Analysis and Probability Seminar

The seminar is joint for the division of Analysis and Probability and its main themes are Mathematical Physics, Probability Theory and Harmonic and Functional Analysis. The seminar is encouraged to be aimed at a broader audience, but it may be of a more specialised nature when indicated in the last line of the abstract. The talks in the seminar are usually 60 minutes, including questions.

At the moment, the seminar takes place via Zoom. See schedule below for links to Zoom meetings.

Should you have any questions or suggestions, feel free to email one of the organizers Jakob Björnberg (jakobbj 'at', Probability Theory), Erik Broman (broman 'at', Probability Theory) or Genkai Zhang (genkai 'at', Analysis).

Coming seminars 

2021-04-27 at 13:15
Augusto Teixera (IMPA, Brazil)

Phase transition for percolation on randomly stretched lattices
In this talk we study the existence/absence of phase transitions for Bernoulli percolation on a class of random planar graphs. More precisely, the graphs we consider have vertex sets given by Z^2 and we start by adding all horizontal edges connecting nearest neighbor vertices. This gives us a disconnected graph, composed of infinitely many copies of Z, with the trivial behavior p_c(Z) = 1. We now add to G vertical lines of edges in {X_i}xZ, where the points X_i are given by an i.i.d. integer-valued renewal process with inter arrivals distributed as T. This graph G now looks like a randomly stretched version of the nearest neighbor graph on Z^2. In this talk we show an interesting phenomenon relating the existence of phase transition for percolation on G with the moments of the variable T. Namely, if E(T^{1+eps}) is finite, then G almost surely features a non-trivial phase transition. While if E(T^{1-eps}) is infinite, then p_c(G) = 1.

This is a joint work with Hilário, Sá and Sanchis.

The seminar will be held via zoom. Members of the department will receive the link and password via mail. Others interested are welcome to attend and can get the link by contacting one of the organizers (please inform us who you are).

2021-05-04 at 13:15
Balazs Rath (Budapest)

The seminar will be held via zoom. Members of the department will receive the link and password via mail. Others interested are welcome to attend and can get the link by contacting one of the organizers (please inform us who you are).

2021-06-01 at 13:15
Jacob van den Berg (CWI, Amsterdam)

The seminar will be held via zoom. Members of the department will receive the link and password via mail. Others interested are welcome to attend and can get the link by contacting one of the organizers (please inform us who you are).

2021-06-08 at 13:15
Nina Gantert (Munich)

The seminar will be held via zoom. Members of the department will receive the link and password via mail. Others interested are welcome to attend and can get the link by contacting one of the organizers (please inform us who you are).

Previous seminars

2021-02-23 at 13:15
Noam Berger (Munich)

Harnack inequalities for difference equations with random balanced coefficients.
We consider difference equations with balanced i.i.d. coefficients which are not necessarily elliptic, and prove a (large scale) elliptic Harnack inequality with an optimal Harnack constant for non-negative solutions of such equations. We then turn to prove a parabolic Harnack inequality in the same setting, and prove it under the additional condition of a (relatively mild) growth condition. We show by example that the growth condition is necessary. I will start the talk with a general background on Harnack inequalities and their significance in (analysis and) probability theory, then I will sketch the main ingredients of the proofs and then I will show a few applications.
The talk is based on joint works with Moran Cohen, David Criens, Jean-Dominique Deuschel and Xiaoqin Guo.

2021-03-02 at 13:15
Jan Felipe van Diejen (Universidad de Talca, Chile)

Quantum eigenfunctions and bispectral duality for Inozemtsev's Toda chain with boundary interactions
In the late nineteen eighties, Inozemtsev observed by means of an explicit Lax-pair representation that the Toda chain remains integrable when coupled to a Pöschl-Teller potential. Previous boundary potentials of Morse type that were known to preserve the integrability of the Toda chain arise in his picture as limiting cases. In this talk we report on some recent results in collaboration with Erdal Emsiz concerning the eigenfunctions of the quantum version of Inozemtsev's Toda chain, with an emphasis on bispectrality.

2021-03-09 at 13:15
Irina Pettersson (Chalmers)

Multiscale analysis of myelinated axons
Starting from the late 1990s, peripheral nerve stimulation has been a commonly used method for localizing nerves before the injection of local anaesthetic. It has been used both for a single-injection technique, and in a combination with ultrasound guidance during the insertion of continuous nerve block catheters.
A neuron is a basic structural unit of the nervous system, and one needs to know how a signal propagates along neurons to be able to simulate the excitation. We focus on the multiscale modeling of a myelinated axon. Taking into account the microstructure with alternating myelinated parts and nodes Ranvier, we derive an effective nonlinear cable equation describing the potential propagation along a single axon. Cable equations used in electrophysiology are traditionally formulated based on an equivalent circuits consisting of a capacitor in parallel with a conductor. Such models, however, do not take into account the geometry of the myelin sheath. I will also discuss some recent results about the multiscale analysis of a bundle consisting of many myelinated axons.
(Joint work with V. Rybalko and C. Jerez-Hanckes.)

2021-03-16 at 13:15
Gabor Pete (Budapest)

The Free Uniform Spanning Forest is disconnected in some virtually free groups, depending on the generating set
The uniform random spanning tree of a finite graph is a classical object in probability and combinatorics. In an infinite graph, one can take any exhaustion by finite subgraphs, with some boundary conditions, and take the limit measure. The Free Uniform Spanning Forest (FUSF) is one of the natural limits, but it is less understood than the wired version, the WUSF. Taking a finitely generated group, several properties of WUSF and FUSF have been known to be independent of the chosen Cayley graph of the group: the average degree in WUSF and in FUSF; the number of ends in the components of the WUSF and of the FUSF; the number of trees in the WUSF. Lyons and Peres asked if this latter should also be the case for the FUSF. In joint work with Ádám Timár, we give two different Cayley graphs of the same group such that the FUSF is connected in one of them but has infinitely many trees in the other. Since our example is a virtually free group, this is also a counterexample to the general expectation that such "tree-like" graphs would have connected FUSF. Many open questions are inspired by the results.

2021-03-23 at 13:15
Nizar Demni (Marseille)

Probabilistic aspects of magnetic Laplacians with uniform fields on certain Riemann surfaces
I shall introduce new discrete probability distributions arising from the spectral resolutions of magnetic Laplacians on the plane and on the hyperbolic disc. These probability distributions are shown to be quasi-infinitely divisible and give rise to new Levy processes. Time permitting I shall also mention the connections of the aforementioned magnetic Laplacians with sub-Riemannian geometry and zeros of random Gaussian series.

2021-03-30 at 13:15
Henna Koivusalo (Bristol)

Sizes of some random limsup and liminf sets
The classical random covering set is the limsup set of balls of random centres. More precisely, given an i.i.d sequence of uniformly distributed centres (\omega_n), and a sequence of radii (r_n) converging to 0, denote by B_n the balls of radius r_n and centres \omega_n. The random covering set E is the set of points that belong to infinitely many of these balls, or \cap_k \cup_{n>k} B_n. This set has been studied for at least a hundred years -- in fact, the Borel-Cantelli lemma was first proven in this context -- and for example, it is known exactly when E has full covering or full measure, and what its dimension is.
We will suggest a far less-studied counterpart to random covering sets: with the centres and radii as above, we study the uniform covering set U, the liminf set of points \cup_{k=1}^n B_{k,n}=\cup_{k=1}^n B(\omega_k, r_n). This set compares to uniformly approximable numbers in Diophantine approximation. We will give conditions on U having full covering, full/null measure, and give bounds on its dimension.
The new results are joint with Lingmin Liao and Tomas Persson.

2021-04-06 at 13:15
Eusebio Gardella (Munster)

The classification problem for free ergodic actions.
One of the basic problems in Ergodic Theory is to determine when two measure-preserving actions of a group on the atomless Borel probability space are orbit equivalent. When the group is amenable, classical results of Dye and Ornstein-Weiss show that any two such actions are orbit equivalent. Thus, the question is relevant only in the non-amenable case. In joint work with Martino Lupini, we showed that for every nonamenable countable discrete group, the relations of conjugacy and orbit equivalence of free ergodic actions are not Borel, thereby answering questions of Kechris. This means that there is in general no method, or uniform procedure, that allows us to determine when two actions of a nonamenable group are conjugate/orbit equivalent. It is a non-classification result, which rules out the existence of any classification theorems which use "nice" (Borel) invariants. The statement about conjugacy also solves the nonamenable case of Halmos' conjugacy problem in Ergodic Theory, originally posed in 1956 for ergodic transformations. The main conceptual innovation is the notion of property (T) for triples of groups, for which a cocycle superrigidity theorem à la Popa can be established. In combination with induction methods developed by Epstein, this is used to obtain a large family of free ergodic actions of the given nonamenable group which have pairwise distinct 1-cohomology groups. No previous knowledge on group amenability will be assumed, and all relevant definitions will be introduced in the course of the presentation.

2021-04-13 at 13:15
Isabelle Tristani (ENS, Paris)

Incompressible Navier-Stokes limit of the Boltzmann equation.
In this talk, we are interested in the link between strong solutions of the Boltzmann and the Navier-Stokes equations. The problem of justifying the connection between mesoscopic and macroscopic equations has been extensively studied. Here, we propose a different approach, intertwining fluid mechanics and kinetic estimates. It enables us to prove convergence of smooth solutions of the Boltzmann equation to solutions to the fluid dynamics equations when the Knudsen number goes to zero. We do not require any smallness at initial time, and our result is valid for any initial data (well prepared or not) in the case of the whole space. We also prove that the time of existence of the solution to the Boltzmann equation is bounded from below by the existence time of the fluid equation as soon as the Knudsen number is small enough. This is a joint work with Isabelle Gallagher.

2021-04-20 at 13:15
Eviatar Procaccia (Technion, Israel)

Dimension of Stationary Hastings Levitov
The Hastings-Levitov process is a planar aggregation process defined by a composition of conformal maps, in which at every time a new particle attaches itself to the existing cluster at a point which is determined by the harmonic measure. The main advantage of this model is that its direct connection to complex analysis makes it tractable. The main disadvantage is non-physical behaviour of the particle sizes. In this talk I will present a new half-plane variant of the Hastings-Levitov model, and will demonstrate that our variant, called the Stationary Hastings-Levitov, maintains the tractability of the original model, while avoiding the non-physical behavior of the particle sizes. Thus this model can be seen as a tractable off-lattice Diffusion Limited Aggregation (DLA). Our main result concerns exact growth bounds and fractal dimension.
Based on joint work with Noam Berger and Amanda Turner


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