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for events the day of Thursday, November 7, 2019.

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Thursday, November 7, 2019

11:00 am in 241 Altgeld Hall,Thursday, November 7, 2019

An even parity instance of the Goldfeld conjecture

Ashay Burungale (Caltech)

Abstract: We show that the even parity case of the Goldfeld conjecture holds for the congruent number elliptic curve. We plan to outline setup and strategy (joint with Ye Tian).

12:00 pm in 243 Altgeld Hall,Thursday, November 7, 2019

Trees, dendrites, and the Cannon-Thurston map

Elizabeth Field (Illinois Math)

Abstract: When $1 \to H \to G \to Q \to 1$ is a short exact sequence of three word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from $H$ to $G$ extends continuously to a map between the Gromov boundaries of $H$ and $G$. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point $z$ in the Gromov boundary of $Q$ an ``ending lamination'' on $H$ which consists of pairs of distinct points in the boundary of $H$. We prove that for each such $z$, the quotient of the Gromov boundary of $H$ by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who prove that in the case where $H$ is a free group and $Q$ is a convex cocompact purely atoroidal subgroup of Out($F_n$), one can identify the resultant quotient space with a certain $\mathbb R$-tree in the boundary of Culler-Vogtmann's Outer space.

1:00 pm in 347 Altgeld Hall,Thursday, November 7, 2019

Theoretical and Empirical Advances in Large-Scale Species Tree Estimation

Tandy Warnow (Computer Science, University of Illinois)

Abstract: The estimation of the "Tree of Life" -- a phylogeny encompassing all life on earth--is one of the big Scientific Grand Challenges. Maximum likelihood (ML) is a standard approach for phylogeny estimation, but estimating ML trees for large heterogeneous datasets is challenging for two reasons: (1) ML tree estimation is NP-hard (and the best current heuristics can use hundreds of CPU years on relatively small datasets, just to find local optima), and (2) the statistical models used in ML tree estimation methods are much too simple, failing to acknowledge heterogeneity across genomes or across the Tree of Life. These two "big data" issues -- dataset size and heterogeneity -- impact the accuracy of phylogenetic methods and have consequences for downstream analyses. In this talk, I will describe a new graph-theoretic "divide-and-conquer" approach to phylogeny estimation that addresses both types of heterogeneity. Our protocol operates as follows: (1) we divide the set of species into disjoint subsets, (2) we construct trees on the subsets (using appropriate statistical methods), and (3) we combine the trees together using auxiliary information, such as a matrix of pairwise distances. I will present three such strategies (all published in the last year) that operate in this fashion, and that improve the theoretical and empirical performance of phylogeny estimation methods. One of the main applications of this work is species tree estimation from multi-locus data sets when gene trees can differ from the species tree due to incomplete lineage sorting. This talk is largely based on joint work with my PhD students, Erin Molloy and Vladimir Smirnov (Illinois).

2:00 pm in 243 Altgeld Hall,Thursday, November 7, 2019

Noncommutative strong maximals and almost uniform convergence in several directions

Jose Conde Alonso (Universidad Autónoma de Madrid)

Abstract: We revisit Miguel de Guzmán's proof of the strong maximal theorem and we give an argument than can be generalized to noncommutative probability spaces. As is usual in this context, one difficulty that we must face is the definition of the operator itself: in principle, one cannot make sense of the supremum of a sequence of positive operators, something that can be done pointwise in the case of functions. We shall also discuss applications to almost everywhere convergence of sequences of averaging operators in von Neumann algebras.

4:00 pm in 245 Altgeld Hall,Thursday, November 7, 2019

A survey of Sperner theory

Richard Stanley   [email] (MIT and University of Miami)

Abstract: Let $X$ be a collection of subsets of an $n$-element set $S$ such that no element of $X$ is a subset of another. In 1927 Emanuel Sperner showed that the number of elements of $X$ is maximized by taking $X$ to consist of all subsets of $S$ with $\lfloor n/2\rfloor$ elements. This result started the subject of \emph{Sperner theory}, which is concerned with the largest subset $A$ of a finite partially ordered set $P$ that forms an \emph{antichain}, that is, no two elements of $A$ are comparable in $P$. We will give a survey of Sperner theory, focusing on some connections with linear algebra and algebraic geometry.