Discrete Mathematics 2010-2022

10:30 opening
10:40-11:30 main talk 1: Christopher Frei (Manchester)
11:30-10:40 Math.Video
10:45-12:15 PhD talk 1: JunSeok Oh (KFU Graz)
12:15-12:25 Math.Video
12:25-13:30 Lunch buffet
13:45-14:15 PhD talk 2: Shu-Qin Zhang (MU Leoben)
14:15-14:25 Math.Video
14:30-15:00 PhD talk 3: Irene de Parada (TU Graz)
15:00-15:10 Math.Video
15:10-15:40 Coffee break
15:40-16:30 main talk 2: Silke Rolles (TU München)
16:30-16:40 Math.Video

A more detailed programme will follow.

In this master-seminar, linearly reinforced random walks are explained and some results are presented.

Tamari lattice is a partial order defined on objects counted by Catalan numbers, such as binary trees and Dyck paths. It is a well-studied object in combinatorics, with several generalizations. In this talk, we will see how intervals in these Tamari-like lattices are related to planar maps. More precisely, we discovered a bijection between non-separable planar maps and intervals in the generalized Tamari lattice, which naturally extends to a bijection between bridgeless planar maps and intervals in the original Tamari lattice. We will also discuss the consequences of these bijections in enumerative and structural studies of the related objects.

We are going to derive spectral estimates for several classes of magnetic Laplacians. They include the magnetic Laplacian on three-dimensional regions with Dirichlet boundary conditions as well as the magnetic Laplacian defined in $\mathbb{R}^3$ with the local change of the magnetic field. We establish two-dimensional Berezin-Li-Yau and Lieb-Thirring-type bounds in the presence of constant magnetic fields and, using them, get three-dimensional estimates for the eigenvalue moments of the corresponding magnetic Laplacians. Also we derive separately the Lieb-Thirring bounds for the magnetic Schroeodinger operators defined on two dimensional circle with radially symmetric magnetic field and electric potentials.

We formulate and prove a Khintchine-type law for inhomogeneous Diophantine approximation. The denominators form a lacunary sequence of integers, while the size of the set of well-approximable numbers is given with respect to a probability measure with Fourier coefficients of a prescribed logarithmic decay rate.

(Remark: Agamemnon Zafeiropoulos is a new member of the Institute of Analysis and Number Theory, who started here as a Postdoc researcher in November 2017.)

We prove that, for arbitrary Dirichlet $L$-functions $L(s;\chi_1),\ldots, L(s;\chi_n)$ (including the case when $\chi_j$ is equivalent to $\chi_l$ for $j\ne k$), suitable shifts of type $L(s+i\alpha_jt^{a_j}\log^{b_j}t;\chi_j)$ can simultaneously approximate any given analytic functions on a simply connected compact subset of the right open half of the critical strip, provided the pairs $(a_j,b_j)$ are distinct and satisfy certain conditions. Moreover, we consider a discrete analogue of this problem where $t$ runs over the set of positive integers.

On Facebook, people with high number of connections tend to be connected more likely than randomness would suggest, while in biological networks vertices with high number of connections tend to be connected less likely than randomness would suggest. In terms of network science, the first network is assortative, while the second is disassortative.

Degrees (number of connections) do not tell if a network is assortative or disassortative. The Joint Degree Matrix (JDM) of a network (graph) counts number of edges between the sets of degree $i$ and degree $j$ vertices, for any $i,j$. The JDM realization problem asks whether a graph exists with prescribed number of connections (degree) at the vertices, and with prescribed number of edges between the sets of degree $i$ and degree $j$ vertices, for any $i,j$. The JDM realization problem is well understood. The usual measure for assortativity, the assortativity coefficient, the Pearson correlation coefficient of degree between pairs of linked nodes, is computable from the JDM.

A further generalization of the JDM is the following.
Given a set $W$ and numbers $d(w)$ associated with $w\in W$, and a $W_i:i\in I$ partition of $W$, with numbers $c(W_i,W_j)$ associated with unordered pairs of partition classes, the Partition Adjacency Matrix (PAM) realization problem asks whether there is a simple graph $G$ on the vertex set $W$ with degrees $d_G(w)=d(w)$ for $w\in W$, with exactly $c(W_i,W_j)$ edges with endpoints in $W_i$ and $W_j$; and the PAM construction problem asks for such a graph, if they exist. (These problems are conjectured to be NP-hard.) The bipartite version of these problems are more restricted: $I=I_1\cup I_2$ and $c(W_i,W_j)=0$ whenever $i,j\in I_1$ or $i,j\in I_2$.

We provide algebraic Monte-Carlo algorithms for the bipartite Partition Adjacency Matrix realization and construction problems, which run in polynomial time, say, when $|I|$ is fixed. When the algorithms provide a positive answer, they are always correct, and when the truth is positive, the algorithms fail to report it with small probability.

A tanglegram of size $n$ is a triplet $(L,R,M)$ where $L$ and $R$ are rooted binary trees with $n$ leaves each, and $M$ is a perfect matching between the two sets of leaves. Two tanglegrams $(L_1,R_1,M_1)$ and $(L_2,R_2,M_2)$ are the same if there is a pair of tree-isomorphisms $(\phi,\psi)$ mapping $L_1$ to $L_2$ and $R_1$ to $R_2$ such that matched pairs of leaves get paired to matched pairs of leaves. Tanglegrams are used in phylogenetics, where for example they can represent the phylogenetic trees of parasites and hosts, where the matching gives which parasite infects which host.

A tanglegram layout (i.e. the way tanglegrams are usually drawn) is as follows: draw the two rooted binary trees in the plane with straight lines and without crossing edges such that the leaves of $L$ are on the line $x=0$ and $L$ is drawn in the semi-plane $x\le 0$, the leaves of $R$ are drawn on the line $x=1$ and $R$ is drawn in the semi-plane $x\ge 1$, and the edges of the matching are drawn with straight line. The crossing number of a layout is the number of unordered pairs of matching edges that cross and the tangle crossing number of a tanglegram is the minimum crossing number over all of its layouts. The tangle crossing number is related to a number of biologically important quantities, e.g. the number of times parasites switched hosts. I will present some results about the tangle crossing number, including a Kuratowski type theorem.

In this master seminar, self-avoiding walks on infinite graphs are discussed,
with focus on Cayley graphs and quasi-transitive graphs.
The connective constant is the exponential growth rate of the number of self-avoiding walks of length n. Its computation for lattices is a difficult problem coming from Statistical Physics. In the talk, some basic properties, recent results, and computations are presented.

We study arithmetic progressions $\{a,a+b,a+2b,\dots,a+(\ell-1) b\}$, with $\ell\ge 3$, in random subsets of the initial segment of natural numbers
[$n$]$:=\{1,2,\dots, n\}$. Given $p\in[0,1]$ we denote by [$n$]$_p$ the random subset of [$n$] which includes every number with probability $p$, independently of one another. The focus lies on sparse random subsets, i.e.\ when $p=p(n)=o(1)$ with respect to $n\to\infty$.

Let $X_\ell$ denote the number of distinct arithmetic progressions of length $\ell$ which are contained in [$n$]$_p$. We determine the limiting distribution for $X_\ell$ not only for fixed $\ell\ge 3$ but also when $\ell=\ell(n)\to\infty$ sufficiently slowly. Moreover, we prove a central limit theorem for the joint distribution of the pair $(X_{\ell},X_{\ell'})$ for a wide range of $p$. Our proofs are based on the method of moments and combinatorial arguments, such as an algorithmic enumeration of collections of arithmetic progressions.

These results are joint work with Y.~Barhoumi-Andr\'eani and H.~Liu (Warwick).

Abstract.
After surveying the related results in Pascal triangle we turn
our attention to Pascal pyramid to consider horizontal rays. The newest
result desribes linearly recurrent sequences with rational function coefficients as the sum of elements located along some specific rays. In this way we obtain, for instance, the central Delannoy numbers. The application of the theorem proves many recurrence relations conjectured in The On-Line Encyclopedia of Integer Sequences of Sloane.

Traditionally, algebraic structures are presented by stating generators and relations between words over these generators. There are, however, alternatives to this way of presentation. One of these is the use of automata. Although this approach does not work for every group, the class of groups admitting automaton presentations, the so-called automaton groups, have received quite some attention in research since many groups answering important questions in group theory (such as the Milnor Problem and the Burnside Problem) turn out to be automaton groups. Starting with groups, this interest seems to extend more and
more also to automaton semigroups as it often turns out to be much easier to obtain undecidability results for automaton semigroups than it is for automaton groups.

In this talk, we are going to introduce automaton semigroups and groups, and look into known results as well as open problems concerning decision problems in this area.

\hskip 5pt 14:00 Wolfgang L. Wendland (Universität Stuttgart)
\hskip 1.1cm On Neumann's series and the double layer potential

14:45 Martin Neumüller (Johannes Kepler Universität Linz)
\hskip 1.1cm On Space Time Methods

15:30 Kaffeepause

16:00 Matthias Taus (MIT)
\hskip 1.1cm Fast and accurate methods for wave propagation

16:45 Sergej Rjasanow (Universität des Saarlandes)
\hskip 1.1cm Alternative effective numerical methods for partial differential
equations:
\hskip 1.1cm Differences and Bridges

In this report on the bachelor thesis, we present results on the convergence
to stationarity of lamplighter random walks on some finite graphs.

(Due to a master thesis defense, the talk might start with a small delay.)

I will present recent results on the asymptotics in singular limits of low-lying eigenvalues of self-adjoint operators defined in corner domains. As a model case, I will present the Robin Laplacian with a large Dirichlet parameter.

Firstly, I will give results for the regular case. The asymptotics is given through an effective semi-classical Hamiltonian, defined on the boundary, involving the mean curvature. We deduce from these results a Faber-Krahn inequality for the regular case, rising the question of optimizing the mean curvature of an open set of fixed volume.

Secondly, I will focus on the analysis in n-dimensional corner domains, in which the the singularities of the boundary modify the asymptotics. I will present the recursive class of corner domains and associated singular chains. The asymptotics of the first eigenvalues is obtained through a minimization process over the tangent geometries, and a multi-scale analysis provides an estimate of the remainder.

We consider the following generalization of the classical ``cow path problem''. Suppose we are sending out $k$ robots from $0$ to search the real line at constant speed (with turns) to find a target at an unknown location; $f$ of the robots are faulty (of so-called crash type), meaning that they fail to report the target although visiting its location. The goal is to find the target in time at most $\lambda |d|$, if the target is located at $d$, $|d|\ge 1$, for $\lambda$ as small as possible. We show that it cannot be achieved for $\lambda < 2\frac{(1+\rho)^{1+\rho}}{\rho^\rho} + 1$ where
$\rho := 2\frac{f+1}{k}-1$, which is tight due to earlier work. This also gives some better than previously known lower bounds for so-called Byzantine-type faulty robots (that may, deceitfully, actually wrongly report a target).

(Joint work with Andrey Kupavskii.)

Vortragende:[5mm]

Donnerstag, 19.10.2017
09:00-09:30: Eröffnung durch VR Bischof und Dekan Ernst
09:30-10:00: Harald Niederreiter
Donald Knuth’s problem and Robert Tichy’s solution
10:00-10:45 János Pintz
Some conjectures of Erdös and Turán on consecutive
primegaps
10:45-11:15 Kaffeepause
11:15-12:00 Kálmán Györy
S-parts of values of binary forms and decomposable forms
Mittagspause
14:15-15:00 Yuri Bilu
Effective bounds for singular units
15:00- 15:45 Pietro Corvaja
The Hilbert Property for algebraic varieties
15:45-16:15 Kaffeepause
16:15-17:00 Clemens Fuchs
Diophantine triples and linear recurrences of Pisot type
17:15-18:00 A. V.[5mm]

Freitag, 20.10.2017
09:00-09:45 Klaus Schmidt
Entropy and periodic points of algebraic actions of discrete groups
09:45-10:30 Vitaly Bergelson
Ramsey Theory at the Junction of Additive and Multiplicative Combinatorics
10:30-11:00 Kaffeepause
11:00-11:45 István Berkes
On the uniform theory of lacunary series
Mittagspause
14:00-14:45 Michael Drmota
Digital Expansions and Uniform Distribution
14:45-15:15 Kaffeepause
15:15-16:00 Gerhard Larcher
On Weyl Products and Irregularities of Distribution[5mm]
Aktuelle Informationen unter:
http://www.math.tugraz.at/Tichy60}

Let $F$ be a binary form of degree $r \geq 3$, integer coefficients, on-zero discriminant, and such that the largest irreducible factor of $F$ has degree $d$. For a positive number $Z$ and a positive integer $k \geq 2$ put R_{F,k}(Z)$ for the number of $k$-free integers in the interval $[-Z,Z]$ which is representable by $F$. We shall give an asymptotic formula for $R_{F,k}(Z)$ when $k > \min\{7d/18, \lceil d/2 \rceil - 2\}$. This is joint work with C.L. Stewart.

Speiser in 1935 showed that the Riemann hypothesis is equivalent to the first derivative of the Riemann zeta function having no zeros on the left-half of the critical strip. This result shows that the distribution of zeros of the Riemann zeta function is related to that of its derivatives. The number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been investigated by Berndt, Levinson, Montgomery, and Akatsuka. Berndt, Levinson, and Montgomery investigated the general case, meanwhile Akatsuka gave sharper estimates under the truth of the Riemann hypothesis. This result is further improved by Ge. In the first half of this talk, we introduce these results and generalize the result of Akatsuka to higher-order derivatives of the Riemann zeta function.

Analogous to the case of the Riemann zeta function, the number of zeros and many other properties of zeros of the derivatives of Dirichlet L-functions associated with primitive Dirichlet characters were studied by Yildirim. In the second-half of this talk, we improve some results shown by Yildirim for the first derivative and show some new results. We also introduce two improved estimates on the distribution of zeros obtained under the truth of the generalized Riemann hypothesis. We also extend the result of Ge to these Dirichlet L-functions when the associated modulo is not small. Finally, we introduce an equivalence condition analogous to that of Speiser’s for the generalized Riemann hypothesis, stated in terms of the distribution of zeros of the first derivative of Dirichlet L-functions associated with primitive Dirichlet characters.

After the talk there will be coffee & cake.

For a typical character $\pmod p$, a classical result of Polya and Vinogradov
implies cancellation for character sums longer than $p^{1/2}$. Burgess' bound
allows us to go further, implying cancellation for character sums longer
than $p^{1/4}$. But in practice, one often needs to bound shorter character
sums, and no such bounds are known for a general character. I will describe
recent work (joint with Elijah Fromm) in which we show that even a mild
improvement of the Polya-Vinogradov inequality would imply cancellation in
character sums as short as $p^{0.00001}$, thus significantly improving the
Burgess bound.

The space of images will be considered as a Riemannian manifold, where the
underlying Riemannian metric simultaneously measures the cost of image
transport and intensity variation, introduced by Trouve and Younes as the
metamorphosis model.
A robust and effective variational time discretization of geodesics paths
will proposed and a variational scheme for a time discrete exponential map
will investigated.
The approach requires the definition of a discrete path energy consisting of
a sum of consecutive image matching functionals over a set of image
intensity maps and pairwise matching deformations.
The talk will present existence and convergence results and discuss
applications in image morphing and image animation.

We will investigate spectral properties of $H+V_\lambda$, where $H=-i\alpha\cdot\nabla+m\beta$ is the free Dirac operator in ${\mathbb R}^3$, $m>0$ denotes the mass and $V_\lambda$ is an electrostatic shell potential (which depends on a parameter $\lambda\in{\mathbb R}$) located on the boundary of a smooth domain in ${\mathbb R}^3$. I will present an isoperimetric-type inequality for the admissible range of $\lambda$s for which the coupling $H+V_\lambda$ generates pure point spectrum in $(-m,m)$. That the ball is the unique optimizer of this inequality will also be discussed. This is a joint work with N. Arrizabalaga and L. Vega.

Being motivated by the study of negative-index metamaterials, we will discuss the definition and the spectral properties of the operators given by the differential expressions $\text{div}\,h\,\text{grad}$ in a bounded domain $U$ with a function $h$ which is equal to $1$ on a part of $U$ and to a constant $b<0$ on the rest of $U$. We will see how the properties of such operators depend on the parameter $b$ and the geometry of $U$. In particular, for the critical case $b=-1$ one can have a non-empty essential spectrum, and our results extend the constructions of Behrndt and Krejcirik for symmetric rectangles (2014) to arbitrary smooth geometries. The proof features an interplay between the machinery of boundary triples and the microlocal analysis. Based on a joint work with Claudio Cacciapuoti and Andrea Posilicano (University of Insubria).

Abstract: In joint work with C. Elsholtz and I. Shparlinski we study the equidistribution of multiplicatively defined sets such as squarefree numbers or
quadratic non-residues in sets which are defined in an additive way, for example sumsets, Hilbert cubes or
sets having digit restrictions. In particular, we show that if one fixes any proportion of less than 2/5 of
the digits of all numbers of a given binary bit length, then the remaining set still has the asymptotically
expected number of squarefree integers.

Yu. N. Mukhin asked in 1984 in the Kourovka Notebook (9.32) to classify all  locally compact groups in which for any two closed subgroups X and Y their set theoretic product XY is a closed subgroup.

In joint work with K. H. Hofmann and F. G. Russo the class of “near abelian” groups has been introduced and extensively discussed. As a byresult we can offer a complete answer to Mukhin’s question.

In this talk I will highlight the concepts and present the classification result.

Abstract:
An integer $n$ is said to be $y-$smooth if its largest prime factor
$P(n)$ is less than $y$. As usual, we denote by $S(x,y)$ the set of
$y-$smooth numbers up to $x$,
$$ S(x,y)= \{ 1\leq n\leq x, P(n)\leq y\}.$$

In this talk, we provide an asymptotic formula for the number of
integers $n$ in $S(x,y)$ such that $S_{q}(n) \equiv l \mod m$ for $l \in
\mathbb{Z}$ and $m \geq2$, where $S_q(n)$ denotes the sum of the digits
in base $q$ of the integer $n$. Also, we show that the sequence $\big(
\alpha S_{q}(n)\big)_{n \in S(x,y)}$ is uniformly distributed modulo 1
if and only if $\alpha $ is irrational.
Furthermore, we study the number of ordered pairs $(a,b) \in A\times B$
such that $P(a+b)\leq y$ and $S_{q}(a+b) \equiv l \mod m$, $(l \in
\mathbb{Z}$ and $m \geq2)$, for a given sets of integers $A$ and $B$.
Finally, we discuss sums of the form
$$ \sum_{n \in S(x,y)\atop{S_{q}(n) \equiv l \mod m}} f(n-1), $$
where $f$ is a multiplicative function, $l \in \mathbb{Z}$ and $m \geq2$.

Prime ideals and prime spectra of rings play a central role both in
Commutative Ring Theory and Algebraic Geometry, being them foundation of
Scheme Theory, for instance. Order properties and topological properties
of spaces of prime ideals have been useful to characterize some classes of
rings. From the topological point of view, since the 60s it has been of
interest to put in evidence conditions that a topological space must
satisfy in order to be homeomorphic to the prime spectrum of some
(commutative) ring (with 1). This was the main subject of the PhD thesis
of M. Hochster, where he proved that the topological spaces that are
homeomorphic to the prime spectrum of a ring - also called spectral
spaces - are precisely the spaces $X$ satisfying the following axioms:

\begin{enumerate}

\item $X$ is compact;

\item $X$ admits a basis of open and compact subspaces that is
closed under finite intersections.

\item $X$ is sober, that is, every irreducible closed
subspace of $X$ has a unique generic point.

\end{enumerate}

While for some class of spectral spaces, like Riemann-Zariski spaces (see
\cite{fi-fo-lo}), a class of rings realizing Hochster corrispondence was
explicitly found, for several other spaces naturally arising in
Commutative Ring Theory it is non trivial to understand if they are
spectral because, in particular, it can be not so easy to verify condition
(3) of Hochster's characterization.

A goal of this survey talk is to present some new perspective about the
study of spectral spaces and, in particular, a criterion, based on
ultrafilters, to decide if a topological space is spectral (see
\cite{fi}). Some recent new examples will be discussed.

\begin{thebibliography}{99}

\bibitem{fi} C. A. Finocchiaro, Spectral spaces and ultrafilters.
Comm. Algebra 42 2014, no. 4, 1496--1508.

\bibitem{fi-fo-lo} C. A. Finocchiaro, M. Fontana, K. A. Loper, The
constructible topology on spaces of valuation domains. Trans. Amer.
Math. Soc. 365 2013, no. 12, 6199--6216.

\end{thebibliography}

The Tur\'an function $ex(n,F)$ denotes the maximal number of edges in an $F$-free graph on $n$ vertices. However once the number of edges in a graph on $n$ vertices exceeds $ex(n,F)$, many copies of $F$ appear. We study the function $h_F(n,q)$, the minimal number of copies of $F$ in a graph on $n$ vertices with $ex(n,F)+q$ edges. The value of $h_F(n,q)$ has been extensively studied when $F$ is colour critical. In this paper we consider a simple non-colour-critical graph, namely the bowtie and establish bounds on $h_F(n,q)$ when $q=o(n^2)$.

This is joint work with Mihyun Kang and Oleg Pikhurko.

Sehr geehrte Damen und Herren,

im Rahmen der Besetzung einer §99 Professoren-Laufbahnstelle aus Finanz- und Versicherungsmathematik am Institut für Statistik finden sechs Bewerbungsvorträge statt, zu denen wir Sie herzlich einladen.


Montag, 26.6.2017

9:00-9:30 Paul Krühner (TU Wien): Density bounds for Ito processes and their applications in acturial and mathematical finance,

10:30-11:00 Julia Eisenberg (Univ. Graz): The effects of exchange rates on some actuarial risk measures,

12:00-12:30 Michaela Szölgyenyi (WU Wien): Numerical methods for SDEs in finance and insurance mathematics

Mittwoch, 28.6.2017

9:00-9:30 Peter Hieber (Univ. Ulm): Retirement products: Recent developments,

10:30-11:00 Stefan Thonhauser (TU Graz): On the solution of optimization problems in insurance

Montag, 3.7.2017

10:30-11:00 Christa Cuchiero (Univ. Wien): Polynomial processes in stochastic portfolio theory

A natural number is called $k$-almost prime if it has exactly $k$ prime factors, counted with multiplicity. In this talk I consider the asymptotic counting of such numbers (mostly with $k\leq 3$) with additional constraints put on the prime factors. E.g., if we take $k=2$ and write $n=pq$ with $p$ and $q$ primes of similar size we obtain the so called RSA-integers that play an important role in cryptography. We also consider some examples with $k=3$ inspired by cryptography and the study of coefficients of cyclotomic polynomials. The talk is based on two papers with coauthors P. Moree (accepted in 2016), and Florian Luca, Robert Osburn and Alisa Sedunova (2017+).

Learning and games comprise some of the most sophisticated behaviors
of agents, and examples of situations combining the two keep coming
up. I will focus on three recent instances in my research: How can
you learn successfully if your data can be of varying quality and is
provided by strategic agents who would rather work less on improving
the data quality? If the points to be classified are strategic, what kind of classifier can best anticipate their gaming of the system? Finally, recent results suggest that learning dynamics in games typically fail to converge to the Nash equilibrium. Is gym this bad news for learning dynamics? Or perhaps for the Nash
equilibrium?

Recently Marcus, Spielman & Srivastava (2015) and Marcus (2016) studied polynomial convolutions and Hadamard products that are inspired by free probability. These convolutions capture the expected characteristic polynomials of random matrices. We explore analogues of these convolutions in the setting of Max-Plus Algebra. In this setting, the max-permanent replaces the determinant and the maximum is the analogue of the expected value. Our results resemble those of Marcus et al. Moreover, whereas in the classical setting Marcus et al provide bounds on the roots of the convolution of polynomials, we get exact description of the roots of the convolution of characteristic polynomials in the Max-Plus setting. A brief introduction to operations in Max-Plus Algebra will be given.

This is a joint work with Franz Lehner and Aljosa Peperko.

In this talk I will give some overview on risk theoretic modeling and recent developments.
Classical approaches try to describe insurance risks by means of a stochastic process and quantify them by a risk functional,
typically a short-fall probability or a function depending on a possible deficit. Since these approaches are based on a static parameter choice,
they are oversimplifying in many situations and are not able to cover many realistic phenomena. We will put a focus on controllable extensions
of classical models, which lead to stochastic optimization problems, and touch some model alternatives.
For example such extensions comprise investments in a financial market, risk control via reinsurance or shareholder participation.
Using concrete examples we can discuss different solution procedures and related mathematical questions.
Finally, some relations to other fields of applied mathematics are highlighted.

Abstract.}
In 1977, Volker Strassen presented a deterministic and rigorous algorithm for solving the problem to compute the prime factorization of natural numbers $N$. His method is based on fast polynomial arithmetic techniques and runs in time $\widetilde{O}(N^{1/4})$, which has been state of the art for the last forty years. In this talk, we discuss the core ideas for improving the bound by a superpolynomial factor. The runtime complexity of our algorithm is of the form
\[
\widetilde{O}\left(N^{1/4}\exp(-C\log N/\log\log N)\right).
\]

The concept of linear representations (aka realizations or linearizations) provides some very powerful tool to deal with non-commutative rational functions, namely elements in the universal field of fractions for the ring of non-commutative polynomials (in finitely many variables). While these methods are mostly of purely algebraic origin, they are also nicely compatible with the analytic machinery of (operator-valued) free probability. This theory and the underlying notion of free independence were invented around 1985 by D. Voiculescu, originally for operator-algebraic purposes. It can be seen as a highly non-commutative analogue of classical probability theory and has deep connections to many other fields of mathematics, especially to random matrix theory. In my talk, I will explain how this fascinating interplay leads to explicit algorithms for the computation of distributions and Brown measures, respectively, of evaluations of non-commutative rational functions in freely independent random variables. As we will see, this can be used to determine the asymptotic eigenvalue distribution of certain random matrix models. Furthermore, some concrete examples will show that these algorithms are easily accessible for numerical computations. This is based on joint works with S. Belinschi, J. W. Helton, and R. Speicher.

Classical Waring--Goldbach problem concerns representability of all large integers satisfying a certain local condition as sums of fixed number of $k$-th powers
of prime numbers where $k \geq 1$. For instance Goldbach's conjecture states that
every even number $\geq 4$ can be expressed as a sum of two primes. Denote by
$H(k)$ the least integer $s$ such that every sufficiently large positive integer satisfying the aforementioned local condition may be expressed as a sum of
$s$ $k$-th powers of primes. Following the pioneering work of Vinogradov (1937) (which
yields $H(1) \leq 3$), Hua (1938-1959) showed that $H(k) \leq 2^k+1$. He then reduced
his bound to $H(k) \leq 4k \log k(1 + o(1))$ for every large $k$. In this talk, we shall
look at Waring--Goldbach problem with primes chosen from Piatetski Shapiro
sequences; sequences of the form $\left\{ \lfloor n^c \rfloor \right\}_{n=1}^\infty$ where $c > 1$. Such sequences are
known to contain infinitely many primes when $1 < c < 1.18$.

We formulate and prove a Khintchine-type law for inhomogeneous Diophantine approximation. The denominators form a lacunary sequence, and the probability measure which gives the size of the set of well approximable numbers has Fourier coefficients with a prescribed logarithmic decay rate.

In this talk, I will explain how we can extend the previous work of Mauduit and Rivat about the Rudin--Shapiro sequence for all sequences which count blocks of digits. At first I will consider the case when the length of the block is constant, and then the case when it is a non decreasing function. If time permits, I will present briefly new results (work in progress with Olivier Robert) about the orthogonality between the Möbius function, and some functions related to polynomials along binary digits (like the Rudin--Shapiro sequence).

In the past decade there has been a lot of progress in analysis on metric measure spaces based on ideas from optimal transport. We discuss how some of these ideas can be developed for Markov chains on discrete spaces, using a discrete analogue of the Kantorovich-Wasserstein metric. In particular we present a discrete notion of Ricci curvature based on geodesic convexity of the entropy, which allows us to obtain discrete functional inequalities, such as spectral gap and logarithmic Sobolev inequalities. We also discuss recent applications to interacting particle systems. This is based on joint works with Matthias Erbar (Bonn), Prasad Tetali (Georgia Tech), and Max Fathi (Toulouse).