Summer 2015 Seminar Schedule

Location: 4421 (CS Thesis Room)
Time:  Tuesdays 4pm - 5pm (unless otherwise noted)
Organizer: Delaram Kahrobaei  (faculty adviser)
Student organizers: Jonathan Gryak


 Date  Topic
Kelsey Horan (CS, GC)
Title: Applications of Group Theory to Data Science
Bianca Sosnovski (Math and CS, QCC)
Title: A new platform semigroup for the Tillich-Zemor hash function scheme
Abstract: In this talk, I will present a new candidate as platform for the Tillich-Zemor general scheme of hash functions (1994) with the goal of improving the security of the scheme. The underlying platform is the noncommutative (semi)group of linear functions under composition. This a joint work with V. Shpilrain.
Jordi Delgado (Math, UPC Barcelona)
Title: On Algorithmic Recognition of Certain Classes of Groups
I will introduce the standard framework for recognition problems in group theory, then move to a more specific setting: namely group extensions by infinite cyclic groups.

For this family, I will review some folklore facts and previous partial results which we use to provide an argument that implies the undecidability of the BNS invariant (a nice geometrical invariant on which I would also like to say a few words).

This is joint work with B. Cavallo, D. Kahrobaei, and E. Ventura.
07/21 Armin Weiß (Universität Stuttgart, Germany)
Title: Amenability of Schreier Graphs and Strongly Generic Algorithms for the Conjugacy Problem, Part 1 - Amenability of Schreier graphs of HNN extensions and Amalgamated Products

In various occasions the conjugacy problem in finitely generated amalgamated products and HNN extensions can be decided efficiently for elements which cannot be conjugated into the base groups. This observation asks for a bound on how many such elements there are. Such bounds can be derived using the theory of amenable graphs:

We examine Schreier graphs of amalgamated products and HNN extensions. For an amalgamated product $G =H\star_AK $ with $[H:A]\geq [K:A]\geq 2$, the Schreier graph with respect to $H$ or $K$ turns out to be non-amenable if and only if $[H:A]\geq 3$. Moreover, for an HNN extension of the form $G = \left< \, H,b \;\middle| \; bab^{-1}=\phi(a),
a \in A \, \right>$, we show that the Schreier graph of $G$ with respect to the subgroup $H$ is non-amenable if and only if $A\neq H \neq \phi(A)$.

As application of these characterizations we show that under certain conditions the conjugacy problem in fundamental groups of finite graphs of groups with free abelian vertex groups can be solved in polynomial time on a strongly generic set. Furthermore, the conjugacy problem in groups with more than one end can be solved with a strongly generic algorithm which has essentially the same time complexity as the word
problem. These are rather striking results as the word problem might be easy, but the conjugacy problem might be even undecidable. Finally, our results yield another proof that the set where the conjugacy problem of the Baumslag group $\mathrm{\bf{G}}_{1,2}$ is decidable in polynomial time is also strongly generic.

The first talk will cover the characterization theorem of non-amenable Schreier graphs.
07/28 Armin Weiß (Universität Stuttgart, Germany)
Title: Amenability of Schreier Graphs and Strongly Generic Algorithms for the Conjugacy Problem, Part 2 - Strongly Generic Algorithms for the Conjugacy Problem

Abstract: See abstract for 07/21

In the second talk we will focus on its applications to the conjugacy problem.
Serena Yuan (Math, NYU)
Title: A New Cryptographic Hash Function Involving Modified Cayley Graphs
Alexander Wood (CS, GC)
Title: Generic Case Complexity in Group Theory