Rhein-Main Kolloquium Stochastik

Schwerpunkt Stochastik


Frühere Vorträge

Donnerstag, 25. April 2013: Stochastikkolloquium / Oberseminar Stochastik

12:00 Uhr: Prof. Dr. Stas Volkov, Universtät Lund

The simple harmonic urn

The simple harmonic urn is a discrete-time stochastic process approximating the phase portrait of the simple harmonic oscillator. This urn is a version of a two-colour generalized Polya urn with negative-positive reinforcements, and in a sense it can be viewed as a "marriage" between the Friedman urn and the OK Corral model, where we restart the process each time it hits a border by switching the colours of the balls. We show the transience of the process using various couplings with birth and death processes and renewal processes. It turns out that the simple harmonic urn is just barely transient, as a minor modification of the model makes it recurrent, and for a good reason, as the embedded urn process is, in fact, a Lamperti-type random walk. We also show links between this model and oriented percolation, as well as a few other interesting processes. This is joint work with E. Crane, N. Georgiou, R. Waters and A. Wade.

Raum 110, Robert-Mayer-Str. 10

Mittwoch, 24. April 2013: Oberseminar Stochastik

14:15 Uhr: Prof. Dr. Coja-Oghlan, Universtät Frankfurt

Chasing the k-SAT threshold

Let F be a random Boolean formula in conjunctive normal form over n Boolean variables with m clauses of length k. The existence of a (non-uniform) sharp threshold for the satisfiability of such formulas is well known [Friedgut 1999]. However, despite considerable effort the precise location of this phase transition remains unknown for any k>2. The best previous upper and lower bounds differ by an additive $k\ln 2/2$ [Achlioptas, Peres 2003]. In this talk I present an improved lower bound, which reduces the gap to ~0.19. The proof is inspired by the cavity method of statistical mechanics.

Raum 308, Robert-Mayer-Str. 6-8

Freitag, 23. November 2012: Rhein-Main-Kolloquium Stochastik in Frankfurt

15:15 Uhr: Igor Kortchemski, Universtät Paris 11

Galton-Watson trees conditioned to be large (Abstract)

16:45 Uhr: Patrick Hoscheit, CERMICS, Universtät Paris-Est

Record processes on the Continuum Random Tree (Abstract)

Raum 302 (Hilbert-Raum), Robert-Mayer-Str. 8

Montag, 16. Juli 2012: Oberseminar Stochastik

11:00 Uhr: Luis Gorostiza, Centro de Investigación y Estudios Avanzados

Oscillatory fractional Brownian motions and related processes arising from particle systems (Abstract)

Raum 711 (groß), Robert-Mayer-Str. 10

Donnerstag, 28. Juni 2012: Rhein-Main-Kolloquium Stochastik in Frankfurt

16:00 Uhr: Thomas G. Kurtz, University of Wisconsin - Madison

Particle representations and limit theorems for stochastic partial differential equations (Abstract)

17:30 Uhr: Steven N. Evans, University of California at Berkeley

Uplift under Lévystan: Lipschitz minorants of Lévy processes (Abstract)

Raum 711 (groß), Robert-Mayer-Str. 10

Mittwoch, 30. Mai 2012: Oberseminar Stochastik

14:15 Uhr: Birte Muhsal, Universität Karlsruhe

Change-point methods for multiple structural breaks and regime-switching models

Change-point analysis is concerned with detecting structural breaks in time ordered data. In this talk a method based on moving sums is introduced, which tests for at least one structural break at significance level $\alpha$ and simultaneously estimates the number and locations of change-points. Asymptotic properties, such as consistency of the change-point estimators, in case of changes in the mean in an otherwise stationary sequence of random variables are investigated. Moreover results for corresponding regime-switching models are deduced, which in contrast to the classical change-point model allow for random change-points as well as a random and asymptotically unbounded number of structural breaks.
The talk is based on joint work with Claudia Kirch.

Raum 711 (klein), Robert-Mayer-Str. 10

Mittwoch, 23. Mai 2012: Oberseminar Stochastik

14:15 Uhr: Mamadou Ba (Université de Provence, Marseille)

Branching processes with competition and a generalized Ray Knight Theorem

15:15 Uhr: Boubacar Bah (Université de Provence, Marseille)

A lookdown model with selection

Raum 711 (klein), Robert-Mayer-Str. 10

Mittwoch, 09. Mai 2012: Oberseminar Stochastik

14:15 Uhr: Matthias Riedel, Goethe-Universität Frankfurt am Main

Price-setting of market makers: A filtering problem with an endogenous filtration

We study the price-setting problem of market makers under perfect competition in continuous time. Thereby we follow the classic Glosten-Milgrom model (1985) that defines bid and ask prices as the expectation of a true value of the asset given the market makers partial information that includes the customers trading decisions. The true value is modeled as a Markov process that can be observed by the customers with some noise at Poisson times. We analyze the price-setting problem in a mathematically rigorous way by solving a non-standard filtering problem with an endogenous filtration that depends on the bid and ask price process quoted by the market maker. Under some conditions we show existence and uniqueness of the price processes. This result would even be new in the static case.

Raum 711 (klein), Robert-Mayer-Str. 10

Freitag, 02. März 2012: Oberseminar Stochastik

14:15 Uhr: Götz Kersting, Goethe-Universität Frankfurt am Main

Den Zufall erleben - mit dem Computer

Raum 711 (groß), Robert-Mayer-Str. 10

Mittwoch, 08. Februar 2012: Oberseminar Stochastik

14:15 Uhr: Max Stroh, Goethe-Universität Frankfurt am Main / B. Metzler seel. Sohn & Co. KGaA

Continuous time trading of a small investor in a limit order market

We provide a mathematical framework to model continuous time trading in limit order markets of a small investor whose transactions have no impact on the order book dynamics. The investor can continuously place market and limit orders. A market order is executed immediately at the best currently available price whereas a limit order is stored in the book until it can be executed at its limit price. The limit orders can be chosen from a continuum of limit prices. We show how elementary strategies (hold limit orders with only finitely many different limit prices and rebalance at most finitely often") can be extended in a suitable way to general continuous time strategies containing orders with infinitely many different limit prices. The general limit buy order strategies are predictable processes with values in the set of nonincreasing demand functions (not necessarily left- or right-continuous in the price variable). It turns out that our strategy set of limit and market orders is closed, but limit orders can turn into market orders when passing to the limit, and any element can be approximated by a sequence of elementary strategies.

(Link zum Paper)

Raum 711 (klein), Robert-Mayer-Str. 10

Mittwoch, 21. Dezember 2011: Oberseminar Stochastik

14:15 Uhr: Kevin Leckey, Goethe-Universität Frankfurt am Main

Asymptotic properties of Hoppe trees

The Hoppe tree is a family of randomly growing tree models, where the growth dynamic is given by the evolution of the Hoppe urn as follows: The distinguished ball that initializes the Hoppe urn corresponds to the root of the Hoppe tree. In the Hoppe urn this balls has probability proportional to a parameter theta>0 to be drawn, all other balls have probability proportional to 1. Whenever a ball is drawn it is placed back to the urn together with a new ball which corresponds to a node being inserted in the Hoppe tree as a child of the node that corresponds to the ball drawn. For the special case theta=1 the Hoppe tree coincides with the random recursive tree.

We give asymptotic results on mean, variance and limit laws for the depth of nodes, the height, the path length and the number of leaves of the Hoppe tree and clarify the dependence upon the parameter theta.

Raum 711 (klein), Robert-Mayer-Str. 10

Mittwoch, 07. Dezember 2011: Berufspraxiskolloquium

14:15 Uhr: Miriam Friderichs und Wolfgang Baumann, Towers Watson GmbH (Köln)

Mathematiker berichten aus der Berufspraxis: Der Mathematiker als beratender Aktuar

Was macht ein Aktuar und wieso sind wir in der Nichtversicherungswelt so unbekannt? Bei dem Wort Unternehmensberatung denken die meisten Menschen an Berater, die eine kurze Zeit ins eigene Unternehmen kommen und dann ein Gutachten verfassen, welche Arbeitsplätze man besser besetzten könnte oder streichen sollte. In Wirklichkeit gibt es aber auch eine große Nachfrage nach aktuarieller Beratung. Dies ist unter anderem in den Bereichen der Personenversicherung (Lebens- und Krankenversicherung, Pensionsversicherung) und Schadenversicherung der Fall. Wir wollen hier einen kurzen Einblick geben, in welchen Bereichen Versicherer aktuarielle Beratung in Anspruch nehmen. Ebenso wollen wir verdeutlichen, welche Voraussetzungen man für aktuarielle Beratung mitbringen sollte, welche Entwicklungsmöglichkeiten es gibt und was man nach 1.5 Berufsjahren schon alles gelernt aber auch noch zu lernen hat.

Im Bereich der Schadenversicherung werden wir die 3 großen Kernbereichen vorstellen und den Zusammenhang zwischen deterministischen und stochastischen Modellen verdeutlichen.

Raum 711 (klein), Robert-Mayer-Str. 10

Freitag, 25. November 2011: Rhein-Main-Kolloquium Stochastik in Frankfurt

15:15 Uhr: Marie Albenque, École Polytechnique

Convergence of stack triangulations (Abstract)

16:45 Uhr: Nicolas Broutin, INRIA Rocquencourt

Cutting down trees, and putting them back together (Abstract)

Raum 711 (groß), Robert-Mayer-Str. 10

Freitag, 18. November 2011: Gastvortrag

14:15 Uhr: Dr. Christel Kamp, Paul-Ehrlich-Institut Langen

Evolution und Ausbreitung von Pathogenen - kann man Pathogene mit Mathematik dingfest machen?

Bio-medizinische Fragestellungen bestimmen die Forschungsschwerpunkte am Paul-Ehrlich-Institut. Die Komplexität biologischer Systeme bietet dabei nicht nur Herausforderungen in experimenteller Hinsicht - ebenso spannende Herausforderungen ergeben sich bei der Entwicklung mathematischer Modelle, die tieferen Einblick in die Dynamik biologischer Systeme gewähren können. Von besonderem Interesse sind dabei die Wechselwirkungen zwischen Pathogenen und ihrem Wirt sowohl für das Pathogen im einzelnen Wirt als auch bei seiner Ausbreitung auf der Populationsebene. Exemplarisch wird ein Netzwerkmodel zur Beschreibung von Epidemien vorgestellt, mit dem die Wechselwirkungen von epidemischem Geschehen und der Struktur des Transmissionsnetzwerkes beleuchtet werden. Jedoch findet auch bereits in einer einzelnen Infektion eine komplexe ko-evolutionäre Dynamik statt, die sich aus der Interaktion zwischen dem Pathogen und dem Wirtsorganismus, insbesondere der adaptiven Immunantwort, ergibt. Diese Dynamik wird in einem einfachen Quasispezies-Modell untersucht. Vor diesem Hintergrund wird ein Ausblick gegeben auf weiter gehende stochastische Modelle insbesondere zur HIV Evolution im Patienten.

Raum 902, Robert-Mayer-Str. 10

Mittwoch, 02. November 2011: Oberseminar Stochastik / Diplomandenseminar

14:15 Uhr: Thomas Wissen, Goethe-Universität Frankfurt am Main

Zur Eigenwertverteilung zufälliger Matrizen

Raum 711 (klein), Robert-Mayer-Str. 10

Freitag, 28. Oktober 2011: Oberseminar Stochastik / Diplomandenseminar

14:15 Uhr: Stephan Gufler, Goethe-Universität Frankfurt am Main

Exkursionen der Wright-Fisher-Diffusion und eine Pfadzerlegung

Raum 711 (klein), Robert-Mayer-Str. 10

Donnerstag, 13. Oktober 2011: Stochastik-Kolloquium

15:15 Uhr: Olivier Hénard, École des Ponts ParisTech

Two viewpoints on measure valued processes and applications

We first introduce superprocesses in the classical framework. These measure valued processes are models for the spatial evolution of a (big) branching population. We derive representation of the superprocess conditioned on non extinction in remote time, when the branching mechanism is allowed to exhibit some spatial dependence. We then introduce the lookdown framework, after Donnelly and Kurtz, and show how previous representations may be (partly) recovered by a simple change of filtration. We also derive similar results for constant size population models, namely the Generalized Fleming Viot process, without additional effort. The first part is based on a joint work with Jean-François Delmas.

Raum 711 (klein), Robert-Mayer-Str. 10

Freitag, 07. Oktober 2011: Stochastik-Kolloquium

17:15 Uhr: Ellen Baake, Universität Bielefeld

Neues vom Moran-Modell mit (Mutation und) Selektion

Wir betrachten die Ahnenlinien einzelner Individuen, die aus der Gleichgewichtsverteilung des Moran-Modells mit Mutation und Selektion gezogen werden, und interessieren uns fuer die stationaere Typenverteilung der Vorfahren (die `Ahnenverteilung'). Dazu gibt es bereits einige Resultate (von Fearnhead (2002) und Taylor (2007)), die allerdings etwas unbefriedigend sind, weil die Interpretation im Sinne des zugrundeliegenden Teilchensystems fehlt. Wir versuchen nun, eben diesen Zusammenhang herzustellen, und im Fall von Selektion allein (ohne Mutation) gelingt das bereits vollstaendig. Im Fall mit Mutation sind viele Fragen offen, aber bereits jetzt ergeben sich erhebliche Vereinfachungen in dem (urspruenglich recht technischen) Zugang von Taylor.

Raum 711 (klein), Robert-Mayer-Str. 10

Freitag, 23. September 2011: Gastvortrag

14:15 Uhr: Cornelia Pokalyuk, Albert-Ludwigs-Universität Freiburg

The rate of adaptation in structured populations

We consider the process of adaptation in an asexual population distributed onto several islands. New beneficial mutations arise at constant rate u_b and each mutation has the same selective advantage s > 0. We assume that within islands beneficial mutations fix successively, i.e. the time of local fixation of a mutation is short as compared to the waiting time to the next mutation. In order to compute the rate of adaptation, we introduce an approximate but simpler model, which can be simulated efficiently in many parameter ranges and leads to accurate numerical results. In the simple model, mutations fix instantly within an island, and migrants may take over the destination island if they are fitter than the residents. In the special case that the population is distributed onto two equally large islands with population size N, we provide an analytical result for the rate of adaptation. In this case, the analytical results describe the transition of adaptation ranges from 2 N u_b s2 for small migration rates to 4 N u_b s2 for large migration rates.

Raum 711 (klein), Robert-Mayer-Str. 10

08. - 09. September 2011: Frankfurt Seminar on Random Discrete Structures

Raum 711 (groß), Robert-Mayer-Str. 10

Vortragsreihe von Brooks Ferebee (Frankfurt) über Quanteninformatik


06. Juli 2011, 14:15 Uhr:

Was ist ein Quantenbit?

Raum 711 (klein), Robert-Mayer-Str. 10

13. Juli 2011, 14:15 Uhr:

Quantenalgorithmen: Schnelles Suchen, Dichte Kodierung, Teleportation

Raum 711 (klein), Robert-Mayer-Str. 10

20. Juli 2011, 14:15 Uhr:

Ist der Mond noch da, wenn keiner hinschaut? Spukhafte Fernwirkung und die Bellschen Ungleichungen.

Raum 711 (groß), Robert-Mayer-Str. 10

Mittwoch, 22. Juni 2011: Stochastik-Kolloquium

14:15 Uhr: Constantin Tudor, Universtät Bukarest

On some fractional processes (Abstract)

Raum 711 (klein), Robert-Mayer-Str. 10

Freitag, 13. Mai 2011: Mathematisches Kolloquium Frankfurt & Rhein-Main-Kolloquium Stochastik - Colloquium on Analysis and Probability

Raum 711 (groß), Robert-Mayer-Str. 10


15:00: Nathanaël Berestycki (Univ. of Cambridge)

Asymptotic behaviour of near-critical branching Brownian motion

15:50: Coffee and Tea

16:15: Henry Berestycki (EHESS Paris and Univ. of Chicago)

Generalized principal eigenvalues of elliptic operators in unbounded domains and applications

17:15: Julien Berestycki (Univ. Paris VI)

Branching Brownian motion seen from its tip


Henri Berestycki: Generalized principal eigenvalues of elliptic operators in unbounded domains and applications

I will present notions that extend to unbounded domains the classical one of principal eigenvalue of an elliptic operator (non necessarily self-adjoint) with Dirichlet condition in a bounded domain. Starting with a formulation introduced in earlier work with L. Nirenberg and S. Varadhan, I will give new definitions and describe various properties of these generalized eigenvalues as well as a characterization of the maximum principle. These notions have many applications to semi-linear elliptic or parabolic equations in heterogeneous media. In particular, the study of general FKPP type equations involve such quantities. I report here on joint work with Luca Rossi.

Julien Berestycki: Branching Brownian motion seen from its tip

It has been conjectured at least since a work of Lalley and Sellke (1987) that the branching Brownian motion seen from its tip (e.g. from its rightmost particle) converges to an invariant point process. Very recently, it emerged that this can be proved in several different ways (see e.g. Brunet and Derrida, Arguin et al.). The main goal of the present talk is to give a complete description of the limit object and an alternative proof of the convergence. As conjectured by Brunet and Derrida and proved in Arguin et al., the structure of this extremal point process turns out to be a certain Poisson point process with exponential intensity in which each atom has been decorated by an independent copy of an auxiliary point process. Here, we give an explicit construction of this decoration point process.

Nathanaël Berestycki: Asymptotic behaviour of near-critical branching Brownian motion

Consider a system of particles that perform branching Brownian motion with negative drift \sqrt(2- \eps) and are killed upon hitting zero. Initially, there is just one particle at x. Kesten (1978) proved that the system survives if and only if \eps>0. In this talk I shall describe recent joint work with Julien Berestycki and Jason Schweinsberg concerning the limiting behaviour of this process as \eps tends to 0. In particular we establish sharp asymptotics for the limiting survival probability as a function of the starting point x. Moreover, the limiting genealogy between individuals from this population is shown to have a characteristic time scale of order \eps^{-3/2}. When time is measured in these units we show that the geometry of the genealogical tree converges to the Bolthausen-Sznitman coalescent. This is closely related to a set of conjectures by Brunet, Derrida and Simon.

Mittwoch, 19. Januar 2011

13:30 Uhr: Götz Kersting (Frankfurt)

External lengths in Kingman's coalescent and a puzzling urn model

In this paper we prove asymptotic normality of the total length of external branches in Kingman's coalescent. The proof uses an embedded Markov chain, which can be described as follows: Take an urn with $n$ {\em black} balls. Empty it in $n$ steps according to the rule: In each step remove a randomly chosen pair of balls and replace it by one {\em red} ball. Finally remove the last remaining ball. Then the numbers $U_k$, $0 \le k \le n$, of red balls after $k$ steps exhibit an unexpected property: $(U_0, \ldots,U_n)$ and $(U_n, \ldots,U_0)$ are equal in distribution. The proof is given via another urn model. Joint work with Svante Janson.

Raum 711 (groß), Robert-Mayer-Str. 10

Freitag, 21. Januar 2011: Rhein-Main-Kolloquium Stochastik in Frankfurt

15:15 Uhr: Amandine Veber (Ecole Polytechnique, Paris)

Spatial Lambda-Fleming-Viot process : genealogies in the presence of recombination (Abstract)

16:45 Uhr: Peter Mörters (University of Bath)

Typical distances in ultrasmall random networks (Abstract)

Raum 711 (groß), Robert-Mayer-Str. 10

Freitag, 6. August 2010

11:00 Uhr: Robert Knobloch (University of Bath)

Martingales associated with killed fragmentation processes.

We introduce homogenous fragmentation processes whose blocks are killed when being sufficiently small. In this context we consider the probability of extinction of this process as a function of the initial value of the fragmentation and derive certain properties of that function. Our main results are concerned with associated additive and multiplicative martingales. The approach is based on defining killed spectrally negative Lévy processes by resorting to intrinsic subordinators. Our motivation for considering the aforementioned killed fragmentation processes stems from their connection with FKPP travelling waves in the setting of fragmentations.

Raum 711 (klein), Robert-Mayer-Str. 10

9. Juni 2010

14 Uhr c.t.: Dr. Guillaume Voisin (Universität Orleans)

Pruning of random trees: an overview.

Prunings have been studied first for Galton-Watson trees, they can be viewed as percolations. It is possible to create a tree-valued process depending on the parameter of percolation. Continuum random trees can be defined as limits of Galton-Watson trees. Several prunings have been defined for continuum random trees, independently of the discrete case. Recently a convergence of discrete to continuous pruning has been studied.

Raum 711 (groß), Robert-Mayer-Str. 10

12. Mai 2010

16 Uhr c.t.: Prof. W. Szpankowski (Purdue University)

Analytic information theory and beyond

Analytic information theory aims at studying problems of information theory using analytic techniques of computer science and combinatorics. Following Hadamard's precept, we tackle these problems by complex analysis methods such as generating functions, Mellin transform, Fourier series, saddle point method, analytic poissonization and depoissonization, and singularity analysis. This approach lies at the crossroad of computer science and information theory. In this talk, we concentrate on one facet of information theory (i.e., source coding better known as data compression), namely the redundancy rate problem. The redundancy rate problem for a class of sources is the determination of how far the actual code length exceeds the optimal (ideal) code length. In a minimax scenario one finds the maximal redundancy over all sources within a certain class while in the average scenario one computes the average redundancy over all possible sources. The redundancy rate problem is typical of a situation where second-order asymptotics play a crucial role (since the leading term of the optimal code length is known to be the entropy of the source). This problem is an ideal candidate for ``analytic information theory''. We survey our recent results on the redundancy rate problem obtained by analytic methods. In particular, we present our findings for the redundancy of Shannon codes, Huffman codes, minimax redundancy for memoryless sources, Markov sources, and renewal processes. In the second part of the talk, we discuss the limitations of classical Shannon information theory, and argue that there is need for post-Shannon information theory.

Raum 711 (groß), Robert-Mayer-Str. 10

23. April 2010

15:15 Uhr: Prof. Jan Swart, (UTIA Prague)

Intertwining of Markov processes and the contact process on the hierarchical group

Two Markov processes X and X' are called intertwined if the conditional distribution of X(t) given the history of X' up to time t depends only on X'(t). An interesting feature of such a coupling is that the transition rates of the process X' may depend on the state of X, yet the process X', on its own, has the Markov property. Such couplings were first discovered by Rogers and Pitman and have more recently been used by Diaconis and Miclo in the study of strong stationary times of Markov processes. In the present talk, I will show how the technique of intertwining Markov processes can be used in the study of the contact process on the hierarchical group, where sites infect other sites at a rate depending on their hierarchical distance, and sites become healthy with a constant recovery rate. If the infection rates decay too fast as a function of the hierarchical distance, then we show that the critical recovery rate is zero. On the other hand, we derive sufficient conditions on the speed of decay of the infection rates for the process to exhibit a nontrivial phase transition between extinction and survival.
This is joint work with Siva Athreya (Bangalore).

16:45 Uhr: Prof. Silke Rolles, (TU München)

Bayesian analysis for reversible Markov chains

We introduce a natural conjugate prior for the transition matrix of a reversible Markov chain. The prior arises from random walk with reinforcement in the same way the Dirichlet prior arises from Polya's urn.
The talk is based on joint work with Persi Diaconis.

Raum 711 (groß), Robert-Mayer-Str. 10


21. April 2010

14:15 Uhr: Prof. Götz Kersting

External lengths in Kingman's coalescent and a puzzling urn model

Raum 711 (groß), Robert-Mayer-Str. 10

Rhein-Main-Kolloquium Stochastik

12. Februar 2010

15:15 Uhr: Prof. Dirk Metzler, Biozentrum, Universität München

Efficient parameter estimation in population genetic models of complex demography

Given population genetic data from related populations or species we aim to estimate parameters like population split times, population growth rates and migration rates. Maximum-Likelihood and Bayesian methods based on importance sampling and MCMC are expected to give the best results but often need several months of computer run-time for just one dataset. Moreover, some dataset require models which are not incorporated in the available implementations of these methods, and the development of such a software package may also take years due to the complex data structures involved. We discuss approaches for heuristics which are fast and easy to implement.

16:45 Uhr: Prof. Jean-François Delmas, École des Ponts, Paris-Marne-la-Vallée

Most recent common ancestor and bottleneck in a simple size-varying population model

We consider a simple model of a neutral population with evolving size. (This model corresponds to the usual continuous state space branching process conditioned to non extinction in a stationary regime.) For this model, we compute the size of the population at the time of Most Recent Common Ancestor (MRCA). We show that it is in mean less than 1/3 of the mean size of the population under the stationary regime, and that it is less than the current population size with probability 11/16. This elementary example provides an explanation of the bottlenck effect at the time of MRCA.

Raum 711 (groß), Robert-Mayer-Str. 10


14 Uhr diesmal s.t., Raum 711 (groß), Robert-Mayer-Str. 10

27. Januar 2010    Dr. Arno Siri-Jégousse, Université René Descartes, Paris

Estimation of mutation rates using coalescent theory

In many models from population genetics, genealogies of the population can be represented, when appropraiely renormalized, by a coalescent process. This representation is very useful to study mutation rates, because the number of mutant types in the population is closely linked to the total length of the coalescent tree. We give, in the Beta-coalescent case, asymptotic results on this length when initial population size grows, leading to a convergence result on the total number of mutations that appeared in the tree.

Raum 711 (groß), Robert-Mayer-Str. 10

30. September 2009    Prof. Etienne Pardoux (Universite de Provence, Marseille)

Binary trees, the exploration process and the Ray-Knight theorem

14 Uhr c.t., Raum 711 (groß), Robert-Mayer-Str. 10

21. September 2009    Prof. Dr. Hsien-Kuei Hwang (Academia Sinica, Taipeh)

Asymptotic variances for random digital trees

14 Uhr c.t., Raum 711 (groß), Robert-Mayer-Str. 10

4. September 2009    Dr. Benjamin Staude und Prof. Dr. Stefan Rotter    (Freiburg)

Higher-order correlations in large neuronal populations

10 Uhr c.t., Raum 711 (groß), Robert-Mayer-Str. 10

Spiking neurons are known to be quite sensitive for the higher-order correlation structure of their respective input populations (Kuhn et al. 2003). What is the role of these correlations in cortical information processing? A prerequisite to answering this question is an appropriate framework to describe and effectively estimate the correlation structure of neuronal populations. Approaches available thus far suffer from the combinatorial explosion of the number of parameters that grows exponentially with the number of recorded neurons. As a consequence, methods that go beyond pairwise correlations and aim for estimating genuine higher-order effects require vast samples, rendering them essentially inapplicable to populations of more than ~10 neurons. Here, we discuss the compound Poisson process as an intuitive and flexible model for correlated populations of spiking neurons. Based on this generative model, we present novel estimation techniques to infer the correlation structure of a neural population from sampled spike trains (Ehm et al. 2007; Staude et al. 2009). Our techniques can provide conclusive evidence for higher-order correlations in rather large populations of ~100 neurons, based on sample sizes that are compatible with current physiological in vivo recording technology.

Kuhn A, Aertsen A, Rotter S. Higher-order statistics of input ensembles and the response of simple model neurons. Neural Computation 15(1): 67-101, 2003
Ehm W, Staude B, Rotter S. Decomposition of neuronal assembly activity via empirical de-Poissonization. Electronic Journal of Statistics 1: 473-495, 2007
Staude B, Rotter S, Grün S. CuBIC: cumulant based inference of higher-order correlations in massively parallel spike trains. Under review

1. Juli 2009    Dr. Arleta Szkoƚa    (MPI für Mathematik in den Naturwissenschaften, Leipzig)

Operational meaning of entropic functionals and their generalizations on state spaces of operator algebras

14 Uhr c.t., Raum 711 (groß), Robert-Mayer-Str. 10

Abstract and References

24. Juni 2009    Dr. Simona Grusea    (Marseille)

Measures for the exceptionality of gene order in conserved genomic regions

14 Uhr c.t., Raum 711 (groß), Robert-Mayer-Str. 10

The goal of this work is to find "good" measures for quantifying the exceptionality of the order of the orthologs in conserved genomic regions between two different species. Here "good" means both biologically relevant and computationally accessible. We propose three measures based on the transposition distance in the permutation group for measuring the gene order conservation in orthologous gene clusters found by the reference region approach. We obtain analytic expressions for their distribution in the case of a random uniform permutation, i.e. under the null hypothesis of random gene order. Our results can be used to increase the power of the significance tests for gene clusters which take into account only the proximity of the orthologous genes and not their order.

19. Juni 2009    Prof. Dr. Andreas Orth    (Institut für professionelle Informatikanwendungen, FH Frankfurt, und Umesoft GmbH)

Multivariate Modellierung der Immunrekonstitution bei Kindern nach Stammzelltransplantation

14 Uhr c.t., Hilbertraum (Raum 302, Robert-Mayer-Str. 8)

Es wird die Frage erörtert, ob farbzytometrische Messdaten, die in regelmäßigen Abständen routinemäßig bei Kindern nach Stammzelltransplantation erhoben werden, für prognostische und therapeutische Zwecke verwendet werden können. Die (glücklicherweise) geringe Patientenzahl erlaubt bisher nur explorative Ansätze zum Hypothesengenerieren - fördert aber bereits interessante Ergebnisse zu Tage. Ein Ziel des Vortrags ist es, einfache multivariate Modelle vorzustellen, die bisher für die Modellierung verwendet wurden, ein zweites, Impulse dahin gehend zu erhalten welche Modelle und Methoden in Zukunft für eine durchzuführende konfirmatorische Studie verwendet werden sollen.

10. Juni 2009    Prof. Dr. Eva Herrmann    (Institut für Biostatistik und Mathematische Modellierung, Goethe-Universität)

Statistische Analysen mathematischer Modelle zur Virushepatitis

14 Uhr c.t., Raum 711 (groß), Robert-Mayer-Str. 10

8. Mai 2009    Prof. Dr. Olav Kallenberg    (Dept. of Mathematics, Auburn University)

Iterated Palm conditioning and some Slivnyak-type theorems for Cox and cluster processes

14 Uhr c.t., Hilbertraum (Raum 302, Robert-Mayer-Str. 8)

Under suitable regularity conditions, the operations of conditioning on a $\sigma$-field and of forming the Palm distributions with respect to a random measure commute. This principle of iterated conditioning is used to prove some multivariate versions for randomizations and Cox processes of Slivnyak's celebrated theorem - the fact that the reduced Palm distributions of a Poisson process agree with the original distribution. That leads in turn to some useful representations for the Palm measures of certain conditional cluster processes, such as those arising in the context of spatial branching and superprocesses.

17.12.2008    Sandra Kliem    (Department of Mathematics, UBC Vancouver)

Degenerate Stochastic Differential Equations for Catalytic Branching Systems

12:30-14 Uhr, Raum 612

First we shall discuss uniqueness of the martingale problem corresponding to a degenerate SDE which models catalytic branching networks. This work is an extension of a paper by Dawson and Perkins to arbitrary catalytic branching networks. Next, we investigate the long-term behaviour of a particular system of SDEs for $d \geq 2$ types, involving catalytic branching and mutation between types. It can be shown that the overall sum of masses converges to zero but does not hit zero in finite time a.s. Finally, results on the relative behaviour of types are given to obtain some insight on how the process approaches zero.