# AP-seminar 2018

18-01-23
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 measure 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 interested 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. Download 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.

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

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.

18-08-28 (at 13:15)
Sven Raum (Stockholms universitet)
C*-superrigidity of 2-step nilpotent groups
A discrete group is called C*-superrigid, if it can be recovered from its reduced group C*-algebra.  An important question on this notion asks whether all torsion-free discrete groups are C*-superrigid, paralleling Kaplansky's unit conjecture for complex group rings. However, until 2017 the only know such examples of torsion-free C*-superrigid groups were abelian.

In this talk, I will introduce the topic of C*-superrigidity assuming only a background in functional analysis and basic group theory. I will then proceed to describe developments since 2017 and finish my talk by a discussion of my recent result with Caleb Eckhardt, showing that all finitely generated, torsion-free, 2-step nilpotent groups are C*-superrigid.  This result provided the first natural class of non-abelian torsion-free groups known to have this property.

The talk is aimed at an audience familiar with basic functional analysis and group theory.

18-10-02
Przemysław Ohrysko
Inversion problem in measure and Fourier-Stieltjes algebras

In my talk, I will present the recent developments on the inversion problem in measures in Fourier-Stieltes algebras proposed by N. Nikolski in the paper ‘In search of the invisible spectrum’. The framework within the Banach algebra of all complex-valued Borel regular measures is as follows: suppose that we are given a measure on a locally compact Abelian group with norm one and infimum of the modulus of its Fourier-Stieltjes transform greater than some fixed constant. What is the minimal value of this constant assuring the invertibility of the measure? What can be said about the inverse? The non-triviliaty of this problem is justified via existence of the Wiener-Pitt phenomenon. Similar questions for Fourier-Stieltjes algebras (associated with non-commutative groups) will be discussed. The presented results can be found on arxiv.org in the preprint with the same title.

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

18-10-09

Gunnar Carlsson (Stanford University)
Persistent Homology, Theory and  Applications
Persistent homology has been developed since roughly 2000 as a tool for "measuring" the shape of data sets and as a tool for feature generation for certain kinds of unstructured data.  Ordinary homology has been used for many years to study the shape of spaces where one has complete information about the space, and persistent homology adapts these techniques to sampled situations.  I will survey the subject with examples.

The talk is intended to introduce the methods of persistent homology to a general audience, so no particular knowledge of topology is required. I will assume an elementary knowledge of linear algebra.

18-10-15 (Monday at 13:15 in MVH:12)
Lashi Bandara (University of Potsdam)
First-order elliptic boundary value problems beyond self-adjoint induced boundary operators
The Bär-Ballmann framework is a comprehensive framework to consider  elliptic boundary value problems (and also their index theory) for first-order elliptic operators on manifolds with compact and smooth boundary. A fundamental assumption in their work is that the induced operator on the boundary is symmetric. Many operators satisfy this requirement including the Hodge-Dirac operator as well as the Atiyah-Singer Dirac operator. Recently, there has been a desire to study more general operators with the quintessential example being the  Rarita-Schwinger Dirac operator, which is an operator that fails to satisfy this hypothesis.

In this talk, I will present recent work with Bär where we dispense the symmetry assumption and consider general elliptic operators. The ellipticity of the operator still allows us to understand the spectral theory of the induced adapted operator on the boundary, modulo a lower order additive perturbation, as bi-sectorial operator. We use a mixture of methods coming from pseudo-differential operator theory, bounded holomorphic functional calculus, semi-group theory as well as methods arising from the resolution of the Kato square root problem to recover many of the results of the Bär-Ballman framework.

The talk is aimed at an audience familiar with geometry (manifolds and bundles), some operator theory (adjoints, maximal operator, elliptic first-order).

18-10-18 (Thursday at 09:00 in MVL:14)
David Fajman (University of Vienna)
Stability of the Milne model for the Einstein-Vlasov system
The Milne model is the only cosmological model known to be future stable for the vacuum Einstein equations with vanishing cosmological constant. We discuss the generalization of this result to the non-vacuum setting where massive collisionless particles are present. In particular, we focus on methods to control matter of this type in cosmological spacetimes with slow expansion. This is joint work with L. Andersson.

The talk is aimed at an audience familiar with mathematical physics.

18-10-30
Andrew McKee
Multipliers and approximation properties
A locally compact group has several associated Banach algebras, including the group C*-algebras and the Fourier algebra. A natural question is: how are properties of the group reflected in the properties of these algebras, and vice versa? I will explain how this question can be answered for certain properties using the notion of a Herz-Schur multiplier, going back to work of Haagerup and his collaborators. In the second half of the talk I will explain joint work with Skalski, Todorov and Turowska, extending these techniques to study properties of C*-algebras associated to group actions.

The talk is aimed at an audience familiar with the basics of groups and abstract harmonic analysis.

18-11-06
Genkai Zhang
Convexity of energy functions on Teichmüller space

The length function of a geodesic of fixed homotopy type on Riemann surfaces defines a function on Teichmüller space, and has been studied extensively. In particular it is geodesic convex and plurisubharmonic. We find a general variational formula for energy function E for harmonic maps u from a Riemannian manifold to Riemann surfaces and prove its convexity. (Joint work with I. Kim and X. Wan.)

This talk is aimed at an audience familiar with some basic analysis and Riemann geometry.

18-11-08 (Thursday at 15:15 in MVL:14)
Eero Saksman (University of Helsinki)
On imaginary chaos
We consider random imaginary multiplicative chaos $e^{i{beta X}$  for general log-normal Gaussian fields $X$. In particular, we will describe regularity properties of $\cos (\Beta X)$ and growth of moments or general log-normal Gaussian fields. The talk is based on joint work with Janne Junnila (University of Helsinki) and Christian Webb (Aalto University).

The talk will contain both probability theory and analysis and so could be of interest to anyone in the division. Some familiarity with probability will be assumed.

18-11-13
Ivar Lyberg (International Institute of Physics)
The six vertex model with various boundary conditions

The six vertex model with fixed boundary conditions is an example of a model where different phases may coexist. The study of the arctic curves, that is to say, the curves that separate the different phases, is today a very active topic of research. In such cases where the partition function can be written as a determinant, it has sometimes been possible to find algebraic equations for these curves in the thermodynamic limit. For example, with Domain Wall Boundary Conditions, a disordered phase coexists with ferromagnetic phases when the parameter ∆ = 0. The arctic curve in this case is an ellipse.

I will discuss new numerical results on three variations of Domain Wall Boundary Conditions, namely Partial Domain Wall Boundary Conditions, Reflective End Boundary Conditions and Half Turn Boundary Conditions. I will present conjectures on the nature of the arctic curves in the two latter cases.

The talk is aimed at an audience familiar with mathematical physics.

18-11-20

Lijia Ding (Fudan University, Shanghai China)

Toeplitz operators on higher Cauchy-Riemann spaces over the unit ball

In this talk, we investigate  some algebraic properties of  Toeplitz operators over higher Cauchy-Riemann spaces C_{\alpha,m} on the unit ball B^d. We first discuss the Berezin transform on higher Cauchy-Riemann spaces. By making use of Berezin transform, we completely characterize (semi-)commuting Toeplitz operators   with bounded pluriharmonic symbols  over higher Cauchy-Riemann space C_{\alpha,m}. Moreover, when d ≥ 2 we show that compact  products of finite Toeplitz operators  with a class of bounded pluriharmonic symbols only happen in the trivial case. (This is a joint work with my Ph. D. supervisor prof. Kai Wang.)

18-12-04
Rami Ayoush (Institute of Mathematics of Polish Academy of Sciences)
Martingale approach to Sobolev embedding theorems and Uncertainty Principle
During the talk I will discuss two probabilistic analogs of theorems connected with Sobolev spaces. First concerns the problem of generalizing Hardy-Littlewood-Sobolev inequality: we investigate which subspaces of L_{1} are mapped by Riesz potential I_{\alpha} to L_{\frac{d}{d-\alpha}}. The second analog corresponds to estimates of Hausdorff dimension of vector-valued measures under restrictions on its Fourier transform. It turns out that both problems are dependent on Van Schaftingen's cancelling condition (and its 'higher degree' versions). Joint work with D. Stolyarov and M. Wojciechowski.

The talk is aimed at an audience working in PDEs and harmonic analysis but can interest also probabilists.

18-12-11
Karen Strung
Optimal transport and unitary orbits in C*-algebras
Two areas of mathematics which have received substantial attention in recent years are the theory of optimal transport and the Elliott classification programme for C*-algebras. In this talk, we combine these to address a classical problem of Weyl: when can the distance between unitary orbits of normal operators be computed by spectral data? Here, the settings for our operators are certain well-behaved C*-algebras which arise from Elliott's classification programme.

In the first part of my talk, I will introduce C*-algebras and their classification. In the second part, I will discuss how we combine classification machinery with the theory of optimal transport to find sufficient conditions under which the distance between unitary orbits of two normal elements can be computed by tracial states. This is based on joint work with Jacelon and Vignati.

The first part of the talk is aimed at a general analysis audience, while the second half assumes familiarity with basic functional analysis.

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