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Title: Contracting Self-similar Groups in Group-Based Cryptography Speaker: Arsalan Akram Malik Time: 2:00pm–3:00pm Place: CMC 118
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.
No seminar this week
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
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.
Title: Isomorphic Derived Graphs Speaker: Greg McColm Time: 2:00pm–3:00pm Place: CMC 118
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.
Title: An Introduction to Sub-Riemannian geometry Speaker: Thomas Bieske Time: 2:00pm–3:00pm Place: CMC 118
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.
Title: Diagonal Actions of Groups Acting on Rooted Trees Speaker: Dima Savchuk Time: 2:00pm–3:00pm Place: CMC 118
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.
Seminar is cancelled due to Hurricane Milton.
Title: Instanton Floer homology and Dehn surgery, Part II Speaker: Zhenkun Li Time: 2:00pm–3:00pm Place: CMC 118
Title: Instanton Floer homology and Dehn surgery Speaker: Zhenkun Li Time: 2:00pm–3:00pm Place: CMC 118
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.
Title: Introduction to 4-manifolds and symplectic topology, Part II Speaker: Sümeyra Sakallı Time: 2:00pm—3:00pm Place: CMC 118
Title: Introduction to 4-manifolds and symplectic topology, Part I Speaker: Sümeyra Sakallı Time: 2:00pm—3:00pm Place: CMC 118
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.