Stochastisches Kolloquium

Mittwoch, 10. Februar 2016

14:15 Uhr: Timo Hirscher (Chalmers University, Göteborg)
Segregating Markov chains
Abstract:
Dealing with finite Markov chains in discrete time, we often focus
on convergence behavior and try to make different copies of the chain
meet as fast as possible and then stick together. There is, however, a very
peculiar kind of Markov chains for which two copies started
in different states can be coupled to meet almost surely in finite time, yet
their distributions keep a total variation distance bounded away from 0.
We show that the supremum of total variation distance kept in this context
equals 1/2.
This is joint work with Anders Martinsson.
Raum 711 (groß), RobertMayerStr. 10

RheinMain Kolloquium Stochastik

Freitag, 05. Februar 2016

15:15 Uhr: René Schilling (TU Dresden)
Level sets of Feller processes
Abstract:
We give brief introduction to Feller processes and their symbols and we then use symbols and indices
of Feller processes to give bounds on the Hausdorff dimension of the level (and collision) sets of a
class of Feller processes. This extends analogous results for Levy processes by Hawkes (1974)
(for level sets) and Taylor (1966) and Jain and Pruitt (1969) (for collision sets).
This is a joint work with Victoria Knopova from Kiev.
16:15 Uhr: Kaffee und Tee
16:45 Uhr: Jean Bertoin (Universität Zürich)
Compensated Fragmentations
Abstract:
Fragmentation processes have been introduced first by Kolmogorov as a model for the disintegration
a mass as time passes. When the set of times at which dislocations occur remains discrete, they
can be described in terms of a branching random walk. Roughly speaking, we shall consider the
situation when the intensity of dislocations is so high that the entire mass is almost instantaneously
reduced to dust. Following the classical work of Paul Lévy, who showed that certain diverging series
of random variables can be turned convergent after a proper compensation, we shall see how the
instantaneous shattering of the mass can nonetheless be prevented by incorporating a suitable
exponential growth of the fragments. This yields a new class of fragmentation processes, which
appear naturally in the study of large random planar maps.
TU Darmstadt, Ehemaliges Maschinenhaus, Magdalenenstr. 12, Geb. S105, Raum 23



RheinMain Kolloquium Stochastik

Freitag, 22. Januar 2016

15:15 Uhr: Amin CojaOghlan (Universität Frankfurt)
Limits of discrete distributions and Gibbs measures on random graphs
Abstract:
We construct continuous embeddings of discrete probability distributions via the theory of graph limits.
Moreover, we show various alternative descriptions and an approximation theorme related to the Szemeredi
regularity lemma. As an application we study Gibbs measures of sparse random factor graphs.
Based on joint work with Will Perkins and Kathrin Skubch.
16:15 Uhr: Kaffee und Tee
16:45 Uhr: Lutz Warnke (University of Cambridge)
The phase transition in boundedsize Achlioptas processes
Abstract:
Perhaps the best understood phase transition is that in the component structure of the uniform random
graph process introduced by Erdös and Rényi around 1960.
Since the model is so fundamental, it is very interesting to know which features of this phase transition are specific
to the model, and which are `universal', at least within some larger class of processes.
Achlioptas process, a class of variants of the ErdösRényi process that are easy to define but difficult
to analyze, have been extensively studied from this point of view.
Here, settling a number of conjectures and open problems, we show that all `boundedsize' Achlioptas
processes share many key features of the ErdösRényi phase transition (in particular the asymptotic behaviour of the size of the largest component above and below the critical window). We do not expect this to hold for Achlioptas processes in general.
This is joint work with Oliver Riordan.
Universität Frankfurt, Institut für Mathematik, RobertMayerStr. 10, Raum 711 (groß), 7. Stock

RheinMain Kolloquium Stochastik

Freitag, 04. Dezember 2015

15:15 Uhr: Olivier Garet (Université de Lorraine, Nancy)
Essential hitting times for the contact process
Abstract:
The goal of this talk is to present the notion of essential hitting times. We introduced it in order
to prove an asymptotic shape theorem for the contact process in random environment. Further, it appeared
that this tool was also useful to prove large deviations estimates and could be applied to several
interacting particle systems. Essential hitting times also have a role to play in the further talk,
by Regine Marchand.
16:15 Uhr: Kaffee und Tee
16:45 Uhr: Régine Marchand (Université de Lorraine, Nancy)
The number of open paths in supercritical oriented percolation
Abstract:
In this talk, I will explain how we can obtain the asymptotic behavior of the number of open paths with
length n in supercritical oriented percolation, with the help of the essential hitting times. Oriented
percolation is a very simple example of a random growth model with possible extinction for which
essential hitting times can be useful.
Universität Mainz, Institut für Mathematik, Staudingerweg 9, Raum 05432 (Hilbertraum), 5. Stock

Stochastisches Kolloquium

Mittwoch, 22. Juli 2015

10:00 Uhr: Fima Klebaner (Monash University, Melbourne)
Escape from the boundary in Markov population processes
Abstract:
tba
11:00 Uhr: Jason Schweinsberg (University of California at San Diego)
Rigorous results for a population model with selection
Abstract:
We consider a model of a population of fixed size $N$ in which each individual
acquires beneficial mutations at rate $\mu$. Each individual dies at rate one,
and when a death occurs, an individual is chosen with probability proportional to
the individual's fitness to give birth. We obtain rigorous results for the rate
at which mutations accumulate in the population, the distribution of the fitnesses
of individuals in the population at a given time, and the genealogy of the population.
Our results confirm predictions of Desai and Fisher (2007), Desai, Walczak,
and Fisher (2013), and Neher and Hallatschek (2013).
Raum 711 (groß), RobertMayerStr. 10

RheinMain Kolloquium Stochastik

Freitag, 10. Juli 2015

15:15 Uhr: Loïc Chaumont (Université Angers)
Time change in multitype branching processes with application to mutations
Abstract:
From the breadth first search algorithm, we construct a bijection between the set of multitype plane
forests and some set of multivariate sequences. This coding allows us to obtain the law of the total
population in a multitype branching forest and a Lamperti type transformation between multitype branching
processes and multivariate random walks. In continuous time, we show that any discrete valued branching
process, with $d$ types can be obtained as a $d^2$dimensional compound Poisson process, time changed by
the integral of this branching process. As an application of these results, we can obtain the law of the
number of mutations in a multitype branching forest and we study its asymptotic behaviour.
16:45 Uhr: Ron Doney (University of Manchester)
A NASC for the strong renewal theorem via local large deviations
Abstract:
Suppose (S_{n},n\geq 0) is an integervalued random walk such that S_{n}/a_{n} converges in distribution
to a stable law of index \alpha\in(0,1)\cup(1,2). A new upper bound for P(S_{n}=x) when x/a_{n}\to\infty
is given, and this is then used, in the case \alpha\in(0,1), to establish a NASC for the strong renewal
theorem. This fills a gap in the literature which has existed since the 1963 paper by Garsia and Lamperti.
TU Darmstadt, ehemaliges Maschinenhaus, Magdalenenstr. 12 (Gebäude S105), Raum 22

Extra Kolloquium Stochastik Darmstadt

Mittwoch, 08. Juli 2015

ab 16:45 Uhr: Teerunde in Raum 244 des Mathematikgebäudes (S2/15), Schlossgartenstaße 7
17:15 Uhr: Andreas Greven (Erlangen)
Evolution von Genealogien in Populationsmodellen
Abstract:
tba
TU Darmstadt, Raum S214 /24

Stochastisches Kolloquium

Dienstag, 07. Juli 2015

11:00 Uhr: Prof. Kristan Schneider (Mathematik, Hochschule Mittweida)
Schätzung von MarkerHäufigkeiten und der Verteilung von KoInfektionen bei Malaria
Abstract:
The number of coinfections of a pathogen (multiplicity of infection or MOI) is a relevant parameter
in epidemiology as it relates with transmission intensity. Notably, such a quantity can be built into
a metric in the context of disease control and prevention. Having applications to malaria in mind,
we introduce a maximumlikelihood (ML) framework to estimate MOI and pathogenlineage frequencies
(which are genetically characterized, e.g., alleles or haplotypes in a short nonrecombining region).
Assuming specifically that infections are rare and independent events, the number of infections per host
follows a conditional Poisson distribution. Under this assumption, existence and uniqueness of the ML
estimate can be proved. The estimate is asymptotically unbiased, consistent and efficient and found
by a simple recursion. Moreover, we provide explicit formulas for asymptotic confidence intervals,
and show that profilelikelihood based confidence intervals exists, which are found by a simple
twodimensional recursion. We will discuss the finite sample properties of the ML estimate and
the profilelikelihood confidence intervals, which were investigated by MonteCarlo simulation.
Finally, we illustrate the methods on three malaria data sets. The statistical framework itself,
however, is not limited to malaria.
Raum 310, RobertMayerStr. 6

Oberseminar Stochastik
Stochastisches Kolloquium

Mittwoch, 01. Juli 2015

14:15 Uhr: Stephan Gufler (Universität Frankfurt)
Genealogies and sampling
Abstract:
This talk will be about neutral evolving genealogies as treevalued processes,
constructed from the lookdown model. By decomposing genealogical trees and sampling
from marked metric measure spaces, the case with dust can be included.
15:15 Uhr: Steven N. Evans (Departments of Mathematics and Statistics, University of California at Berkeley)
The fundamental theorem of arithmetic for metric measure spaces
Abstract:
A metric measure space (mms) is simply a complete, separable metric space equipped with a
probability measure that has full support. A fundamental insight of Gromov is that the space
of such objects is much ``tamer'' than the space of complete, separable metric spaces per se
because mms carry within themselves a canonical family of approximations by finite structures:
one takes the random mms that arises from picking some number of points independently at random
and equipping it with the induced metric and uniform probability measure. A natural (commutative
and associative) binary operation on the space of mms is defined by forming the Cartesian product
of the two underlying sets equipped with the sum of the two metrics and the product of the two
probability measures. There is a corresponding notion of a prime mms and an analogue of the
fundamental theorem of arithmetic in the sense that any mms has a factorization into countably
many prime mms which is unique up to the order of the factors. Moreover, a rich Fourier theory
enables one to analyze convolutions of probability measures on the space of mms and obtain
counterparts of classical results in the theory of infinitely divisible and stable probability
measures on Euclidean spaces due to Lévy, Itô, Hinčin, and LePage.
This is joint work with Ilya Molchanov (Bern).
Raum 711 (groß), RobertMayerStr. 10

Oberseminar Stochastik

Mittwoch, 24. Juni 2015

14:15 Uhr: Yasmin Straub (Universität Frankfurt)
GaltonWatson Prozesse in zufälliger Umgebung
Abstract:
Bachelorabschlusspräsentation
15:15 Uhr: Anna Kremer (Universität Frankfurt)
Perpetuities in der Analyse fairer Leader Election Algorithmen
Abstract:
Bachelorabschlusspräsentation
Raum 711 (groß), RobertMayerStr. 10

RheinMain Kolloquium Stochastik

Freitag, 12. Juni 2015

15:15 Uhr: Paul Jenkins (University of Warwick)
New evolutionary models for patterns of genetic variation along a DNA sequence
Abstracts:
Many biological and demographic processes contribute to the patterns of variation we observe in samples of DNA
sequence data: mutation, natural selection, migrations, and so on. Also of fundamental importance is recombination, a
type of rearrangement of genetic material during cell division. Recombination profoundly influences the correlations
between genetic variants seen at different positions along a sequence. Standard stochastic evolutionary models
incorporating recombination include the multilocus WrightFisher diffusion, which traces the frequencies of each genetic
type through time, and its dual genealogical process, the ancestral recombination graph. However, these models are difficult
to work with, statistical inference typically being intractable. In this talk I will describe two simple limiting approximations
to these models as their recombination parameter goes to infinity. The first is via an application to the WrightFisher diffusion
of a central limit theorem for densitydependent population processes, which fully describes the (Gaussian) fluctuations in the
correlation of genetic types at different positions. The second approach uses a coupling argument applied to the ancestral
recombination graph. These results reveal some hidden structure in these models and also have practical implications for
genomic inference, since the likelihoods under these new models are analytically tractable.
16:45 Uhr: Andrej Depperschmidt (Universität Freibug)
Recombination as a treevalued process along the genome
Abstracts:
The ancestral recombination graph (ARG) gives the joint genealogy of a population sample at various loci under recombination.
In other words, one can read off genealogical trees from the ARG at all loci along the genome, giving rise to a treevalued
stochastic process. For a continuous genome, we study this treevalued process in the limit of large population sizes. Encoding
trees as metric measure spaces, we show convergence to a treevalued process with cadlag paths. In addition, we study mixing
properties of the resulting process for loci which are far apart.
This is joint work with Andrej Depperschmidt (Freiburg) and Etienne Pardoux (Marseille).
Universität Mainz, Institut für Mathematik, Staudingerweg 9, Raum 05432 (Hilbertraum), 5. Stock

RheinMain Kolloquium Stochastik

Freitag, 29. Mai 2015

15:15 Uhr: Vincent Vargas (Ecole Normale Supérieure, Paris)
Liouville quantum gravity on the Riemann sphere (Part I)
Abstracts:
This talk is made up of two parts. The first part (Vargas) will explain in details the rigorous construction of the Liouville
quantum field theory on the Riemann sphere based on Polyakov's functional integral. The second part (Rhodes) of the talk will
be more informal, presenting motivations, connections with conformal field theories and conjectures relating this theory to
the scaling limit of random planar maps conformally embedded onto the Riemann sphere.
Based on joint works with F. David, A. Kupiainen, H. Lacoin.
16:15 Uhr: Kaffee und Tee
16:45 Uhr: Rémy Rhodes (Université ParisEst Marne La Vallée)
Liouville quantum gravity on the Riemann sphere (Part II)
Universität Frankfurt, Institut für Mathematik, RobertMayerStr. 10, Raum 711 (groß), 7. Stock

Workshop Mannheim

Donnerstag, 28. Mai 2015
und Freitag, 29. Mai 2015

Abstract und Informationen zum Workshop
Universität Mannheim

Berufspraxiskolloquium

Mittwoch, 27. Mai 2015

16:15 Uhr: Carolin Breitenbach und Nadine Mbenda (PwC Frankfurt)
Mathematiker in der Wirtschaftsprüfung und Beratung
Abstract:
Nach der Krise ist vor der Krise  dies ist auch den nationalen und internationalen Bankenaufsichtsbehörden klar.
Finanzinstitute sind einer Vielzahl von Risiken ausgesetzt und müssen sich zunehmend mit einer Fülle von neuen
Regulierungsvorschriften auseinandersetzen. Bei der Umsetzung und Einhaltung dieser gesetzlichen Vorgaben sind
die Institute auf externe Beratung und Unterstützung angewiesen. Die neuen Anforderungen beziehen sich vermehrt
auch auf die Güte der mathematischen Modelle, die zur Schätzung von bspw. Kredit, Marktpreis,
Liquiditätsrisiken eingesetzt werden. Um die Angemessenheit dieser Modelle beurteilen zu können,
benötigt man neben Kenntnissen der regulatorischen Vorgaben auch ein tiefes grundlegendes Verständnis
von finanzmathematischen Methoden und Modellen.
Das FinanzmatheTeam des Bereichs Risk & Regulation von PwC bietet Unterstützung bei
diversen finanzmathematischen Fragestellungen im Zusammenhang mit Risikomanagement,
Bilanzierung und Handel.
Raum , RobertMayerStr. 10

Extra Kolloquium Stochastik Darmstadt

Mittwoch, 06. Mai 2015

ab 16:45 Uhr: Tee
17:15 Uhr: Erwin Bolthausen (Zürich)
tba
Abstract:
tba
TU Darmstadt

Oberseminar Stochastik

Montag, 27. April 2015

10:15 Uhr: Rebecca Seeger (Universität Frankfurt)
Statistische Tests zur Varianzhomogenität in Erneuerungsprozessen
Abstract:
Masterarbeit
Raum 711 (groß), RobertMayerStr. 10

Stochastisches Kolloquium

Donnerstag, 19. März 2015

16:00 Uhr: Professor F. Alberto Grünbaum (Math. Dept , UC Berkeley )
Timeandband limiting and the bispectral problem: motivation, new applications and open problems
Abstract:
The "bispectral problem" is tied up with several parts of mathematics.
I will try to give a historical introduction going all the way back to Claude Shannon around 1950.
Here are the basic facts:
In signal processing, once you specify a measuring mechanism you arrive at a GLOBAL operator, given by an integral kernel
or a full matrix. One needs to compute many of its eigenfunctions efficiently and economically. The quality of the possible images will
depend on these eigenfunctions and their eigenvalues.
In certain cases this can be done by a MIRACLE: one can exhibit a differential operator of very low order which has the same
eigenfuntions as the global operator. The numerical computation of the eigenfunctions of the differential operator, a LOCAL one,
is (relatively speaking) a trivial matter compared to the initial task. In many cases the initial task cannot be implemented because of
numerical instability.
Raum 711 (klein), RobertMayerStr. 10

RheinMain Kolloquium Stochastik

Freitag, 30. Januar 2015

ab 14:45 Uhr: Kaffee, Tee und Dessert
15:15 Uhr: Zakhar Kabluchko (Münster)
Asymptotic expansions for profiles of branching random walks and random trees
Abstract:
Consider a branching random walk on the lattice $\mathbb Z$. Let $L_n(k)$ be the number of particles which are
located at site $k\in\mathbb Z$ at time $n\in\mathbb N_0$. The function $k\mapsto L_n(k)$ is called the profile of the branching
random walk. It is well known that for large times $n$ the profile has approximately Gaussian shape. We will derive a complete
asymptotic expansion for the profile which is similar to the classical Edgeworth expansion for sums of i.i.d. random variables but
contains additional terms involving the derivatives of the Biggins martingale. This expansion allows to treat several problems on the
branching random walk in a unified way. For example, we will obtain a.s.\ limit theorems for $L_n(k_n)$ as $n\to\infty$, where $k_n$
is an integer sequence behaving in some regular way. In the simplest case when $k_n=k$ is constant, the existence of the limiting distribution
of $L_n(k)$ depends on certain arithmetic properties of the branching random walk. Using an embedding into a continuoustime branching random
walk similar results can be obtained for binary search trees, random recursive trees and some other families of random trees. This is joint
work with Rudolf Grübel (Hannover).
16:45 Uhr: Remi Monasson (CNRS/Ecole polytechnique)
Statistical physics of the representation(s) of space in the brain
Abstract:
Understanding the mechanisms by which space gets represented in the brain is a fundamental problem in neuroscience.
The experimental discovery of socalled "place cells" and "gridcells" (rewarded by the Nobel Prize in Medecine last Fall),
encoding specific positions in space, provide essential elements in this context. How a spatial chart, that is, a relation
between different points in space, may be built and memorized? In this talk, I will present a model involving binary neurons
(either active or silent), making possible to store one or more spatial charts. In the case of a single map the model is
equivalent to a latticegas system, and undergoes a phase transition for decreasing temperatures from a vapor phase to a
liquid phase (LebowitzPenrose theory). When more than one maps are stored the model is an extension of the Hopfield autoassociative
memory model, in which the memory items are Ddimensional continuous attractors rather than fixed points. The model may be solved
with the help of the (non rigorous) techniques of the statistical physics of disordered systems, and shows a combination of ferromagnetic,
paramagnetic, and glassy phases, depending on the control parameters. I will discuss the dynamical features of the system, with an emphasis
on the transitions between maps, in relationship with recent teleportation experiments on living rats.
Universität Frankfurt, Institut für Mathematik, RobertMayerStr. 10, Raum 711 (groß), 7. Stock

Extra Kolloquium Stochastik Darmstadt

Donnerstag, 29. Januar 2015

16:15 Uhr: Alex Novikov (Sydney)
First passage time problems: the martingale approach revisiteds
Abstract:
tba
17:15 Uhr: Vitali Wachtel (Augsburg)
Asymptotics of invariant measures of recurrent Markov chains with asymptotically zero drift
Abstract:
The study of nonnegative Markov chains with asymptotically
zero drift has been initiated
by Lamperti, who provided recurrence/transience criteria for such
chains. We shall concentrate on
the case of recurrent chains and present different approaches (Lyapunov
functions and harmonic functions)
to the problem of the asymptotic behaviour of invariant measures.
TU Darmstadt, S215 (Mathebau), Raum 401

RheinMain Kolloquium Stochastik

Freitag, 23. Januar 2015

15:15 Uhr: Olivier Zindy (Paris 6)
PoissonDirichlet statistics for the extremes of logcorrelated Gaussian fields
Abstract:
Gaussian fields with logarithmically decaying correlations, such as branching Brownian motion and the 2D Gaussian
free field, are conjectured to form a new universality class of extreme value statistics (notably in the work of
Carpentier & Ledoussal and Fyodorov & Bouchaud). This class is the borderline case between the class of IID random variables,
and models where correlations start to affect the statistics. In this talk, I will describe a general approach based on
rigorous works in spin glass theory to describe features of the Gibbs measure of these Gaussian fields. I will focus on a
model de fined on the periodic interval [0;1]. At low temperature, we show that the normalized covariance of two points
sampled from the Gibbs measure is either 0 or 1. This is used to prove that the joint distribution of the Gibbs weights
converges in a suitable sense to that of a PoissonDirichlet variable. This is joint work with LouisPierre Arguin.
16:45 Uhr: LouisPierre Arguin (Université de Montréal)
Gaussian fields and the maxima of the Riemann Zeta function on the critical line
Abstract:
A recent conjecture of Fyodorov, Hiary & Keating states that the maxima of the Riemann Zeta function on a
bounded interval of the critical line behave similarly to the maxima of a specific class of Gaussian fields,
the socalled logcorrelated Gaussian fields. These include important examples such as branching Brownian motion
and the 2D Gaussian free field. In this talk, we will highlight the connections between the number theory problem
and the probabilistic models. We will outline the proof of the conjecture in the case of a randomized model of the
zeta function. We will discuss possible approaches to the problem for the function itself. This is joint work
with D. Belius (NYU) and A. Harper (Cambridge).
Universität Mainz, Institut für Mathematik, Staudingerweg 9, Raum 05432 (Hilbertraum)

Oberseminar Stochastik

Mittwoch, 10. Dezember 2014

14:15 Uhr: Henriette Arlt (Universität Frankfurt)
Modellierung von Orderbuchprozessen aus Verzweigungsbäumen
Abstract:
Masterarbeit
Raum , RobertMayerStr. 10

RheinMain Kolloquium Stochastik

Freitag, 05. Dezember 2014

15:15 Uhr: XueMei Li (University of Warwick)
Stochastic homogeneization on Lie groups
Abstract:
I will discuss a family of reduced equation on the Lie group and their rate of convergence and the
identification of the limiting objects. These equations are reduced from a family of naturally arising
stochastic differential equations in the context of group actions.
16:45 Uhr: Martin Hairer (University of Warwick)
Weak Universality of the KPZ equation
Abstract:
tba
TU Darmstadt, Institut für Festkörperphysik, Hochschulstraße, Gebäude S204, Raum 213

Oberseminar Stochastik

Montag, 29. September 2014

14:15 Uhr: Nor Jaafari (Universität Frankfurt)
Konvergenzraten bei zufälligem kSAT mit mäßig wachsendem k
Abstract:
Masterarbeit
Raum , RobertMayerStr. 10

Oberseminar Stochastik

Mittwoch, 10. September 2014

14:15 Uhr: Xiao Yin (Universität Frankfurt)
Austauschbare Zufallsfelder
Abstract:
Masterarbeit
Raum 711 (klein), RobertMayerStr. 10

Oberseminar Stochastik

Mittwoch, 03. September 2014

14:15  14:55 Uhr: Kathrin Skubch (Universität Frankfurt)
Pfadlängen in scale free trees
Abstract:
Masterarbeit
15:00  15:40 Uhr: Andrea Kuntschik (Universität Frankfurt)
Konvergenzraten für PolyaUrnen
Abstract:
Masterarbeit
Raum 711 (klein), RobertMayerStr. 10

Oberseminar Stochastik

Freitag, 25. Juli 2014

10:15 Uhr: Stefan Albert (Universität Frankfurt)
Ein MultipleFilterTest zur Detektion von Varianzänderungen in Erneuerungsprozessen
Abstract:
Masterarbeit
11:15 Uhr: Benjamin Straub (Universität Frankfurt)
Optimierung von Phasen und Ratenparametern in einem stochastischen Modell neuronaler Feueraktivität
Abstract:
Masterarbeit
Raum , RobertMayerStr. 10

Oberseminar Stochastik

Montag, 21. Juli 2014

14:15 Uhr: Carolin Breitenbach (Universität Frankfurt)
Asymptotik der totalen Astlänge im XiKoaleszenten mit Staub
Abstract:
Masterarbeit
Raum 711 (klein), RobertMayerStr. 10

Oberseminar Stochastik

Dienstag, 15. Juli 2014

10:15 Uhr  14 Uhr: Gaby Schneider & Team (Universität Frankfurt)
Abschlusspräsentation des Statistischen Praktikums
Programm mit den Abstracts:
http://www.math.unifrankfurt.de/~ismi/schneider/StatPrakt14.html
Raum 711 (klein), RobertMayerStr. 10

Stochastisches Kolloquium

Mittwoch, 09. Juli 2014

14:15 Uhr: Matthias Meiners (TU Darmstadt)
Solutions to multivariate smoothing equations
Abstract:
In several models of applied probability such as cyclic Pólya urns, mary
search trees, and fragmentation processes, limiting distributions of
quantities of interest are solutions to smoothing equations in the complex
plane. Further, the stationary solutions of certain 3dimensional
kinetictype evolution equations satisfy smoothing equations with random
similarities as coefficients.
In my talk, I will consider smoothing equations in dimension d with
random similarities as coefficients. This is a unified framework which
contains all examples listed above. The main focus of the talk is on
the problem of determining all solutions to these equations and to
compare the set of solutions with the corresponding ones in one
dimension.
The talk is based on ongoing joint research with Sebastian Mentemeier
(Wroclaw).
Raum 711 (klein), RobertMayerStr. 10

RheinMain Kolloquium Stochastik

Freitag, 27. Juni 2014

15:00 Uhr: Markus Heydenreich (Leiden)
Spontaneous breaking of rotational symmetry in the presence of defects
Abstract:
The formation of crystals, and in particular melting and freezing transitions, are not yet mathematically
understood. It is expected that crystallization phenomena are intricately connected with the breaking of
certain symmetries. In this talk, we consider a simple twodimensional model of crystallization with random
defects in thermal equilibrium. We prove a strong form of spontaneous breaking of rotational symmetry at low
temperatures.
(Based on joint work with Franz Merkl and Silke Rolles).
16:30 Uhr: Erwin Bolthausen (Zürich)
Exit distributions for random walks in anisotropic
random environments (joint work with Erich Baur and Ofer Zeitouni)
Abstract:
Most results on nonballistic (and nonreversible) random walks in random environments
use an isotropy condition for the random environment. We present a recent joint result with
Erich Baur on an anisotropic case in dimension 3 and above. We also discuss briefly the difficult
twodimensional case which is work in progress (with Erich Baur and Ofer Zeitouni).
TU Darmstadt, Fachbereich Mathematik, Schloßgartenstr. 7 (Gebäude S215), Raum: tba

Stochastisches Kolloquium

Donnerstag, 12. Juni 2014

16:15 Uhr: Stas Volkov (Universität Lund, Schweden)
5x+1: how many go down?
Abstract:
I will talk about how probabilistic methods of analyzing randomlylabeled trees can
provide an important insight on the 5x+1 version of the famous Collatz problem (3x+1).
Though no rigorous results about number theory will be proved in my talk, a number of
properties of the trees with random labels will be rigorously established. The work is
inspired by Yakov Sinai's earlier work on the statistic of 3x+1 problem.
Raum 308, RobertMayerStr. 68

RheinMain Kolloquium Stochastik

Freitag, 6. Juni 2014

15:15 Uhr: Volker Betz (TU Darmstadt)
Long cycles of spatial random permutations
Abstract:
Consider a locally finite set X of points in R^d, and the set S of permutations mapping X to X.
Roughly speaking, a model of spatial random permutations is a probability measure on S which
suppresses permutations where typical distances between points and their images are large.
Mathematically, such a measure has to be constructed via finite volume approximations, and
then the existence of an infinite volume limit has to be proved. In the first part of my talk
I will show how to do this when X is a regular lattice.
The most interesting feature of spatial random permutations is the presence of a phase transition
in dimension 3 and higher:
when the penalization of long distances between points and their images is strong enough, the
cycle C_x starting from a given point x is known to be finite with probability one. When the penalization
strength is below a critical value, it is believed (and seen numerically) that C_x is infinite with positive
probability. This phenomenon is of physical relevance since it is closely connected with condensation in the
interacting Bose gas. Mathematically, it is so far only understood in the annealed variant of the model.
In the second part of my talk, I will give an overview on what is known about the existence of infinite cycles.
16:45 Uhr: Steffen Dereich (Universität Münster)
Condensation in preferential attachment models with fitness
Abstract:
A popular model for complex networks is the preferential attachment model which gained popularity
in the end of the 90's since it gives a simple explanation for the appearance of power laws
in real world networks. Mathematically, one considers a sequence of random graphs that is
built dynamically according to a simple rule. In each step a new vertex is added and linked
randomly by a random or deterministic number of edges to the vertices already present in the system.
In this process, links to vertices with high degree are preferred. A variant of the model, additionally,
assigns each vertex a random positive fitness (say a $\mu$distributed value) which has a linear impact
on its attractivity in the network formation.
Such network models show a phase transition for compactly supported $\mu$.
In the condensation phase, in the limit, there is a comparably small set of vertices
(the condensate) that attracts a constant fraction of new links established by new vertices.
This condensation effect was observed for the first time by Bianconi and Bara\'asi in 2001, where
it was coined BoseEinstein phase due to similarities to BoseEinstein condensation. The fitness of the
vertices in the condensate gradually converges to the essential supremum of $\mu$ and in the talk we discuss
the dynamics of this process.
Universität Frankfurt, Institut für Mathematik, RobertMayerStr. 10, Raum 711 (groß), 7. Stock

RheinMain Kolloquium Stochastik

Freitag, 23. Mai 2014

15:15 Uhr: Jiri Cerny (Wien)
Vacant set of random walk on discrete torus
Abstract:
We prove a phase transition in the behaviour of certain macroscopic observable on the vacant set of random
walk on the ddimensional discrete torus, d>2. To this end we discuss a technique of coupling of Markov
chains so that their ranges almost coincide all the time.
This is a joint work with A. Teixeira (IMPA)
16:45 Uhr: Noam Berger (TU München)
Local CLT for ballistic random walk in random environment
Abstract:
We prove a local version of the quenched CLT for strongly ballistic random walk in random environment
in dimension 4 and higher.
Joint work with Ron Rosenthal and Moran Cohen.
Universität Mainz, Institut für Mathematik, Staudingerweg 9, Raum 05432 (Hilbertraum)

Stochastisches Kolloquium

Mittwoch, 23. März 2014

14:00 Uhr: Peter Ney (University of Wisconsin, Madison)
Large deviations of Markov chains
15:00 Uhr: Vladimir Vatutin (Steklov Institut, Moskau)
Macroscopic and microscopic structure of the family tree of decomposable branching process
Raum 711 (klein), RobertMayerStr. 10

Oberseminar Stochastik

Mittwoch, 12. März 2014

14:00 Uhr: Christina Diehl (Universität Frankfurt)
Stabile Verteilungen von Längen in BetaKoaleszenten
15:00 Uhr: Christopher P. Imanto (Universität Frankfurt)
Die Fixierungswahrscheinlichkeit eines LambdaWrightFisherProzesses mit Selektion
Raum 711 (groß), RobertMayerStr. 10

Stochastisches Kolloquium

Montag, 24. Februar 2014

12:45 Uhr: Ted Cox (Syracuse University)
Convergence of finite voter model densities
Raum 711 (groß), RobertMayerStr. 10

RheinMain Kolloquium Stochastik

Freitag, 31. Januar 2014

15:30 Uhr: Zhan Shi (Paris)
Biased random walks on trees
Abstract:
I am going to make some elementary discussions on asymptotic properties of randomlybiased random walks on
supercritical GaltonWatson trees, and present a couple of unanswered questions. Joint work with Yueyun Hu.
17:00 Uhr: Anton Bovier (Bonn)
Extremal Processes in Branching Brownian Motions
Abstract:
I review the constuction of the process of extremes of branching Brownian motion.
Using a time change between the branching mechanism and the Brownian motion, one can construct
a larger class of ``variable speed'' branching Brownian motions, introduced by Derrida and Spohn.
I present some recent results (obtained with Lisa Hartung) on the extremal processes for some cases.
TU Darmstadt, Fachbereich Mathematik, Schloßgartenstr. 7 (Gebäude S215), Raum 51

Stochastisches Kolloquium Extratermin

Donnerstag, 23. Januar 2014

14:30 Uhr: Dr. Tatjana Tchumatchenko und Sabrina Münzberg (Frankfurt)
Informationstheoretische Analyse von neuronalen Netzwerken
Abstract:
Unser Gehirn besteht aus einem Netzwerk von Neuronen, die untereinander über digitale Strompulse,
sogenannte Spikes, kommunizieren. Der Austausch von Spikes ermöglicht einen Austausch von Information.
Wir sind daran interessiert festzustellen, wie viel Information im Netzwerk kodiert ist.
Mit Hilfe informationstheoretischer Ansätze werden wir zunächst die Menge der Information analytisch bestimmen,
die eine SpikeAntwort über den eingehenden Stimulus eines Neurons enthält. Diese analytische Quantifizierung
basiert auf der Mathematik von LevelCrossings stochastischer Prozesse.
Der Vortrag wird sich auf den Fall des einzelnen Neurons konzentrieren und einen Ausblick auf die
Analyse des Netzwerkes geben.
Raum 109d, RobertMayerStr. 68

RheinMain Kolloquium Stochastik

Freitag, 13. Dezember 2013

13:30  14:30 Uhr: JeanFrançois Marckert (Bordeaux)
Compact convexes of the plane and probability theory
Abstract:
This is a common work with David Renault (LaBRI).
We revisit the connections between compact convexes
of the plane and probability measures.
First, we will discuss the existing models of random
compact convexes of the plane, and the lack of models
of smooth random convexes.
But, the main point we want to discuss is a deep
bijection attributed to GaussMinkowski (!), between the set of
probability measures $\mu$ on $[0,2\pi]$ such that $\int_0^{2\pi}
e^{ix}d\mu(x)=0$ and compact convexes of the plane with
length 1. This deep connection was not stated in terms of
probability theory,and was seen only from a geometrical
perspective.
Introducing probabilistic tools in this domain appeared to be really
powerful.
We show that some natural operations on convexes  for
example, the Minkowski sum  have natural translations in terms of
operations on probability measures. Further applications are
provided, as a new notion of convolution of convexes, and the proof
that a polygonal curve associated with a sample of $n$ random
variables (satisfying $\int_0^{2\pi} e^{ix}d\mu(x)=0$) converges to
a convex associated with $\mu$ at speed $\sqrt{n}$, result much
similar to the convergence of empirical process in statistics.
Finally, we present some models of smooth random convexes and
simulations.
14:30  15:00 Uhr: Kaffee
15:00  16:00 Uhr: Rudolf Grübel (Hannover)
From trees to functions to ultrametric spaces, and back
Abstract:
The famous Harris correspondence provides a very useful
link between simply generated random trees and random
functions on the unit interval. I will
 describe two attempts (2009, 2014) to obtain an
analogue for search trees,
 discuss some current work, some of it joint with
Steve Evans and Anton Wakolbinger, on the relation
to ordered ultrametric spaces and IDLA models.
Universität Frankfurt, Institut für Mathematik, RobertMayerStr. 10, Raum , 1. Stock

Stochastisches Kolloquium Extratermin

Dienstag, 10. Dezember 2013

14:15 Uhr: Peter Kratz (Marseille)
Anwendungen großer Abweichungen in der Epidemiologie
Abstract:
Wir betrachten Kompartimentmodelle zur Beschreibung der Ausbreitung von Krankheiten.
Es ist bekannt, dass deterministische und entsprechende stochastische Modelle dieser Art
über ein Gesetz der großen Zahlen miteinander verbunden sind. Wir betrachten große Abweichungen
des stochastischen Modells vom deterministichen Modell und erhalten ein Resultat über den
Austrittszeitpunkt des stochastischen Prozesses aus dem Anziehungsgebiet eines stabilen Gleichgewichts
des deterministischen Modells.
Angewandt auf konkrete epidemiologische Modelle erhalten wir dadurch zum Beispiel den Zeitpunkt an dem
eine Krankheit ausstirbt.
Raum 711 (klein), RobertMayerStr. 10

Stochastisches Kolloquium

Mittwoch, 27. November 2013

14:15 Uhr: Nicolas Broutin (INRIA Rocquencourt)
The scaling limit of a minimum spanning tree of a complete graph
Abstract:
Consider the minimum spanning tree of the complete graph with $n$ vertices, when edges are assigned
independent random weights. We show that, when this tree is endowed with the graph distance renormalized
by $n^{1/3}$ and with the uniform measure on its vertices, the resulting space converges in distribution
as $n\to\infty$ to a random limiting metric space in the GromovHausdorffProkhorov topology. We show
that this space is a random binary real tree and has Minkowski dimension three almost surely. In particular,
its law is mutually singular with that of the Brownian continuum random tree or any rescaled version thereof.
Our approach relies on a coupling between the minimum spanning tree problem and the ErdosRenyi random graph,
and the explicit description of its scaling limit in the socalled critical window.
Raum 711 (klein), RobertMayerStr. 10

RheinMain Kolloquium Stochastik

Freitag, 22. November 2013

15:15 Uhr: Martin Zerner (Tübingen)
On the forward branching process for onedimensional excited random walks
Abstract:
We consider onedimensional excited random walks in random environments and explain how their relation to certain
branching processes with migration can be used to obtain various recurrence/transience properties of these walks.
This talk is partially based on joint work with Elena Kosygina.
16:1516:45 Uhr: Kaffee und Tee
16:45 Uhr: Artem Sapozhnikov (Leipzig)
Quenched invariance principle for simple random walk in correlated percolation models
Abstract:
Let S be a random subgraph of Z^d. I will discuss a set of conditions on the distribution of S under
which the quenched invariance principle holds for the simple random walk on S. Examples of models satisfying
the conditions include Bernoulli percolation, random interlacements and its vacant set, level sets of the
Gaussian free field.
This is a joint work with E. Procaccia and R. Rosenthal, based on an earlier joint work with A. Drewitz and B. Ráth.
Im Anschluss gemeinsame Nachsitzung
Universität Mainz, Institut für Mathematik, Staudingerweg 9, Raum 05432 (Hilbertraum)

Stochastisches Kolloquium

Mittwoch, 21. August 2013

14:15 Uhr: Markus Kuba (Wien)
Urnenmodelle mit Mehrfachziehungen
Abstract:
tba.
Raum 711 (klein), RobertMayerStr. 10

Oberseminar Stochastik

Dienstag, 13. August 2013

14:15 Uhr: Benjamin Dadoun (ENS Cachan)
A statistical view on exchanges in Quickselect
Abstract:
In this talkt we study the number of key exchanges required by Hoare's FIND algorithm (also called Quickselect)
when operating on a uniformly distributed random permutation and selecting an independent uniformly distributed
rank. After normalization we give a limit theorem where the limit law is a perpetuity characterized by a recursive
distributional equation. To make the limit theorem useable for statistical methods and statistical experiments we
provide an explicit rate of convergence in the KolmogorovSmirnov metric, a numerical table of the limit law's
distribution function and an algorithm for exact simulation from the limit distribution. We also investigate the
limit law's density. This case study provides a program applicable to other cost measures, alternative models for
the rank and more balanced choices of the pivot element such as medianof$2t+1$ versions of Quickselect.
Raum 711 (groß), RobertMayerStr. 10

Stochastisches Kolloquium

Montag, 15. Juli 2013

14:00 Uhr: Vi LE (Universität Marseille)
Height and total mass of a forest of genealogical trees with competition
Abstract:
We consider a discrete model of population with interaction where the birth and death rates are non linear functions
of the population size. After proceeding to renormalization of the model parameters, we obtain in the limit of large
population that the population size evolves as a diffusion solution of the SDE
${Z}_{\mathrm{t}}^{\mathrm{x}}=\; x\; +{\int}_{0}^{\mathrm{t}}f({Z}_{\mathrm{s}}^{\mathrm{x}})\; ds\; +\; 2{\int}_{0}^{\mathrm{t}}\sqrt{{Z}_{\mathrm{s}}^{\mathrm{x}}}{dW}_{\mathrm{s}}^{\mathrm{x}}$.
There is a natural way of describing the genealogical tree of the discrete
population. The genealogy in the continuous limit is described by a real tree
(in the sense of Aldous). In both the discrete and the continuous case, we
study the height and the total mass of the genealogical tree, as an (increasing)
function of the initial population. We show that the expectation of the height
of the tree remains bounded as the size of the initial population tends to
infinity iff
${\int}_{1}^{\mathrm{\infty}}1/\; f(x)\; dx\; <\; \infty $,
while the expectation of the total mass of the tree remains
bounded as the size of the initial population tends to infinity
iff
${\int}_{1}^{\mathrm{\infty}}x\; /\; f(x)\; dx\; <\; \infty $.
Raum 711 (klein), RobertMayerStr. 10

RheinMain Kolloquium Stochastik

Freitag, 12. Juli 2013

15:15 Uhr: Francesco Caravenna (Mailand)
Scaling limits and universality for random pinning models
Abstract:
We consider the socalled random pinning model, which may be described as a Markov chain that receives a random
reward/penalty each time it visits a given site. When the return time distribution of the Markov chain has a polynomial
tail, with exponent larger than 1/2, the model is said to be disorderrelevant, since an arbitrarily small amount of
external randomness (quenched disorder) changes radically the critical properties of the model. In this regime, we show
that the partition function of the model, under an appropriate weak coupling scaling limit, converges to a universal
quantity, given by an explicit Wiener chaos expansion. This quantity can be viewed as the partition function of a
universal "continuum random pinning model", whose construction is part of our approach.
(Joint work with Nikos Zygouras and Rongfeng Sun)
16:45 Uhr: Dima Ioffe (Technion Haifa + Universität Bonn)
An invariance principle for random walks with prewetting
Abstract:
I shall discuss an invariance principle for a class of 1+1 SOS interfaces (walks) in potential fields.
The interfaces are constrained to stay above the wall. The potential field is \lambda X^a with a\geq 1. The case a=1
corresponds to tilting areas below interfaces. Limiting objects, as tends to zero, are reversible diffusions with
drifts given by logderivatives of ground states for the associated singular SturmLiouville operators. For a=1 such ground state
is a rescaled Airy function, and the corresponding diffusion was derived by Ferrari and Spohn in the context of scaling
limits for Brownian motions conditioned to stay above circular barriers.
Based on a work in progress with Senya Shlosman and Yvan Velenik.
Universität Mainz, Institut für Mathematik, Staudingerweg 9, Raum 05432 (Hilbertraum)

Oberseminar Stochastik

Mittwoch, 03. Juli 2013

14:15 Uhr: Franziska Wandtner (Universität Frankfurt)
Simulationen und ihre Rolle für das Verständnis von Stochastik
Abstract:
tba.
Raum 308, RobertMayerStr. 68

Stochastisches Kolloquium

Mittwoch, 26. Juni 2013

14:15 Uhr: Henning Sulzbach (Universität Frankfurt)
Gaussians limits for improved Quickselect variants
Abstract:
Despite of its weak worst case behaviour, Quickselect (or FIND) has become one of the most popular selection algorithms.
We discuss and analyse (in a certain sense) optimal Medianofk Quickselect routines with increasing subsample size k introduced
by Martinez and Roura (2001/02). Here, the complexity of the algorithm measured in terms of the number of key comparisons converges
in the Skorokhod space to a Gaussian process with interesting covariance function. We discuss several properties of the limit process.
Our results are based on a new approach relying on the contraction method towards functional limit theorems for processes satisfying
distributional recurrences where uniform convergence can not be expected and jumps persist in the limit.
If time permits, we will also consider (in a different sense) optimal adapted Quickselect variants where no results beyond asymptotic
expansions of first order are known.
Raum 308, RobertMayerStr. 68

Mathematisches Kolloquium

Donnerstag, 20. Juni 2013

ab 16:00 Uhr: Tee
16:15 Uhr: Steven N. Evans (Departments of Statistics and Mathematics, University of California at Berkeley. U.S.A.)
Optimal transport of measures and metagenomics
Abstract:
Metagenomics attempts to sample and study all the genetic
material present in a community of microorganisms in environments
that range from the human gut to the open ocean. This enterprise is
made possible by highthroughput pyrosequencing technologies that
produce a ``soup'' of DNA fragments which are not a priori associated
with particular organisms or with particular locations on the genome.
Statistical methods can be used to assign these fragments to locations
on a reference phylogenetic tree using preexisting information about
the genomes of previously identified species. Each metagenomic sample
thus results in a cloud of points on the reference tree. In seeking to
answer questions such as what distinguishes the vaginal microbiomes
of women with bacterial vaginosis from those of woment who don't,
one is led to consider statistical methods
for distinguishing between such clouds. I will discuss joint work
with Erick Matsen from the Fred Hutchinson Cancer Research Center
in which we use ideas based on distances between probability measures
that go back to Gaspard Monge's 1781 ``M\'emoir sur la th\'eorie
des d\'eblais et des remblais'' as well as some familiar objects
(e.g. reproducing kernel Hilbert spaces) from the world of
Gaussian processes.
Raum 302 (HilbertRaum), RobertMayerStr. 10

Stochastisches Kolloquium

Mittwoch, 19. Juni 2013

14:15 Uhr: Christian Böinghoff (Universität Frankfurt)
Conditional Limit Theorems for Intermediately Subcritical Branching Processes in Random Environment
Abstract:
Branching processes in random environment (BPRE) are a model for the development of a population of individuals, which are exposed to
a random environment. More precisely, it is assumed that the offspring distribution of the individuals varies in a random fashion,
independently from one generation to the other. Given the environment, all individuals reproduce independently according to
the same mechanism.
As it turns out, there is a phase transition within the subcritical regime, i.e. the asymptotics of the survival probability and the
behavior of the branching process, conditioned on survival, changes fundamentally. In the talk, we will describe the behavior of
the BPRE in the intermediately subcritical case, which is at the borderline of the phase transition. In this case, the BPRE conditioned
on survival consists of periods with supercritical growth, alternating with subcritical periods of small population sizes.
This kind of 'bottleneck' behavior appears under the annealed approach only in the intermediately subcritical case. Our results
include the asymptotics of the survival probability and conditional limit theorems for the (properly scaled) random environment and
the BPRE, conditioned on nonextinction.
The proofs rely on a construction of the conditioned branching tree going back to Geiger (1999) and Lyons, Pemantle and Peres (1995). Using this
construction and a conditional limit theorem for the (properly scaled) random environment, simulations are feasible in the case
of geometric offspring distributions. These are used to illustrate our results.
15:15 Uhr: Vincent Bansaye (Ecole Polytechnique, Palaiseau)
On the extinction of continuous state branching processes with
catastrophes
Abstract:
We consider continuous state branching processes (CSBP's for short) with additional multiplicative jumps, which we call
catastrophes. Informally speaking, the dynamics of the CSBP is perturbed by
independent random catastrophes which cause negative (or positive) jumps to the original process. These jumps
are described by a Lévy process with paths of bounded variation.
We construct this class of processes CSBP in random environment as the unique solution of a SDE and
characterize their Laplace exponent as the solution of a backward ODE. We can then study their asymptotic behavior
and establish whether the process becomes extinct. For a class of processes for which extinction and absorption coincide,
we determine the speed of extinction of the process.
Finally, we apply these results to a cell infection model, which was a motivation for considering such
CSBP's with catastrophes. If some time remains, we will speak a little about a more general class of CSBP in random
environment.
Raum 308, RobertMayerStr. 68

Oberseminar Stochastik

Mittwoch, 12. Juni 2013

14:15 Uhr: Götz Kersting (Universität Frankfurt)
Das Ranking von Webseiten bei Google (Probevortrag)
Probevortrag für die Night of Science.
Raum 308, RobertMayerStr. 68

Stochastisches Kolloquium Extratermin

Mittwoch, 12. Juni 2013

10:15 Uhr: Steven N. Evans (Departments of Statistics and Mathematics, University of California at Berkeley, U.S.A.)
Unseparated pairs and fixed points in random permutations
Abstract:
``Smoosh shuffling'' is a naive physical mechanism for randomizing a deck of cards. In order to measure the effectiveness of smoosh shuffling,
Persi Diaconis and his collaborators counted the number of cards in the shuffled deck that are still immediately followed by their successor in
the original deck and compared the observed distribution of this statistic with the corresponding distribution derived from computer simulations
of perfect shuffles. It turns out that the distribution of this statistic was already considered by William Allen Whitworth in his 1867 book
``Choice and Chance'', where he computed the probability that there are no such cards. I will show how it is possible to derive the explicit
distribution of the statistic using a variety of methods ranging from elementary enumeration, through coupled constructions of random permutations
using the ``Chinese restaurant process'', to combinatorial bijections using the {\em transformation fondamentale} of Foata and Sch{\"u}tzenberger.
Some natural extensions of this problem lead to questions about the asymptotics of the number of fixed points of the commutator of a given permutation
with a uniformly distributed random one. I will show how such problems can be solved using a relatively simple instance of the exchangeable pairs
version of Stein's method. This is joint work with Persi Diaconis (Stanford University) and Ron Graham (University of California at San Diego).
Raum 302 (HilbertRaum), RobertMayerStr. 10

Stochastisches Kolloquium

Mittwoch, 29. Mai 2013

14:15 Uhr: Linglong Yuan (Université Paris 13, Villetaneuse)
Smalltime behavior of Beta ncoalescent
Abstract:
Beta$(2\alpha, \alpha)$ ncoalescent with $1\le\alpha\le2$ serves as an important tool to model the genealogy
of a sample of $n$ individuals from a large population which allows one individual to have many children in the
next generation with a large probability. A good understanding of the "shape" of this process helps the biologists
to well describe the genealogical trees. This talk will give some results to know the behavior of the process at
"small times". The "small time" here is of the order of one uniformly chosen external branch length, which is
therefore the first object to study. More generally, we will pick $k (k\geq1)$ individuals randomly and see their
asymptotic external branch lengths. At the moment of the coalescence of individual 1, the quantity so called "minimal
clade size" in biology which is the total number of offspring at time $0$ of those individuals coalesced has been proved
to have a limit theorem. Moreover, the largest total number of offspring at time $0$ of one individual at that moment
enjoys also a nice limit behavior. Another interesting quantity is the probability for individual $1$ to be coalesced
with $k$ individuals for any $k\geq 1$.
The proofs of these results rely heavily on the connections between Beta$(2\alpha, \alpha)$ ncoalescent and
Beta$(2\alpha, \alpha)$coalescent. The latter can be considered as the limit of the former when $n$ tends to $+\infty$.
Beta$(2\alpha, \alpha)$coalescent has been well studied thanks to its rich relations with other processes, such as the
Beta flemingviot process, the ranked coalescent process, CSBP, etc. The usage of the connections to get properties of
Beta$(2\alpha, \alpha)$ ncoalescent from Beta$(2\alpha, \alpha)$coalescent seems new and this point of view can be
applied to other coalescent processes.
Raum 308, RobertMayerStr. 68

RheinMain Kolloquium Stochastik

Freitag, 24. Mai 2013

15:15 Uhr: JeanFrançois Le Gall (Université ParisSud)
The Brownian map: A universal model of random geometry
Abstract:
Consider a triangulation of the sphere chosen uniformly at random among all triangulations with a fixed
number of faces (two triangulations are identified if they correspond via an orientationpreserving homeomorphism of the sphere).
We equip the vertex set of this triangulation with the usual graph distance. When the number of faces tends to infinity, the
(suitably rescaled) resulting metric space converges in distribution, in the sense of the GromovHausdorff distance, towards a
random compact metric space called the Brownian map. This result, which confirms a conjecture of Schramm in 2006, holds with the
same limit for much more general random graphs drawn on the sphere, including the socalled quadrangulations of the sphere. The
Brownian map thus appears as a universal model of a random surface, which is homeomorphic to the sphere but has Hausdorff dimension
4. We will discuss the construction of the Brownian map, and present some ingredients of the proof of our main convergence result.
The behavior of GaltonWatson trees whose offspring distribution is critical and has finite variance, conditioned on having a fixed
large size, has drawn a lot of attention. We will be interested in what happens outside of this typical framework. More precisely,
what can be said if the conditioning on having a fixed size is replaced by the conditioning on having a fixed number of leaves? What
happens if the offspring distribution is not critical?
16:45 Uhr: Alison Etheridge (University of Oxford)
Modelling natural selection
Abstract:
The basic challenge of mathematical population genetics is to understand the relative importance of the different
forces of evolution in shaping the genetic diversity that we see in the world around us. This is a problem that has
been around for a century, and a great deal is known. However, a proper understanding of the role of a population's
spatial structure is missing. Recently we introduced a new framework for modelling populations that evolve in a spatial
continuum. In this talk we briefly describe this framework before outlining some preliminary results on the importance
of spatial structure for natural selection.
Universität Frankfurt, Institut für Mathematik, RobertMayerStr. 10, Raum 711 (groß), 7. Stock

Oberseminar Stochastik

Mittwoch, 22. Mai 2013

14:15 Uhr: Michael Messer (Universität Frankfurt)
A multiple filter test for change point detection in renewal processes with varying variance
Abstract:
Nonstationarity of the event rate is a persistent problem in modeling time series of events, such as neuronal spike trains.
Motivated by a variety of patterns in neurophysiological spike train recordings, we define a general class of renewal processes. This class
is used to test the null hypothesis of stationary rate versus a wide alternative of renewal processes with finitely many rate changes
(change points). Our test extends ideas from the filtered derivative approach by using multiple moving windows simultaneously.
To adjust the rejection threshold of the test we use a Gaussian process, which emerges as the limit of a cadlag filtered derivative process.
We analyze the benefits of our multiple filter test and study the increase in test power against a single window test. We also develop a multiple
filter algorithm, which can be used when the null hypothesis is rejected in order to estimate the number and location of change points. In a
sample data set of spike trains recorded from dopamine midbrain neurons in anesthetized mice in vivo, the detected change points agreed
closely with visual inspection, and in over 70% of all spike trains which were classified as rate nonstationary, different change
points were detected by different window sizes.
This work was supported by the LOEWESchwerpunkt `Neuronale Koordination Forschungsschwerpunkt Frankfurt'.
The manuscript is available at arXiv:1303.3594 [stat.AP].
Raum 308, RobertMayerStr. 68


Frühere Vorträge
