Seminar 12.01.22 Mirek – Virtually

Title: On recent developments in pointwise ergodic theory

Speaker:  Mariusz Mirek – Rutgers University

Abstract: This will be a survey talk about recent progress on pointwise convergence problems for multiple ergodic averages along polynomial orbits and their relations with the Furstenberg-Bergelson-Leibman conjecture.

Zoom link:

Meeting ID: 919 4381 2487

Password: Mixing

Link of recorded talk:

Seminar 11.10.22 Kuca – Virtually

Title: Multiple ergodic averages along polynomials and joint ergodicity

Speaker:  Borys Kuca – University of Crete

Abstract: Furstenberg’s dynamical proof of the Szemerédi theorem initiated a thorough examination of multiple ergodic averages, laying the grounds for a new subfield within ergodic theory. Special attention has been paid to averages of commuting transformations with polynomial iterates owing to their central role in Bergelson and Leibman’s proof of the polynomial Szemerédi theorem. Their norm convergence has been established in a celebrated paper of Walsh, but for a long time, little more has been known due to obstacles encountered by existing methods. Recently, there has been an outburst of research activity which sheds new light on their limiting behaviour. I will discuss a number of novel results, including new seminorm estimates and limit formulas for these averages. Additionally, I will talk about new criteria for joint ergodicity of general families of integer sequences whose potential utility reaches far beyond polynomial sequences. The talk will be based on two recent papers written jointly with Nikos Frantzikinakis.

Zoom link:

Meeting ID: 919 4381 2487

Password: Mixing

Link of recorded talk:

Seminar 09.08.22 Dilsavor – Virtually

Title: Statistics of periodic points and a positive proportion Livsic theorem

Speaker:  Caleb Dilsavor – Ohio State University

Abstract: The connection between the Ruelle-Perron-Frobenius operator and the statistics of a Hölder observable g with respect to an equilibrium state has a rich history, tracing back to an exercise in Ruelle’s book. A somewhat lesser known, but related, statistical theorem studied first by Lalley, and later by Sharp using the RPF operator, states that the periods of g grow approximately linearly with respect to length, with square rootoscillations chosen according to a normal distribution whose variance is equal to the (dynamical) variance of g. This result is known for aperiodic shifts of finite type, but surprisingly it is still notknown in full generality for their Hölder suspensions. I will describe a tentative result that fills in this gap, along with joint work with James Marshall Reber which uses this result to deduce a strengthening of Livsic’s theorem not previously considered: if a positive-upper-density proportion of the periods of g are zero, then g is in fact a coboundary.

Zoom link:

Meeting ID: 919 4381 2487

Password: Mixing

Link of recorded talk:

Seminar 09.01.22 Bersudsky – In person

Title: On the image in the torus of sparse points on expanding analytic curves

Speaker: Michael Bersudsky (OSU)

Abstract: It is known that the projection to the 2-torus of the normalised parameter measure on a circle of radius R in the plane becomes uniformly distributed as R grows to infinity. I will discuss the following natural discrete analogue for this problem. Starting from an angle and a sequence of radii {Rn} which diverges to infinity, I will consider the projection to the 2-torus of the n’th roots of unity rotated by this angle and dilated by a factor of Rn. The interesting regime in this problem is when Rn is much larger than n so that the dilated roots of unity appear sparsely on the dilated circle.

Seminar program for Fall 2022

Our seminar continues with a mixture of in person and virtual talks. As usual, we meet on (most) Thursdays at 3.00pm EST unless otherwise noted. In person talks will be in MW154.

For virtual talks, the Zoom link can be obtained from the co-organizer, and responsible for the virtual component of the seminar, Andreas Koutsogianis.

The following is our current schedule; more talks might be announced soon.

August 25: In peron – Dmitri Scheglov

September 1: In person – Michael Bersudsky

September 8: Virtually – Caleb Dilsavor

September 15: In person – Andrey Gogolev

September 22: In person – Tomasz Downarowicz

September 29: No talk

October 6: Virtually – Yunied Puig de Dios

October 13: No seminar, Fall break

October 20: Virtually – Jiajie Zheng (postponed for a future date)

October 27: In person –  Andreas Koutsogiannis

November 3: In person – Michał Misiurewicz

November 10: Virtually – Borys Kuca

November 17: No talk

November 24: No seminar, Thanksgiving break

December 1: Virtually – Mariusz Mirek

December 8: In person – Martin Leguil