Discrete Mathematics 2010-2022

Abstract:

In this talk, we discuss recent results concerning spectra of exponents of Diophantine approximation. Weighted homogeneous-inhomogeneous transferences, and joint spectra of an exponent and its uniform counterpart. Included are some recent pictures of the Himalayas.

Abstract:

We discuss the mutual behavior of the irrationality measure functions for two real numbers and some related problems. In particular we will formulate some new results about the function associated with the Minkowski diagonal continued fraction and with the functions related to the second best approximations, and introduce some multidimensional generalizations.

Abstract:

The Schatten classes $S_p$ ($0< p \leq \infty$), consisting of all compact linear operators on a Hilbert space for which the sequence of their singular values belongs to the sequence space $l_p$, are one of the most important classes of unitary operator ideals. Their analysis, particularly in the finite-dimensional setting, has a long tradition in asymptotic geometric analysis and the local theory of Banach spaces.
In [Studia Math. 80, 63--75, 1984], Saint Raymond studied the volumetric properties of unit balls in finite-dimensional real and complex Schatten $p$-classes and his results are used frequently in the literature.
He obtained an asymptotic formula for their volume, which contains an unknown limiting constant for which he provided both lower and upper bounds. We determine the exact limiting constant and as an application compute the precise asymptotic volume ratio of Schatten $p$-classes as the dimension tends to infinity. This extends Saint Raymond's estimate in the case of the nuclear norm ($p=1$) to the full regime $1\leq p \leq \infty$ with exact limiting behavior. (Joint work with Z. Kabluchko and C. Thäle)

Abstract:

Recently Marcus, Spielman and Srivastava studied certain combinatorial polynomial convolutions that preserve real-rootedness and capture expectations of characteristic polynomials of unitarily invariant random matrices, thus providing a link to free probability.
We show that with respect to these convolutions, the subset of diagonal matrices and the subset of principally balanced matrices form a ``convolvent pair''.
We also explore analogues of these types of convolutions in the setting of max-plus algebra. Our results resemble those of Marcus et al., however, in contrast to the classical setting we obtain an exact and simple description of all roots of the convolutions.
Joint work with Franz Lehner, Aljoša Peperko and Octavio Arizmendi.

Abstract:

We consider general self-adjoint polynomials in several independent
random matrices whose entries are centered and have constant
variance. Under some numerically checkable conditions, we establish
the optimal local law, i.e., we show that the empirical spectral
distribution on scales just above the eigenvalue spacing follows the
global density of states which is determined by free probability
theory. First, we give a brief introduction to the linearization
technique that allows to transform the polynomial model into a linear
one, which has simpler correlation structure but higher
dimension. After that we show that the local law holds up to the
optimal scale for the generalized resolvent of the linearized model,
which also yields the local law for polynomials. Finally, we show how
the above results can be applied to prove the optimal bulk local law
for two concrete families of polynomials: general quadratic forms in
Wigner matrices and symmetrized products of independent matrices with
i.i.d. entries. This is a joint work with Laszlo Erdös and Torben
Krüger.

Abstract:

The eigenvalue density of many large Hermitian random matrices is
well-approximated by a deterministic measure on \mathbb{R}, the self-consistent
density of states. In the case of an $N\times N$ random matrix with
nontrivial expectations of its entries or a nontrivial correlation
among them, this measure is obtained from the matrix Dyson equation on
$N\times N$ matrices. The matrix Dyson equation generalizes scalar- or
vector-valued Dyson equations that have been studied previously. In
this talk, we will show that the self-consistent density of states is
real-analytic apart from finitely many square root edges and cubic
root cusps. We will also explain how detailed information about these
singularities can be used to prove Tracy-Widom fluctuation for the
eigenvalues close to the square root edges of the associated
self-consistent density of states. This is joint work with László
Erdos, Torben Krüger and Dominik Schröder.

Abstract:

It is well-known that the Pell equation $a^2-db^2=1$ in integers has a
non-trivial solution if and only if $d$ is positive and not a perfect
square. If one considers the polynomial analog, i.e., for fixed $D \in
\mathbb{C}[X]$, the equation $A^2-DB^2=1$, the matter is more
complicated. Indeed, for the existence of a solution the clear
necessary conditions that the degree of $D$ must be even and that $D$
cannot be a perfect square are not sufficient. While the case of
degree two is analogous to the integer case, there are non-square
polynomials of degree 4 such that the corresponding Pell equation is
not solvable. On the other hand, as in the integer case, once we have
a non-trivial solution, we have infinitely many and we call minimal
solution a solution $(A,B)$ with $A$ of minimal degree.

In joint work with Laura Capuano and Umberto Zannier we showed that
there exist equations $A^2-DB^2=1$, with $(A,B)$ minimal solution, for
any choice of degrees deg$D \geq 4$ even and deg$A \geq $deg$D/2$.

Abstract:

Abstract: It goes back to Lagrange that a real quadratic irrational
has always a periodic continued fraction. Starting from decades ago,
several authors proposed different definitions of a p-adic continued
fraction, and the definition depends on the chosen system of residues
mod p. It turns out that the theory of p-adic continued fractions has
many differences with respect to the real case; in particular, no
analogue of Lagranges theorem holds, and the problem of deciding
whether the continued fraction is periodic or not seemed to be not
known until now. In recent work with F. Veneziano and U. Zannier we
investigated the expansion of quadratic irrationals, for the p-adic
continued fractions introduced by Ruban, giving an effective criterion
to establish the possible periodicity of the expansion. This
criterion, somewhat surprisingly, depends on the ‘real’ value of the
p-adic continued fraction.

Abstract:

Abstract:


Let $(R, \mathfrak{m})$ be a regular local ring of dimension $d \geq 2$. A \it local monoidal transform \rm of $R$ is a ring of the form $$R_1= R \left[ \frac{\mathfrak{p}}{x} \right]_{\mathfrak{m}_1}$$ where $ \mathfrak{p} $ is a prime ideal generated by regular parameters, $x \in \mathfrak{p}$ is a regular parameter and $ \mathfrak{m}_1 $ is a maximal ideal of $ R[\frac{\mathfrak{m}}{x}] $ lying over $ \mathfrak{m}. $ If $\mathfrak{p}= \mathfrak{m}$ the ring $R_1$ is called a \it local quadratic transform\rm.

Recently, several authors studied the rings of the form $ S= \cup_{n \geq 0}^{\infty} R_n $ obtained as infinite directed union of iterated local quadratic transforms of $R$, and call them \it quadratic Shannon extension\rm.
A directed union of local monoidal transforms of a regular local ring is said \it monoidal Shannon extension\rm.

Here we study features of monoidal Shannon extensions and more in general of directed unions of Noetherian UFDs.

(L. Guerrieri, Directed unions of local monoidal transforms and GCD domains (2018) arXiv:1808.07735 )

Abstract:

In the first part of the talk, I will speak about the normal structure of classical and classical-like groups. The description of the normal subgroups for various classes of concrete groups, and especially for classical groups over rings, has been one of the central themes of group theory in the last two centuries, right after Galois introduced the notion of normal subgroups. In the second part of the talk, I will speak about weighted Leavitt path algebras. Weighted Leavitt path algebras are algebras associated to weighted graphs. They generalise in a natural way the usual Leavitt path algebras and also Leavitt's algebras of module type $(n,k)$ where $n,k>0$.

Abstract:

This talk consists of two sections. In the first section,
we prove that if $D$ is an almost P$v$MD, then $D$ is of finite $t$-character
if and only if each nonzero $t$-locally finitely generated $w$-ideal of $D$ is of finite type.
As a corollary, we
get that if $D$ is an almost Pr\"ufer domain, then $D$ is of finite character
if and only if each locally finitely generated ideal of $D$ is finitely generated.

In the second section, we consider to
what extent conditions on the homogeneous elements or ideals of a graded integral domain $R$ carry
over to all elements or ideals of $R$.
For instance, we prove that $R$ is a gr-$t$-quasi-Pr\"ufer domain if and only if $R$ is a $t$-quasi-Pr\"ufer domain.
However, we show that homogeneously $tv$-domains and homogeneously $w$-divisorial domains are not equal to $tv$-domains and $w$-divisorial domains, respectively.


This talk is based on joint works with G. W. Chang and P. Sahandi.

Abstract:

Abstract: This talk contains two parts. In the first section of this talk, we define the set of integer-valued polynomials over the subsets of matrix rings. We do this on full matrix ring, upper triangular matrix ring and upper triangular matrix ring with constant diagonal. We present some examples to show that these sets may be not rings.
Then, we introduce some cases that the set of integer-valued polynomials over subsets of matrix ring is a ring. Furthermore, we consider some properties of these rings as Noetherian property and Krull dimension.
In the second section, we generalize the ring of integer-valued polynomials over upper triangular matrices and define the set of integer-valued polynomials over some cases of block matrices. Then, we show that this set is a ring. It solves the open problem of integer-valued polynomials on algebras in a special case of block matrix algebras.

Abstract:

\noindent Let $G$ be a group. A non-empty subset $S$ of $G$ is `product-free}' if $ab \not\in S$ for all $a, b \in S$. We call such a set `locally maximal}' in $G$ if it is not properly contained in any other product-free subset of $G$.
Locally maximal product-free sets were first studied in 1974 by Street and Whitehead, who analysed some properties of these sets, and introduced the concept of filled groups.
We say a locally maximal product-free subset $S$ of $G$ `fills}' $G$ if
$G^{\ast}\subseteq S \cup SS$ (where $G^{\ast}=G\setminus \{1\}$), and $G$ is called a `filled group}'
if every locally maximal product-free set in $G$ fills $G$.
In this talk, we shall consider questions like:
(a) for a given positive integer k, which finite groups contain a locally maximal product-free set of size k?; (b) how many finite groups are filled?.

Abstract:

\noindent
It is well known that Gauss Elimination produces a factorization into elementary matrices of any invertible matrix over a field. A classical problem, studied since the middle of the 1960's, is to characterize integral domains different from fields that satisfy the same property. As a partial answer, in 1993, Ruitenburg proved that in the class of B\'ezout domains, any invertible matrix can be written as a product of elementary matrices if and only if any singular matrix can be written as a product of idempotents.

\noindent
In this talk, after giving an overview of the classical results on these factorization properties, we will present some recent developments on the topic. In particular, we will consider products of elementary and idempotent matrices over special classes of non-Euclidean PID's and over integral domains that are not B\'ezout.

\vskip 0.5cm

\textsc{References}

\vskip 0.1cm

\begin{itemize}{\footnotesize

\item[\textnormal{[1]}] L.~Cossu, P.~Zanardo, U.~Zannier,
Products of elementary matrices and non-Euclidean principal ideal domains}, J. Algebra 501: 182–205, 2018.
\item[\textnormal{[2]}] L.~Cossu, P.~Zanardo,
Factorizations into idempotent factors of matrices over Pr\"ufer domains}, accepted for publication on Communications in Algebra, 2018.


}\end{itemize}

\vskip 0.5cm

\noindent{\footnotesize
\textsc{Department of Mathematics ``Tullio Levi Civita'', University of Padova}
{Via Trieste, 63}
{35121, Padova}
{Italy}
}

\vskip 0.2cm
\noindent{\footnotesize{E-mail address}:
lcossu@math.unipd.it}
}

Abstract:

The theory of time harmonic electromagnetic waves propagation involves always the structure of the obstacle, which have in most cases thin geometry. In the case of perfectly conducting obstacle coated with a thin dielectric layer some difficulties linked to the numerical simulation appear. To overcome this problem many authors gave approximations of an impedance operator for perfectly conducting obstacle coated by thin shell of dielectric material. In the present work, we calculated approximations until the third order of the impedance operator for perfectly conducting obstacle coated by two contrasted thin layers of dielectric materials, the approach used is that of Bendali et al. [1], when they wrote the boundary condition on the perfect conductor in terms of Taylor expansion in the thickness of the thin layer.

[1] A. Bendali, M. Fares, K. Lemrabet and S. Pernet, Recent Developments in the Scattering of an Electromagnetic Wave by a Coated Perfectly Conducting Obstacle.
Waves 09 (2009), Pau, France.

Abstract:

On a countable tree T, allowing vertices with infinite degree, we consider an arbitrary stochastic irreducible nearest neighbour transition operator $P$. We provide a boundary integral representation for general eigenfunctions of $P$ with complex eigenvalue $\lambda$. This is possible whenever $\lambda$ is in the resolvent set of $P$ as a self-adjoint operator on a suitable l(2)-space and the on-diagonal elements of the resolvent do not vanish at $\lambda$.

When $P$ is invariant under a transitive group action, the latter condition
holds for all $\lambda$ in the resolvent set except possibly 0. These results extend and complete previous results by Cartier, by Figa'-Talamanca and Steger, and by Woess.

Furthermore, for those eigenvalues, we provide an integral representation of $\lambda$-polyharmonic functions of any order n, that is, complex functions $f$ on T for which $(\lambda I - P)^n f=0$. This is a far-reaching extension of work of Cohen et al.

We can also provide an analogous result for polyharmonic functions on the unit disk with respect to the hyperbolic Laplacian, based on old results of Helgason.

This is joint work with Massimo Picardello (Rome).

Abstract:

Abstract:
Im my talk I would like to put a little footbridge between clinical environment and mathematical modelling society in the topic of aortic mechanics.
The biggest artery of our organizm has it’s own life. It may not be treated as a viscoelastic tube but we may have to take into account some physiological considerations.
Aorta by itself is a living organ with internal metabolism and high level of protein production and turnover. In modelling perspective it may be quite difficult to bridge the scale between molecular level and cellular level. I propose to avoid that by organising model into subdomains representing basic structural unit of the aorta. By degenerating properties of single components of the subunit we may simulate the pathology on the level of whole organ. It may also help in explanation of the ageing processes of the aorta and acconying
changes of mechanical properties.

Abstract:

Abstract: Unlike the real line, the $d$-dimensional space $\mathbb{R}^d$, for $d \geq 2$, is not canonically ordered. As a consequence, such fundamental and strongly order-related univariate concepts as quantile and distribution functions, and their empirical counterparts, involving ranks and signs, do not canonically extend to the multivariate context. Palliating that lack of a canonical ordering has remained an open problem for more than half a century, and has generated an abundant literature, motivating, among others, the development of statistical depth and copula-based methods. We show that, unlike the many definitions that have been proposed in the literature, the measure transportation-based ones introduced in Chernozhukov, Galichon, Hallin and Henry (2017) enjoy all the properties (distribution-freeness and the maximal invariance property that entails preservation of semiparametric efficiency) that make univariate quantiles and ranks successful tools for semiparametric statistical inference. We therefore propose a new center-outward} definition of multivariate distribution and quantile functions, along with their empirical counterparts, for which we establish a Glivenko-Cantelli result---the quintessential property of all distribution functions.
Our approach, based on results by McCann (1995), is geometric rather than analytical and, contrary to the Monge-Kantorovich one in Chernozhukov et al. (2017) (which assumes compact supports, hence finite moments of all orders), does not require any moment assumptions. The resulting ranks and signs are shown to be strictly distribution-free, and maximal invariant under the action of a data-driven class of (order-preserving) transformations generating the family of absolutely continuous distributions; that maximal invariance, in view of a general result by Hallin and Werker (2003), is the theoretical foundation of the semiparametric efficiency preservation property of ranks. The corresponding quantiles are equivariant under the same transformations.

Abstract:

Abstract:

Let $K$ be a number field with ring of integers $O_K$. Recently, Loper and Werner introduced the following ring which generalizes the classical definition of ring of integer-valued polynomials: ${\rm Int}_{\mathbb Q}(O_K)=\{f\in \mathbb Q[X] \mid f(O_K)\subseteq O_K\}$. If $K=\mathbb Q$ then we get the classical ring ${\rm Int}(\mathbb Z)$ of polynomials with rational coefficients mapping $\mathbb Z$ into $\mathbb Z$. Loper and Werner prove that ${\rm Int}_{\mathbb Q}(O_K)$ is a Pr\"ufer domain, which is stricly contained in ${\rm Int}(\mathbb Z)$ if $K$ is a proper extension of $\mathbb Q$. Here, we show that in case $K,K'$ are Galois extensions of $\mathbb Q$, then ${\rm Int}_{\mathbb Q}(O_K)={\rm Int}_{\mathbb Q}(O_{K'})$ if and only if $K=K'$. We also characterize a basis for ${\rm Int}_{\mathbb Q}(O_K)$ as a $\mathbb Z$-module when $K/\mathbb Q$ is a tamely ramified Galois extension. This is a joint work with Bahar Heidaryan and Matteo Longo.

We also give the following new generalization: for any number fields $K,K'$, if ${\rm Int}_{\mathbb Q}(O_K)={\rm Int}_{\mathbb Q}(O_{K'})$, then $K,K'$ are conjugated over $\mathbb Q$.

Abstract:

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\maketitle
\thispagestyle{empty}

Let $K$ be a number field with ring of integers $O_K$. Recently, Loper and Werner introduced the following ring which generalizes the classical definition of ring of integer-valued polynomials: ${\rm Int}_{\mathbb Q}(O_K)=\{f\in \mathbb Q[X] \mid f(O_K)\subseteq O_K\}$. If $K=\mathbb Q$ then we get the classical ring ${\rm Int}(\mathbb Z)$ of polynomials with rational coefficients mapping $\mathbb Z$ into $\mathbb Z$. Loper and Werner prove that ${\rm Int}_{\mathbb Q}(O_K)$ is a Pr\"ufer domain, which is stricly contained in ${\rm Int}(\mathbb Z)$ if $K$ is a proper extension of $\mathbb Q$. Here, we show that in case $K,K'$ are Galois extensions of $\mathbb Q$, then ${\rm Int}_{\mathbb Q}(O_K)={\rm Int}_{\mathbb Q}(O_{K'})$ if and only if $K=K'$. We also characterize a basis for ${\rm Int}_{\mathbb Q}(O_K)$ as a $\mathbb Z$-module when $K/\mathbb Q$ is a tamely ramified Galois extension. This is a joint work with Bahar Heidaryan and Matteo Longo.

We also give the following new generalization: for any number fields $K,K'$, if ${\rm Int}_{\mathbb Q}(O_K)={\rm Int}_{\mathbb Q}(O_{K'})$, then $K,K'$ are conjugated over $\mathbb Q$.

%\vskip3cm
%
%\addcontentsline{toc}{section}{Bibliography}
%\begin{thebibliography}{99}
%\bibliographystyle{plain}
%\bibitem{ChabPolCloVal} J.-L. Chabert, On the polynomial closure in a valued field}, J. Number Theory 130 (2010), 458-468.
%\bibitem{LW} K. A. Loper, N. Werner, Pseudo-convergent sequences and Pr\"ufer domains of integer-valued polynomials}, J. Commut. Algebra 8 (2016), no. 3, 411-429.
%\bibitem{PerPrufer} G. Peruginelli, Pr\"ufer intersection of valuation domains of a field of rational functions}, sottomesso (2017), Arxiv: \href{https://arxiv.org/abs/1711.05485}{https://arxiv.org/abs/1711.05485}.
%\end{thebibliography}

\end{document}

Abstract:

Abstract:

Spherical designs are discrete point sets on the sphere which provide exact equal-weight cubature formulas for polynomials up to a certain degree, i.e. the average of any polynomial over the points of a design is equal to the average over the whole sphere. Several years ago, Bondarenko, Radchenko, and Viazovska solved a long-standing conjecture of Korevaar and Meyers, showing that there exist designs of degree t on the d-dimensional sphere, which have $t^d$ points. We shall discuss properties of spherical designs and present the proof of the aforementioned conjecture.

This is the fourth lecture in a mini-series of four lectures. All lectures are independent and fully self-contained. No prior knowledge of any of the subjects involved in the talk will be assumed.

Abstract:

Unit norm tight frames are objects that play an important role in functional analysis, discrete geometry, signal processing, and even quantum physics. These are collections of unit vectors, which behave like orthonormal bases: they satisfy an analog of Parseval's identity and every vector can be exactly reconstructed from its projections onto the elements of the frame. Tight frames have a number of interesting properties -- in particular, they minimize a certain discrete energy. This object is closely intertwined with the following interesting question in discrete geometry: can one draw N lines in a d-dimensional Euclidean space, so that the angle between any two is the same? For which values of N, d, and the angle is this possible? What is the maximal cardinality of a set of equiangular lines for a given dimension? We shall survey the main results and conjectures on these topics.

This is the third lecture in a mini-series of four lectures. All lectures are independent and fully self-contained. No prior knowledge of any of the subjects involved in the talk will be assumed.

Abstract:

Two most standard ways to measure the quality of a point set on the sphere are discrepancy and energy. In the former, one compares the proportion of points in certain subdomains to their area, while in the latter one views points as electrons that repel according to a certain force. We shall talk about various energy minimization problems on the sphere, when minimal energy induces uniform distribution, how the structure of the function affects minimizers, special point sets that arise as minimizers (tight frames, spherical designs), connections to spherical harmonics, Gegenbauer polynomials, and positive definite functions etc. Then we shall discus discrepancy on the sphere: known bounds, methods, constructions, as well as relations to energy.

This is the second lecture in a mini-series of four lectures. All lectures are independent and fully self-contained. No prior knowledge of any of the subject involved in the talk will be assumed.

Abstract:

We investigate the model selection properties of the Lasso estimator in finite samples with no conditions on the regressor matrix X. We show that which covariates the Lasso estimator may potentially choose in high dimensions (where the number of explanatory variables p exceeds sample size n) depends only on X and the given penalization weights. This set of potential covariates can be determined through a geometric condition on X and may be small enough (less than or equal to n in cardinality) so that the Lasso estimator acts as a low-dimensional procedure also in high dimensions. Related to the geometric conditions in our considerations, we also provide a necessary and sufficient condition for uniqueness of the Lasso solutions.

Abstract:

The discrepancy function is one of the most standard tools designed to measure equidistribution of points in the unit square: it compares the actual vs. expected number of points in axis-parallel rectangles. Various norms of this function provide different information about the uniformity of a point distribution. Lower bounds for the discrepancy, which states that, no matter how one distributes points, their discrepancy cannot be too small, have come to be known collectively as "irregularities of distribution". The methods of proof for such bounds most of the times arrive from harmonic analysis (e.g. Haar wavelets with product structure) and have been actively employed in the past by Roth, Schmidt, Halasz, Beck etc. We shall discuss these methods, known results and techniques, as well as connections of the problem to questions of harmonic analysis, probability, and approximation theory, which have been discovered more recently.

This is the first lecture in a mini-series of four lectures. All lectures are independent and fully self-contained. No prior knowledge of any of the subject involved in the talk will be assumed.

Abstract:

Abstract:

In this talk we will look at a new variant of the polynomial method which
was first used to prove that sets avoiding 3-term arithmetic progressions in groups like $\mathbb{Z}_4^n$ (Croot, Lev and myself) and $\mathbb{Z}_3^n$ (Ellenberg and Gijswijt) are exponentially small (compared to the size of the group).

We will discuss lower and upper bounds for the size of the extremal subsets,
including some recent bounds found by Elsholtz and myself. We will also mention some further applications of the method, for instance, the solution of the Erd\H{o}s-Szemer\'edi sunflower conjecture.



---------
Ab 15.45 Saft, Kekse und Caf\'{e} im Sozialraum des Instituts für Analysis und Zahlentheorie, Kopernikusgasse 24, 2. Stock.

Abstract:

Abstract: A set of $m$ positive integers is called a Diophantine
$m$-tuple if the product of any two elements in the set increased by 1
is a perfect square. One of the question of interest is how large those
sets can be. Very recently He, Togb\' e and Ziegler proved the folklore
conjecture that there does not exist a Diophantine quintuple.

There is also stronger version of that conjecture which states that
every Diophantine triple can be extended to a quadruple, with a larger
element, in a unique way. That conjecture is still open. In this talk we
study the two families of Diophantine pairs and consider their
extension. More precisely we prove the mentioned conjecture for the
triples $\{a, b, c\}$, where $a$ and $b$ are positive integers defined
by $a=KA^2$, $b=4KA^4+4\varepsilon A$ with $K, A$ positive integers and
$\varepsilon \in \{\pm1\}$ and $c$ is given by $c=c_{\nu}^{\tau}$, where
\begin{align*}
c_{\nu}^{\tau}=\frac{1}{4ab}\left\{(\sqrt{b}+\tau
\sqrt{a})^2(r+\sqrt{ab})^{2\nu}+(\sqrt{b}-\tau
\sqrt{a})^2(r-\sqrt{ab})^{2\nu}-2(a+b)\right\}
\end{align*}
with $\nu$ a positive integer and $\tau \in \{\pm\}$.

This is joint work with Mihai Cipu and Yasutsugu Fujita.

Abstract:

Abstract: A set of $m$ positive integers is called a Diophantine
$m$-tuple if the product of any two elements in the set increased by 1
is a perfect square. One of the question of interest is how large those
sets can be. Very recently He, Togb\' e and Ziegler proved the folklore
conjecture that there does not exist a Diophantine quintuple.

There is also stronger version of that conjecture which states that
every Diophantine triple can be extended to a quadruple, with a larger
element, in a unique way. That conjecture is still open. In this talk we
study the two families of Diophantine pairs and consider their
extension. More precisely we prove the mentioned conjecture for the
triples $\{a, b, c\}$, where $a$ and $b$ are positive integers defined
by $a=KA^2$, $b=4KA^4+4\varepsilon A$ with $K, A$ positive integers and
$\varepsilon \in \{\pm1\}$ and $c$ is given by $c=c_{\nu}^{\tau}$, where
\begin{align*}
c_{\nu}^{\tau}=\frac{1}{4ab}\left\{(\sqrt{b}+\tau
\sqrt{a})^2(r+\sqrt{ab})^{2\nu}+(\sqrt{b}-\tau
\sqrt{a})^2(r-\sqrt{ab})^{2\nu}-2(a+b)\right\}
\end{align*}
with $\nu$ a positive integer and $\tau \in \{\pm\}$.

This is joint work with Mihai Cipu and Yasutsugu Fujita.

Abstract:

In my talk I will present results that come from a joint work with Michael Bennett and Adela Gherga from the University of British Columbia. We studied Goormaghtigh's equation:
\begin{equation}\label{eq}
\frac{x^m-1}{x-1} = \frac{y^n-1}{y-1}, \; \; y>x>1, \; m > n > 2.
\end{equation}
There are two known solutions $(x, y,m, n)=(2, 5, 5, 3), (2, 90, 13, 3)$ and it is believed that these are the only solutions. It is not known if this equation has finitely or infinitely many solutions, and not even if that is the case if we fix one of the variables. It is known that there are finitely many solutions if we fix any two variables. Moreover, there are effective results in all cases, except when the two fixed variables are the exponents $m$ and $n$. If the fixed $m$ and $n$ additionally satisfy $\gcd(m-1, n-1)>1$, then there is an effective finiteness result. My co-authors and me showed that if $n \geq 3$ is a fixed integer, then there exists an effectively computable constant $c (n)$ such that $\max \{ x, y, m \} < c (n)$ for all $x, y$ and $m$ that satisfy Goormaghtigh's equation with $\gcd(m-1,n-1)>1$. In case $n \in \{ 3, 4, 5 \}$, we solved the equation completely, subject to this non-coprimality condition.


\vspace{1cm}Remark:} After the talk a light lunch will be served in the social room of the institute.

Abstract:

We study generalised prime systems $\mathcal{P}$ and generalised integer systems $\mathcal{N}$ obtained from them. The asymptotic distribution of generalised integers is deduced assuming that the generalised prime counting function $\pi_\mathcal{P}(x)$ behaves as $\pi_\mathcal{P}(x)=b ~\textup{li}(x)+O(x^\alpha)$ for some $b>0$ and $\alpha\in(0,1)$.

Remark:} Laima Kaziulyte will work as a guest researcher at the Institute of Analysis and Number Theory from September 2018 -- January 2019. She is a PhD student at Vilnius University.

Abstract:

Abstract: Let l be a positive integer. We discuss improved average
bounds for the l-torsion of the class groups for some families of number
fields, including degree-d-fields for d between 2 and 5. The
improvements are based on refinements of a technique due to Ellenberg
and Venkatesh. This is joint work with Martin Widmer.

Abstract:

Today’s data are increasingly complex and dynamic. Online transactions, financial markets or social media produce information that is highly interconnected and changes instantaneously, resulting in data shapes such as spikes and troughs, or lines or curves, all of which capture the speed – or the dynamics – at which the information is changing. Functional shape analysis takes a collection of curves and attempts to extract common features among the curves with the goal of better explaining – and eventually predicting – the dynamics captured inside the data. In this presentation, we will give an overview of some recent opportunities and challenges of shape analysis. Understanding dynamics has become particularly important in online markets where prices, bids or opinions change continuously. We will provide an overview over techniques that allow us to capture, characterize, segment and predict shapes of online markets. Being able to predict shapes is particularly important in applications where market penetration or diffusion curves are desired and where restrictive parametric approaches are too inaccurate. We will illustrate these techniques on several examples from virtual stock markets, crowd-funding and box office results.

Abstract:

Given a dimension $k \ge 2$ and a probability $p$, define the binomial random $k$-dimensional simplicial complex $\mathcal{G}_p$ as the downward-closure of the binomial random $(k+1)$-uniform hypergraph, in which each hyperedge is present with probability $p$ independently. For each $j \le k$, call a $k$-dimensional simplicial complex $\mathbb{F}_2$-cohomologically $j$-connected} if all cohomology groups for dimensions 1 up to $j$ with coefficients in the two-element field $\mathbb{F}_2$ vanish, and if furthermore the zero-th cohomology group is isomorphic to $\mathbb{F}_2$.

For each $j \le k$, we prove the existence of a sharp threshold for $\mathbb{F}_2$-cohomological $j$-connectedness of $\mathcal{G}_p$. A similar result has been proved for a different model of random complexes by Linial and Meshulam (2006) and by Meshulam and Wallach (2009). In addition, we prove a hitting time result, relating $\mathbb{F}_2$-cohomological $j$-connectedness with the disappearance of the last minimal obstruction. As a corollary, we deduce an analogous hitting time result for the Linial-Meshulam model, a result which has previously only been known for $k = 2$. Finally, we determine the limiting probability for $\mathbb{F}_2$-cohomological $j$-connectedness when $p$ lies in the critical window around the threshold.

In this talk, we focus on the main intuition and proof ideas behind these results.

Joint work with Oliver Cooley, Nicola del Guidice, and Mihyun Kang.

Abstract:

We study random unlabelled $k$-trees by combining the colouring approach by Gainer-Dewar and Gessel (2014) with the cycle pointing method by Bodirsky, Fusy, Kang and Vigerske (2011). Our main applications are Gromov--Hausdorff--Prokhorov and Benjamini--Schramm limits that describe their asymptotic geometric shape on a global and local scale as the number of hedra tends to infinity.

This is the joint work with Benedikt Stufler.

Abstract:

Let $q$ be a positive integer $q>1$, and let $\chi$ be a Dirichlet character modulo $q$. Let $L(s, \chi)$ be the attached Dirichlet $L$-functions,
and let $L^\prime(s, \chi)$ denote its derivative with respect to the complex variable $s$. In this talk, we survey certain known results on the evaluation of values of Dirichlet $L$-functions and of their logarithmic derivatives at $1+it_0$ for fixed real number $t_0$.
We also give a new asymptotic formula for the $2k$-th power mean value of $\left|(L^\prime/L)(1+it_0, \chi)\right|$ when $\chi$ runs over all Dirichlet characters modulo $q>1$, for any fixed real number $t_0$. This is joint work with professor Kohji Matsumoto.

Abstract:

We investigate the genus of the Erd\H{o}s-R\'enyi random graph $G(n,m)$, providing a thorough description of how this relates to the function $m=m(n)$, and finding that there is different behaviour depending on which region $m$ falls into.

We then also show that the genus of a fixed graph can increase dramatically if a small number of random edges are added - we thus call this the ``fragile genus'' property.

This is joint work with Mihyun Kang and Michael Krivelevich.

Abstract:

The indecomposable decompositions of persistence modules play an important role in understanding their structure. Especially in the case of persistence of a filtration, the indecomposables (the building blocks) are intervals, and have the interpretation as lifespans of topological features. We discuss the computation of indecomposable decompositions of some more general persistence modules. In particular, we focus on the 2 by n commutative grid (called commutative ladders) and provide an algorithm for the representation finite case ($n<5$) via matrix problems, from which persistence diagrams can be obtained. We also describe some problems and complications in the representation infinite case.

Short bio:
Emerson is currently a postdoctoral researcher in the Topological Data Analysis Team of the Center for Advanced Intelligence Project at RIKEN (Japan). Research interests include topological data analysis, representation theory, and computation. Website:
https://emerson-escolar.github.io/index.html

Abstract:

Inspired by the works of Forman on discrete Morse theory, we give a combinatorial adaptation of stratified Morse theory of Goresky and MacPherson, referred to as the discrete Stratified Morse theory. We relate the topology of a finite simplicial complex with the critical simplices of a discrete stratified Morse function on the complex. We provide an algorithm that constructs such a function on any finite simplicial complex equipped with a real-valued function. We also discuss on-going work that study the relation to classical stratified Morse theory. This is joint work with Kevin Knudson (Univ. Florida).

Abstract:

Abstract:

In my talk I will discuss results coming from a joint work with Mike
Bennett and Adela Gherga from the University of British Columbia. We show
that if $n \geq 3$ is a fixed integer, then there exists an effectively
computable constant $c (n)$ such that if $x, y$ and $m$ are integers
satisfying \[ \frac{x^m-1}{x-1} = \frac{y^n-1}{y-1}, \; \; y>x>1, \; m > n,
\] with $\gcd(m-1,n-1)>1$, then $\max \{ x, y, m \} < c (n)$. Prior to our
work, to obtain finiteness results for this equation, two of the variables
$x, y, m$ and $n$ had to be fixed. Furthermore, in case $n \in \{ 3, 4, 5
\}$, we completely solve this equation subject to the same constraint on
the exponents. This extends the result of Yuan who completely solved the
equation in the case $n=3$ and $m$ is odd.

Abstract:

We discuss an effective method of computing traces, determinants, and $\zeta$-functions for some classes of linear operators and apply this to the concrete case of Sturm-Liouville operators.

To illustrate the formalism, we will sketch a spectral theorist's computation of $\pi$, Jacobi's classical transformation formula for one-dimensional theta functions (utilizing the heat equation on the circle), and sketch a derivation of a formula for Ap\'ery's constant, $\zeta(3)$, employing a trace formula.

The talk will minimize technicalities and be accessible to students.

This is based in part on recent joint work with Klaus Kirsten, Lance Littlejohn, and Hagop Tossounian.

Abstract:

Eine ergodische Markovkette hat 'cutoff', wenn die Aproximation der stationären Verteilung innerhalb eines kurzen Zeitfensters von 'schlecht' auf 'gut' übergeht. Dies geht auf Ideen von Persi Diaconis zurück.
In den beiden Kurzvorträgen werden das Cutoff-Phänomen erklärt und Beispielklassen vorgestellt.

Abstract:

Manuel Kauers is professor for algebra at Johannes Kepler University. He is working on computer algebra and its applications to discrete mathematics. Together with Doron Zeilberger and Christoph Koutschan, Kauers proved two famous open conjectures in combinatorics using large scale computer algebra calculations. The first concerned Gessel's lattice path conjecture, the second was the so-called q-TSPP conjecture. In 2009, Kauers received the Austrian Start-proze, and in 2016, with Ch. Koutschan and D. Zeilberger he received the David P. Robbins prize of the American Mathematical Society.

Abstract of the talk: We give an overview of some recent developments in the
area of counting restricted lattice walks. This his a hot topic in the combinatorics community which has attracted the interest of a lot of people. We will limit ourselves to aspects in which computer algebra has helped making some progress.

Abstract:

Let $N(b)$ be the set of real numbers which are normal to base $b$. A well-known result of H. Ki and T. Linton is that $N(b)$ is $\boldsymbol{\Pi}^0_3$-complete. We show that the set $N^\perp(b)$ of reals which preserve $N(b)$ under addition is also $\boldsymbol{\Pi}^0_3$-complete. We use the characterization of $N^\perp(b)$ given by G. Rauzy in terms of an entropy-like quantity called the noise}. It follows from our results that no further characterization theorems could result in a still better bound on the complexity of $N^\perp(b)$. We compute the exact descriptive complexity of other naturally occurring sets associated with noise. One of these is complete at the $\boldsymbol{\Pi}^0_4$ level. Finally, we get upper and lower bounds on the Hausdorff dimension of the level sets associated with the noise.

Abstract:

13:30-13:40 Eröffnung
13:40-14:00 E. STADLOBER: Erinnerungen an Ulrich Dieter
14:00-14:20 S. HÖRMANN: Zur Prognose von funktionalen Zeitreihen
14:20-14:40 S. THONHAUSER: Über die optimale Wahl von Rückversicherung

14:40-15:30 Kaffeepause

15:30-16:00 N. HASELGRUBER: Strategien der Stichprobenplanung zur Verifikation von Zuverlässigkeitszielen
16:00-16:30 Th. MENDLIK: Warum man den Klimawandel nicht genau vorhersagen kann, uns aber die Statistik dennoch hilft
16:30-17:00 M. PIFFL: Modell-Basierte Flotten Validierung
17:00-17:30 P. SCHEIBELHOFER: Statistische Methoden für die fortgeschrittene Prozesskontrolle in der Halbleiterfertigung

Abstract:

We consider the inverse scattering problem of determining a singular perturbation supported on a bounded and closed surface in 3D from the knowledge of the scattering data. Different approaches to this problem and possibly some recent result will be presented.

This talk is based on joint work-in-progress with A. Posilicano.

Abstract:

We consider the direct scattering problem and provide a corresponding representation of the scattering matrix for the couple $(A_{0},A)$, where $A_{0}$ and $A$ are semi-bounded self-adjoint operators in $L^{2}(M,{\mathscr B},m)$ such that the set $\{u\in D(A_{0})\cap D(A):A_{0}u=Au\}$ is dense. Applications to the case in which $A_{0}$ corresponds to the free Laplacian in $L^{2}({\mathbb R}^{n})$ and $A$ describes the Laplacian with self-adjoint boundary conditions on rough compact hypersurfaces are given.

The talk is based on joint work with Andrea Mantile.

Abstract:

The radially symmetric unbounded surface $\Sigma\subset\mathbb{R}^3$ is defined via the mapping $\mathbb{R}^2\ni x\mapsto (x,f(|x|))$, where a sufficiently smooth profile function $f\colon \mathbb{R}_+\rightarrow \mathbb{R}_+$ satisfies $f(0) = 0$. The radially symmetric layer $\Omega := \{x\in\mathbb{R}^3\colon {\rm dist}\,(x,\Sigma) < a\} \subset\mathbb{R}^3$ with $a > 0$ can be viewed as a tubular neighbourhood of $\Sigma$. Under rather non-restrictive extra assumptions on $f$, it was proven by Duclos, Exner, and Krej\v{c}i\v{r}\'{i}k in 2001 that the essential spectrum of the self-adjoint Dirichlet Laplacian on $\Omega$ coincides with $[\frac{\pi^2}{4a^2},\infty)$ and that its discrete spectrum below $\frac{\pi^2}{4a^2}$ is infinite.

First, we will re-visit the case of the conical layer ($f(t) = kt$, $k >0$).
Spectral asymptotics for such a layer was obtained by Dauge, Ourmi\`{e}res-Bonafos, and Raymond in 2015. We will discuss what happens with the spectrum, upon replacement the usual Laplacian by its magnetic counterpart with a magnetic field of an Aharonov-Bohm-type. It turns out that there are two regimes, the transition between which is abrupt. The discrete spectrum remains infinite for the weak magnetic field, but its flux enters the leading term in the spectral asymptotics. In the strong field regime, the discrete spectrum completely disappears.

Second, we will consider generalized parabolic layers
($f(t) = kt^{1+\alpha}$, $\alpha, k > 0$).
In this setting, we, for the first time, compute the spectral asymptotics.
The power $\alpha$ influences the law of accumulation for the eigenvalues. In particular, for the conventional parabolic layer ($\alpha = 1$) the spectral asymptotics is similar to the one for the Hamiltonian of the hydrogen atom.

These results are obtained in collaborations with Pavel Exner,
David Krej\v{c}i\v{r}\'{i}k, Thomas Ourmi\`{e}res-Bonafos.

Abstract:

In the first part of the talk we investigate the Dirichlet Laplacian in a straight twisted tube of a non-circular cross section with a local perturbation of the twisting velocity. It is known that the essential spectrum covers the half-line. Under some additional assumptions on the twisting velocity perturbation the non-emptiness of the discrete spectrum is also guaranteed (P. Exner and H. Kovarik; 2005). In the second part of the talk we study the Dirichlet Laplacian on a straight tube but already with exploding to infinity twisting velocity. Then the spectrum is becoming purely discrete (D. Krejcirik; 2015). In both cases we investigate the upper bounds for eigenvalue moments.

Abstract:

The basic goal of topological data analysis is to apply algebraic topology
tools to understand and describe the shape of data. In this context,
homology is one of the most relevant topological descriptors. Persistent
Homology (PH) tracks homological features along an increasing one-parameter
sequence of spaces and its descriptors are well-appreciated for comparing
and recognizing changes in the shape of data.
The focus of this talk is on a generalization of Persistence Theory to the
analysis of multivariate data, called Multiparameter Persistent Homology
(MPH).


At the moment, MPH requires both computational optimizations towards the
applications to real-world data, and theoretical insights for finding and
interpreting suitable descriptors. We address the problem of reducing
computational costs in MPH by proposing a fully discrete preprocessing
algorithm. In doing so, we propose a new notion of optimal preprocessing
algorithm for MPH and show that our proposed algorithm satisfies it for low
dimensional domains. We provide experimental results in comparing our
approach to other similar approaches, and in evaluating the impact of our
approach on MPH computations.

Abstract:

Abstract.}
Let $a,k\in\mathbb{N}$. For the $k-1$-th iterate of the exponential function $x\mapsto a^x$, also known as tetration, we write
\[
^k a:=a^{a^{.^{.^{.^{a}}}}}.
\]
In this talk, we show how an efficient algorithm for tetration modulo natural numbers $N$ may be used to factorize $N$. In particular, we prove that the problem of computing the squarefree part of integers is deterministically polynomial-time reducible to modular tetration.

Abstract:

Abstract:
In this talk, we explore the connection between the ranks of elliptic curves and the length of certain rational sequences contained in the coordinates of rational points. The structure of the presentation is the following. First, we give a lazy survey of rank records and the methods used in obtaining them, especially the algebraic approach of Mestre. In the middle part, we talk about the past and recent progress in the construction of elliptic curves admitting long arithmetic and geometric progressions. Finally, we present some related new results as part of my research project at TU Graz.

Abstract:

The aim of this talk is to give a simple geometric condition
that guarantees the existence of spectral gaps of the discrete Laplacian
on periodic graphs. For proving this, we analyse the discrete magnetic
Laplacian on the finite quotient and interpret the vector potential as a
Floquet parameter. We develop a procedure of virtualising edges and
vertices that produces matrices whose eigenvalues (written in ascending
order and counting multiplicities) specify the bracketing intervals
where the spectrum of the Laplacian is localised.

Abstract:

Kronecker sequences build an important class of one dimensional low-discrepancy sequences. They can also be realized as orbits of circle rotations, which are in in turn the simplest class of interval exchange transformations (IETs). Hence, it is natural to ask whether there exist more general IETs which also yield low-discrepancy sequences. In the case of n=3 intervals, the criteria of circle rotation can easily be carried over. For more than three intervals the dynamic of IETs is much more complex. In this talk, it is shown how to obtain low-discprancy sequences involving certain IETs with an arbitrary number of intervals.

Abstract:

It is well-known that the spectrum of self-adjoint periodic differential operators has the form of a locally finite union of compact intervals called bands. In general the bands may overlap. A bounded open interval (a,b) is called a gap in the spectrum σ(H) of the operator H if (a,b)∩σ(H)=Ø with a,b belonging to σ(H).

The presence of gaps in the spectrum is not guaranteed: for example, the spectrum of the Laplacian in ℝn has no gaps. Therefore the natural problem is a construction of periodic operators with non-void spectral gaps. The importance of this problem is caused by various applications, for example in physics of photonic crystals.

Another interesting question arising in this area is how to control the location of spectral gaps via a suitable choice of coefficients of the underlying operators or/and via a suitable choice of geometric parameters of the medium. In the talk we give an overview of the results, where this problem is studied for various classes of periodic differential operators. In a nutshell, our goal is to construct an operator (from some given class of periodic operators) such that its spectral gaps are close (in some natural sense) to predefined intervals.

Abstract:

Lambda-calculus is a well-known model of computation, whose statistical properties have been recently under investigation. From a syntactic point of view, the terms of lambda-calculus (lambda-terms) are interesting combinatorial structures: the underlying structure is a Motzkin tree to which are added edges from some unary nodes towards leaves, or equivalently whose nodes are colored, according to some specific set of rules. While trees are usually easy to enumerate, this enriched version is actually a class of directed graphs, and their combinatorial study gives rise to many interesting problems.

While the enumeration and statistical analysis of unrestricted lambda-terms still remains an open problem, different models, either redefining the notion of size of the lambda-term or restricting the set of lambda-terms under investigation (such as terms of restricted height or restricted node arity), have proved more amenable to study. We present enumeration and statistical behaviour for these models, using tools from analytic combinatorics.

(No previous knowledge either of lambda-terms or of analytic combinatorics is assumed.)

Abstract:

In view of the interplay of the DK `Discrete Mathematics' with the Field of Expertise `Information, Communication & Computing' of TU Graz, the last talk of this semester's seminar comes from a sister institue within the FoE. There are clear relations with the topics of the DK.

Abstract. Information dimension of random variables was introduced by Alfred Renyi in 1959. Only recently, information dimension was shown to be relevant in various areas in information theory. For example, in 2010, Wu and Verdu showed that information dimension is a fundamental limit for lossless analog compression. Recently, Geiger and Koch generalized information dimension from random variables to stochastic processes. They showed connections to the rate-distortion dimension and to the bandwidth of the process. Specifically, if the process is Gaussian, then the information dimension equals the Lebesgue measure of the support of the process' power spectral density. This suggests that information dimension plays a fundamental role in sampling theory.

The first part of the talk reviews the definition and basic properties of information dimension for random variables. The second part treats the information dimension of stochastic processes and sketches the proof that information dimension is linked to the process' bandwidth.

Abstract:

Spinal groups form a family of branch groups acting on d-regular rooted trees containing many interesting examples, including Grigorchuk's family of groups of intermediate growth. Because of its natural action on the tree, it is interesting to know how their Schreier graphs look like. In this talk we will define the spinal family, provide some examples, describe the Schreier graphs in general and find some of the properties they exhibit, like how many ends do they have, how do they partition into isomorphism classes and under which conditions is the action linearly repetitive or Boshernitzan, properties related to symbolic dynamics.

Abstract:

The $f$-vector of a $d$-dimensional (convex) polytope $P$ is defined as
\begin{equation*}
f(P) = (f_0(P), f_1(P), \ldots, f_d(P))
\end{equation*}
where $f_i(P)$ is the number of $i$-dimensional faces of $P$.
The question is whether a given vector of non-negative numbers is the $f$-vector of a polytope.
For $d\ge 4$, finding a complete characterization of $f$-vectors is an open problem.
However, the so-called $g$-Theorem gives a description for $f$-vectors of simplicial and simple polytopes.

The purpose of this talk is to give a summary of the lectures of a summer school at MSRI on the $g$-Theorem and related topics.

Abstract:

We investigate the bottom of the spectra of infinite quantum graphs, i.e., Kirchhoff Laplacians on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Moreover, we establish a connection with the combinatorial isoperimetric constant, which enables us to prove a number of criteria for a quantum graph to be uniformly positive or to have purely discrete spectrum. If time permits, we'll demonstrate our findings by considering trees, antitrees and Cayley graphs.

Abstract:

The notion of quantum graph refers to a graph considered as a one-dimensional simplicial complex and equipped with a differential operator (``Hamiltonian'').
From the mathematical point of view, quantum graphs are interesting because they are a good model to study properties of quantum systems depending on geometry and topology of the configuration space. They exhibit a mixed dimensionality being locally one-dimensional but globally multi-dimensional of many different types.

We will review the basic spectral properties of infinite quantum graphs (graphs having infinitely many vertices and edges). In particular, we will discuss recently discovered, fruitful connections between quantum graphs and discrete Laplacians on graphs.

Based on a joint work with P. Exner, M. Malamud and H. Neidhardt