Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, March 23, 2021.

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Tuesday, March 23, 2021

11:00 am in Zoom (contact Olivia Beckwith at obeck@illinois.edu for details),Tuesday, March 23, 2021

Modular zeros in the character table of the symmetric group

Sarah Peluse (Princeton)

Abstract: In 2017, Miller conjectured, based on computational evidence, that for any fixed prime $p$ the density of entries in the character table of $S_n$ that are divisible by $p$ goes to $1$ as $n$ goes to infinity. Iíll describe a proof of this conjecture, which is joint work with K. Soundararajan. I will also discuss the (still open) problem of determining the asymptotic density of zeros in the character table of $S_n$, where it is not even clear from computational data what one should expect.

2:00 pm in Zoom Meeting (email daesungk@illinois.edu for info),Tuesday, March 23, 2021

Phylogenomics: Inverting Random Trees

Sebastien Roch (University of Wisconsin-Madison)

Abstract: Phylogenomic analysis, in particular the estimation of species phylogenies from genome-scale data, is a common step in modern evolutionary studies. This estimation is complicated by the fact that genes evolve under biological processes that produce discordant trees. Such processes include horizontal gene transfer (HGT), incomplete lineage sorting (ILS), and gene duplication and loss (GDL), all of which can be modeled using specialized random tree distributions. I will survey some recent results regarding the analysis of these probabilistic models. Specifically their identifiability, or "invertibility," will be discussed as well as the asymptotic properties of species tree estimation methods (time permitting). Based on joint works with Max Bacharach, Brandon Legried, Erin Molloy, Elchanan Mossel, Allan Sly, Tandy Warnow, Shuqi Yu.

2:00 pm in Zoom,Tuesday, March 23, 2021

Extremal results for convex geometric hypergraphs

Jacques Verstraete (University of California, San Diego)

Abstract: A convex geometric hypergraph or cgh is a hypergraph whose vertex set comprises the vertices of a convex polygon in the plane. The extremal problem consists in determining, given an $r$-uniform cgh $F$, the maximum number of edges $\mbox{ex}(n,F)$ in an $r$-uniform cgh on $n$ vertices that does not contain $F$. In the case of graphs, this problem has a rich history with applications to a variety of problems in combinatorial geometry. In this talk, we survey some of the known results for convex geometric graphs, and determine tight bounds for the extremal function for most configurations consisting of a pair of triangles, using a variety of different methods. For instance, we show that the maximum number of triangles that can be formed amongst $n \geq 3$ points in the plane so that any two of the triangles share an interior point is $n(n - 1)(n + 1)/24$ if $n$ is odd and $n(n - 2)(n + 2)/24$ if $n$ is even. Our results generalize old work of Kupitz, Perles, Capoyleas-Pach, Brass, Brass-Karolyi-Valtr and recent work of Aronov-Dujmovic-Morin-Ooms-da Silveira, and answer recent questions of Frankl, Holmsen and Kupavskii.

Joint work with Z. Furedi, T. Jiang, A. Kostochka, D. Mubayi and J. O'Neill.

For Zoom information, please contact Sean at SEnglish (at) illinois (dot) edu.