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Mathematisches Kolloquium

Anand Srivastav (Kiel): Recent Advances in the Maker Breaker Subgraph Game

The triangle game introduced by Chvátal and Erdős (1978) is one of the old and famous combinatorial games. For n , q ∈ N, the ( n , q )-triangle game is played by two players, called Maker and Breaker, on the complete graph K _ n . Alternately Maker claims one edge and thereafter Breaker claims q edges of the graph. Maker wins the game if he can claim all three edges of a triangle. Otherwise Breaker wins. Chvátal and Erdős (1978) proved that for q < sqrt( n /2), Maker has a winning strategy, while for q > 2 sqrt( n ), Breaker wins. So, the threshold bias must be in the interval [sqrt(1/2)sqrt( n ) , 2 sqrt( n )]. Since then, the problem of finding the exact constant (and an associated Breaker strategy) for the threshold bias of the triangle game has been one of the interesting open problems in combinatorial game theory. In fact, the constant is not known for any graph with a cycle and we do not even know if such a constant exists. Balogh and Samotij (2011) slightly improved the Chvátal-Erdős constant for Breaker’s winning strategy from 2 to 1.935 with a randomized approach. Thereafter, no progress was made. In this work, we present a new deterministic strategy for Breaker leading to his win if q > sqrt(8/3) sqrt( n ), for sufficiently large n . This almost matches the Chvátal-Erdős bound of sqrt(1/2)sqrt( n ) for Maker's win (Glazik, Srivastav, Europ. J. Comb. 2022). In contrast to previous (greedy) strategies we introduce a suitable non-linear potential function on the set of nodes. By keeping the potential small, Breaker picks edges that neutralize the most ‘dangerous’ nodes with incident Maker edges blocking Maker triangles. A characteristic property of the dynamics of the game is that the total potential is not monotone decreasing. In fact, the total potential of the game may increase, even for several turns, but finally Breaker’s strategy prevents the total potential of the game from exceeding a critical level, which results in Breaker’s win. We further survey recent results for cycles of length k , and a general potential function theorem (Sowa, Srivastav 2023). This is joint work with Christian Glazik, Christian Schielke and Mathias Sowa, Kiel University.

Ort: Freie Universität Berlin Institut für Informatik Takustr. 9 14195 Berlin Great Lecture Hall (Ground Floor)

10.07.2023 | 16:00 s.t.

Ralf Kornhuber (FU-Berlin): Neural networks, Fredholm integral equations and all that jazz …

The industrial revolution started with the invention of the steam engine in the 19th century and has made physical work redundant to a large extend. Data Science and Artificial Intelligence (AI) might have the potential to play a similar role for intellectual work.   There is a huge overlap of Data Science and AI with mathematics, which on one hand comes with unprecedented social responsibility of mathematics and on the other hand with lots of opportunities for application and extension of existing mathematical concepts and results. In this talk, I will give three examples. First I will present some recent ideas on neural network training by Fredholm integral equations (joint work with P. Gelß and A. Issgali).  Then I will rely on recent work of other authors to discuss the curse of dimensionality in neural network approximation, and to finally sketch a backward error attack on deep learning.

Ort: Seminarraum 019 Arnimallee 3 14195 Berlin

05.07.2023 | 16:30

Tibor Szabó (FU Berlin): Topology at the North Pole

In the max-min allocation problem a set P of players are to be allocated disjoint subsets of a set R of indivisible resources, such that the minimum utility among all players is maximized. We study the restricted variant, also known as the Santa Claus problem , where each resource has an intrinsic positive value, and each player covets a subset of the resources. Bezakova and Dani showed that this problem is NP-hard to approximate within a factor less than 2, consequently a great deal of work has focused on approximate solutions. The principal approach for obtaining approximation algorithms has been via the Configuration LP (CLP) of Bansal and Sviridenko. Accordingly, there has been much interest in bounding the integrality gap of this CLP. The existing algorithms and integrality gap estimations are all based one way or another on the combinatorial augmenting tree argument of Haxell for finding perfect matchings in certain hypergraphs. Here we introduce the use of topological tools for the restricted max-min allocation problem. This approach yields substantial improvements in the integrality gap of the CLP. In particular we improve the previously best known bound of 3.808 to 3.534. The talk represents joint work with Penny Haxell.

Ort: Freie Universität Berlin Institut für Informatik Takustr. 9 14195 Berlin Great Lecture Hall (Ground Floor)

26.06.2023 | 16:00 s.t.

Günter Rote (Freie Universität Berlin): Grid Peeling and the Affine Curve-Shortening Flow

Grid Peeling is the process of taking the integer grid points inside a convex region and repeatedly removing the convex hull vertices. By contrast, the Affine Curve-Shortening Flow (ACSF) is defined as a particular deformation of a smooth curve. It has been observed in 2017 by Eppstein, Har-Peled, and Nivasch, that, as the grid is refined, Grid Peeling converges to the Affine Curve-Shortening Flow. As part of the M.Ed. thesis of Moritz Rüber, we have investigated the grid peeling process for special parabolas, and we could observe some striking phenomena. This has lead to the precise value of the constant that relates the two processes. With Morteza Saghafian from IST Austria, we could prove the convergence of grid peeling for the class of parabolas with vertical axis.

Ort: Freie Universität Berlin Institut für Informatik Takustr. 9 14195 Berlin Great Lecture Hall (Ground Floor)

05.06.2023 | 16:00 s.t.

Max von Kleist (FU-Berlin Antrittsvorlesung): Mathematics for public health

Public health is concerned with measures that improve the general health and prevent infections. In my talk, I will give an overview and outlook of our current work and explain how data science in conjunction with mathematical modeling and simulation can be utilized to guide public health decisions. In particular, I will present approaches that utilize primary and secondary data of SARS-CoV-2 to permanently monitor and assess the pandemic. Moreover, I will give examples where these approaches supported the choice of containment and testing strategies in 2020/21. I will then give some insight into our ongoing work in the field of HIV-1 prevention, the mathematical methods developed along the way, and illustrate how this work is used to quantify risk reduction, to develop guidelines, as well as to a posteriori assess the impact of interventions on the HIV pandemic.

Ort: Seminarraum 019 Arnimallee 3 14195 Berlin

01.02.2023 | 16:30

Marita Thomas (FU-Berlin Antrittsvorlesung): Modeling and Analysis of Bulk-Interface Processes

Heterogeneous materials can be seen as bulk-interface systems. They consist of distinct bulk components with different material properties meeting at thin interfacial layers forming lower-dimensional substructures of the system. In many applications the properties of interfaces strongly impact the functionality of the whole system and, in turn, interfaces are strongly affected by processes taking place in the bulk material. Interfaces thus follow their own evolution laws in interaction with bulk processes. In this talk I discuss a general thermodynamical modeling framework for bulk-interface processes and, in particular, apply it to problems related to heat conduction and fracture in elastic composites. Here, a challenge in the modeling and in the analysis lies in the change of the material geometry with the progressing fracture and in the constraint that in many materials crack growth is a unidirectional process, since the crack cannot heal. Models suited to handle these challenges and thus suited to describe dynamic fracture processes in elastic solids with the aid of non-smooth constraints will be introduced. Recent results on their mathematical analysis will be presented.

Ort: Seminarraum 019 Arnimallee 3 14195 Berlin

18.01.2023 | 16:30

Claudia Schillings (FU-Berlin Antrittsvorlesung): Quantification of uncertainty for inverse and optimization problems

Approaches to decision making and learning mainly rely on optimization techniques to achieve “best” values for parameters and decision variables. In most practical settings, however, the optimization takes place in the presence of uncertainty about model correctness, data relevance, and numerous other factors that influence the resulting solutions. For complex processes modeled by nonlinear ordinary and partial differential equations, the incorporation of these uncertainties typically results in high or even infinite dimensional problems in terms of the uncertain parameters as well as the optimization variables. We will discuss methods which can be shown to be robust with respect to the number of parameters and are therefore suitable for this setting.

Ort: Seminarraum 019 Arnimallee 3 14195 Berlin

14.12.2022 | 16:30

Milena Hering (Edingburgh): Embedding of Algebraic Varieties and Toric Vector bundles

Algebraic varieties are geometric objects that can be described as the zero locus of polynomial equations. While the relationship between geometry and algebra is fundamental to algebraic geometry, it still remains quite mysterious. I will explain some aspects that are known about it, as well as some open questions.  And how toric vector bundles enter the equation.

Ort: Seminarraum 019 Arnimallee 3 14195 Berlin

30.11.2022 | 16:30

Arend Bayer (Edingburgh): Derived Categories, Wall-crossing and Birational Geometry

Birational geometry studies maps between algebraic varieties defined by rational functions. Recently, derived categories, stability conditions and wall-crossing have led to an entirely new approach to fundamental open questions in birational geometry. I will survey these developments, with an emphasis on Hyperkaehler varieties and cubic fourfolds.

Ort: Seminarraum 019 Arnimallee 3 14195 Berlin

23.11.2022 | 16:30

Ana Djurdjevac (FU-Berlin Antrittsvorlesung): Randomness and PDEs: Analysis, Numerics and Applications

We will first consider interacting particle systems that provide powerful models that are useful in many application areas such as sociology (agents), molecular dynamics (proteins) etc. The first model that we will define is a non-linear stochastic PDE that provides a faithful representation of the evolution of the empirical density of a given particle system. This model has a direct applications in the opinion dynamics that will be discussed. Furthermore, we will explain difficulties in numerical approximations of these problems. Instead of considering many particles, next we will consider just one Brownian particle, but which is now evolving on a random domain. Using the rough path analysis, we will investigate different scaling regimes of this system. As a natural question in this setting is how to present a Gaussian random fields on a sphere. One way to do this is using the so-called spherical harmonics. We will discuss the advantages of this approach and challenges in its generalizations to an arbitrary manifold.

Ort: Seminarraum 019 Arnimallee 3 14195 Berlin

09.11.2022 | 16:30

Imre Bárány (Rényi Institute, Budapest): Cells in the box and a hyperplane

It is well known that a line can intersect at most 2 n −1 cells of the  n × n chessboard. What happens in higher dimensions: how many cells of the  d -dimensional [0, n ]^ d box can a hyperplane intersect? We answer this question asymptotically. We also prove the integer analogue of the following fact. If  K,L are convex bodies in  R ^d and  K ⊂ L , then the surface area  K is smaller than that of  L . This is joint work with Péter Frankl.

Ort: Chemistry building Arnimallee 22 14195 Berlin Hörsaal A

30.05.2022 | 16:00 s.t.

János Pach (Rényi Institute, Budapest): Facets of Simplicity

We discuss some notoriously hard combinatorial problems for large classes of graphs and hypergraphs arising in geometric, algebraic, and practical applications. These structures are of bounded complexity: they can be embedded in a bounded-dimensional space, or have small VC-dimension, or a short algebraic description. What are the advantages of low complexity? I will suggest a few possible answers to this question, and illustrate them with classical examples.

Ort: Chemistry building Arnimallee 22 14195 Berlin Hörsaal A

30.05.2022 | 14:15

Anna-Laura Sattelberger (Leipzig): Algebraic and Topological Data Analysis

16:30-16:50 Uhr Lehrprobe zum Thema The Theorem of Carathéodory  (auf Englisch) ca. 17:00-17:40 Uhr Fachvortrag: Algebraic and Topological Data Analysis Algebraic analysis investigates linear differential equations with polynomial coefficients by encoding them as ideals in the Weyl algebra D . In this talk, I present several applications of this theory in the sciences. Among others, I present how maximum likelihood estimation - a technique from statistics for the inference of data - can be tackled in terms of D -modules. The second part of my talk is about the development of algebraic tools for topological data analysis. This area of research extracts intrinsic information of data with methods from (algebraic) topology. The main tool is persistent homology. While the one-parameter case is fully described by so-called "barcodes" associated to the data, one encounters a lack of a corresponding invariant in the multivariate case. I give insights into an ongoing project with Wojciech Chachólski and René Corbet, in which we construct stable invariants for multipersistence modules.

Ort: Die Veranstaltung wird virtuell via Webex-Meetings stattfinden. https://fu-berlin.webex.com/fu-berlin/j.php?MTID=me8550a880d3f0007933e9fd90ac1e89b Meeting-Kennnummer (Zugriffscode): 188 230 6744 Meeting Passwort: Villa

24.08.2021 | 16:30 s.t. - 17:50

Alessio D’Alì (Osnabrück): Constructing Koszul Gorenstein algebras from Cohen-Macaulay simplicial complexes

14:15-14:35 Uhr Lehrprobe zum Thema The Theorem of Carathéodory  (auf Englisch) ca. 14:45-15:25 Uhr Fachvortrag: Constructing Koszul Gorenstein algebras from Cohen-Macaulay simplicial complexes My main area of interest is combinatorial commutative algebra, a topic that sits at the crossroads between algebra, combinatorics and topology. The main aim of this talk is to discuss a joint project with Lorenzo Venturello (KTH Stockholm) relating Koszul Gorenstein algebras and Cohen-Macaulay simplicial complexes. Koszul algebras are quadratic algebras satisfying desirable homological properties and arising naturally in many geometric and combinatorial contexts: for instance, the coordinate rings of Veronese, Segre and Grassmannian varieties (in their natural embeddings) are all Koszul, and so is the Stanley-Reisner ring of any flag simplicial complex. However, the Koszul property is hard to control and to check in general, and many conjectures about the general behaviour of Koszul algebras are currently open. Starting from a flag pure simplicial complex Δ, we propose a construction of a (non-monomial) Gorenstein ring R_Δ which is Koszul if and only if Δ is Cohen-Macaulay, thus providing a bridge between these two worlds. On a more combinatorial level, the very same correspondence also yields that R_Δ has a Gröbner basis of quadrics if and only if Δ is shellable. As an application, we provide counterexamples to an algebraic generalization of a conjecture by Charney and Davis about flag homology spheres.

Ort: Die Veranstaltung wird virtuell via Webex-Meetings stattfinden. https://fu-berlin.webex.com/fu-berlin/j.php?MTID=m5497d01eaea770dee34bfc7e2751dddb Meeting-Kennnummer (Zugriffscode): 188 591 7254 Meeting Passwort: Villa

24.08.2021 | 14:15 s.t. - 15:35

Marvin Anas Hahn (Leipzig): Die tropische Geometrie von monotonen Hurwitz-Zahlen

12:00-12:20 Uhr Lehrprobe zum Thema The Theorem of Carathéodory  (auf Englisch) ca. 12:30-13:10 Uhr Fachvortrag: Die tropische Geometrie von monotonen Hurwitz Zahlen Hurwitz-Zahlen sind wichtige enumerative Invarianten in der algebraischen Geometrie. Sie zählen verzweigte Abbildungen zwischen Riemannschen Flächen. Äquivalent enumerieren sie Faktorisierungen in der symmetrischen Gruppe. Hurwitz-Zahlen wurden in den 1890er Jahren von Adolf Hurwitz eingeführt und wurden in den 1990er Jahren durch enge Verbindungen zur sogenannten Gromov-Witten-Theorie zu zentralen Objekten der enumerativen algebraischen Geometrie. Dieses Zusammenspiel zwischen Hurwitz und Gromov–Witten-Theorie ist ein aktives Forschungsfeld und brachte u.a. die gefeierte  ELSV–Formel  hervor. Im letzten Jahrzehnt sind viele Varianten von Hurwitz-Zahlen eingeführt und untersucht worden. Insbesondere die Frage nach Verbindungen zwischen diesen Varianten von Hurwitz Zahlen und Gromov–Witten-Theorie ist von großem Interesse. Sogenannte  monotone Hurwitz-Zahlen , die der Theorie der Zufallsmatrizen entstammen, gehören zu den meistuntersuchten Varianten von Hurwitz-Zahlen. Dieser Vortrag ist ein Fortschrittsbericht unseres größeren Programms, in welchem wir die Verbindungen zwischen monotonen Hurwitz-Zahlen und Gromov-Witten-Theorie durch kombinatorische Methoden der tropischen Geometrie untersuchen und dessen langfristiges Ziel ein Beweis der noch offenen Vermutung einer ELSV – Typ Formel für doppelte monotone Hurwitz-Zahlen ist. Der Vortrag basiert zum Teil auf gemeinsamen Arbeiten mit Reinier Kramer und Danilo Lewanski.

Ort: Die Veranstaltung wird virtuell via Webex-Meetings stattfinden. https://fu-berlin.webex.com/fu-berlin/j.php?MTID=m315d6e41aa6c6bf480ab721b7c39ffb2 Meeting-Kennnummer (Zugriffscode): 188 485 7830 Meeting Passwort: Villa

24.08.2021 | 12:00 s.t. - 13:20

Giulia Codenotti (Frankfurt): The flatness constant and its relatives

16:30-16:50 Uhr Lehrprobe zum Thema The Theorem of Carathéodory  (auf Englisch) ca. 17:00-17:40 Uhr Fachvortrag: The flatness constant and its relatives The lattice width of a convex body is a parameter measuring how thin the body is in lattice directions. In each fixed dimension, the flatness constant is the supremum of the widths of a special class of convex bodies, those which are hollow. In this talk we will explore certain generalizations and restrictions of the flatness constant obtained by changing the class of convex bodies whose width we study: hollow lattice polytopes, for example, or those convex bodies which do not contain a certain polytope. We will see how these modified flatness constants have connections and motivations in different fields, like integer linear programming, lattice polytopes, and symplectic geometry.

Ort: Die Veranstaltung wird virtuell via Webex-Meetings stattfinden. https://fu-berlin.webex.com/fu-berlin/j.php?MTID=m76775164d53f79a640ff25283a4e97d8 Meeting-Kennnummer (Zugriffscode): 188 706 6091 Meeting Passwort: Villa

11.08.2021 | 16:30 s.t. - 17:50

Jorge Olarte (TU Berlin): Valuated matroids and regions of the tropical Grassmannian

14:15-14:35 Uhr Lehrprobe zum Thema The Theorem of Carathéodory  (auf Englisch) ca. 14:45-15:25 Uhr Fachvortrag: Valuated matroids and regions of the tropical Grassmannian A valuated matroid is essentially a matroid polytope regularly subdivided into matroid polytopes. In tropical geometry, valuated matroids take the role of linear spaces, hence their importance. The tropical Grassmannian is the space of valuated matroids which are realizable, that is, they arise as tropicalizations of a classical linear space. Certain regions in the tropical Grassmannian have deep connections to certain types of matroid, such as positroids and transversal matroids. In this talk we will discuss three regions of interest: the positive part, the image of the tropical Stiefel map and the tropical symplectic Grassmannian.

Ort: Die Veranstaltung wird virtuell via Webex-Meetings stattfinden. https://fu-berlin.webex.com/fu-berlin/j.php?MTID=m1ce0f308fc47db6efd6567ee88a715ca Meeting-Kennnummer (Zugriffscode): 188 591 5127 Meeting Passwort: Villa

11.08.2021 | 14:15 s.t. - 15:35

Marta Panizzut (TU Berlin): Polytopes meet polynomials: realization spaces and tropical varieties

12:00-12:20 Uhr Lehrprobe zum Thema The Theorem of Carathéodory  (auf Englisch) ca. 12:30-13:10 Uhr Fachvortrag: Polytopes meet polynomials: realization spaces and tropical varieties Many exciting research topics lie at the interface between discrete, tropical, and algebraic geometry. In this talk I will present examples of such topics based on some of my research projects. The first part introduces the study of algebraic degrees of realizations of polytopes satisfying some geometric constraints. The second one focuses on the analysis through the lens of tropical geometry of models of cubic surfaces, matroid polytopes, and their subdivisions. Throughout the talk, I will highlight how discrete and algebraic methods fruitfully interact and provide new insights and computational tools.

Ort: Die Veranstaltung wird virtuell via Webex-Meetings stattfinden. https://fu-berlin.webex.com/fu-berlin/j.php?MTID=me9ce01b68aa839f549f35db21c67be58, Meeting-Kennnummer (Zugriffscode): 188 466 2732. Meeting Passwort: Villa

11.08.2021 | 12:00 s.t. - 13:20

Mathe. Kolloquium: Brigitte Lutz-Westphal (FU-Berlin): Die „Sachanalyse“ als Basis der Unterrichtsvorbereitung

Welche Art fachlicher Betrachtungen braucht man, um guten Mathematikunterricht gestalten zu können?  Hat das etwas mit den Inhalten des Lehramtsstudiums zu tun?

Ort: Meeting-Link: https://fu-berlin.webex.com/fu-berlin/j.php?MTID=mf61fc1d8b90f1a30f6cf8efe3ec10e9c Meeting-Kennnummer: 121 770 8911 Passwort: Zweiundvierzig

16.06.2021 | 16:15

Mathe. Kolloquium: Jörg Fandrich (FU-Berlin): Was ist guter Unterricht?

Ein Blick aus der Physik mit Seitenblicken auf die Mathematik Irgendetwas läuft schief im deutschen Physikunterricht. Gäbe es einen Preis für das unbeliebteste Schulfach, so wäre das Fach Physik ein heißer Anwärter auf den Sieg. Für die anderen „harten MINT-Fächer“ (Chemie, Mathematik) sieht es nicht viel besser aus. Die Unis spüren das, es fehlt an Nachwuchs. Doch das ist nicht das einzige Problem - auch die Abbrecherquoten in diesen Fächern sind überdurchschnittlich hoch. Offen­sichtlich kommt nicht nur der schulische MINT-Unterricht bei den Lernenden nicht besonders gut an, sondern auch der universitäre. Das muss sich ändern! Wir brauchen Unterricht, der den Lernenden Spaß macht, der ihr Interesse weckt und sie zur aktiven Auseinandersetzung mit den Inhalten anregt. Von zentraler Bedeutung beim Wandlungsprozess des MINT-Unterrichts ist hierbei der „Top-down-Ansatz“: Wir Universitäten dürfen nicht nur fordern, dass sich der Unterricht in den Schulen verbessert, sondern wir müssen guten Unterricht selbst aktiv vorleben! Da Unterricht jedoch immer auch abhängig von der Lerngruppe sowie den allgemeinen Rahmenbedingungen ist, lässt sich „guter Unterricht per se“ nicht definieren. Es lassen sich jedoch Merkmale benennen, die erfolgreichen Unterricht begünstigen und die von der Mehrzahl der Lernenden als positiv wahrgenommen werden. Der Vortrag wirft einen frischen Blick auf MINT-Unterricht und gibt Anregungen, wie guter Unterricht aussehen kann. Er stellt Leitlinien vor, die bei der Planung von „interessantem“ Unterricht helfen können. Nicht alles, was hier vorgestellt wird, ist neu – doch oft gehen gute Ideen im Alltagsstress unter, obwohl die Umsetzung gar nicht so viel Mühe machen würde. Alle vorgestellten Ansätze, Konzepte und Methoden werden durch Beispiele illustriert.

Ort: Meeting-Link: https://fu-berlin.webex.com/fu-berlin/j.php?MTID=mf61fc1d8b90f1a30f6cf8efe3ec10e9c Meeting-Kennnummer: 121 770 8911 Passwort: Zweiundvierzig

02.06.2021 | 16:15

Mathe. Kolloquium: Prof. Dr. Petra Skiebe-Corette (FU-Berlin): NatLab - ein Schülerlabor in der Fachausbildung

Prof. Dr. Petra Skiebe-Corrette leitet seit 2004 das Schülerlabor  NatLab in der Biologie. Sie erklärt in einem Impulsvortrag, was ein  Schülerlabor ist und welche Ziele NatLab verfolgt.  Im Anschluss können wir mit ihr diskutieren, warum hier  Lehrveranstaltungen in der Fachausbildung angeboten werden und nicht in der Didaktik, und wir erfahren, warum NatLab  mittlerweile mit zwei SFBs kooperiert. https://www.bcp.fu-berlin.de/natlab

Ort: Webex: https://fu-berlin.webex.com/fu-berlin/j.php?MTID=m73a3301d67540de4e73e2859ca7f18bc Meeting-Kennnummer: 121 451 9876 Passwort: Zweiundvierzig

28.04.2021 | 16:15