## The 2022 BUGCAT Conference will be in-person

## November 5-6, 2022

Graduate students of all levels and faculty are invited to register to give a 30 minute talk. Talks may be expository or on current research.

If you would like to submit a talk, please use our Title and Abstract Submission Form. The deadline for submitting a talk is October 18th, 2022.

To request funding, please use our Funding Request Form. The deadline for requesting funding is October 7th, 2022.

To attend the conference, please register by October 18th, 2022.

## Keynote Speakers

An end-periodic map of an infinite-genus surface is a homeomorphism that, "near infinity", looks like a translation. These naturally arise in the theory of foliations of 3-manifolds, as the first-return maps to certain leaves along a transverse flow. In particular Thurston pointed out how they arise in a fibred 3-manifold when the suspension flow of the fibration is transverse to a non-fibre surface.

End-periodic maps have been studied deeply by several authors, including Handel-Miller and Cantwell-Conlon-Fenley. We give a somewhat new point of view on this theory by showing that they can always be obtained from the kind of fibered situation that Thurston considered. I will try to describe the background of this story and explain what comes out of the new construction. This is joint work with Michael Landry and Sam Taylor.

It is known that the greatest common divisor of two Fibonacci numbers is again a Fibonacci number. It is called the strong divisibility property. However, this property does not hold for every second order recursive sequence. We give a characterization of GFPs that satisfy the strong divisibility property. We also give formulas to evaluate the gcd of GFPs that do not satisfy the strong divisibility property.

In a celebrated paper from 1948, André Weil proposed a beautiful connection between algebraic topology and the number of solutions to equations over finite fields: the zeta function of a variety over a finite field is simultaneously a generating function for the number of solutions to its defining equations and a product of characteristic polynomials of endomorphisms of cohomology groups. The ranks of these cohomology groups are the number of holes of each dimension of the associated complex manifold.

This talk will describe the Weil conjectures and then enrich the zeta function to have coefficients in a group of bilinear forms. The enrichment provides a connection between the solutions over finite fields and the associated real and complex manifolds. It is formed using A1-homotopy theory. No knowledge of A1-homotopy theory is necessary. The new work in this talk is joint with Margaret Bilu, Wei Ho, Padma Srinivasan, and Isabel Vogt and is available:

https://arxiv.org/abs/2210.03035.