# Seminars and Colloquia

## Algebra Seminar

Apr 6

#### Algebra Seminar

01:20 pm

Exley Science Center Tower ESC 618

Cameron Hill, Wesleyan Some thoughts on probability measures as varieties. Abstract: There are special infinite structures (graphs and hyper-graphs) that reflect the asymptotic first-order properties of their finite induced substructures very, very accurately. Exactly what makes these infinite structures special can be described model-theoretically in terms super-simplicity or in terms of higher amalgamation properties of the finite substructures. Using machinery from functional analysis, I have proved that model-theoretic specialness yields certain nice probability measures with which you then perform your asymptotic analyses. I dont fully understand what these measures are really like, and I imagine that some algebraic geometry might provide a more concrete transformation from higher amalgamation properties to measures. I will describe a possible set up for this.

Feb 16

#### Algebra Seminar

01:20 pm

Exley Science Center Tower ESC 618

Wai Kiu Chan, Wesleyan Warings problem for integral quadratic forms Abstract : For every positive integer n , let g ( n ) be the smallest integer such that if an integral quadratic form in n variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of g ( n ) squares of integral linear forms. As a generalization of Lagranges Four-Square Theorem, Mordell (1930) showed that g (2) = 5 and later that year Ko (1930) showed that g ( n ) = n + 3 when n 5. More than sixty years later, M.-H. Kim and B.-K. Oh (1996) showed that g (6) = 10, and later (2005) they showed that the growth of g ( n ) is at most an exponential of n . In this talk, I will discuss a recent improvement of Kim and Oh's result showing that the growth of g ( n ) is at most an exponential of $\sqrt{n}$. . This is a joint work with Constantin Beli, Maria Icaza, and Jingbo Liu.

Nov 3

#### Algebra Seminar

01:20 pm

Exley Science Center Tower ESC 618

Jonathan Huang, Wes A Macdonald formula for zeta functions of varieties over finite fields Abstract : We provide a formula for the generating series of the zeta function Z ( X , t ) of symmetric powers Sym n X of varieties over finite fields. This realizes Z ( X , t ) as an exponentiable motivic measure whose associated Kapranov motivic zeta function takes values in W ( R ) the big Witt ring of R = W ( ). We apply our formula to compute Z (Sym n X , t ) in a number of explicit cases. Moreover, we show that all -ring motivic measures have zeta functions which are exponentiable. In this setting, the formula for Z ( X , t ) takes the form of a MacDonald formula for the zeta function.