Seminar 03.04.21 Merriman

Title: Using modular surfaces to generate continued fractions

Speaker: Claire Merriman – The Ohio State University

Abstract: Continued fractions are frequently studied in number theory, but they can also be described geometrically. I will give both pictorial and algebraic descriptions of the flows that describe continued fraction expansions. This talk will focus on continued fractions of the form $a_1\pm\frac{1}{a_2\pm\frac{1}{a_3\pm\ddots}}$, where the $a_i$ are odd. I will show how to describe these continued fractions as geodesic on the hyperbolic plane, and how they cross cells of the Farey tessellation.

Meeting ID: 980 3359 0349

Recorded talk: https://osu.zoom.us/rec/play/FNCFPum1mokl6Bnf8uJ76iRehQRPNq5Op3VMXBDbNz7lAPb5qGWwnud4KJJucCuZQhrufoMV3d7X7MbK.icd0xSMxEksuitng?continueMode=true&_x_zm_rtaid=TxJ_aekJTv69MOkCQOL-dA.1614920606908.f7398a7dd87a1fb116574333eca30d89&_x_zm_rhtaid=489

Seminar 02.25.21 Richter

Title: Additive and geometric transversality of fractal sets in the integers

Speaker: Florian K. Richter – Northwestern University

Abstract: Using the language of fractal geometry and dynamical systems, Furstenberg proposed a series of conjectures in the 1960s that explore the relationship between digit expansions of real numbers in distinct prime bases. While his famous x2 x3 conjecture remains open, recent solutions to some of his “transversality conjectures” have shed new light on old problems. In this talk we explore analogues questions in the discrete setting of the integers, with the aim of understanding the independence of sets of integers that are structured with respect to different prime bases. This is based on joint work with Daniel Glasscock and Joel Moreira.

Meeting ID: 980 3359 0349

Recorded talk: https://osu.zoom.us/rec/play/Rkd9fe3_RRhHi58HtHE0Gjv6Ls831YiiVREOlnvsMCZh1MAQ2pwLNXdLTF04YV1bASAUr0xCj6JIoVO_.ukxDRCKORfRDV69r?continueMode=true&_x_zm_rtaid=SeyN37XMRTCLsJefPOamKg.1614300556080.85186464ed81cb1aa81402c607077979&_x_zm_rhtaid=566

Seminar 02.18.21 Glasscock

Title: Recent progress on a question of Katznelson concerning topological recurrence

Speaker: Daniel Glasscock – UMass Lowell

Abstract: Katznelson’s question is a longstanding open question at the intersection of topological dynamics, combinatorial number theory, and harmonic analysis: Is every set of Bohr recurrence a set of topological recurrence?  Equivalently, does the set of differences A-A of a set of integers A with bounded gaps contain the iterated difference set (B-B)-(B-B) of a set B of positive upper density?  In this talk, I will survey what little is known about Katznelson’s question and explain some recent progress achieved in joint work with Andreas Koutsogiannis and Florian Richter.

Meeting ID: 980 3359 0349

Seminar 02.11.21 Donoso

Title: Topological and combinatorial aspects of finite topological rank systems

Speaker: Sebastián Donoso – University of Chile

Abstract: In this talk, I will review recent results in the class of finite topological rank minimal subshifts. Such systems are the ones that can be represented with a Bratteli diagram (and a Vershik map on it) where the number of vertices at each level is uniformly bounded. I will analyze their correspondence with the $\mathcal{S}$-adic subshifts and their complexity word function.