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

Speaker: Alexei Miasnikov (Stevens Institute of Technology)

Title: Elementary classification questions for groups and algebras II: Associative and Lie algebras.

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 O. Kharlampovich.

Speaker: Paul Kim Long V. Nguyen (UH Leeward Community College)
Title: $&#;92;Sigma^0_3$-completeness of subdirect irreducibility of lattices

Speaker: Anna Barry (UBC)

Title: Vortex Crystals and the (1+N)-Vortex Problem.