Seminars and Colloquia

Algebra Seminar

Apr 6

Algebra Seminar

01:20 pm

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

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

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.

Sep 27

Algebra Seminar

04:15 pm

Bweong-Kweon Oh (Seoul National University): The number of representations of squares by integral quadratic forms Abstract: Let f be a positive definite integral ternary quadratic form and let r ( k , f ) be the number of representatives of an integer k by f . We say that the genus of f is indistinguishable by squares if for any integer n , r ( n 2 , f ) = r ( n , f ) for any quadratic form f in the genus of f . In this talk, we will give some examples of non trivial genera of ternary quadratic forms which are indistinguishable by squares. Also, we give some relations between indistinguishable genera by squares and a conjecture by Cooper and Lam, and we resolve their conjecture completely. This is a joint work with Kyoungmin Kim.

Sep 20

Algebra Seminar

04:15 pm

Christopher Rasmussen (Wes):A (Necessarily Incomplete) Introduction to Frobenioids Abstract: A man who knows a little is sometimes more dangerous than a man who knows nothing at all. In his approach to proving the ABC Conjecture, Mochizuki relies on the concept of a Frobenioid, which in his own words is ``a sort of a category-theoretic abstraction of the theory of divisors on [models of global fields].'' In the present talk, we will attempt to carefully introduce the notion of a Frobenioid and provide a small amount of context. Nothing will be assumed beyond a basic knowledge of category theory and some standard algebra.

Feb 5

Algebra Seminar

01:15 pm

Michael Wijaya, Dartmouth College: A function-field analogue of Conway's topograph Abstract: In "The Sensual (Quadratic) Form", Conway introduced a new visual method to display values of a binary quadratic form Q(x,y)=ax^2+bxy+cy^2 with integer coefficients. This topograph method, as he calls it, leads to a simple and elegant method of classifying integral binary quadratic forms and answering some basic questions about them. In this talk, I will present an analogue of Conway's topograph method for binary quadratic forms with coefficients in F_q[T], where q is an odd prime power. The constructions will take place on the Bruhat-Tits tree of SL(2), which is an analogue of the real hyperbolic plane.

Dec 4

Algebra Seminar, Michael Kelly (University of Michigan): " Uniform Dilations in High Dimensions"

01:10 pm

Abstract: It is a theorem of Glasner that given an infinite subset X of the torus R/Z and an epsilon greater than 0 there exists a positive integer n such that any interval of length epsilon in R/Z contains a point of the set nX (that is, nX is epsilon-dense in R/Z). The set nX is called a dilation of X by n. Alon and Peres have shown that the dilation factor n can be chosen to be a prime or n=f(m) for some integral polynomial f with degree(f)>0 and integer m. We will discuss various developments on these sorts of topics and I'll present joint work with Le Thai Hoang where we consider this phenomenon in higher dimensions.

Nov 6

Algebra Seminar, Christelle Vincent (UVM): "Compuiting equations in hyperelliptic curves whose Jacobian has CM"

01:10 pm

Abstract: It is known that given a totally imaginary sextic field with totally real cubic subfield (a so-called CM sextic field) there exists a non-empty finite set of abelian varieties of dimension 3 that have CM by this field. Under certain conditions on the field and the CM-type, this abelian variety can be guaranteed to be principally polarized and simple. In this talk, we begin by reviewing quickly the situation for elliptic curves with complex multiplication, which is the dimension 1 case of the work we present. We then move to the dimension 3 case, and present an algorithm that takes as input such a field and CM-type, and outputs a period matrix for such an abelian variety. We then check computationally if the abelian variety is the Jacobian of a hyperelliptic curve, and compute an equation for the curve if this is the case. This is joint work with J. Balakrishnan, S. Ionica and K. Lauter.