewood-Paley-Stein functions for the associated heat and Poisson semigroups.
This is joint work with Hong-Quan Li, Shanghai.

abstract.drift.pdfDownload Abstract



The talk is intended for a general audience of analysts.


18-02-20
Przemys
ław Ohrysko

Spectrally reasonable measures

Let us recall that the measure on the circle group is said to have a natural spectrum iff its spectrum is equal to the closure of the image of the Fourier-Stieltjes transform. As the set of all measures does not possess good algebraic properties (it is not closed under addition, in particular) we introduced the concept of spectrally reasonable measures, i. e. a measure perturbing any measure with a natural to spectrum to a measure with a natural spectrum. It appears that the collection of all such measures has a Banach algebra structure. In my talk I will describe a wide class of examples of measures from this class including absolutely continuous ones. Moreover, I will prove that (except trivial cases) no discrete measure is reasonable. The presentation is based on the paper: ‘Spectrally reasonable measures’, St. Petersburg Math. J. 28 (2017), 259-271 written in collaboration with Michał Wojciechowski.

 

The talk is aimed at everybody familiar with the basics of harmonic analysis and the Banach algebra theory.

 



18-03-13

Ramiz Reza (Indian Institute of Science, Bangalore)

Curvature inequalities for operators in the Cowen-Douglas class of a planar domain


Chalmers Abstract.pdf
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The talk is aimed at an audience with background and interests in Operator Theory and Complex Analysis.



18-03-20

Sergey Zuev

Self-decomposable point processes

Self-decomposable (SD) point processes constitute the class of point processes arising as a limit of superpositions of independent point processes. They are subclass of infinitely divisible point processes and contain the class of stable point processes. We characterise SD for the most general scheme involving independent branching operation on points, in particular, thinning. We show that a (regular) point process is SD iff it is representable as a series of branching operations applied to independent realisations of another point process whose scaled distribution correspond to the Levy measure.

The talk is aimed at a general audience familiar with basic probability.



18-04-10
Simone Calogero

Self-gravitating elastic balls


A body is said to be self-gravitating if the only forces acting on the body are due to its internal strain and to its own self-generated gravitational field. The body is in equilibrium when these two forces balance each other at any point of the body. Many astrophysical objects (e.g. stars) are modelled as static self-gravitating bodies. Mathematically they are described as solutions to non-linear systems of elliptic PDEs. In this talk I will show how to prove existence of solutions using variational methods when the body is made up of elastic matter.

The talk is intended for a general audience of analysts.



18-05-22
Johan Tykesson

Generalized divide and color models

In this talk, we consider the following model: one starts with a finite or countable set V, a random partition of V and a parameter p in [0,1]. The corresponding Generalized Divide and Color Model is the {0,1}-valued process indexed by V obtained by independently, for each partition element in the random partition chosen, with probability p, assigning all the elements of the partition element the value 1, and with probability 1-p, assigning all the elements of the partition element the value 0.

A very special interesting case of this is the ``Divide and Color Model'' (which motivates the name we use) introduced and studied by Olle Häggström.

Some of the questions which we study here are the following. Under what situations can different random partitions give rise to the same color process? What can one say concerning exchangeable random partitions? What is the set of product measures that a color process stochastically dominates? For random partitions which are translation invariant, what ergodic properties do the resulting color processes have?

The motivation for studying these processes is twofold; on the one hand, we believe that this is a very natural and interesting class of processes that deserves investigation and on the other hand, a number of quite varied well-studied processes actually fall into this class such as the Ising model, the stationary distributions for the Voter Model, random walk in random scenery and of course the original Divide and Color Model.


The talk is based on joint work with Jeff Steif.


The talk is aimed at a general probability audience, but is probably accessible to a general analysis audience as well.



18-05-24 (Thursday at 15:15 in MVL:14)
​Artur Alho
Generic blow up for solutions of the wave equation towards Big Bang singularities
We study the behaviour of smooth solutions to the wave-equation $\Box_g\phi=0$ near spacelike cosmological singularities. We consider as fixed backgrounds the explicit spatially homogenous flat Friedmann-Lemaitre-Robertson-Walker (flat FLRW) and Kasner metrics, and provide generic conditions on initial data given at a spacelike Cauchy hypersurface, such that the solutions blow up at polynomial, and logarithmic rates respectively.
This is joint work with Grigorios Fournodavlos (Cambridge) and Anne Franzen (Lisbon).

The talk is aimed at an audience familiar with Analysis and Geometry.

 

18-05-29

Perla Sousi (University of Cambridge)

Capacity of random walk and Wiener sausage in 4 dimensions

In four dimensions we prove a non-conventional CLT for the capacity of the range of simple random walk and a strong law of large numbers for the capacity of the Wiener sausage. This is joint work with Amine Asselah and Bruno Schapira.


This talk is primarily for a probability audience but could be of interest to a wider analysis audience.


18-06-08 (Friday at 13:15 in Pascal)
Eric Carlen (Rutgers University)

Inequalities for $L^p$-norms that sharpen the triangle inequality and complement Hanner's Inequality

In 2006 Carbery raised a question about an improvement on the na\"ive norm inequality $\|f+g\|_p^p \leq 2^{p-1}(\|f\|_p^p + \|g\|_p^p)$ for two functions in $L^p$ of any measure space. When $f=g$ this is an equality, but when the supports of  $f$ and $g$ are disjoint the factor $2^{p-1}$ is not needed. Carbery's questiona concerns a proposed interpolation between the two situations for $p>2$. The interpolation parameter measuring the overlap is $\|fg\|_{p/2}$. We prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for {\it all } $p$.

This is joint work with Rupert Frank and Elliott Lieb



The talk is aimed at a general analysis and probability audience.


18-06-13 (Wednesday at 15:15 in MVL:14)
Christian Hagendorf (Université Catholique de Louvain)
On the transfer-matrix of the supersymmetric eight-vertex model
The topic of this talk is the square-lattice eight-vertex model with weights $a,b,c,d$ obeying the relation $(a^2+ab)(b^2+ab)=(c^2+ab)(d^2+ab)$. In this case, the model possesses a hidden supersymmetry. I will show how to use this supersymmetry to prove that the transfer matrix of the model for $L=2n+1$ vertical lines and periodic boundary conditions along the horizontal direction possesses the doubly degenerate eigenvalue $\Theta_n = (a+b)^{2n+1}$. This proves a conjecture by Stroganov.

The talk is aimed at an audience working in mathematical physics.


18-06-21 (Thursday at 15:15 in MVL:14)
Robin Deeley (University of Colorado, Boulder)

Minimal dynamical systems and groupoids with prescribed K-theory

I will speak about joint work in progress with Ian Putnam and Karen Strung. The goal of the project is to study the existence of minimal dynamical systems and more generally minimal equivalence relations. In particular, I will discuss the following question: given a compact Hausdorff space does there exist a minimal homeomorphism on it? Although the answer is no, a similar question has a positive answer for any finite CW-complex. This question and the question of which C*-algebras can be realized as groupoid C*-algebras related to minimal dynamical systems are the motivation for our constructions.

 

The prerequisites for the first half of the talk are basic metric space theory and equivalence relations. For the second half some knowledge of C*-algebra theory (in particular crossed products) will be required.


18-06-28 (Thursday at 10:00 in MVL:14)
Paola Rioseco (Universidad Michoacana de San Nicolás de Hidalgo)

Mixing phenomena in relativistic kinetic gas distributions around black holes
Phase space mixing is a well-known phenomena which is the driving effect in many different physical fields, including galactic dynamics (as revealed by Lynden-Bell) and plasma physics (as displayed recently by Villani and Mouhot in the context of Landau damping).

As a first approximation to understand this phenomenon, I will explain the mixing and why it occurs for the simplest case of a potential in one dimension and for gas particles that satisfy the Vlasov equation. Subsequently, I will discuss some applications for the evolution of inhomogeneities in dark matter halos. Finally, I show that the same phenomena occurs for a collisionless, relativistic kinetic gas which is trapped in the gravitational potential of a (rotating) Kerr black hole.

This talk is aimed at an audience familiar with kinetic theory.

Published: Mon 12 Dec 2016. Modified: Thu 28 Jun 2018