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Vorträge 2021

Lehrprobe und Fachvortrag Anna-Laura Sattelberger: 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 zum Thema: 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

Lehrprobe und Fachvortrag 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. - 15:25

Lehrprobe und Fachvortrag 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. - 13:10

Lehrprobe und Fachvortrag Giulia Codenotti: 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. - 17:40

Lehrprobe und Fachvortrag Jorge Olarte: 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 zum Thema: 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. - 15:25

Lehrprobe und Fachvortrag Marta Panizzut 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 zum Thema: 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. - 13:10