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.

Toshio Horiuchi (Ibaraki University)
Improved Kato's inequalities for a quasilinear elliptic operator and related topics
Let $1<p<\infty$, $p^* =\max[1,p-1]$  and let $\Omega$ be a bounded domain of {\boldmath $R^{ \it N}$}$ (N\ge 1)$. In this paper, we consider a class of  second order quasilinear elliptic operators $\mathcal A$  in $\Omega$ with a growth order of  degree $p-1$, which  include the $p$-Laplace operator $\Delta_p$. We establish improved Kato's inequalities, the inverse maximum and the strong maximum principle under suitable assumptions. To this end, we introduce a notion of admissible class of functions. We also construct a counter example to the admissible class, which was originally  constructed by J.Serrin to give a pathological solution to a certain  Dirichlet boundary value problem.
We further prove the existence and partial uniqueness results of  the admissible solution for the following boundary value problem with a measure data:
$-\mathcal Au = \mu  in \Omega$  and  $u= 0 on \Omega$.

The talk is aimed at a general analysis audience.

17-12-05 at 14:15-15:00
Adam Rennie
(University of Wollongong)
Continuum models for topological insulators

Everyone ``knows'' that the answers we obtain from continuum or discrete models of topological features of condensed matter systems should be the same. Reasons for caring include finite samples: for these the discrete model loses the topological information.


In this talk I will discuss the mathematical resolution of some of these folk theorems. The methods for computing the quantum numbers also provide a natural framework for the study of disorder and localisation in these systems, but much remains to be done. Joint work with Chris Bourne. Funded by the Gothenburg Centre for Advanced Studies in Science and Technology.


The talk is aimed at a general audience from analysis and mathematical physics.

17-12-05 at 15:15-16:00
Bram Mesland (University of Bonn)
Index theory and topological phases of aperiodic solids
This talk will zoom in on a particular case of the setting discussed by Rennie. To model a solid material we look at certain discrete subsets of Euclidean space, the so called Delone sets. To a Delone set we associate a dynamical system coming from the translation action along vectors in the ambient space. This dynamical system has good properties when the underlying set is aperiodic and has finite local complexity. Its discrete invariants can be described by the K-theory of the groupoid C*-algebra of the dynamical system to obtain Chern number formulas. if time allows we discuss strong and weak phases, the bulk-boundary correspondence and the extension of phases to regions of dynamical localisation. Joint work with Chris Bourne.
This talk is aimed at a general functional analysis and mathematical physics audience.

Previous seminars in 2017:

17-02-08 (MVL:15)
Jean-Michel Bismut (Université Paris-Sud)
Analytic torsion, the hypoelliptic Laplacian and the trace formula
On a compact odd dimensional manifold equipped with a flat unimodular vector bundle, the Reidemeister torsion is a combinatorial invariant. Analytic torsion is a spectral invariant of the corresponding Hodge Laplacian. The Cheeger-Müller theorem asserts that these two invariants coincide.

A proof was given by Zhang and ourselves based on the Witten deformation of Hodge theory that is associated with a Morse function f. In a first part of the talk, we will review some aspects of the proof.

In a second part of he talk, when replacing f by the energy functional on the loop space, the corresponding Hodge Laplacian is exactly the hypoelliptic Laplacian that was considered in the first talk. A result by Lebeau and ourselves asserts that elliptic and hypoelliptic torsion coincide.

If one pushes the deformation parameter to infinity, on manifolds with negative curvature, one obtains a formal version of the Fried conjecture, that relates the values at 0 of the Ruelle zeta function associated with the geodesic flow and analytic torsion.

In a last part of our talk, we will describe our approach to Selberg’s trace formula as a form of a Lefschetz formula. In the case of locally symmetric spaces, a different version of the hypoelliptic Laplacian does appear, and the deformation is essentially isospectral. This leads to an explicit evaluation of semisimple orbital integrals, which is compatible with Selberg’s trace formula.

The talk is aimed at an audience familiar with geometric analysis and probability theory.

Jakob Björnberg
Probabilistic representations of quantum spin system
This talk is an introduction to some of my current research on models for phase transition. I will discuss examples of quantum spin systems, such as the Heisenberg model, which can be represented probabilistically. Connections to percolation-theory and the random-transposition random walk will be mentioned.

The talk is aimed at the whole division.

Alexandr Usachev
Singular traces in noncommutative geometry: history and recent advances
In the present talk we discuss recent advances in the theory of singular (Dixmier) traces and their role in noncommutative geometry.

The first part of the talk is a historical survey. We outline J. Dixmier's original construction of singular traces and their employment in noncommutative geometry. Here we also discuss the issues arising in a computation of singular traces.

The second (more technical) part of the talk consists of a recent revision of Dixmier's ideas and relations between singular traces and the residues of $\zeta$-function.

The talk is aimed at a general analysis audience.

17-02-28 at 1530
Thomas Bäckdahl
Tools for decay estimates for metric perturbations on black hole spacetimes
The Kerr spacetime is a solution to the Einstein equations describing a stationary rotating black hole in vacuum. It is conjectured to be unique and dynamically stable. The stability problem is one of the central open problems in general relativity. It deals with a quasi-linear PDE system in a non-trivial geometry, and requires substantial new techniques for its solution. In this talk I will discuss some of the difficulties and methods to tackle these problems. Integrated decay estimates, conservation laws and symmetry operators play an important role as well as finding gauge invariant quantities for metric perturbations and their hyperbolic evolution equations. The methods are built from the geometry of the problem, in particular the (hidden) symmetries of the spacetime.

The talk is aimed at a general audience from analysis and mathematical physics.

Lyudmila Turowska
Essential spectral synthesis and compact operator synthesis
W. Arveson in his fundamental paper (Ann. Math 1974) discovered an interplay between invariant subspaces, operator algebras theory and spectral synthesis in harmonic analysis. The notion of operator synthesis was proposed. It provided a powerful tool to study different questions in harmonic analysis, operator theory, theory of multipliers and so on. In this talk we shall discuss sets that are operator synthetic "modulo compact operators" or "modulo Schatten ideals" and their analogs in harmonic analysis. Different examples of such sets, an analog of Malliavin's theorem and applications to operator equations will be presented.

The talk is aimed at an analysis audience with general knowledge in functional analysis.

17-03-09 (Thursday at 10 in MVL14)
Calum Robertson (Monash University)
Post-Newtonian cosmological approximations
The Newtonian limit of General Relativity (GR) remains an important tool in relativistic dynamics. It entails using the Newtonian theory of gravity, coupled with suitable non-relativistic matter, as an approximate model for some gravitating system. The system can be extended, as it is in the cosmological setting, or isolated, as it is in the case of stellar systems and galaxies.

Post-Newtonian (PN) models add relativistic corrections to the Newtonian limit model in order to obtain more accurate results. In certain situations, PN models are both highly accurate and require much less computational effort than that required by the GR model.

In this project, we ask how the cosmological Einstein-Euler initial value problem must be set up so as to ensure that a well-defined PN model for its evolution actually exists, with a measurable “distance” between the solutions of the PN and GR models.

This talk is aimed to an audience familiar with Riemannian geometry.

Ksenia Fedosova
Analytic torsion of hyperbolic orbifolds
The main objects of this talk are hyperbolic orbifolds, i.e. quotients of a hyperbolic space by discrete groups, which possibly contain elements of finite order. To every finite-dimensional representation of such a group we associate an orbibundle. We consider the varying orbibundle over a fixed orbifold. The main question of the talk is: how does the analytic torsion of the orbibundle change and what kind of geometric information about the orbifold can we obtain from it? We also discuss possible applications of the proposed answer to the study of the cohomology of arithmetic groups.

Our main method is the Selberg trace formula. We show that though its geometric side cannot be used to calculate the analytic torsion for a given orbibundle explicitly, it suits well for studying its asymptotic behaviour.

The talk is aimed at an audience familiar with spectral geometry.

17-03-23 (Thursday 1515 in MVH11)
Eero Saksman (University of Helsinki)
On Gaussian multiplicative chaos and the Riemann zeta function
We recall some basic properties of Gaussian multiplicative chaos and describe its connection to the functional statistics of the Riemann zeta function on the critical line (and to that of random unitary matrices). The talk is based on joint work with Christian Webb (Aalto University).

This talk is aimed to both probabilists and analysts.

Stein Bethuelsen (Technical University of Munich)
On recent progress for random walks in dynamic random environment
Random walk in random environment (RWRE) models are natural extensions of the classical simple random walk model where the transition kernel of the random walk is modified according to a random environment. Such models are the probabilistic description of what in the PDE literature are known as stochastic homogenization of discrete linear elliptic equations.

In the first half of the talk, I will give an (brief) overview of results obtained for RWRE models, with a focus on cases where the environment itself is evolving dynamically with time. For such models, the mixing propertiesof the dynamics play a central role and limiting theorems for the random walk (such as LLN and CLT) have been proven mainly under strong (uniform) mixing assumptions on the dynamical environment (such as cone mixing).

In the second half of the talk I will present a new approach to obtain limiting theorems for random walks in dynamic random environment. The approach applies to a general class of models under a mixing assumption which is strictly weaker than cone mixing. This approach goes by studying the invariant laws of the “environment as seen from the walker”-process and is based on joint work with Florian Völlering (Bath University). If time allowsit, I will also discuss some more concrete examples where the environment is taken by the supercritical contact process.

The talk is aimed at a general probability audience.

17-03-30 (Thursday 10:00 in MVL14)
Mahmood Alaghmandan
A representation theorem for regular bounded Borel measures on locally compact groups
In this talk we learn about a representation theorem for bounded Borel measures on a locally compact groups, the motivation behind it, and its generalization using the theory of locally compact quantum groups.

This talk is based on a collaboration with Lyudmila Turowska (Gothenburg) and Ivan Todorov (Belfast). The talk is aimed for audiences with no background in non-commutative Harmonic Analysis and Operator Theory.

Daniel Ueltschi (University of Warwick)
Universal behaviour of loop soups in 3D
Many classical and quantum spin systems can be described by models of random loops, which are 1D objects living in 3D space. They typically have a low temperature phase with macroscopic loops, and the joint distribution of their lengths is universally given by Poisson-Dirichlet distributions. I will define some examples of loop soup models, and explain the conjectures and the heuristics.

The talk is aimed at the whole division.

Chris Hopper (Aalto University)
Partial Regularity for Holonomic Minimisers of Quasiconvex Functionals
We prove partial regularity for local minimisers of certain strictly quasiconvex integral functionals, over a class of Sobolev mappings into a compact Riemannian manifold, to which such mappings are said to be holonomically constrained. Several applications to variational problems in condensed matter physics with broken symmetries will also be discussed related to the manifold constraint condition.

The talk is aimed at a general analysis audience.

Michael Björklund
Approximate lattices
The aim of this talk is to discuss some recent work (joint with T. Hartnick, Technion) on large discrete almost groups in locally compact second countable groups, which we call approximate lattices. We establish some basic properties of these objects using ergodic-theoretical techniques.

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

17-04-19 (Wednesday 1000, MVL14)
Alexander Holroyd (Microsoft Research)
Embedding Percolation
Percolation is concerned with the existence of an infinite path in a random subgraph of a given graph H. We can rephrase this as the existence of an injective graph homomorphism (or an injective 1-Lipschitz map) from the infinite line Z+ to the random subgraph. What happens if we replace Z+ with another graph G? Answering this for various choices of G and H will lead us to a surprising range of topics, including topological combinatorics, first-passage percolation, and queueing theory. Based on joint works with Dirr, Dondl, Grimmett, Martin and Scheutzow.

Mikhail Chebunin (Novosibirsk State University)
Functional central limit theorem for infinite urn models
The talk is aimed at an audience working with probability theory.

17-05-09 (kl. 1515-1700)
Grigori Rozenblum
The finite rank property for Toeplitz operators in Bergman type spaces and related problems in analysis
For several years I was studying the question: what should the symbol F be so that the Toeplitz operator T_F in the Bergman type space B have finite rank. The typical example is the case of the classical Bergman space B^2\subset L^2 of analytic functions on the unit disk, while the operator T_F is defined as associating the function PFu \in B^2 with the function u\in B^2, where P is the Bergman projection P:L^2\to B^2.

I am going to recall the basic results of the last 10 years and explain some new ones, the latter being related to the solvability of some partial differential equations in spaces of distributions.

We also discuss Toeplitz operators in higher poly-Bergman and poly-Fock spaces.

The talk is aimed to a general analysis audience.

Tamás F. Görbe (University of Szeged)
Integrable many-body systems of Calogero-Ruijsenaars type
Calogero-Ruijsenaars type models describe interacting particles that move in one spatial dimension (e.g. on a line or circle). In these systems, the equations of motion can be solved exactly due to the presence of many independent commuting first integrals. Moreover, the Calogero-Ruijsenaars systems occupy a central place among integrable models and serve as a busy crossroad between various subfields in mathematics and physics.


The talk is aimed at a general analysis audience.

Ragnar Freij-Hollanti (Aalto University)
Private Information Retrieval from Coded Data via Reed-Solomon Codes
In distributed data storage, Private Information Retrieval (PIR) addresses the problem of how to retrieve data items without disclosing their identity, using random queries. Mathematically, this reduces to designing a family of probability distributions on a product space, that share the same projection to some prescribed sets of coordinates, but whose supports have pairwise large Hamming distance. We present a general framework for PIR from arbitrary linearly coded databases, and show that, under certain technical assumptions, our scheme is asymptotically capacity-achieving precisely if the storage code is an evaluation code on projective line, known as a Reed-Solomon code. This is joint work with Salim El Rouayheb, Oliver Gnilke, Camilla Hollanti, David Karpuk, and Razane Tajjedine.

The scheme constructions exploit basic ideas from combinatorial algebraic geometry, whereas the capacity results are derived from information theoretical arguments. However, the theory will be developed as needed, and the talk as a whole will be accessible to a general probablity theory audience.

17-05-30 (1515-1700)
Florian Vasilescu (University of Lille)
Integral Representations of Semi-Inner Products in Function Spaces
Various spaces of measurable functions are usually endowed with semi-inner products expressed in terms of positive measures. Trying to give answers to the inverse problem, we present integral representations for some semi-inner products on function spaces of measurable functions, obtained either directly or by adapting and extending techniques from the theory of moment problems.

The talk is aimed at a general functional analysis audience

17-06-08 (Thursday 1100, MVL15)
Bram Mesland (University of Bonn)
Hecke operators, KK-theory and arithmetic groups
Cohomology of arithmetic groups and its structure as a Hecke module plays a prominent role in modern number theory. Classically the cohomology of an arithmetic group \Gamma can studied geometrically through its action on the associated global symmetric space X. In low dimensions, such actions produce noncompact hyperbolic manifolds as quotient spaces, as well as dynamically complicated actions on the boundary of X. In recent joint work with Haluk Sengun (Sheffield), we show that KK-theory provides a natural framework for the Hecke module structure on K-theory and K-homology of C*-algebras associated to the action of \Gamma on X and its boundary.

The talk is aimed at a general audience in global analysis or number theory, with some familiarity with cohomology or K-theory.

17-06-08 (Thursday 1515, MVL14, joint with the statistics seminar)
Jie Yen Fan (Monash University)
Limit theorems for age, type, and population size dependent populations

17-06-13 (1000 in MVL:14)
Steven Pankavich (Colorado School of Mines)
Large-time Behaviour of Collisionless Plasmas
The motion of a collisionless plasma, a high-temperature, low-density, ionized gas, is described by the Vlasov-Maxwell system. In the presence of large velocities, relativistic corrections are meaningful, and when magnetic effects are neglected this formally becomes the relativistic Vlasov-Poisson system. Similarly, if one takes the classical limit as the speed of light tends to infinity, one obtains the non-relativistic Vlasov-Poisson system. In either setting, we study the long-time dynamics of these systems of PDEs and describe differing mechanisms that drive their behaviour, including dispersion, velocity averaging, Landau damping, charge repulsion, and charge cancellation. Finally, we contrast the behaviour when considering the cases of classical versus relativistic velocities, electrostatic versus electromagnetic interactions, and monocharged (i.e., single species of ion) versus neutral plasmas.

The talk is aimed at a general audience familiar with Partial Differential Equations, and will be especially suitable for those within an interest in Kinetic Theory.

Nico Spronk (University of Waterloo)
On convoluters on $L^p$

Mariel Saez (Universidad Catolica de Chile)
Fractional Laplacians and extension problems: the higher rank case (Joint with Maria del Mar Gonzalez)
The aim of this talk is to define conformal operators that arise from an extension problem of co-dimension two. To this end we interpret and extend results of representation theory from a purely analytic point of view.

In the first part of the talk I will give definitions and interpretations of the fractional Laplacian and the conformal fractional Laplacian in the general framework of representation theory on symmetric spaces and also from the point of view of scattering operators in conformal geometry.

In the second part of the talk I will show constructions of boundary operators with good conformal properties that generalise the fractional Laplacian in $\mathbb R^n$ using an extension problem in which the boundary is of co-dimension two. Then we extend these results to more general manifolds that are not necessarily symmetric spaces.

The talk is aimed at a general analysis audience, but it contains aspects that may be of interest for the probability audience (since there is a direct interpretation of the above in terms of probability theory).

Thursday 17-09-07 at 15:15 in MV:L15
Tobias Hartnik (Technion - Israel Institute of Technology)
From stability of the dilogarithm to bounded cohomology and back
Rogers’ dilogarithm is the unique continuous function on the interval (0,1) which satisfies a reflection symmetry at 1/2 and a functional equation called the Spence-Abel-5-term equation. We show that if a function satisfies these two equations up to a bounded error, then it is at bounded distance from the dilogarithm; in fact, the bound is linear in the initial error. This stability result is obtained by interpreting the functional equation in cohomological terms. While the result is purely in terms of functions on the interval (0,1), the proof uses in an essential way the action of PSL_2(R) on the hyperbolic plane and its boundary. Joined work with Andreas Ott (Heidelberg).

17-09-12 at 16:15
Marc Pollicott (University of Warwick)
The Selberg zeta function for higher Teichmuller spaces
For compact surfaces of constant curvature -1 the Selberg zeta function is a complex function defined in terms of the lengths of closed geodesics, and has an analytic extension to the entire complex plane. The zeta function can also be defined in terms of representations in SL(2,R) of the Fundamental Group of the surface. Higher Teichmuller deals with representations of the group in, for example, SL(3,R). In this setting we define, by analogy, a Selberg zeta function and consider its properties.

No prior knowledge of zeta functions nor Higher Teichmuller theory will be assumed.

Peter Sjögren
Weak type (1,1) for a normal Ornstein-Uhlenbeck semigroup

The Ornstein-Uhlenbeck operator in Rn is linear and elliptic with constant second-order coefficients. But the first- order coefficients are linear in the coordinates and such that they cause a drift inwards. This operator generates a semigroup, and we study the corresponding maximal operator. The relevant measure here is a gaussian measure, which replaces Lebesgue measure. Assuming only that the semigroup is normal, i.e., commutes with its adjoint, we prove that the maximal operator is of weak type (1,1) for the gaussian measure. This extends earlier results by several authors. The first step in the proof is a transformation of variables which gives the semigroup a reasonable, explicit expression.

Except for the end which is technical, this talk should be understandable for a rather general audience of analysts.

17-10-06 at 13:15 in MV:L14
Steffen Winter (Karlsruhe Institute of Technology)
Geometric functionals of fractal percolation

Fractal percolation is a family of random sets suggested by Mandelbrot in the seventies to model certain aspects of turbulence. They are known to undergo a very sharp phase transition from a totally disconnected to a percolating regime, when the continuous parameter of the models passes some threshold. The exact values of these percolation thresholds remain unknown until today, and the known rigorous upper and lower bounds are still rather far from each other.

In the recent physics literature (see e.g. Mecke, Neher, Wagner, J. Stat. Mech. 2008;  Klatt, Schröder-Turk, Mecke, J. Stat. Mech. 2017) the idea is explored that the sharp topological transition at the threshold in percolation models should be visible in geometric functionals such as the (expected) Euler characteristic of these sets. Indeed, simulations suggest a close relation between percolation thresholds and the zeros of the Euler characteristic (as a function of the model parameter) in many percolation models. Motivated by the desire to find better bounds on percolation thresholds for fractal percolation, we study the expectations of some geometric functionals of theconstruction steps of fractal percolation (or rather their rescaled limits). These functionals are closely related to fractal curvatures. We obtain explicit formulas for some of these limit functionals including some rescaled Euler characteristic and compare them to the known bounds for percolation thresholds.

Joint work in progress with Michael Klatt.

The talk is aimed at the whole division. No preknowledge on percolation or fractal geometry is assumed.

Lukás Malý
Trace and extension theorems for BV and Sobolev-type functions on domains in metric spaces

In the general Dirichlet problem, one starts with a domain, prescribes boundary values, and looks at the set of functions on the interior of the domain whose trace on the boundary matches the prescribed boundary values. For domains in metric measure spaces, we investigate the class of functions on the boundary that can be extended to functions of some specified regularity on the interior. Under some rather mild requirements on regularity of the boundary, we find a linear extension operator from a certain Besov class on the boundary to the BV (or to the Newton–Sobolev class) on the interior of the domain. This operator can then be used to find BV or Newton–Sobolev extensions of Lp boundary data provided that the boundary is endowed with a codimension-p regular Hausdorff measure.

We will also discuss the converse problem of establishing that generic BV and Sobolev-type functions defined inside the domain allow for a reasonable notion of a trace on the boundary. In particular, we will look into how smoothness (in terms of a Besov-type seminorm) and integrability of the trace depend on the codimension of the boundary and on the integrability of “gradients” of the Sobolev-type functions. Under somewhat stronger requirements on regularity of the domain and its boundary, we will see that the aforementioned extension results are sharp.

The talk is (partly) based on a joint project with P. Lahti, N. Shanmugalingam, and M. Snipes.


The seminar is aimed at a general analysis audience. No prior knowledge of analysis on metric spaces or of classical trace theorems will be assumed.


Miroslav Englis (Czech Academy of Sciences, Prag)

Bergman kernels and asymptotic expansions on Jordan-Kepler manifolds

We discuss the Bergman kernels and their high-power, or Tian-Yau-Zelditch,  expansions on the Jordan-Kepler manifolds of elements of a fixed rank in a bounded symmetric domain in $\bold C^n$. This generalizes the 2009 result of Gramchev and Loi for the special case of the ordinary Kepler manifold, consisting of rank 1 elements in the Lie ball $IV_n$.

(Joint work with H. Upmeier, Marburg.)

The talk is aimed at an audience familiar with harmonic and complex analysis.

ław Ohrysko

Non-separability of the Gelfand space of measure algebras

The classical Wiener-Pitt phenomenon asserts that there exists $\mu\in M(\mathbb{R})$ such that $|\widehat{\mu}|(x)\geq c > 0$ for $x\in\mathbb{R}$ ($\widehat{\mu}$ is the Fourier-Stieltjes transform of a measure) but $\mu$ is not invertible in $M(\mathbb{R})$. Later this observation was rigorously proven and extended to all (non-discrete) locally compact Abelian groups implying that the dual group of a locally compact non-discrete Abelian group (identified with evaluations of Fourier-Stieltjes transform) is not dense in the Gelfand space of a measure algebra on this group. However, it was an open problem if any other countable dense subset exists in the Gelfand space of a measure algebra (this would give a procedure to calculate the spectra of particular measures). In this talk I will present the detailed proof of the non-separability of the Gelfand space of the measure algebra on the circle group and indicate how to extend this result to arbitrary locally compact Abelian groups. The talk is based on the paper written in collaboration with Michal Wojciechowski and Colin C. Graham.

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

Sigurður Stefánsson (University of Iceland)
Geometry of large Boltzmann outerplanar maps

A planar map is a planar graph drawn in the sphere such that no edges cross.  A planar map which has the property that there is a face in the map such that all the vertices lie on the boundary of that face is called outerplanar. We investigate a class of random outerplanar maps which are defined by assigning non-negative weights to each face of a map. An important observation is that outerplanar maps may be viewed as trees in which each vertex is a dissection of a polygon. Dissections of polygons are further in bijection with trees which allows us to relate the random outerplanar maps to the model of simply generated trees which is understood in detail. Our main contribution is showing that for certain choices of weights the random outerplanar maps, with a rescaled graph metric, converge towards the so called random stable looptree which was recently introduced by Curien and Kortchemski.

This is a joint work with Benedikt Stufler and is based on arXiv:1710.04460.

The talk is aimed at a general probability audience.

17-11-14 at 15:15-17:00 (2*45 mins)
Magnus Goffeng
Geometry and magnitude

Around a decade ago, Leinster introduced the notion of magnitude as a generalization of the Euler characteristic of a finite category. It has since been extended to an invariant of compact metric spaces where it captures several geometric features. The ideas coming from magnitudes have also found applications in new definitions of diversity in biology.


What I will consider in this talk is the magnitude function of a Euclidean domain: the magnitude of the domain rescaled by a variable R>0. There are surprisingly few computations of magnitude that have been done. It was nevertheless conjectured by Leinster-Willerton that for convex domains, the magnitude function is a polynomial in R where the coefficients are the intrinsic volumes of the convex body (e.g. volume, surface volume, total mean curvature,…, Euler characteristic). We prove an asymptotic version of this conjecture and show that the magnitude function extends meromorphically to the complex plane. Based on joint work with Heiko Gimperlein.

The talk will be 2*45 minutes. The first half overviews the results and area, and is aimed at a general analysis audience. The second half discusses the proofs and is aimed at an audience familiar with global analysis.

17-11-17 at 13:15

Cliff Gilmore (University of Helsinki)

Minimal growth of frequently hypercyclic harmonic functions

Linear dynamics has been a rapidly evolving area since the 1990s.  It lies at the intersection of operator theory and topological dynamics, and its central property is hypercyclicity.  In the first part of this talk I will give a general introduction to linear dynamics and I will demonstrate that many natural continuous linear maps turn out to be hypercyclic.


The second part of the talk is concerned with the stronger property of frequent hypercyclicity, which has fundamental connections to ergodic theory.  We will identify sharp growth rates, in terms of the L^2-norm on spheres, of harmonic functions that are frequently hypercyclic for the partial differentiation operator.  This answers a question posed by Blasco, Bonilla and Grosse-Erdmann (2010).


This is joint work with Eero Saksman and Hans-Olav Tylli.


This talk is aimed at an audience with a general analysis background.

17-11-24 (Friday) at 13:15 in MVL:14
Mohammud Foondun (University of Strathclyde)​ 
Approximating fractional stochastic heat equations
In this talk, I will describe one of the  motivations behind a series of results by various groups of people. A general aim behind these works is to go beyond existence-uniqueness results and establish qualitative properties like "intermittency".  I will show how one can approximate a class of fractional stochastic heat equation by a system of interacting SDEs. This will enable us to deduce results about the SPDE from the system of SDEs. This in turn will give us various qualitative properties like "intermittency" and boundedness properties.

The talk is aimed at a general probability audience.

Published: Mon 12 Dec 2016. Modified: Mon 04 Dec 2017