# Matematiska kollokviet

Matematiska kollokviet är en seminarieserie vid Matematiska institutionen som vänder sig till en bred matematisk publik. Alla intresserade är välkomna.

Kollokviet organiseras av Anders Björn, Milagros Izquierdo, Vladimir Kozlov och Hans Lundmark vid avdelningen Matematik och tillämpad matematik.

## Tid och lokal

Seminarietiden är vanligtvis varje onsdag kl 13:15-14:15 i Hopningspunkten som ligger i B-huset, ingång 23, plan 2, Campus Valla i Linköping.

## Kommande seminarier

### Onsdag 29 januari 2020, Evgeniy Lokharu, Matematiska institutionen (MAI), Linköpings universitet

Titel: Nonexistence of subcritical solitary waves

Tid och plats: Onsdag 29 januari 2020, Hopningspunkten, kl 13:15-14:15

Sammanfattning: We prove the nonexistence of two-dimensional solitary gravity water waves with subcritical wave speeds and an arbitrary distribution of vorticity. This is a longstanding open problem, and even in the irrotational case there are only partial results relying on sign conditions or smallness assumptions. As a corollary, we obtain a relatively complete classification of solitary waves: they must be supercritical, symmetric, and monotonically decreasing on either side of a central crest. The proof introduces a new function related to the so-called flow force which has several surprising properties. In addition to solitary waves, our nonexistence result applies to "half-solitary" waves (e.g. bores) which decay in only one direction.

This is a joint work with Vladimir Kozlov (MAI) and Miles H. Wheeler (University of Bath, UK).

## Tidigare seminarierVisa/dölj innehåll

### Onsdag 22 januari 2020, Elina Rönnberg, Matematiska institutionen (MAI), Linköpings universitet

Seminariet arrangeras ihop med Tvärvetenskapliga seminarier på MAI.

Titel: Efficient use of hardware resources in avionic systems

Sammanfattning: A key ingredient when designing an avionic system – i. e. the electronic system of an aircraft – is to make sure that it always can be trusted. In modern integrated modular avionic systems, different aircraft functions share hardware resources on a common avionic platform. For such architectures, it is necessary to create a spatial and temporal partitioning of the system to prevent faults from propagating between different functions. One way to establish a temporal partitioning is through pre-runtime scheduling.

While the avionic systems are growing more and more complex, so is the challenge of scheduling them. Scheduling of the system has an important role when a new avionic system is developed. Typically, functions are added to the system over a period of several years and a scheduling tool is used both to determine if the platform can host the new functionality and, in case this is possible, to create a new schedule.

In this talk, I will discuss a design case from Saab Aeronautics and present an optimisation-based scheduling tool that we have developed. From an optimisation point of view, the problem can be described as a rich multiprocessor scheduling problem that also includes a communication network to be scheduled. Results are presented for practically relevant large-scale instances with up to 60 000 tasks.

### Onsdag 15 januari 2020, Andrew Ross Winters, Matematiska institutionen (MAI), Linköpings universitet

Titel: My numerical scheme crashed, now what?

Sammanfattning: Numerical methods to approximate the solution of partial differential equations are a powerful tool to model problems we otherwise could not. But can they tell us something even when they fail?

We present why numerical methods can break, and how to fix them. To do so, we motivate the discussion from a physical perspective, which we then translate and inspect with the language of mathematics.

### Onsdag 18 december 2019, Antonio F. Costa, UNED, Madrid, Spanien

Titel: Concepts in Geometry and Topology Illustrated Using Decorations of Islamic Art

Sammanfattning: Islamic art, because of its abstract character, is particularly well suited to exemplify some mathematical concepts. We'll start by remembering one of the most well-known and controversial subject: the appearance in this art of all possible euclidean planar crystalline symmetry. Other recent discoveries, as the presence of quasi-crystals, and examples of relations with mathematical concepts will also be exhibited.

### Onsdag 11 december 2019, Joakim Arnlind, Matematiska institutionen, Linköpings universitet

Titel: Projective modules over the noncommutative cylinder

Sammanfattning: I will give an introduction to the noncommutative cylinder, which is a simple example of a non-compact noncommutative manifold. Finitely generated projective modules over a noncommutative algebra correspond to vector bundles in classical geometry, and we present explicit projectors generating the so called K-theory of the noncommutative cylinder. Furthermore, as everyone might not be familiar with the basic ideas of noncommutative geometry, I will try to motivate and explain several of the concepts as they appear in this context.

### Onsdag 4 december 2019, Hans Lundmark, Matematiska institutionen, Linköpings universitet

Titel: Peakon solutions of the Novikov and Geng–Xue equations

Sammanfattning: Peakons (short for peaked solitons) are solutions of a particular form admitted by certain integrable partial differential equations. These solutions consist of a train of peak-shaped waves that interact with each other in a nonlinear way. The most well-known of these PDEs with peakon solutions is the Camassa–Holm shallow water equation from 1993, but there are several others, such as the Degasperis–Procesi equation and two of its close mathematical relatives which I will focus on in particular in this talk, namely the Novikov equation and the Geng–Xue equation. All these equations are similar in many respects, but they also have interesting differences, for example regarding how regular the solutions need to be, and how solutions can be continued past a singularity where some kind of breakdown occurs. Explicit formulas for the peakon solutions are known, and with their aid one can for example study in detail the kind of wave-breaking that takes place when a positive-amplitude peakon collides with a negative-amplitude antipeakon. This is particularly interesting for the Novikov equation, whose peakon-antipeakon solutions display a much wider array of behaviours than usual, including the possibility of several peakons and antipeakons travelling together in breather-like clusters. The Geng–Xue equation is interesting in a different way. It is a two-component system, with many possible inequivalent configurations depending on the order in which the peakons appear in the two components. The solution formulas describing an arbitrary configuration are very intricate and have been derived only recently, relying not only on the usual inverse spectral techniques, but also (crucially) on a certain limiting procedure for turning peakons into “ghostpeakons” with amplitude zero. This talk is based on joint works with Jacek Szmigielski, Marcus Kardell, Budor Shuaib and Andy Hone.

### Måndag 25 november 2019, Yasunao Hattori, Shimane University, Japan

Titel: Interaction between domain theory and topology

Sammanfattning: In the talk, I will give a brief survey on the interaction between domain theory and topology. Recall that the domain theory studies order structures on (continuous) partial ordered sets (posets). A domain (= a continuous directed complete poset) $D$ is called a computational model for a topological space $X$ if the set $M(D)$ of maximal elements of $D$ with the Scott topology is homeomorphic to $X$. A space $X$ is said to be domain-representable if $X$ has a computational model. In 1981 Weihrauch and Schreiber introduced a set $B^+(X; d)$ of formal balls for a metric space $X$, and showed that $B^+(X; d)$ is a computational model for $X$ if $X$ is complete, i.e. every complete metric space is domain-representable. So I will suggest some results on the domain-representable spaces (in particular, the real line) and show a relationship among several topologies on the sets of (generalized) formal balls for (generalized) metric spaces.

### Onsdag 20 november 2019, Mario Natiello, Lunds universitet

Titel: Winged promises or biological contamination? Modelling genetic diffusion in the RIDL-SIT technique

Sammanfattning: Recently, the RIDL-SIT technology has been field-tested for control of Aedes aegypti. The technique consists of releasing genetically modified mosquitoes carrying a “lethal gene”. In 2016 the World Health Organisation (WHO) and the Pan-American Health Organization (PAHO) recommend to their constituent countries to test the new technologies proposed to control Aedes aegypti populations. However, issues concerning effectiveness and ecological impact have not been thoroughly studied so far. In order to study these issues, we develop an ecological model compatible with the information available as of 2016. It presents an interdependent dynamics of mosquito populations and food in an homogeneous setting. Mosquito populations are described in an stochastic compartmental setup in terms of reaction norms depending on the available food in the environment. The development of the model allows us to indicate some critical biological knowledge that is missing and could (should) be produced. Hybridisation levels, release numbers during and after intervention and population recovery time after the intervention as a function of intervention duration and target are calculated under different hypotheses with regard to the fitness of hybrids and compared with two field studies of actual interventions. The minimal model should serve as a basis for detaile models when the necessary information to construct them is produced. For the time being, the model shows that nature will not clean the non-lethal introgressed genes.

Joint work with H.G. Solari, Buenos Aires, Argentina.

### Onsdag 13 november 2019, Sara Maad Sasane, Lunds universitet

Titel: Monotone smoothing splines with bounds

Sammanfattning: Splines are functions that are used to interpolate between data points. We distinguish between interpolating splines and smoothing splines. Interpolating splines are curves that interpolate between the data points and at the same time are as smooth as possible. The name comes from the drawing tool wooden spline, that was previously used to construct ships and aeroplanes.

Smoothing splines are used when there are measuring errors, and it is not desirable to force the curve to pass exactly through the data points. Instead, the aim is to find a smooth curve which comes close to these points while being as little bent as possible (in a sense that wíll be made precise in the talk).

Monotone smoothing splines are curves that solve a similar minimization problem but where the feasible set of functions also satisfy a monotonicity condition. I will discuss this problem from a calculus of variations point of view, and show that it can be reformulated as a finite dimensional problem which can be solved with optimization techniques.

### Onsdag 30 oktober 2019, Sebastián Reyes Carocca, Universidad de la Frontera (Temuco), Chile

Titel: On Riemann surfaces and Jacobian varieties with automorphisms

Sammanfattning: Let a be an integer greater than 2. A classification of compact Riemann surfaces of genus g with a(g-1) automorphisms is known under the assumption that g-1 is a prime number. In this talk we shall discuss some recent results concerning the same classification problem for a=3 and when g-1 is assumed to be the square of a prime number. We also show interesting relations which induces the corresponding group action on the associated Jacobian varieties. This is a joint work (in progress) with Angel Carocca.

### Onsdag 23 oktober 2019, Erik Lindgren, Uppsala universitet

Titel: Nonlinear nonlocal equations

Sammanfattning: In this talk, I will discuss some classes of nonlocal or fractional partial differential equations. In particular, I will describe recent developments for fractional versions of equations such as the p-Laplace equation.

### Onsdag 16 oktober 2019, Marc Mars, University of Salamanca, Spanien

Titel: Kerr-de Sitter spacetime and conformal infinity

Sammanfattning: In this talk I will present several results concerning the characterization of the Kerr-de Sitter spacetime in terms of the asymptotic data at null infinity. This is relatively recent joint work with Paetz and Senovilla, and partially Simon. The first part of the talk is intended to be introductory: after reviewing the Kerr-de Sitter metric and recalling standard results on the initial value problem in General Relativity, I will discuss the initial value problem at past null infinity for the EFE with positive cosmological constant, as well as the notion of asymptotic Killing initial data. In the second part I will present the characterization results at infinity of Kerr-de Sitter based on a previous local spacetime characterization of this metric.

### Onsdag 9 oktober 2019, Sergey Vakulenko, St. Petersburg, Ryssland

Titel: Centralized Networks, Robotics and Biology

Sammanfattning: This work is conjoint with Prof. A. Weber and I. Morozov (Bonn University).

We consider a special class of networks, which can appear in biological and economical applications. The topological structure of interactions in these networks reflect so-called free-scale structure, these networks include central nodes having many connections and satellites having a few connections. The key assumption is that the satellites do not interact with each other.

We show that these networks are capable to generate any finite dimensional attractors and exhibit complex bifurcations. These analytical results can be applied to robotics, to obtain a compact description of human body motions. We show that typical motions are determined by 2-3 leading frequencies.

### Onsdag 2 oktober 2019, Jürgen Rossmann, Universität Rostock, Tyskland

Titel: On the nonstationary Stokes system in a cone

Sammanfattning: The talk is concerned with the problem
\begin{eqnarray*}
&& u_t -\Delta u +\nabla p = f, \ -\nabla u =g \quad \mbox{in }K\times {\Bbb R}, \\
&& u(x,t)=0 \ \mbox{ for }x\in \partial K, \quad u(x,0)=0 \ \mbox{for }x\in K,
\end{eqnarray*} where $K$ is a cone in ${\Bbb R}^3$ with vertex at the origin. The speaker concentrates on solvability and regularity assertions for this problem in weighted Sobolev spaces. Here, he works out the differences and similarities with the heat equation. A feature of the Stokes system is that the bounds for the weight parameter $\beta$ in the solvability and regularity results depend on the eigenvalues of two different operator pencils. In many other parabolic problems, one has to consider only one operator pencil.
The major part of the talk deals with the Dirichlet problem for the parameter-depending
system $(s-\Delta)\, U + \nabla P= F, \quad -\nabla\cdot U=G \ \mbox{ in }K$ which arises after applying the Laplace transform to the original problem. The speaker presents theorems on the existence and uniqueness of weak and strong solutions of this problem in weighted Sobolev spaces and describes the behavior of the solutions near the vertex of the cone and at infinity. It turns out that the behavior at infinity is completely different from what is known for the heat equation.
The results of the talk are published in common papers with Vladimir Kozlov.

### Onsdag 25 september 2019, Paul Tod, University of Oxford, Storbritannien

Titel: Penrose's Weyl Curvature Hypothesis and his Conformal Cyclic Cosmology

Sammanfattning: Penrose’s Weyl Curvature Hypothesis, which dates from the late 70s, is a hypothesis, motivated by observation, about the nature of the Big Bang as a singularity of the space-time manifold. His Conformal Cyclic Cosmology is a remarkable suggestion, made a few years ago and still being explored, about the nature of the universe, in the light of the current consensus among cosmologists (and the Nobel Committee) that there is a positive cosmological constant. I shall review both sets of ideas within the framework of general relativity, emphasise how the second set solves a problem posed by the first, and say something about predictions of CCC.

### Onsdag 18 september 2019, Jana Björn, Matematiska institutionen, Linköpings universitet

Titel: Geometric analysis on Cantor sets, trees and hyperbolic spaces

Sammanfattning: This is a joint work with A. Björn, J.T. Gill and N. Shanmugalingam. Consider an infinite network represented by a weighted rooted tree which we equip with a metric and measure structure enabling first-order Sobolev spaces and harmonic and p-harmonic functions. This is a special case of a procedure called uniformization, due tp Bonk, Heinonen and Koskela. The visual boundary of the tree at infinity is an utrametric space and can be regarded as a Cantor type set.

In this setting, we show that the trace of the Sobolev space is exactly a Besov space with an explicit smoothness exponent. This, in particular, means that such Besov boundary data have harmonic extensions to the whole tree and it is possible to solve the Dirichlet and obstacle problems with such boundary data. These harmonic extensions can be seen as potentials or stationary flows in the network.

Similar considerations can be done on more general hyperbolic spaces.

If time permits, mappings between pairs of such trees and between their boundaries will also be considered. It turns out that quasisymmetries between two Cantor sets exactly extend to rough quasiisometries between their generating trees, and vice versa.

### Onsdag 11 september 2019, Elin Götmark, Chalmers och Göteborgs universitet

Sammanfattning: What is the shortest travel distance between two cities as the crow flies? How can you determine your position on the Earth by means of the sun or stars? The answer relies on spherical trigonometry. This is old mathematics that first began to develop in Hellenistic times, but most of us do not encounter it in our university courses today. This talk will be accessible to undergraduate students.

### Torsdag 29 augusti 2019, Francis Seuffert, University of Pennsylvania, Philadelphia, USA

Titel: A Qualitative Description of Extremals for Morrey's Inequality

Sammanfattning (PDF)

### Onsdag 21 augusti 2019, Petros Petrosyan, Yerevan State University, Armenien

Titel: Generalizations of interval edge-colorings of graphs

Sammanfattning: An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an \emph{interval $t$-coloring} if all colors are used and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. The concept of interval edge-coloring of graphs was introduced by Asratian and Kamalian more than 30 years ago and was motivated by the problems in scheduling theory. In the last 10 years different types of variations and generalizations of interval edge-colorings were studied. In this talk we will give a survey of the topic and present a recent progress in the study of interval edge-colorings and their various generalizations.

### Onsdag 5 juni 2019, Peter Schenzel, Martin Luther University Halle-Wittenberg, Tyskland

Titel: On families of blowups of the real affine plane

Sammanfattning: TBA

### Tisdag 28 maj 2019, Victor Falgas-Ravry, Umeå universitet

Titel: Bridg'it revisited

Samanfattning: In the popular game of Bridg'it, two players - let us call them Alice and Bob - play in alternating turns by placing bridges on a rectangular grid-like board. On each of their turns, Alice adds a red bridge while Bob adds a blue bridge. Alice's aim is to build a connected path of red bridges from the left-handside of the board to the right-hand side, while Bob for his part tries to hinder her by assembling a connected path of blue bridges from the top side of the board to the bottom side.

Bridg'it was introduced by David Gale in the 1950s, and was instantly and completely solved: for each board we know both the identity of the winning player and an explicit winning strategy. But what happens if the players were allowed to place more than one bridge on each of their turns - say, for example that Alice places two bridges and Bob three?
Such variants of the game turn out to be considerably more difficult to analyse than the original Bridg'it. In this talk, I will describe some special cases where one can find explicit winning strategies (as well as some elementary cases that remain stubbornly open!).

Joint work with A. Nicholas Day.

### Onsdag 22 maj 2019, German Zavorokhin, Russian Academy of Sciences, Sankt Petersburg

Titel: On elastic waves in a wedge

Samanfattning: The existence of waves propagating along the edge of the elastic wedge was established by many authors by physically rigorous arguments on the base of numerical computations. In this talk the mathematically rigorous proof of the existence of a symmetric mode in an elastic solid wedge for all allowable values of the Poisson ratio and arbitrary openings close to π will be presented. A radically new effect—the presence of a wave localized in a vicinity of the edge of a wedge with an opening larger than a flat angle—has been found.

This is a joint work with S.A. Nazarov and A.I. Nazarov.

### Onsdag 15 maj 2019, Olga Balkanova, Chalmers och Göteborgs universitet

Titel: Prime geodesic theorems

Sammanfattning: Prime geodesic theorem for a hyperbolic manifold M provides an asymptotic formula for the number of primitive closed geodesics on M of length at most X as X grows. Similarly to the prime number theorem, the major open problem is to prove the best possible estimate for the error term. I will describe the most recent results in this direction for M being the modular surface and the Picard manifold.

### Onsdag 8 maj 2019, Lilian Matthiesen, KTH

Titel: Correlations of multiplicative functions and rational points in families

Sammanfattning: In the first part of this talk I will discuss an asymptotic result on certain correlations of multiplicative functions and its background. In the simplest instance, these correlations take the form $\sum_{n,d < x} h_1(n) h_2(n+d) ... h_{r+1}(n+rd)$ where  $h_1, ..., h_{r+1}$ are multiplicative functions. I will describe a set of conditions under which such correlations can be evaluated asymptotically as well as examples of functions satisfying these conditions.

The second part of the talk will be about joint work with Dan Loughran on a question of Serre. By combining the analytic result on multiplicative functions mentioned above with ideas from algebraic geometry, we obtain (under suitable conditions) correct-order lower bounds for the number of varieties in a family over Q which have a rational point.

### Onsdag 24 april 2019, Mieczysław Mastyło, University of Poznań, Polen

Titel: Interpolation of isomorphisms and Fredholm operators

Sammanfattning: We will discuss the stability of isomorphisms between Banach spaces generated by abstract interpolation methods which include, as special cases, the real and complex methods up to equivalence of norms. A by-product of our results is that interpolated isomorphisms satisfy uniqueness-of-inverses. We also will present novel results on the stability of Fredholm property of operators on interpolation spaces.

The talk is based on joint work with Irina Asekritova and Natan Kruglyak.

### Tisdag 16 april 2019, Maya Stoyanova och Silvia Boumova, Sofia University, Sofia, Bulgarien

Maya Stoyanova:

Titel: Next levels universal bounds for spherical codes: lifting the Levenshtein framework

Sammanfattning: We introduce a framework based on the Delsarte-Yudin linear programming approach for improving some universal lower bounds for the minimum energy of spherical codes of prescribed dimension and cardinality, and universal upper bounds on the maximal cardinality of spherical codes of prescribed dimension and minimum distance. Our results can be considered as next level universal bounds as they have the same general nature and imply, as the first level bounds do, necessary and sufficient conditions for their local and global optimality. We explain in detail our approach in the most common case. Our model examples include the cases of 24 points and 120 points on S3. In particular, we derive a new proof that the 600-cell is universally optimal, and completely characterize the optimal polynomials of degree at most 17 for the Delsarte-Yudin linear programming lower bounds by finding two new polynomials that, together with Cohn-Kumar's polynomial, form the vertices of the convex hull that consists of all optimal polynomials. Our framework provides a conceptual explanation of why polynomials of degree 17 are needed to handle the 600-cell via linear programming.

Silvia Boumova:

Titel: A Diophantine Transport Problem from 2016 and it solutions by Elliot in 1903

Sammanfattning: The main results are application of the method, used to solve problems from the classical theory of invariants, from the theory of algebras with polynomial identities and noncommutative invariant theory.
We start with a concrete transport problem (about how to transport profitably a group of persons or objects) and have generalized it.
Another approach to this problem is suggested using powerful tool given by Elliot in 1903 and was further developed by MacMachon in his "$\Omega$-Calculus" or Partition Analysis. The "$\Omega$-Calculus" was improved by developing better algorithms and effective computer realizations by Andrews, Paule, and Riese, and Xin.
The idea of Elliot is to find generating functions and formulae for solutions of homogeneous Diophantine equations and inequalities. Following Elliot's idea and we study infinite series which can be represented as rational functions (converges to rational functions) with denominators a product of terms of the form of one minus multivariate monomial and call such functions nice rational functions. Going back to the originals, we see that the results given there provide algorithms to compute the multiplicity series for nice rational functions in any number of variables.

### Måndag 15 april 2019, Magnus Goffeng, Chalmers och Göteborgs universitet

Titel: A problem of magnitude

Sammanfattning: An invariant that has attracted quite some attention in the last decade is the magnitude of a compact metric space. Magnitude gives a way of encoding the size of a metric space. In many ways it resembles a capacity. In this colloquium I will give a short introduction to magnitude and present some recent results for compact metric spaces of geometric origin (i.e. domains in Euclidean space or manifolds). One of the results states that the magnitude recovers geometric invariants such as volume and certain integrals of curvatures. Based on joint work with Heiko Gimperlein and Nikoletta Louca.

### Onsdag 10 april 2019, Sergey Nazarov, Matematiska institutionen, Linköpings universitet, och St Petersburg, Ryssland

Titel: The threshold resonances in the spectrum of a waveguide

Sammanfattning: A threshold resonance is due to the appearance of a stabilizing, in particular, bounded solution in a cylindrical waveguide.Various near-threshold anomalies will be discussed caused by these stabilizing solutions. In particular, a perturbation of the waveguide wall may lead to an eigenvalue which, either belongs to the discrete spectrum, or is embedded into the continuouis spectrum. This effect is well-known for scalar problems about acoustic and quantum waveguides where the eigenvalue is always situated below the threshold. It will be shown that in vectorial problems of the elasticity theory an eigenvalue can move from the threshold in both directions, upwards or downwards. Classification of the threshold resonances will be given as well.

### Onsdag 3 april, 2019, Julia Brandes, Chalmers och Göteborgs universitet

Titel: Diophantine problems via Fourier analysis

Sammanfattning: Questions concerning the solubility or otherwise of Diophantine equations go back to antiquity, and their study has been a driving force not only in number theory, but has also left a distinct mark on several other parts of mathematics. Since the groundbreaking work of Hardy and Ramanujan in the early 20th century, Fourier-analytic methods have played a central role in the understanding of Diophantine equations in many variables. We will give an overview of the underlying ideas and main results, as well as point out some of the greatest current challenges in the field.

Titel: Coherent functors and asymptotic stability

Sammanfattning: We study coherent functors on the category of $A$-modules where $A$ is a commutative noetherian ring. In this talk, we will show that the sets of associated prime ideals of the modules $F(M/a^n M)$ where $F$ is a coherent functor, $a$ is an ideal of $A$ and $M$ is a finitely generated A-module, are independent of $n$ for large $n$. A (dual) result in the context of artinian modules will be proved. We will also discuss results on Hilbert functions related to coherent functors.

### Onsdag 20 mars 2019, Jana Björn, Matematiska institutionen, Linköpings universitet

Titel: Sphericalization and p-harmonic functions on unbounded domains in Ahlfors regular metric spaces

Sammanfattning: We use sphericalization of unbounded metric spaces to transform p-harmonic functions on unbounded domains to p-harmonic functions on bounded ones, for which the theory is much more developed and there are plenty of methods and results. In particular, we consider the Dirichlet problem in unbounded domains, with a particular emphasis on boundary regularity at infinity. As a byproduct, we obtain a result about the p-harmonic measure in Rn.

### Onsdag 13 mars 2019, Milagros Izquierdo, Matematiska institutionen, Linköpings universitet

Titel: Combinatorial Configurations and Dessins d’Enfants

Sammanfattning: In this talk we will discuss relations between Riemann surfaces and combinatorial geometries via dessins d’enfants. We describe how to apply results for Riemann surfaces to graphs.

### Onsdag 20 februari 2019, Daniel J. Fox, Universidad Politécnica de Madrid, Spanien

Titel: Harmonic cubic polynomials satisfying a Hessian equation and commutative nonassociative algebras with nondegenerate invariant trace-form

Sammanfattning: The talk aims to explain a relation between certain partial differential equations and a particular class of commutative nonassociative algebras. Orthogonal equivalence classes of harmonic cubic homogeneous polynomials solving $|\operatorname{Hess} P(x)|^{2} = \varkappa |x|^{2}$ are in bijection with isomorphism classes of commutative nonassociative algebras for which the traces of multiplication operators vanish and the Killing type form given by tracing the product of multiplication operators is a nondegenerate and invariant bilinear form.

There is a surprising range of interesting examples with apparently diverse origins in differential geometry (isoparametric polynomials, Jordan algebras, affine spheres), combinatorics (Steiner triple systems, equiangular tight frames), representation theory (algebras of curvature tensors), and finite group theory and vertex operator algebras (permutation modules, Griess algebras). The talk will describe some of these examples explicitly and indicate some of their common features which appear to make them amenable to classification.

### Onsdag 30 januari 2019, Pavel Exner, Czech Academy of Sciences, Řež, Tjeckien, och Institut Mittag-Leffler, Djursholm

Titel: Leaky quantum graphs and Robin billiards: discrete spectrum and magnetic eﬀects

Sammanfattning: The talk focuses on properties of the discrete spectrum of several operator classes appearing in models of various quantum systems. They include Schrödinger operators with an attractive singular ‘potential’, supported by a geometric complex of codimension one, formally written as −∆−αδ(x−Γ) with α > 0, where Γ is the interaction support. Another class are Hamiltonians describing quantum motion in a region with attractive Robin boundary. We discuss the ways in which spectral properties of such systems are inﬂuenced by the geometry of the interaction support with an attention paid to situations when the coupling constant is large or the geometric perturbation is weak, and asymptotic expansions can be derived. We also discuss eﬀects arising from the presence of a magnetic ﬁeld, in particular, suﬃcient conditions for existence of the discrete spectrum in planar wedges in presence of a homogeneous magnetic ﬁeld, and inﬂuence of an Aharonov-Bohm ﬂux on the so-called Welsh eigenvalues.

## OrganisatörerVisa/dölj innehåll

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Professor

Professor

### Hans Lundmark

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