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Title: Singularities and the monodromy conjecture Speaker: Nero Budur, Katholieke Universiteit Leuven, Belgium Time: 11:30am–12:30pm Place: CMC 204 Sponsor: M. Elhamdadi and M. Saito
The monodromy conjecture of Igusa, Denef, and Loeser claims to relate different aspects of solution sets of multivariate polynomials. It would provide a link between arithmetic, complex analysis, topology, and differential equations. The conjecture is wide open in general. We survey some results related and motivated by the monodromy conjecture.
Title: On the existence of solutions to a class of coagulation-fragmentation equations Speaker: Clovis Oukouomi Noutchie, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa Time: 3:00pm–4:00pm Place: CMC 108 Sponsor: W. Ma
Coagulation-fragmentation equations are a class of mathematical models used to describe the evolution of a population of particles or clusters undergoing aggregation and breakup processes. These equations are widely used in fields such as colloid science, polymer physics, atmospheric science, and astrophysics, among others. In this presentation, we will make use of the theory of perturbation of operators to establish the existence of solutions of a class of coagulation-fragmentation equations. In particular, conservativeness arguments will be explored.
Title: Optimization of Algorithms for Isogeny Evaluation in Cryptography Speaker: Jason LeGrow, Virginia Polytechnic Institute and State University Time: 3:00pm–4:00pm Place: CMC 108 Sponsor: J. Biasse
Key establishment is a cryptographic primitive which allows two users to establish a shared secret piece of information (the “key”) in the presence of eavesdroppers. I will discuss key establishment from a classical perspective, discuss the need for post-quantum key establishment, and introduce the isogeny-based key establishment protocols SIDH (which was recently broken) and CSIDH from a high level. Then I will discuss algorithmic aspects of these protocols, and develop a systematic approach to optimizing algorithms to compute the isogenies required in SIDH and CSIDH. Finally, I’ll present the results of these optimizations and discuss some open questions in the area.
Title: Group-based Cryptography in the Quantum Era Speaker: Delaram Kahrobaei Time: 2:00pm–2:50pm Place: CMC 108 Sponsor: J. Biasse
In this talk we present an overview of the current state-of-the-art in post-quantum group-based cryptography. We describe several families of groups that have been proposed as platforms, with special emphasis in polycyclic groups and graph groups, dealing in particular with their algorithmic properties and cryptographic applications. In the end we discuss several open problems in this direction.
For reference, see this AMS article.
Title: Generalized Cyclotomic Mappings: Switching Between Polynomial, Cyclotomic, and Wreath Product Form Speaker: Qiang Wang, Carleton University Time: 3:00pm–4:00pm Place: CMC 108 Sponsor: X. Hou
In this talk, I focus on so-called index \(d\) generalized cyclotomic mappings of a finite field \(\mathbb{F}_q\), which are functions \(\mathbb{F}_q\to\mathbb{F}_q\) that agree with a suitable monomial function \(x\to ax^r\) on each coset of the index \(d\) subgroup of \(\mathbb{F}_q^*\). We discuss two important rewriting procedures in the context of generalized cyclotomic mappings and present applications in the study of cycle structures of cyclotomatic permutation polynomials.