Day of Informal Groups and Dynamics Seminars
Day of Informal Groups and Dynamics Seminars
Series of Four Talks
Pedro Fernández Espinosa - Universidad Pedagógica y Tecnológica de Colombia at 11:00AM
Title: Integer Sequences Arising from Representation Theory
Abstract: Representation theory of finite-dimensional algebras has deep connections with algebra, geometry, and combinatorics. One intriguing phenomenon is that the enumeration of certain homological and combinatorial objects naturally produces remarkable integer sequences.
The first part of this talk will provide a gentle introduction to representations of quivers and finite-dimensional algebras, with particular emphasis on exceptional modules and exceptional pairs. These objects play a fundamental role in modern representation theory and are closely related to tilting theory, cluster algebras, and derived categories.
In the second part, I will discuss how the enumeration of exceptional pairs over Nakayama algebras fits into this broader picture. In particular, I will show how counting these pairs naturally leads to classical and novel integer sequences. This perspective is closely related to the program of algebraic categorification of integer sequences initiated by Fahr and Ringel, in which classical sequences, such as the Fibonacci numbers, admit representation-theoretic interpretations.
This talk is based on joint research projects with A. M. Cañadas and David Reynoso-Mercado.
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Hari Iyer - Princeton University at 1:30PM
Title: On Shkredov'sÌýWorkÌýon Zaremba's Conjecture
Abstract: Zaremba conjecturedÌýthat there is a constant M such that every denominator q admits a numerator a such that the partial quotients of the continued fraction expansion of a/q are all bounded above by M. We discuss Shkredov's recent strategy for proving Zaremba's conjecture for prime q. This improves on the work of Korobov and others which allow M = O(log q), for instance. Shkredov's new idea deals with a certain critical range of partial quotient denominators by applying the pigeonhole principle to produce large linearÌýcombinationsÌýof them, showing that they "repel" each other. This combined with theÌýAhlfors-David regularity (largeness) of certain Cantor sets relevant to the problem produces lots of a/q withÌýbounded partial quotients.
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Przemek Berk - Nicolaus Copernicus University Toruń, Poland at 3:00PM
Title: Disjointness of Rescalings of Locally Hamiltonian Flows.
Abstract: In this talk I will discuss disjointness of rescalings of locally Hamiltonian flows. More precisely, I will show that for a typicalÌý minimal locally Hamiltonian flow Tt with simple saddles, the flows Tpt and Tqt are disjoint in the sense of Furstenberg as long as p and q are rational and |p| and |q| are distinct. The proof relies on the fact that such flows have special representations given by interval exchange transformations and roof functions with symmetric logarithmic singularities. In this sense, our result is a direct generalization of the result by Kanigowski and myself, for special flows over rotations. I will discuss the techniques used, as well as mention some future projects.
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Mariusz Lemanczyk - Nicolaus Copernicus University ToruÅ„, PolandÌýat 4:30PM
Title: Moebius OrthogonalityÌý- Recent Advancement
Abstract: A role of upgradedÌýorthogonality for topological models of some classes of deterministic measure-preserving systems in proving Sarnak's conjecture for the corresponding classes of zero entropy topological dynamical systems will be discussed. WeÌý especially focus on the logarithmic case and the role of the point spectrum for the systems arising from invariant measures. The talk is based on a joint work with J. Aaronson, A. Danilenko and J. Ku\l aga-Przymus