### Upcoming Talks

#### Vorstellungsvortrag im Rahmen des Habilitationsverfahrens

**Title:**Topological aspects of random discrete objects

**Speaker:**Philipp Sprüssel (Institut für Diskrete Mathematik)

**Date:**05.12.2022, 16:00 - 17:00

**Room:**Seminar room AE06, Steyrergasse 30/EG

**Abstract:**

Since the seminal work of Erdos and Renyi on random graphs in 1959, random discrete objects have been a source of fascinating results. The most exciting results concern thresholds at which major changes in behaviour of the random model occur, such as planarity, connectedness, or the size of the largest component.

In this talk, we focus on topological aspects and properties of random models. How do topological constraints, such as being embeddable on a specific surface, influence the structure of random graphs? What behaviour do higher-dimensional random models, such as random simplicial complexes, exhibit with respect to fundamental properties such as connectedness?

#### Combinatorics Seminar (CHANGED DATE)

**Title:**Degree sequences of random uniform hypergraphs

**Speaker:**Tamás Makai (LMU München)

**Date:**Friday 2nd December 12:15

**Room:**AE06 Steyrergasse 30, EG / Webex

**Abstract:**

Consider the probability that a random graph selected uniformly from the set of $r$-uniform hypergraphs with $n$ vertices and $m$ edges, has a given degree sequence. Previously the value of this probability has been investigated by Kamčev, Liebenau and Wormald, where they examined degree sequences from very sparse to moderately dense hypergraphs when $r=o\left(n^{1/4}\right)$ and the variation of the degrees is small, but exceeds the typical degree variation in random hypergraphs.

We extend their results, by establishing this result for dense hypergraphs, which hold for any value of $r$ and allow for a greater variation on the degrees.

This is joint work with Catherine Greenhill, Mikhail Isaev and Brendan McKay.

Meeting link:

\[

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Meeting number: 2730 500 3129

Password: vQydpg372D4

#### Colloquium Discrete Mathematics and Probability

**Title:**High and low temperature limits of random matrices and random partitions

**Speaker:**Cesar Cuenca (Harvard University)

**Date:**30.11.2022, 13:00 Uhr

**Room:**Hörsaal BE01, Steyrergasse 30

**Abstract:**

In this presentation, I discuss the global-scale behavior of random matrix eigenvalues and random partitions which depend on the "inverse temperature" parameter beta. I will focus on explaining the method of moments, powered by algebraic and combinatorial tools, to prove the Law of Large Numbers in the regimes of high and low temperatures, i.e. when beta converges to zero and to infinite, respectively. This talk is based on joint works with Benaych-Georges--Gorin, and with Dolega--Moll.

#### Colloquium Discrete Mathematics and Probability

**Title:**Decomposition problems

**Speaker:**Matija Bucic (Princeton University)

**Date:**30.11.2022, 08:45 Uhr

**Room:**Hörsaal BE01, Steyrergasse 30

**Abstract:**

One of the most classical classes of combinatorial questions consists of decomposition problems, the scientific study of which dates back to the work of Euler in the 18th century. In a very broad sense, a decomposition problem asks whether we can decompose a large object into isomorphic (in an appropriate sense) copies of a smaller object. In this talk I will give a brief introduction to decomposition problems and discuss two classical conjectures in the area, namely Rota's basis conjecture and the Erdos-Gallai conjecture.

#### Colloquium Discrete Mathematics and Probability

**Title:**Combinatorial generation --- challenges and recent advances

**Speaker:**Torsten Mütze (Warwick University)

**Date:**29.11.2022, 09:00 Uhr

**Room:**Hörsaal BE01, Steyrergasse 30

**Abstract:**

In Mathematics and Computer Science we frequently encounter different kinds of combinatorial objects, such as permutations, partitions, binary trees, spanning trees of a graph, linear extensions of a poset etc.

There are a number of closely related fundamental tasks that we want to perform with these objects, such as counting (how many objects are there?), random sampling (pick one of the objects uniformly at random), or optimization (find the best object subject to some objective function). In this talk the focus is on combinatorial generation, where the goal is to efficiently list all the objects from the class, each object exactly once.

In this talk I give an overview over some long-standing problems and recent advances in this area, highlighting newly discovered connections to related mathematical problems in graph theory, algebra, order theory, geometry, and algorithms.

#### Colloquium Discrete Mathematics and Probability

**Title:**Combinatorial and probabilistic analysis of large discrete objects

**Speaker:**Michael Wallner (TU Wien)

**Date:**28.11.2022, 13:00 Uhr

**Room:**Hörsaal BE01, Steyrergasse 30

**Abstract:**

My research focuses mainly on the large-scale behavior of directed acyclic graphs, Young tableaux, and lattice paths. I am primarily interested in universal phenomena such as specific terms in the asymptotic growth (e.g., critical exponents $n^{-3/2}$ or stretched exponentials $\mu^{n^{1/3}}$) and limit laws (e.g., normal or Mittag-Leffler distributions). I will present several general tools I have developed to prove such phenomena extending methods from Analytic Combinatorics and Probability Theory. They allowed me to solve open problems on, e.g., the number of minimal automata and phylogenetic networks, or the limiting behavior of the sum-of-digits function, periodic Young tableaux and Pólya urns.