Research

Geometry/Topology
(Leaders: ,
, and )

Monday, December 2, 2024

Title: Contracting Self-similar Groups in Group-Based Cryptography
Speaker: Arsalan Akram Malik
Time: 2:00pm–3:00pm
Place: CMC 118

Abstract

We propose self-similar contracting groups as a platform for cryptographic schemes based on simultaneous conjugacy search problem (SCSP). The class of these groups contains extraordinary examples like Grigorchuk group, which is known to be non-linear, thus making some of existing attacks against SCSP inapplicable. The groups in this class admit a natural normal form based on the notion of a nucleus portrait, that plays a key role in our approach. While for some groups in the class the conjugacy search problem has been studied, there are many groups for which no algorithms solving it are known. Moreover, there are some self-similar groups with undecidable conjugacy problem. We discuss benefits and drawbacks of using these groups in group-based cryptography and provide computational analysis of variants of the length-based attack on SCSP for some groups in the class, including Grigorchuk group, Basilica group, and others.

Monday, November 25, 2024

No seminar this week

Monday, November 18, 2024

Title: The central limit theorem for random walks on the lamplighter groups over acylindrically hyperbolic groups
Speaker: Maksym Chaudkhari
Time: 2:00pm–3:00pm
Place: CMC 118

Abstract

Lamplighter groups often exhibit counter-intuitive geometric properties and are known as an important source of counterexamples in geometric group theory. In this talk, we will focus on the asymptotic properties of random walks over lamplighter groups. In particular, we will discuss a version of the central limit theorem for random walks on lamplighters with acylindrically hyperbolic base group. The talk is based on a joint work with Kunal Chawla, Christian Gorski, and Eduardo Silva.

Monday, November 4, 2024

Title: Isomorphic Derived Graphs
Speaker: Greg McColm
Time: 2:00pm–3:00pm
Place: CMC 118

Abstract

Graphs of high symmetry — i.e., graphs with relatively large automorphism groups — can be dealt with via “voltage graphs”: given a graph \(\Gamma\) and an appropriate group of automorphisms \(G\), a voltage graph for \(\Gamma\) is the quotient graph \(\Gamma/G\) with edges labeled with elements of \(G\) serving as instructions for “deriving” \(\Gamma\) from \(\Gamma/G\). Given two appropriately labeled voltage graphs we show a method for determining whether their respective derived graphs are isomorphic.

This is a report on joint work with Nataša Jonoska & Milé Krajčevski.

Monday, October 28, 2024

Title: An Introduction to Sub-Riemannian geometry
Speaker: Thomas Bieske
Time: 2:00pm–3:00pm
Place: CMC 118

Abstract

Motivated by real-world problems, we will begin to explore sub-Riemannian manifolds by highlighting their geometric and topological properties. We will then delve into topics of current research and some open problems.

Monday, October 21, 2024

Title: Diagonal Actions of Groups Acting on Rooted Trees
Speaker: Dima Savchuk
Time: 2:00pm–3:00pm
Place: CMC 118

Abstract

For a group \(G\) acting on a regular rooted \(d\)-ary tree \(T_d\) and on its boundary \(\partial T_d\) we consider the diagonal actions of \(G\) on the powers of \(T_d\) and \(\partial T_d\). For the action of the full group \(\mathrm{Aut}\left(T_d\right)\) of automorphisms of \(T_d\) we describe the ergodic decomposition of its action on \(\left(\partial T_d\right)^n\) for all \(n\geq 1\). To achieve it we analyze the orbits of \(n\)-tuples of elements of vertices of any fixed finite level of \(T_d\). For a subgroup \(G\) of \(\mathrm{Aut}\left(T_d\right)\) the corresponding orbits may be smaller, but sometimes they coincide with the orbits of the full group of automorphisms for all levels. In this case we say that the action of \(G\) on \(\mathrm{Aut}\left(T_d\right)\) is maximally tree \(n\)-transitive. For example, maximal tree \(1\)-transitivity is equivalent to level transitivity of the action of \(G\) on \(T_d\). It follows from the results of \([1,2]\) that Grigorchuk group and Basilica group act maximally tree \(2\)-transitively on \(\partial T_2\). We show that the action of Grigorchuk group on \(\partial T_2\) is, in fact, maximally tree \(4\)-transitive but not maximally \(5\)-transitive. The talk is based on a joint work with Rostislav Grigorchuk and Zoran Šunić.

Monday, October 7, 2024

Seminar is cancelled due to Hurricane Milton.

Monday, September 30, 2024

Title: Instanton Floer homology and Dehn surgery, Part II
Speaker: Zhenkun Li
Time: 2:00pm–3:00pm
Place: CMC 118

Monday, September 23, 2024

Title: Instanton Floer homology and Dehn surgery
Speaker: Zhenkun Li
Time: 2:00pm–3:00pm
Place: CMC 118

Abstract

Dehn surgery is a fundamental tool in dimension three to construct new 3-manifolds out of the old and instanton Floer homology is a powerful tool to study the topology of 3-manifolds. In the first talk, I will present some background knowledge about these objects and in the second talk, I will present some results by my collaborators and I about how to use instanton Floer homology to study Dehn surgeries and related problems in 3-dimensional topology.

Monday, September 16, 2024

Title: Introduction to 4-manifolds and symplectic topology, Part II
Speaker: Sümeyra Sakallı
Time: 2:00pm—3:00pm
Place: CMC 118

Monday, September 9, 2024

Title: Introduction to 4-manifolds and symplectic topology, Part I
Speaker: Sümeyra Sakallı
Time: 2:00pm—3:00pm
Place: CMC 118

Abstract

The topology and geometry of 4-manifolds is one of the main research areas in symplectic and low dimensional topology, and it is also related to topics such as complex surfaces and singularity theory in algebraic geometry. 4-manifolds admitting different structures in different categories are still not completely known and there are many other open problems related to this question. In this sequence of talks I will start with an introduction to 4-manifolds and symplectic topology, then discuss my results, and finish with some interesting open problems.