Singularities

Recently, I have become very interested in the study of singularities of schemes in positive characteristic and their relationship with the singularities appearing in birrational geometry. In particular, my work involves questions about the relationship between log canoncial singularities in characteristic 0 and F-pure singularities in characteristic p.

Log Canonical Thresholds, F-Pure Thresholds, and Non-Standard Extensions

to appear in Algebra and Number Theory.

We present a new relation between an invariant of singularities in characteristic zero (the log canonical threshold) and an invariant of singularities defined via the Frobenius morphism in positive characteristic (the F-pure threshold). We show that the set of limit points of sequences of the form (c_p), where c_p is the F-pure threshold of an ideal on an n-dimensional smooth variety in characteristic p, coincides with the set of log canonical thresholds of ideals on n-dimensional smooth varieties in characteristic zero. We prove this by combining results of Hara and Yoshida with non-standard constructions.

Hilbert-Kunz functions of 2 x 2 Determinantal Rings

(with Irena Swanson )

to appear in Illinois Journal of Mathematics.

We compute the Hilbert-Kunz functions of generic rank 1 matrices via recursion and Groebner basis techniques.

Semi-log canonical vs. F-pure singularities

(with Karl Schwede )

J. of Algebra, 349, (2012), 150-164.

If X is Frobenius split, then so is its normalization and we explore conditions which imply the converse. While this does not occur generally, we show it happens if certain tameness conditions are satisfied on the induced map on the normalization of a scheme from a Cartier linear map. Our result has geometric consequences including a connection between F-pure singularities and semi-log canonical singularities, and a more familiar version of the (F-)inversion of adjunction formula.

Algebra/Number Theory

My work in number theory inspired algebra was motivated by a generalization of Witt vectors due to A. Dress and C. Siebeneicher. The classic Witt vectors are a functor on commutative rings whose values on perfect fields of characteristic p are p-adically complete discrete valuation rings of characteristic 0. The Dress/Siebeneicher construction is a family of functors attached to profinite groups which are defined similarly to the classic Witt vectors. The classic Witt vectors correspond to functor attached to the additive profinite group of p-adic integers. Inspired by the applications of the classic Witt vectors in number theory, my work concerns the structure of the Dress/Siebeneicher functor attached to infinte pro-p groups taken over fields of characteristic p.

On the Structure of Witt-Burnside rings attached to pro-p groups

Submitted

The classical Witt vectors are a ubiquitous object in algebra and number theory. They arise as a functorial construction which takes perfect fields k of characteristic p to p-adically complete discrete valuation rings of characteristic 0 with residue field k and are universal in that sense. Dress and Siebeneicher generalized this construction by producing a functor W_G attached to any profinite group G. The classical case corresponds to choosing G to be the additive group of p-adic numbers. In this work we examine the ring structure of some examples of W_G(k) where G is a pro-p group and k is a field of characteristic p. We will show that the structure is surprisingly more complicated than the classical case.

A Witt-Burnside ring attached to a pro-dihedral group

To appear in International Journal of Number Theory.

This paper explores the Witt-Burnside functor attached to D_{2^\infty} on a field of characteristic 2.

Computational Topology/Geometry

The need for accuracy in computation has never been a more important problem than it is now as high performance computing is employed in more and more domains. Computataional geometry handles the traditional question of how to computationally model objects which agree with our understanding of their geometry. Current applications have demanded more topological considerations of current computational algorithms, asking how can we give a piecewise linear approximation to a manifold that is not only "close" but also topologically the same.

Modeling Time and Topology for Animation and Visualization with Examples on Parametric Geometry

(with Kirk Jordan, E.L.F. Moore, Thomas J. Peters, Alexander Russell )

Theoretical Computer Science, 405, (2008) 41-49.

The art of animation relies upon modeling objects that change over time. A sequence of static images is displayed to produce an illusion of motion, which is frequently trusted to be topologically meaningful. A careful analysis exposes that formal topological guarantees are often lacking. This lack of formal justification can lead to subtle, but significant, flaws regarding topological integrity. A modified approach is proposed that integrates topological rigor with a continuous model of time. Examples will be given for splines widely used in many applications, with particular emphasis upon scientific visualization for molecular modeling. Moreover, the approach of choosing a family of functions and studying their topological properties over time should be broadly applicable to other domains. Prototype animations are available for viewing over the web.

Topological Neighborhoods for Spline Curves: Practice and Theory

(with E.L.F. Moore, Thomas J. Peters, Alexander Russell )

Lecture Notes in Computer Science, 5045,(2008) 149-161.

The unresolved subtleties of floating point computations in geometric modeling become considerably more difficult in animations and scientific visualizations. Some emerging solutions based upon topological considerations for curves will be presented. A novel geometric seeding algorithm for Newtonâ€™s method was used in experiments to determine feasible support for these visualization applications.

Geometric Topology and Visualizing 1-Manifolds

(with Kirk Jordan, Thomas J. Peters, Alexander Russell )

Topological Data Analysis and Visualization: Theory, Algorithms and Applications, pp 1-12, Springer, 2010.

Ambient isotopic approximations are fundamental for correct representation of the embedding of geometric objects in real 3-space, with a detailed geometric construction given here. Using that geometry, an algorithm is presented for efficient update of these isotopic approximations for dynamic visualization with molecular simulations.

Incidence Algebras and Applied Lattice Theory

My early work concerned a method of utilizing lattices in data representation called Formal Concept Analysis. While in graduate school, I was introduced to a noncommutative algebraic structure that one can generate from a lattice called an incidence algebra. For finite lattices, these algebras are subalgebras of the algebra of upper triangular matrices and offer an algebraic context for generalizations of many combinatorial and number theoretic concepts (such as Moebius inversion).

Breadth First Search Graph Partitions and Concept Lattices

(with James Abello, Alex Pogel )

Journal of Universal Computer Science, 10, 8, 2004, 934-954.

We apply the graph decomposition method known as rooted level aware breadth first search to partition graph-connected formal contexts and examine some of the consequences for the corresponding concept lattices. In graph-theoretic terms, this lattice can be viewed as the lattice of maximal bicliques of the bipartite graph obtained by symmetrizing the object-attribute pairs of the input formal context. We find that a rooted breadth-first search decomposition of a graph connected formal context leads to a closely related partition of the concept lattice, and we provide some details of this relationship. The main result is used to describe how the concept lattice can be unfolded, according to the information gathered during the breadth first search. We discuss potential uses of the results in data mining applications that employ concept lattices, specifically those involving association rules.

Visualization of Concept Lattices using Weight Functions

(with Tim Hannan, Alex Pogel )

Conceptual Structures at Work: Contributions to the 12th International Conference on Conceptual Structures, pp. 1-15, Huntsville, AL, Shaker, 2004, 29-23.

Concept lattice layout techniques usually refer exclusively to order structure. In some applications (e.g. epidemiology) the concept lattice must express both the order structure and the support structure, and we propose a method that can use this additional information to help draw the lattice. The method involves a notion of weight function on a lattice -- a strictly order-preserving map from the lattice into the set of non-negative real numbers -- which is used to add a new component to any existing lattice layout method. We highlight the generality of the notion. When aesthetics is a concern, various order-based weight functions are a natural solution to the problem of distension associated with vector sum layout methods. Our most important point is that the use of one particular weight function, the support function, provides improved viewing of association rules (partial implications).