Title: Game-theoretic probability: brief review

Abstract: The standard approach to probabilistic modelling is to assume a probability measure generating the observed outcomes. Game-theoretic probability weakens this assumption but still allows one to obtain many familiar results, such as laws of large numbers and iterated logarithm, central limit theorems, large deviation inequalities, and zero-one laws. It also leads to completely new results.

Speaker: Yuliy Baryshnikov (UIUC)

Title: Billiards with families of periodic orbits and non-holonomic systems.

Abstract

Periodic orbits in typical planar Birkhoff billiards are isolated, but
sometimes come in families (e.g. in ellipses). In this talk I will
explain the context, show how the question about families of periodic
orbits in planar billiard domains can be seem as a problem about
non-holonomic (or control) systems, and prove that planar billiards
cannot have a 2-parameter family of 3-periodic orbits, while spherical
billiard domains can, using this viewpoint.

Speaker: Olga Kharlampovich (CUNY — Hunter College)
Title: Elementary classification questions for groups and algebras I: Groups.

Abstract:
We consider some fundamental model-theoretic questions that can be asked about a given algebraic structure (a group, a ring, etc.), or a class of structures, to understand its principal algebraic and logical properties. These Tarski type questions include: elementary classification and decidability of the first-order theory.

In the case of free groups we proved that two non-abelian free groups of different ranks are elementarily equivalent, classified finitely generated groups elementarily equivalent to a finitely generated free group (also done by Sela) and proved decidability of the first-order theory.

We describe partial solutions to Tarski’s problems in the class of free associative and Lie algebras of finite rank and some open problems. In particular, we will show that unlike free groups, two free associative algebras of finite rank over the same field are elementarily equivalent if and only if they are isomorphic. Two free associative algebras of finite rank over different infinite fields are elementarily equivalent if and only if the fields are equivalent in the weak second order logic, and the ranks are the same. We will also show that for an infinite field the theory of a free associative algebra is undecidable.

These are joint results with A. Miasnikov.

Title: When Prime Numbers Factor

Abstract: We learned in school that prime numbers are numbers that only factor as 1 times themselves. However, this definition assumes that the only allowable factors are positive integers. What does it mean to be prime if we are allowed to write 5 = (-1) x (-5) or 5 = sqrt{5}^2 (where “sqrt” denotes the square root)? How can a prime number factor if we extend our allowable factors beyond the positive integers?

We also learned that every integer can be written as a product of powers of prime numbers in such a way that the prime numbers and their exponents are unique (only their order isn’t unique). For example, 6 = 2 x 3. But if we are also allowed to write 6 = (1+sqrt{-5})(1-sqrt{-5}), the factorization is no longer unique. So how can we re-achieve uniqueness if we again allow factors beyond positive integers?

This talk will explore answers to the above questions and more. No mathematical background beyond high school is required.

Speaker: Bakhadyr Khoussainov (University of Auckland)
Title: A quest for algorithmically random algebraic structures