17/12/2019 from 16:00 to 18:00 in H8.

- 16:15 Marlies Vantomme
- Title: An introduction to classical and virtual knot theory.
- Abstract: Classical knot theory studies how ropes can intertwine in three dimensional space. I will explain how one can mathematically define these knots and how the Reidemeister theorem can be used to study them. In 1999, this notion of a knot was generalised by Louis Kauffman to what he called virtual knots. I will show how these are defined and what the motivation behind them is. Lastly, I will talk about polynomial invariants, which can help to prove that two (classical or virtual) knots are different.
- 17:15 Sophiane Yahiatene
- Title: Reflection factorizations of isometries.

Abstract: In this talk we investigate reflection factorizations of isometries by using results of Brady and Watt from 2002 and McCammond from 2015. First we consider isometries on a finite-dimensional vector space equipped with an anisotropic symmetric bilinear form and sketch a proof of the famous Cartan-Dieudonnè theorem. Afterwards we state a beautiful theorem of Scherk which can be seen as a refinement of the Cartan-Dieudonnè theorem. Finally, if time permitting, we consider euclidean isometries and state some newer results describing their minimal reflection factorizations.

- 16:15 Karsten Schroedter
- Title: The mathematics behind the card game Dobble
- Abstract: The rules of Dobble are simple. Each of the 55 cards has 8 symbols and two cards are placed in the middle of the table. The key property of the deck is that each two cards share exactly one symbol. The player who figures out the common symbol first, receives a point. Despite the simplicity of the rules, it seems to be complicated to create such a deck of cards with the desired property that every two cards have exactly one symbol in common. This is what we want to deal with in this talk. Herein, we will look at smaller examples and upon analysis, we will observe that they naturally give rise to objects in finite geometry. Following this, we will construct decks and determine that it is not possible to form a (complete) Dobble deck with 7 symbols per card.
- 17:15 Bono Adelt
- Title: L-functions attached to modular forms; an introduction.

Abstract: We expand upon our knowledge of modular forms, by considering their associated L-functions L(s,f). We will see that these very naturally arising functions, a priori defined only for real part of s sufficiently large, allow for analytic continuation to the entire complex plane. Furthermore they also have an underlying functional equation, which we will have a look at. These functions play a very important role in modern analytic number theory and are still the subject of many open conjectures, such as the Riemann hypothesis.

- 16:15 Arne Johannsmann
- Title: What's a modular form?
- Abstract: Even though modular forms played a key role in the proof of Fermat's Last Theorem and remain an important topic in modern number theory, most students barely seem aware of their existence. We will try to take a cursory glance at their definition without delving any deeper into the theory. This definition looks odd at first, but we will see that it gives rise to baffling properties which make modular forms interesting objects of study in their own right.
- 17:15 Alvaro Bustos
- Title: Finite automata in unexpected places

Abstract: Abstract: Finite automata are a very simple model of computation, closely related to regular languages (which are sets of strings showing simple patterns). However, due to this very simplicity, they (or objects shaped like them) appear in several other areas of mathematics. We shall introduce finite automata in their role in the theory of computation, before discussing their connections with other branches of mathematics such as dynamical systems and logic.

- 16:15 Philipp Gohlke (Rice University, USA)
- Title: Spectral properties of discrete Schroedinger operators
- Abstract: Schroedinger operators are the "universal tool" to describe the time evolution of quantum mechanical systems. For example, they predict how electrons behave in materials with a given atomic structure. In this setting, the spectrum (basically the set of generalized eigenvalues) of the Schroedinger operator determines the relevant electronic energies. However, there is more spectral information which is important for the dynamics of the system: The "type" of the generalized eigenvalues encodes whether electrons can move freely, are trapped or exhibit some intermediate behaviour. Mathematically, this is captured by the spectral measure which can be split into different parts, reflecting different transport phenomena. If time permits, we will see how the type of the spectral measure depends on structural information (periodic, random, aperiodic,...) about the material.
- 17:15 Enrico Di Gaspero
- Title: The stability-complexity paradox in ecological networks

Abstract: For many years, one of the central themes of population ecology was that the more a trophic web is complex, the more it is stable. This was such a strong belief that it had, on occasion, been accorded the status of mathematical theorem. However, in 1966, Robert May proved that this is not trivial as it may appear at first sight, and actually he showed that sufficiently large or complex networks have a higher probability of being unstable rather than the more simple and small webs. May's work raised an intense debate around this paradox: is it true that complex trophic webs are more unstable? or is there anything we are modelling in the wrong way? The answer to this question is still unclear.

- 16:15 Alexandros Galanakis
- Title: About the 12th Hilbert's problem

Abstract: One of the main goals of algebraic number theory is the study of number fields. Therefore, it makes sense to study Galois groups of number field extensions. A natural question that arises is when these Galois groups are abelian, given a base field. This is the 12th Hilbert's problem. We will see the cases where the base field is the field of rational numbers, or an imaginary quadratic field. - 17:15 Luigi Esercito
- Title: Ancestral lines in population genetics

Abstract: Population genetics is the branch of evolutionary biology focusing on the evolution of populations (rather than speciation). The aim of this talk is to give a glimpse into the field and introduce simple models - underlining in particular the usefulness of a "backward" view of things, using so-called ancestral processes and their duality relationships.

- 16:15 Martina Hofmanova
- Title: Randomness in modeling of fluid motion

Abstract: The phenomenon of hydrodynamic turbulence has puzzled mathematicians, physicists and engineers since the beginning of the twentieth century. Mathematicians are fascinated by the solvability of the corresponding Navier-Stokes system of partial differential equations, physicists are interested in statistical behavior as related to statistical mechanics of turbulence and engineers ask questions like: What are the heat transfer properties of a turbulent flow? What are the forces applied by a fluid to its boundary (be it a pipe or an airfoil)? Of high interest is then also large-time behavior as well as reliable numerical simulations of turbulent flows. Nevertheless, despite the concerted effort of generations of excellent researchers, no complete and satisfactory theory of turbulence is available so far. In a turbulent flow, many interesting physical quantities undergo rapid, random and very chaotic changes. These irregularities are the main obstacles in the rigorous analysis of the Navier-Stokes equations. In addition, the flows tend to amplify even the slightest difference in the initial state, which makes reproduction of experiments very challenging. However, it turns out that certain time-averaged quantities possess a predictable behavior and are perfectly reproducible. All this leads to the need of statistical description of a turbulent flow, the concept of statistical solutions of the Navier-Stokes equations and the use of probability theory in the study of the Navier-Stokes equations. - 17:15 Dirk Frettloeh
- Title: How the zebra got its stripes

Abstract: In 1952 Alan Turing laid one of the foundations of mathematical biology by describing a reaction-diffusion model for the development of patterns in biological organisms. He used a system of nonlinear differential equations to study the solutions of the model. This reaction-diffusion model turned to be rather fruitful, for instance in describing spread of epidemics (e.g., zombie invasions). We describe Turing's original model in detail and sketch some of the later applications.

- 16:15 Julia Sauter
- Title: Every projective variety is a quiver Grassmannian

Abstract: This is a highly contested result proven independently with different explicit constructions by Huisgen-Zimmermann, Hille, Reineke and Ringel. I will explain the definitions of projective varieties and quiver Grassmannians are and show the result. - 17:15 Josh Maglione
- Title: Tools to Tame Tensors

Abstract: Tensors describe diverse structures, including distributive products in algebra, quantum entanglement in particle physics, and measurements and meta-data in statistical models. A natural objective in the study of tensors is to discover properties invariant under basis change. Most of these invariants hide deep problems in algebraic geometry, wild representation theory, and complexity theory. We will briefly overview these topics and discuss recent invariants influenced by (not necessarily associative) algebra.

- 16:15 Benjamin Brück
- Title: Reflections and harmonies: Coxeter groups in music theory

Abstract: I will define Coxeter groups, which generalise the class of finite reflection groups, and give examples of them. I will then explain how these groups can be used to study the structure of pieces of music in what is called "Neo-Riemannian theory". - 17:15 Paula Lins
- Title: Fixed points and twisted conjugacy classes

Abstract: A fixed point of a map \(f: X\to X\) is an element \(x \in X\) such that \(f(x)=x\). Besides mathematics, fixed point theory has applications in variuos fields such as biology, chemistry, economics, engineering, game theory, and physics. We discuss some tools from algebraic topology that are used to obtain information about fixed points of self-maps \(f: X \to X\) of certain spaces. We also introduce the concept of twist conjugacy classes of homomorphisms of groups and present a connection between the existence of fixed points of certain homeomorphisms \(f:X \to X\) and the number of twist conjugacy classes of the induced homomorphism of \(f\) in the fundamental group of \(X\).

- 16:15 Eduard Schesler
- Title: Sigma-Invariants of groups

Abstract: In this talk we will introduce an invariant of groups which is defined via connectivity properties of its so called Cayley graph. We will see how Sigma-Invariants help us unravel some mysteries in group theory. - 17:15 Frederic Alberti
- Title: When zombies attack!

Abstract: Zombies are a staple of pop culture/entertainment and are often portrayed as being brought about through an outbreak or epidemic. Consequently, the classical SIR model from mathematical epidemiology can be adapted to model a zombie attack, using biological assumptions based on popular zombie movies. In this basic model, we will determine equilibria and their stability, and illustrate the outcome with numerical solutions. We then refine the model to include a latent period of zombification, whereby humans are infected, but not infectious, before becoming undead. The model is further modified to explore the effects of a possible quarantine or a cure. Finally, we examine the impact of regular, impulsive reductions in the number of zombies and derive conditions under which eradication can occur. Can we stave off the collapse of society as zombies overtake us all?

- 16:15 Mima Stanojkovski
- Title: Sudoku matrices

Abstract: Sudokus are among our most loyal companions during the holiday season: as the holiday season approaches, why not taking a closer look at them? We will discuss some of their mathematical properties and give a list of related open problems. - 17:15 Dan Rust
- Title: Pairs of pants and their central role in topological quantum field theories

Abstract: Low dimensional topologists have a strange fascination with pants. In 1988, Atiyah (influenced by Segal and Witten) topologically axiomatised what it means to be a quantum field theory, a subject previosuly studied mostly by mathematical physicists. His axiomatisation was simple and yet surprisingly rich. An entire program was initiated to fully understand these TQFTs using techniques from category theory, algebraic topology and conformal geometry. In the case of dimension 2, this axiomatisation is all about sewing together pairs of pants so I'll be drawing lots of them! I'll be happy if by the end of the talk I'm able to hint at how TQFTs can be used to define strong algebraic invariants of knots and links via the so-called 'Khovanov Homology'.

- 16:15 Dawid Kielak
- Title: The Alexander polynomial

Abstract: I will give a gentle (if brief) introduction to knots, and discuss two methods of computing one of the earliest knot invariants, the Alexander polynomial. - 17:15 Fernando Cordero
- Title: Counting self-avoiding paths in the honeycomb lattice

Abstract: The connective constant of an infinite graph G is the asymptotic growth rate of the number of self-avoiding walks on G from a given starting vertex. As a warm up we will see some (of the atypical) examples of graphs for which this constant is relatively easy to compute. In the second part of this talk, we will show that the connective constant of the honeycomb lattice is equal to \(\sqrt{2+\sqrt{2}}\). This value was derived non rigorously by B. Nienhuis in 1982 using the Coulomb gas approach from theoretical physics and proved rigorously only in 2012 by H. Duminil-Copin and S. Smirnov. The proof is based on the construction of an observable with some properties of discrete holomorphicity.

- 16:15 Georges Neaime
- Title: Garside monoids and groups

Abstract: The talk consists of a single definition and an example. - 17:15 Emanuela Gussetti
- Title: An introduction to rough integration

Abstract: The theory of rough paths is an integration theory introduced by Terry Lions, that has its main application in the study of stochastic (partial) differential equations. In the last years there have been huge developements in the theory of controlled rough path of Massimiliano Gubinelli. The aim of this introductory talk is to give some motivation for the rough paths theory: we will define a rough path \(X\) as well as the integration against a rough path, i.e. we want to give a meaning to the expression \(\int Y dX\) for a suitable class of integrands \(Y\), following Gubinelli's approach. We will also discuss some relations between the Itô integral and the rough integral.

- 16:15 Yuri Santos
- Title: Homeomorphisms and dynamics on a line

Abstract: A real interval might be inoffensive at a first glance, but our very first Analysis courses show that there is a lot to be said about (and learn from) such topological spaces and continuous maps between them. Not surprisingly, their self-transformations, i.e. the group of homeomorphisms of an interval carries a rich structure. In this lecture we shall meet a prominent set of homeomorphisms of an interval, namely Richard Thompson's group F, giving a 'modern' description of its elements. We will look at interesting features F is known to have, with a view towards dynamical properties of its elements and a quick mention of trending topics in the social network of Thompson-like groups. - 17:15 Tyrone Cutler
- Title: Counting Homotopy Types of Gauge Groups

Abstract: The gauge group \(\mathcal{G}(P)\) of a principal \(G\)-bundle \(P\rightarrow X\) consists of all the \(G\)-equivariant homeomorphisms \(P\xrightarrow{\cong}P\) that cover the identity on \(X\). Such objects arise naturally in geometry and in physics, and in both areas knowledge of their homotopy-theoretic properties is much prized. Now a key result of Crabb and Sutherland states that if \(G\) is a compact, connected Lie group, and \(X\) is a connected finite complex, then the number of distinct homotopy types amongst all the gauge groups of principal \(G\)-bundles over \(X\) is finite. It is fascinating to observe that this holds in spite of the fact that the number of principal \(G\)-bundle-isomorphism classes over \(X\) may be infinite. In this talk I will explain how it is essentially the algebraic properties of the Lie group \(G\) which give rise to this phenomenon. Following this I will discuss some recent results 'counting' the homotopy types of certain gauge groups, and time permitting I will go into further detail on Kono's enumeration of the number of \(SU(2)\)-gauge groups over \(S^4\).

- 16:15 Alexandros Galanakis
- Title: The Birch and Swinnerton-Dyer conjecture

Abstract: After a brief introduction to the problem of finding rational points on curves we will focus on the case of elliptic curves. An elliptic curve E is a smooth projective algebraic curve of genus one, with at least one rational point. The problem of characterization of the rational points of E appears to be one of the most challenging in mathematics. Due to Mordell, it is known that the group of rational points consists of a torsion part and some copies of the ring of integer. The number of the copies is called rank of the elliptic curve. The B.S-D conjecture provides us with a simple way of computing the rank. Further, we will mention the application of B.S-D conjecture to the congruent number problem. - 17:15 Yasushi Nagai (Montan-Universtität Leoben)
- Title: The role of almost periodicity in mathematical diffraction

Abstract: The theory of mathematical diffraction describes physical diffraction experiment. Ever since quasicrystals were discovered, it is an important problem to decide which models of solid have pure point diffraction spectrum. On the other hand, the theory of almost periodic functions and measures has been developed by many mathematicians in connection to representation theory, the theory of Riemann zeta function and the theory of differential equations. In this talk we will introduce an important result that claims certain almost periodicity of measures are equivalent to their having pure point diffraction spectrum.

- 16:15 Arthur Sinulis
- Title: Concentration of measure and functional inequalities

Abstract: I will present the general concept of a concentration of measure, which was first used in the 70s in the theory of Banach spaces, and was developed in the 90s and 2000s by various authors. To this end, I will introduce some functional inequalities (initially proven in the context of Sobolev spaces) and hint at how to use these to obtain concentration results. I will end the talk by presenting an application: the Johnson-Lindenstrauss theorem, informally stating that almost any random linear mapping from a high to a low-dimensional space is an ε-isometry. - 17:15 Neil Manibo
- Title: Mahler measures, the Lehmer Conjecture, and its consequences

Abstract: Given a polynomial with integer coefficients, its Mahler measure is defined to be its geometric mean on the unit circle. This talk will center on the still open Lehmer's problem, which asks whether such a measure is bounded away from one outside the class of cyclotomic polynomials. In particular, the progression of partial results and proofs on subclasses will be laid out, as well as exhaustive searches for polynomials with small measures. We then mention some of its implications and connections to other gap results in number theory, ergodic theory, and geometric group theory.

- 16:15 Alastair Litterick
- Title: The field with one element: Groups, combinatorics, geometry and zeta functions

Abstract: Linear algebraic groups are an important class of groups with applications and connections throughout mathematics. When considering the finite versions of these groups, such as GL_n(q), it is natural to try and understand properties by counting things, and by seeing how the structure varies with q. Somewhat mysteriously, many aspects of this study make sense "in the limit as q goes to 1" and correspond to known results in combinatorics, even though no field of order 1 exists, and objects like "GL_n(1)" certainly make no sense. This talk will follow the story of this study, from Jacques Tits' original observation in 1957 to an ambitious modern approach to proving the Riemann hypothesis. - 17:15 Philipp Gohlke
- Title: Divergent series

Abstract: What is the sum of all natural numbers? Surely, taking the usual definition of a series we cannot assign it a finite value. In contrast, if interpreted as a formal expression for the Riemann zeta function evaluated at -1, we should answer that it is -1/12. But analytical continuation is not the only way to assign finite values to divergent series. We will see some more examples like smooth summation or the embedding into p-adic numbers, explore pitfalls and connections and discuss some general principles of summation methods. Some applications in physics will be hinted at.

Speakers are invited to choose broad topics -

Everyone is welcome to attend the seminar! Audience members are encouraged to engage actively with the speakers: ask questions, make remarks, propose new directions for the speaker to consider... that is what this seminar is for: teach each other, enrich our networks, and grow together!

- Dan Rust (Founder and Co-organiser), Postdoc at Uni Bielefeld.
- Alex Galanakis (Co-organiser), PhD student at Uni Bielefeld.
- Mima Stanojkovski (Founder and Former Co-organiser), Postdoc at Uni Bielefeld.