Conic divisor classes and Hilbert-Kunz functions

Abstract: Let R be a commutative ring containing a field of positive characteristic p. The map F:R->R, F(a)=a^p, is a ring endomorphism, called the Frobenius endomorphism. We consider R as a module over F^e(R). If R is local or positively graded, the Hilbert Kunz function e -> minimal number of generators of R over F^e(R) is an important invariant of R. We study it for normal affine monoid domain through the decomposition of R into its irreducible components as a module over F^e(R). They are certain divisorial ideals, called conic for a geometric reason. The multiplicities with which each irreducible component appears is given by the number of lattice points in a collection of open cells of a polyhedral complex. Therefore one can apply the theory of Ehrhart functions to it, and establish the basic properties of the Hilbert-Kunz function. For the rings under consideration it turns out to be a quasi-polynomial with rational coefficients whose two topmost coefficients are constant.

De Concini-Procesi arrangement models -- a tour d'horizon

Abstract: Wonderful models for arrangement complements as defined by De Concini & Procesi have attracted a lot of interest in recent years. One of the original motivations being to provide a rational model for the cohomology of arrangement complements, they have triggered a wealth of investigations and applications. We will review the model construction and give an outline of further developments. These include a combinatorial framework for resolutions modelled after the De Concini-Procesi construction process, the study of abstract algebras inspired by cohomology algebras of projective models for hyperplane arrangements, and abelianizations of group actions provided by wonderful arrangement models. Parts of this talk are based on joint work with D. Kozlov and with S. Yuzvinsky.

An introduction to liaison theory

Abstract: The idea of linking space curves using complete intersections was already used by Noether and Macaulay with the purpose of studying a curve by looking at a "simpler" curve. This lead to the the study of liaison theory, a theory that has been largely studied in the last decades and that is a very powerful tool in algebraic geometry nowadays. The aim of this talk is to introduce the audience to this topic providing the basic definitions needed to understand this theory (complete intersection, arithmetically Cohen-Macaulay, arithmetically Gorenstein, Hartshorne-Rao module...), as well as giving the main results and the state of the art.

An Introduction to Dedekind Sums

Abstract: The Dedekind sum is defined for relatively prime positive integers a and b as s(a,b) = sum_{k=1... b-1} ((ka/b)) ((k/b)), where ((x)) = x - [x] - 1/2. This sum and its generalizations have intrigued mathematicians from various areas such as Number Theory, Topology, and Combinatorial Geometry since their introduction by Dedekind in 1892. The most fundamental theorem for the Dedekind sums was already proved by Dedekind: s(a,b) + s(b,a) = - 1/4 + 1/12 ( a/b + 1/ab + b/a ) There are several proofs of this reciprocity law in the literature. We will present an elementary proof based on Pick's Theorem, which relates the area of a polygon with the number of integer points in it. Our proof will also illustrate that Dedekind sums are a natural ingredient of lattice-point enumeration functions.

Traditional Dedekind sums and new polynomial Dedekind sums for weighted lattice point enumeration in polytopes

Abstract: We first give an overview of the history of Dedekind sums, their role in number theory, and their more recent utility in the lattice point enumeration of polytopes. Intuitively, we may think of Dedekind sums as the building blocks of the Discrete volume of a polytope. We next define new types of Dedekind sums which play an important role in evaluating sums of polynomial weights at all lattice points inside rational polytopes. Such sums appear, for example, in the numerical analytic-theory of Euler-MacLaurin summation, and in the combinatorial-geometric theory of lattice polytopes. We find reciprocity laws for the polynomial Dedekind sums in certain cases, which allow us to compute them in linear time. This is joint work with Helaman Ferguson.

T-core partitions

Abstract: A partition of a natural number n is a representation of n as a sum of one or more natural numbers. For example, 7 = 3+2+1+1 is a partition of 7. Corresponding to every partition, there is a diagram, called the Ferrers-Young diagram. Each summand in this left-justified diagram appears as as a row of nodes. A hook is a path in the Ferrers-Young diagram, leading from a HEAD (the rightmost node in a row) to a FOOT (the bottom node in a column.) The hook number is the number of nodes in the hook. If the integer t>=2, then a partition is called "t-core" if no hook length is a multiple of t. Let ct(n) denote the number of t-core partitions of n. t-core partitions arise in the study of the representation theory of the symmetric group. There were numerous papers on t-core partitions in the last two decades of the twentieth century. I will discuss the most significant results on t-core partitions, as well as possible future directions.

Transporting iterative algorithms from Euclidean space to manifolds

Abstract: Newton's method is a well-known example of an iterative algorithm in Euclidean space that computes minima of a real-valued function which is defined on that space. 'Computes' means that for each local minimum of the function there exists a neighborhood in Euclidean space such that the algorithm converges towards the local minimum when started in that neighborhood. Moreover, the convergence is quadratic, meaning that the Euclidean distance to the minimum decreases quadratically in each step. When dealing with real-valued functions that are defined on smooth real manifolds, e.g. the sphere, the design of such algorithms requires considerably more effort. This is mainly due to the fact that in general a manifold has no vector space structure, hence the concept of a straight line (along which a step of the algorithm evolves) does not exist. One traditional approach overcomes this obstacle by endowing the manifold with a Riemannian structure (an inner product on each tangent space that varies smoothly with the base point) and replacing straight lines with geodesics (roughly speaking curves of shortest length, more precisely solutions to a certain differential equation). Generalising Newton's method this way leads to the so-called intrinsic Newton method on Riemannian manifolds which enjoys similar (convergence) properties as its Euclidean counterpart. For concrete situations, e.g. the Rayleigh quotient on spheres, there exists a wealth of highly specialised iterations, e.g. the inverse Rayleigh shift (a method to compute eigenvectors of a positive definite symmetric real matrix), whose convergence properties are usually derived by ad-hoc (setup specific) methods. In this talk I will present a completely different approach to the design of iterative algorithms on manifolds. The main idea is to utilise a known algorithm in Euclidean space which has known properties, and to transfer this algorithm to the manifold in such a way that these properties are preserved. I will explain how the above mentioned algorithms fit as special cases into that general framework.

Polynomial mappings, with emphasize on automorphisms

Abstract: The realm of polynomial mappings, i. e. the study of homomorphisms between polynomial algebras, is a field where most of the techniques of contemporary algebra fail. Its "differential" is the most familiar subject called Linear Algebra, or in the SFSU designation - Math 325. But once we leave the routine world of linear maps and allow polynomial changes of coordinates we immediately run into big open problems of affine algebraic geometry. There our understanding is strikingly limited and in its very embrionic stage. The Jacobian Conjecture, Cancellation Problem, Tame Generation Conjectrure and the like, which try to mimick the linear situation in the nonlinear world, have captured the imagination of algebraists since WWII. The overall picture is that "positive" results can be obtained in small dimensions or for small degrees, but the general conjectures remain quite intractable. I will give a survey of these topics, including the very recent dramatic development(s). The talk should be fully accessible to graduate students and to some extent will serve as an invitation to my graduate algebra course in Spring 2005 (not on exactly the same topics though).

Some geometry and combinatorics of resonant weights

I'll introduce the notion of resonance in the Orlik-Solomon algebra of a complex line arrangement, and talk about some of the interesting combinatorics and geometry that arises. In particular I'll indicate how arrangements that support resonant weights come from nice partitions of multi-arrangements, and yield very special families of projective plane curves.