Algebra Seminar [Hide Abstracts]
Fall 2011

 The ultrafilter topology on spaces of valuation domains and applications Link: View Poster Speaker: Carmelo Finocchiaro (Universita degli Studi "Roma Tre", Italy) Time: Tuesday, September 6, 2011 at 4:00 pm Place: MSB 118 (UConn - Storrs)

 Cox rings of toric vector bundles Link: View Poster Speaker: Milena Hering (University of Connecticut) Time: Tuesday, September 20, 2011 at 4:00 pm Place: MSB 118 (UConn - Storrs) Abstract: The Cox ring of an algebraic variety X comprises all homogeneous coordinate rings of images of X under maps to projective space. It is a basic question whether this ring is finitely generated (it then is called a "Mori dream space"). While toric varieties are Mori dream spaces, Cox rings of vector bundles on toric varieties with a compatible torus action turn out to not be finitely generated. In, fact there is a beautiful relationship between these rings and Cox rings of blow ups of points in projective space, which have been extensively studied in the context of Hilbert's 14th problem.

 A combinatorial formula for cluster variables of rank two Link: View Poster Speaker: Ralf Schiffler (University of Connecticut) Time: Tuesday, September 27, 2011 at 4:00 pm Place: MSB 118 (UConn - Storrs) Abstract: This talk is about the generators of cluster algebras of rank two, the so-called cluster variables. Cluster variables are certain Laurent polynomials in two variables which are defined in a recursive way. Each of these variables corresponds to a (preprojectiv or preinjective) representation of the quiver with two vertices, 1 and 2, and r arrows from 2 to 1, also known as the generalized Kronecker quiver. I will present a direct formula for the cluster variables in terms of certain subpaths of a specific lattice path. This formula is the result of a joint work with Kyungyong Lee.

 Serre's uniformity question and bounds on torsion subgroups of elliptic curves Link: View Poster Speaker: Alvaro Lozano-Robledo (University of Connecticut) Time: Tuesday, October 4, 2011 at 4:00 pm Place: MSB 118 (UConn - Storrs) Abstract: In this talk I will begin by introducing elliptic curves and discuss the structure theorem of the group of rational points (the Mordell-Weil theorem), which says that the group of rational points is a finitely generated abelian group. Next, I will explain what is known about the torsion subgroup on an elliptic curve, and will introduce a question of Serre (the so-called uniformity question). I will end with applications of Serre's question to finding bounds on the size of torsion subgroups.

 Fourth moments and an identity of the divisor function Link: View Poster Speaker: Yichao Zhang (University of Connecticut) Time: Tuesday, October 11, 2011 at 4:00 pm Place: MSB 118 (UConn - Storrs) Abstract: This is based on my Ph.D thesis and a joint work with Henry H. Kim. As a different formulation of Lindelof Hypothesis on the Riemann zeta function, moment problems were introduced. Later, people considered general L-functions, starting with the L-functions associated to modular forms, and moment problems were naturally extended. Moment problems are far from being solved in any case, for example, complete solutions only exist for the second and fourth moment of Riemann zeta function and second moment of L-functions associated to modular forms. For higher moments, even reducing the exponents is much harder. In this talk, we shall see an average version of fourth moments problem for L-functions associated to newforms. We shall also include an identity of the divisor function defined over quaternion algebras, which plays an important role in the proof and is of interest in its own right.

 Computing Tropical Resultants Link: View Poster Speaker: Josephine Yu (Georgia Tech) Time: Monday, October 17, 2011 at 4:20 pm Place: TLS 301 (UConn - Storrs) Abstract: (Please note special day - Monday! - and special room - TLS 301 - and special time 4:20 - 5:15) We fix the supports A=(A_1,...,A_k) of a list of tropical polynomials and define the tropical resultant TR(A) to be the set of choices of coefficients such that the tropical polynomials have a common solution. We show that TR(A) equals the tropicalization of the algebraic variety of solvable systems and that its dimension can be computed in polynomial time. The tropical resultant inherits a fan structure from the secondary fan of the Cayley configuration of A and we present algorithms for the traversal of TR(A) in this structure. We also present a new algorithm for recovering a Newton polytope from the support of its tropical hypersurface. We use this to compute the Newton polytope of the sparse resultant polynomial in the case where TR(A) is of codimension 1. Finally we consider the more general setting of specialized tropical resultants and report on experiments with our implementations. This is joint work with Anders Jensen.

 Automorphisms of Certain Maximal Curves Link: View Poster Speaker: Elisabeth Malmskog (Wesleyan University) Time: Tuesday, November 1, 2011 at 4:00 pm Place: MSB 118 (UConn - Storrs) Abstract: The Hasse-Weil bound restricts the number of points of a curve which are defined over a finite field; if the number of points meets this bound, the curve is called maximal. Giulietti and Korchmaros introduced a curve C_3 which is maximal over F_q^6 and determined its automorphism group. Garcia, Guneri, and Stichtenoth generalized this construction to a family of curves C_n, indexed by an odd integer n at least 3, such that C_n is maximal over F_q^(2n). Rachel Pries, Robert Guralnick, and I determined the automorphism group Aut(C_n) when n > 3; in contrast with the case n = 3, it fixes the point at infinity on C_n. The proof uses ramification groups and results from group theory. I will discuss maximal curves, automorphism groups, and outline our proof.

 A combinatorial description of the Casselman-Shalika formula Link: View Poster Speaker: Ben Salisbury (University of Connecticut) Time: Tuesday, November 8, 2011 at 4:00 pm Place: MSB 118 (UConn - Storrs) Abstract: The Casselman-Shalika formula asserts that a certain integral over a $p$-adic matrix group may be expressed as a product over positive roots corresponding to that matrix group. Recently, for the case when the underlying root system is of finite type A, this formula has been reinterpreted as a sum over the highest weight crystal $B(\lambda+\rho)$. It is known that such crystals have a combinatorial realization in terms of semistandard Young tableaux. In this talk, we explain how the realization of $B(\lambda+\rho)$ in terms of semistandard Young tableaux yields a statistic at each vertex defining a coefficient making this expansion possible, without the need to generate the entire crystal graph. Moreover, these coefficients lead to a definition of a polynomial $H_{\lambda+\rho}(-;q) \in \mathbf Z[q^{-1}]$ which recovers data about the relevant representations.

 A tour through modern deformation theory Link: View Poster Speaker: Jeremy Pecharich (Mount Holyoke College) Time: Tuesday, November 15, 2011 at 4:00 pm Place: MSB 118 (UConn - Storrs) Abstract: Deformation theory went through a transformation partially led by Deligne, Drinfeld, Kontsevich, Quillen and Stasheff in the beginning of the 1990s. The philosophy is that any suitably' nice deformation problem in characteristic 0 is governed by a differential graded Lie algebra. We will discuss this philosophy through examples from complex geometry and algebra. If time permits we will discuss joint work with V. Baranovsky and V. Ginzburg on an extension of Gabber's integrability theorem to the non-filtered case.

 What should e', pi', gamma(1/2)', zeta(3)' be, if numbers' are replaced by `functions'? Link: View Poster Speaker: Dinesh Thakur (University of Arizona) Time: Tuesday, November 22, 2011 at 4:00 pm Place: MSB 118 (UConn - Storrs) Abstract: We will describe old and new analogies discovered between number fields and function fields and discuss applications to the question in the title and to the determination of algebraic relations between these analogs. (This is a joint Colloquium/Algebra seminar. All are welcome.)

 Computing a level-13 modular curve over Q via representation theory Link: View Poster Speaker: Burcu Baran (University of Michigan) Time: Monday, November 28, 2011 at 11:00 am Place: MSB 118 (UConn - Storrs) Abstract: For any $n > 0$, let $X_{ns}(n)$ denote the modular curve over $\mathbb{Q}$ associated to the normalizer of a non-split Cartan subgroup of level $n$. The integral points and the rational points of $X_{ns}(n)$ are crucial in two interesting problems: the class number one problem and the Serre's uniformity problem. In this talk we focus on the genus-3 curve $X_{ns}(13)$. It has no $\mathbb{Q}$-rational cusp (as for any level $n > 2$), so to compute an equation for this curve as a quartic in $P_2(\mathbb{Q})$ we use representation theory. Our explicit description of $X_{ns}(13)$ yields a surprising exceptional $\mathbb{Q}$-isomorphism to another modular curve. We also compute the $j$-function on $X_{ns}(13)$; evaluating it at the known $\mathbb{Q}$-rational points, we obtain the expected CM values.

 Serre's uniformity problem over Q Link: View Poster Speaker: Burcu Baran (University of Michigan) Time: Tuesday, November 29, 2011 at 4:00 pm Place: MSB 118 (UConn - Storrs) Abstract: In this talk I will introduce Serre's uniformity problem over Q, an open problem in the theory of Galois representations of elliptic curves. Past work by Serre, Mazur and Bilu-Parent has led to important progress but has not solved the problem. The remaining and the most dicult part amounts to a problem concerning rational points of modular curves associated to normalizers of non-split Cartan subgroups. I will discuss this case and also briefly introduce my work on these modular curves