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Title: Adventures in the land of \(\mathrm{Homeo}+(R)\) Speaker: Yash Lodha, University of Hawaii at Manoa Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: D. Savchuk
The groups of homeomorphisms of the real line and the circle exhibit a remarkable subgroup structure, and consequently, a plethora of topological, algebraic and combinatorial phenomena. In this talk I shall provide some historical motivation to study such groups, and describe some concrete examples that illustrate the theory. I will also present some recent advances in the area.
Title: A symmetric symbol for triples of modular forms Speaker: Wissam Ghantous, University of Central Florida Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: J. Biasse
We introduce a new quantity based on the Garrett-Rankin \(L\)-function and show that it satisfies symmetry relations when permuting the three input modular forms. We also provide computational evidence confirming that it is indeed cyclic when the modular forms have even weights, and provide counterexamples in the case containing odd weights. To do so, we develop algorithms to compute ordinary projections of nearly overconvergent modular forms as well as certain projections over spaces of non-zero slope. Finally, a curious consequence of our work is an efficient method to calculate certain Poincaré pairings in higher weight.
Title: Tame and wild automorphisms of free algebras Speaker: Ualbai Umirbaev, Wayne State University Time: 2:00pm–3:00pm Place: CMC 130 Sponsor: M. Elhamdadi
The well-known Jung–van der Kulk theorem states that every automorphism of the polynomial algebra \(K[x,y]\) in two variables \(x,y\) over an arbitrary field \(K\) is tame, that is, it is a product of elementary automorphisms. An analogue of this result for free associative algebras in two variables was proven by Makar-Limanov and Czerniakiewicz. Moreover, the automorphism groups of polynomial algebras and free associative algebras in two variables are isomorphic.
The automorphism groups of polynomial algebras and free associative algebras generated by three elements are much more complicated and they admit non-tame automorphisms.
I will talk about these results including the latest results on the automorphisms of polynomial, free associative, free Lie, and Poisson algebras.