Category: Math Colloquium

Gestur Ólafsson, Louisiana State University

Carlos Argaez Garcia, February 11, 2021

Math colloquium

Speaker: Gestur Ólafsson, Louisiana State University

Title: Atomic decomposition of Bergman spaces on Hermitian Symmetric Spaces

Room: Zoom link will be sent by email
Time: Tuesday 16th February, 16:00hrs

Abstract:

Hermitian symmetric spaces, bounded or unbounded, and spaces of functions or distributions on those spaces show up naturally in com- plex analysis, Lie theory, functional analysis and several other parts of mathematics. In this talk we will discuss some resent work on dis- cretization/atomic decomposition of Bergman spaces on those domains and their unbounded realization.

The story goes back to the work of Coifman and Rochberg in the 1980’s where they provided atomic decompositions for Bergman spaces on (the unbounded realization of) bounded symmetric domains as well as on the unit ball. Their atoms were build from the Bergman kernel. One of the shortcomings of their work was that their results did not readily transfer to the bounded realization of the domain except in the case of the unit ball.

By applying representation/coorbit theory we obtain a large family of new atoms, including those of Coifman and Rochberg, for Bergman spaces on bounded symmetric domains. Our approach also allows us to describe the relation between atoms for the bounded and unbounded realizations of the domain thus solving one of the issues raised by Coif- man and Rochberg. If time allows then we will list some open questions for domains of rank higher than one.

Anders Karl Claesson, University of Iceland

Carlos Argaez Garcia, January 12, 2021

Math colloquium

Speaker: Anders Karl Claesson, University of Iceland

Title: On the problem of Hertzsprung and similar problems

Room: Via Zoom. Link to be sent.
Time: Tuesday January 19th, 10:00 hrs

Abstract:

Drawing on a problem posed by Hertzsprung in 1887 (sometimes called the n-kings problem), we say that a permutation w contains the Hertzsprung pattern u if there is factor w(d+1)w(d+2)…w(d+k) of w such that w(d+1)-u(1) = … = w(d+k)-u(k).  Using a combination of the Goulden-Jackson cluster method (which we explain) and the transfer-matrix method we determine the joint distribution of occurrences of any set of (incomparable) Hertzsprung patterns, thus substantially generalizing earlier results by Jackson et al. on the distribution of ascending and descending runs in permutations.  We apply our results to the problem of counting permutations up to pattern-replacement equivalences, and using pattern-rewriting systems—a new formalism similar to the much studied string-rewriting systems—we solve a couple of open problems raised by Linton et al. in 2012.

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Elisa Domínguez-Hüttinger, Universidad Nacional Autónoma de México

Carlos Argaez Garcia, November 7, 2020

Math colloquium

Speakers: Elisa Domínguez-Hüttinger, Universidad Nacional Autónoma de México

Title: A “triple-switch” hybrid mathematical model of epidermal homeostasis.

Room:  Via Zoom. Link to be sent.
Time: Friday 13th  November, 14:00

Abstract:

The epidermis is the outermost layer of the skin. It is a stratified epithelium, constituted by layers of epithelial cells (keratinocytes) with increasing levels of differentiation. The basal layer is formed by undifferentiated cells with proliferative capacity, while the most external layer, termed skin barrier, is formed by terminally differentiated cells that are embedded in a lipid matrix. This skin barrier hinders the invasion of pathogens and other aggressors, protecting the organism from environmental disturbances. Transient environmental perturbations that increase the pathogen load or damage the skin barrier can trigger both immune and tissue remodelling responses, resulting in increased pathogen elimination but also affecting its infiltration rate. Under healthy conditions, this complex feedback control structure effectively counteracts environmental aggressors.  However, perturbations by genetic and environmental factors can lead to the loss of homeostasis and the onset and progression of complex epidermal tissue diseases including atopic dermatitis, psoriasis and skin carcinomas. Characterizing the responses of the epidermis to these perturbations is pivotal to uncover pathogenic mechanisms and improve strategies for diagnosis, prevention and treatment of these diseases.  However, this task is difficult to achieve from a purely experimental or clinical perspective, because these perturbations: (1) Often lead to synergic and non-linear responses that are hard to predict experimentally; (2) Can affect several regulatory processes that operate at different time scales; (3) May result in symptoms that are clinically subtle; and (4) Can affect disease progression  in a history-dependent manner. Here we propose a triple-switch mathematical model that couples a bistable motifs describing the activation of innate and adaptive immune responses as well as the differentiation of skin cells with the dynamically changing tissue level properties. We will show how mathematical analysis of this hybrid model has allowed us to: (1) Characterize the effects of genetic and environmental perturbations on epithelial homeostasis; (2) Identify risk factors that increase the vulnerability to environmental aggressors, and (3) Design new strategies for early detection and prevention of complex epidermal diseases.

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Adam Timar, University of Iceland

Carlos Argaez Garcia, October 23, 2020

Math colloquium

Speakers: Adam Timar, University of Iceland

Title: Uniform Spanning Forests of infinite graphs.

Room:  Via Zoom. Link to be sent.
Time: Friday 30th October, 10:00am

Abstract:

Consider a spanning tree of a given finite graph, chosen uniformly at random. The so-defined Uniform Spanning Tree (UST) has long been an object of interest, since it is intimately connected to harmonic functions on the graph and random walks.
Given an infinite graph G, such as a cubic lattice, one can take its exhaustion by finite graphs G_n and consider a suitably defined limit of the UST of G_n. The limiting measure is called the Uniform Spanning Forest of G. The resulting random object has had crucial importance in statistical physics on 2 dimensional lattices, but also turned out to be interesting when G is a Cayley graphs of a group. Among other uses, it allows for a simple probabilistic interpretation of an intricate geometric parameter.
The talk will be addressed to a general math audience. We will give a brief overview of the above topics, and present some recent results that are joint work with Gábor Pete.

Kristján Jónasson, University of Iceland.

Carlos Argaez Garcia, September 11, 2020

Math colloquium

Speakers: Kristján Jónasson, University of Iceland.

Title: Maximum likelihood estimation of multivariate normal parameters when values are missing.

Room:  Via Zoom. Link to be sent.
Time: Friday 18th September, 10:00am

Abstract:

I have been working on a program to estimate the covariance matrix of a multivariate normal distribution in the presence of missing values via maximum likelihood. Many programs offer to do this by computing pairwise covariances (giving a potentially non-positive-definite matrix). There is a package in R (mvnmle) to do the ML-computation, but it is inefficient on several counts. Matlab’s statistical toolbox has a function mvnmle, and its financial toolbox has ecmnmle which are both quite fast, but they lack flexibility, for example to incorporate REML to eliminate bias, to use regularization (when many values are missing), or to reduce the number of parameters by incorporating some variance structure.

This work is in progress and still unpublished but preliminary results are promising. In the talk I shall tell you a little about the program and the underlying algorithms.

Bobby Cheng, University of Sussex, UK

Carlos Argaez Garcia, September 11, 2020

Math colloquium

Speakers: Bobby Cheng, University of Sussex, UK

Title: Quantum Resonances in Relativistic Systems.

Room:  Via Zoom. Link to be sent.
Time: Friday 9th October, 10:00am

Abstract:

Significant amounts of research have been completed on mathematical quantum resonances in the non-relativistic setting. However success in generalizing these results to the relativistic setting have been limited. In this talk I will describe the work undertaken to study resonances of the Dirac operator, perturbed by an electric potential with certain ‘nice’ properties, and establish two key trace formulae.

Peter Giesl, University of Sussex, UK

Carlos Argaez Garcia, September 11, 2020

Math colloquium

Speakers: Peter Giesl, University of Sussex, UK

Title: Existence and construction of a contraction metric as solution of a matrix-valued PDE.

Room:  Via Zoom. Link to be sent.
Time: Friday 2nd October, 10:00am

 

Abstract:

A contraction metric is a Riemannian metric, with respect to which the distance between adjacent solutions of an ordinary differential equation (ODE) decreases.

A contraction metric can be used to prove existence and uniqueness of an equilibrium of an autonomous ODE and determine a subset of its basin of attraction without requiring information about its location. Moreover, a contraction metric is robust to small perturbations of the system. 

We will prove a converse theorem, showing the existence of a contraction metric for an equilibrium by characterising it as a matrix-valued solution of a certain linear partial differential equation (PDE). This leads to a construction method by numerically solving the matrix-valued PDE using mesh-free collocation. We use and present a recent extension of mesh-free collocation of scalar-valued functions, solving linear PDEs, to matrix-valued ones. Finally, we briefly discuss a method to verify that the computed metric satisfies the conditions of a contraction metric.

This is partly work with Holger Wendland, Bayreuth as well as Sigurdur Hafstein and Iman Mehrabinezhad, Iceland.

Joachim Toft , Department of mathematics, Faculty of Technology, Linnæus University, Växjö, Sweden

Carlos Argaez Garcia, September 11, 2020

Speakers: Joachim Toft , Department of mathematics, Faculty of Technology, Linnæus University, Växjö, Sweden

Title: Analytic pseudo-differential calculus via the Bargmann transform.

Room:  Via Zoom. Link to be sent.
Time: Friday 25th September, 10:00am

Abstract:

The Bargmann transform is a transform which maps Fourier-invariant function spaces and their duals to certain spaces of formal power series expansions, which sometimes are convenient classes of analytic functions.

In the 70th, Berezin used the Bargmann transform to translate problems in operator theory into a pseudo-differential calculi, where the involved symbols are analytic functions, and the corresponding operators map suitable classes of entire functions into other classes of entire functions.

Recently, some investigations on certain Fourier invariant subspaces of the Schwartz space and their dual (distribution) spaces have been performed by the author. These spaces are called Pilipovi ́c spaces, and are defined by imposing suitable boundaries on the Hermite coefficients of the involved functions or distributions. The family of Pilipovi ́c spaces contains all Fourier invariant Gelfand- Shilov spaces as well as other spaces which are strictly smaller than any Fourier invariant non-trivial Gelfand-Shilov space. In the same way, the family of Pilipovi ́c distribution spaces contains spaces which are strictly larger than any Fourier invariant Gelfand-Shilov distribution space.

In the talk we show that the Bargmann images of Pilipovi ́c spaces and their distribution spaces are convenient classes of analytic functions or power series expansions which are suitable when investigating analytic pseudo-differential operators (i. e. Berezin or Wick operators).

We deduce continuity properties for such pseudo-differential operators when the symbols and target functions possess certain (weighted) Lebesgue estimates. We also show that the counter image with respect to the Bargmann transform of these results generalise some continuity results for (real) pseudo-differential operators with symbols in modulation spaces, when acting on other modulation space.

The talk is based on collaborations with Nenad Teofanov and Patrik Wahlberg, and parts of the content of the talk is available at:

N. Teofanov, J. Toft Pseudo-differential calculus in a Bargmann setting, Ann. Acad. Sci. Fenn. Math. 45 (2020), 227–257.

Yadollah Zare, Galatasaray University

Carlos Argaez Garcia, June 9, 2020

Math colloquium

Speakers: Yadollah Zare, Galatasaray University

Title: Center conditions: pull-back of differential equations.

Abstract:

The space of polynomial differential equations of a fixed degree with a center singularity has many irreducible components. We prove that pull-back differential equations form an irreducible component of such a space. This method is inspired by Ilyashenko and Movasati’s method. The main concepts are the Picard-Lefschetz theory of a polynomial in two variables with complex coefficients, the Dynkin diagram of the polynomial and the iterated integral.

Room:  V-158, VR-II
Time: Friday 12th June, 10:00am

Notice that is recommended to keep two meters apart from other attendants.

María Óskarsdóttir, Háskólinn í Reykjavík

Carlos Argaez Garcia, March 5, 2020

Speaker: María Óskarsdóttir, University of Reykjavík

Title: Ranking nodes relative to influence with the Personalized PageRank algorithm applied to fraud detection and credit risk measurement

Room: VRII-258
Time: Thursday 12th March, 10:50hrs

Abstract:

Various phenomena in both the physical and the digital world can be represented with networks, that is, entities that are connected in some way, for example communication, computer, financial and social networks. A central theme in the analysis of networks is finding the most important nodes in a network. The PageRank algorithm was developed to rank webpages in search engines, to find the most important webpages on the internet, but has been applied in numerous others applications. The ranking can be personalized so that nodes which are important relative (or close) to a predefined set of nodes are ranked higher. This approach has been used to identify certain behavior in networks where there is a strong social effect, for example fraud and churn. In this presentation we show how the personalized PageRank algorithm can be extended for two specific types of networks. First, we look at a bipartite network which consists of claims and the involved parties, i.e. policyholders and brokers, with the goal of finding fraudulent insurance claims. Then we consider multiplex networks, in which each node can be connected to another node by more than one type of edge, such as two different networks connecting the same individuals. They arise naturally in lending, as two borrowers can be connected by geographical location, economic activity, and many other relationships. We present a methodology to leverage multiplex networks by a novel multiplex Personalized PageRank algorithm, which we subsequently apply to credit risk assessment.