Interactive visualization of 2-D persistence modules

Michael Lesnick
Columbia University
November 7, 2015
In topological data analysis, we often study data by associating to the data a filtered topological space, whose structure we can then examine using persistent homology. However, in many settings, a single filtered space is not a rich enough invariant to encode the interesting structure of our data. This motivates the study of multidimensional persistence, which associates to the data a topological space simultaneously equipped with two or more filtrations.

Critical mechanical structures: topology and entropy

Xiaoming Mao
University of Michigan
November 7, 2015
Critical mechanical structures are structures at the verge of mechanical instability. These structures are characterized by their floppy modes, which are deformations costing little energy. On the one hand, numerous interesting phenomena in soft matter are governed by the physics of critical mechanical structures, because they capture the critical state between solid and liquid.

Topological and combinatorial methods in Theoretical Distributed Computing

Dmitry Feichtner-Kozlov
Institute for Algebra, Geometry, Topology, and their Applications, University of Bremen
November 7, 2015
In the first half of the talk I will give a very compressed introduction into parts of Theoretical Distributed Computing from the point of view of mathematician. I will describe how to construct simplicial models whose combinatorics contains important information about computability and complexity of standard distributed tasks. In the second part, I will outline our recent progress on estimating the complexity of the so-called Weak Symmetry Breaking task, where we are able to derive some quite surprising results.

Hard Lefschetz theorem and Hodge-Riemann relations for combinatorial geometries

June Huh
Princeton University; Veblen Fellow, School of Mathematics
November 9, 2015
A conjecture of Read predicts that the coefficients of the chromatic polynomial of a graph form a log-concave sequence for any graph. A related conjecture of Welsh predicts that the number of linearly independent subsets of varying sizes form a log-concave sequence for any configuration of vectors in a vector space. In this talk, I will argue that two main results of Hodge theory, the Hard Lefschetz theorem and the Hodge-Riemann relations, continue to hold in a realm that goes beyond that of Kahler geometry.

Pseudo-Anosov constructions and Penner's conjecture

Balazs Strenner
Member, School of Mathematics
November 12, 2015
In this first talk, we give an introduction to Penner’s construction of pseudo-Anosov mapping classes. Penner conjectured that all pseudo-Anosov maps arise from this construction up to finite power. We give an elementary proof (joint with Hyunshik Shin) that this conjecture is false. The main idea is to consider the Galois conjugates of pseudo-Anosov stretch factors.

Algebraic degrees and Galois conjugates of pseudo-Anosov stretch factors

Balazs Strenner
Member, School of Mathematics
November 12, 2015

We consider questions that arise naturally from the subject of the first talk. The have two main results: 1. In genus $g$, the algebraic degrees of pseudo-Anosov stretch factors include all even numbers between $2$ and $6g - 6$; 2. The Galois conjugates of stretch factors arising from Penner’s construction are dense in the complex plane. The techniques may be characterized as asymptotic linear algebra.

THE ART OF DINING: Downton Abbey

Francine Segan
Food Historian
November 13, 2015
Join returning speaker Francine Segan and discover the elaborate etiquette, enchanting entertainments, and dishes Mrs. Patmore would have been proud to send to the table. Vivid descriptions of Lord Grantham-esque dinner parties, cotillions, and elegant picnics will transport you back in time, while you learn all the popular toasts of the era and when it's proper to remove your gloves or tip your hat. Segan will invite you to guess the uses for some now-obsolete objects that Mr. Carson would be shocked to
find you couldn't use properly.

The $\mathrm{SL}(2,\mathbb R)$ action on moduli space

Alex Eskin
University of Chicago; Member, School of Mathematics
November 16, 2015
There is a natural action of the group $\mathrm{SL}(2,\mathbb R)$ of $2 \times 2$ matrices on the unit tangent bundle of the moduli space of compact Riemann surfaces. This action can be visualized using flat geometry models, which allows one to make an analogy with homogeneous spaces, such as the space of lattices in $\mathbb R^n$. I will make the basic definitions, and mention some recent developments. This talk will be even more introductory than usual.