Algebra and Combinatorics Seminar

Boij and Soederberg offered an important shift in perspective when they suggested that graded free resolutions over a polynomial ring were more easily understood when viewed ``up to scalar multiple." We begin by discussing their conjectures and subsequent proofs, which are due to Eisenbud and Schreyer.

We then turn our attention to similar questions for minimal free resolutions over local rings. Over a regular local ring and their hypersurface rings, we completely classify the possible shapes of these resolutions. This illustrates the existence of free resolutions whose Betti numbers behave in surprisingly pathological ways. This is joint work with Daniel Erman, Manoj Kummini, and Steven V Sam.

Electronic transport through chaotic quantum dots exhibits universal properties which experimentally and numerically agree with predictions from random matrix theory (RMT). In particular, systems with broken time-reversal symmetry are well-modeled by uniformly distributed random matrices from U(N). One analytical approach to quantum transport is the semiclassical approximation, which expresses the relevant quantities as sums over the classical scattering trajectories. Correlations between such trajectories are organized diagrammatically and have been shown to yield universal answers for some observables.

We develop a general combinatorial treatment of the semiclassical diagrams by casting them as unicellular maps (graphs embedded on surfaces) and relating them to factorizations of permutations. The expansion of transport quantities corresponds to a genus expansion of the combinatorial generating function. Taking the previously calculated answers for the contribution of a given diagram, we prove agreement between the semiclassical and random matrix approaches to moments of the transmission amplitudes.

The talk will consist of short introductions to the four topics mentioned in the title and will highlight the connections between them.

Based on joint work with Jack Kuipers (Regensburg).

Generalizing the concept of convergency to valued fields, Ostrowski in the 1930s introduced pseudo convergent sequences. In the present paper we classify pseudo convergent sequences in right chain domains R according to the prime ideal P associated to the breadth I of the sequence using an ideal theory developed for right cones in groups. The ring R is I-compact if every pseudo-convergent sequence in R with breadth I has a limit in R, and we construct right chain domains R which are I-compact only for right ideals I in particular subsets B of the set of all right ideals of R.

Krull's perfect valuation rings and then Ribenboim's notion of a valuation ring complete par etages, where B is the minimal set containing the completely prime ideals in a commutative valuation ring, is a special case. For a non discrete right invariant rank one right chain domain R there are exactly two possibilities for the set B if the value group of R is the group of real numbers under addition, there are infinitely many possibilities for B in all other cases.

The set {1, 25, 49} is a 3-term collection of integers which forms an arithmetic progression; the common difference is 24. Hence the set {(1,1), (5,25), (7,49)} is a 3-term collection of rational points on the parabola y = x^2 whose y-coordinates form an arithmetic progression. Similarly, the set {6, 12, 18} is a 3-term collection of integers which also forms an arithmetic progression; the common difference is 6. Hence the set {(6,3), (12,39), (18,75)} is a 3-term collection of rational points on the elliptic curve y^2 = x^3 - 207 whose x-coordinates form an arithmetic progression. Are there other examples such as these? What is the longest progression of rational points on either a quadratic or cubic curve such that either the x- or y-coordinates form an arithmetic progression? In this talk, we give a survey on what's known about arithmetic progressions on algebraic curves. We introduce elliptic curves as a means to show the non-existence of certain arithmetic progressions. We also introduce bielliptic curves in order to settle conjectures of Saraju P. Mohanty. This project is joint work with Alejandra Alvarado.

Let F be a finite field and let W be a finite dimensional symplectic vector space over F (symplectic geometry). In this talk we use a character of the additive group of F and induced data to parametrize all irreducible representations of the Heisenberg group attached to W.

The stable endomorphism ring of a finitely generated module over a finite dimensional symmetric k-algebra is a Z-graded k-algebra. We'll show that it possesses a natural duality between its positive and negative sides. A consequence of this is that if the non-negative part of the endomorphism ring has a regular sequence of length 2, then all products between elements of negative degree are trivial. As a corollary we show this for the Tate-Hochschild cohomology ring of a symmetric k-algebra. This is based on joint work with Petter Bergh and Steffen Oppermann.