Singularities

A great deal of my work concerns singularities in positive characteristic defined by F-splittings. Often, even questions concerning complex schemes are understood via reductions. Two notable themes are invariants, in particular multiplicities and thresholds, and test and multiplier ideals. Typical techniques used to solve these problems arise in either commutative algebra or algebraic geometry.

Derived Canonical Systems

(with Alberto Chiecchio )

Submitted.

Schwede introduced a canonical linear system defined by the action of Frobenius on global sections. We consider a derived variation in hopes of providing more tools to study these canonical linear systems. We classify when such systems are complete for curves and reprove of global generation in positive characteristic based on vanishing of this derived functor.

 

On lower bounds of s-multiplicity

(with William D. Taylor )

submitted.

We explore lower bounds for s-multiplicity, formulate an s-analogue of the Watanabe-Yoshida conjecture and prove it in dimensions at most three and for complete intersections.

 

Semi-log canonical vs. F-pure singularities

(with Karl Schwede )

Journal 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.

 

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

(with Bhargav Bhatt, Daniel Hernandez, Mircea Mustata )

Algebra and Number Theory, 6-7, (2012), 1459-1482.

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 )

Illinois Journal of Mathematics, 57, 1, (2013), 251-277.

We compute the Hilbert-Kunz functions of of ideals of 2 minors of generic matrices via recursion and Groebner basis techniques.

 

The F-pure threshold of a determinantal ideal

(with Anurag K. Singh , Matteo Varbaro )

Proceedings of the 12th ALGA symposium, Bulletin of the Brazilian Mathematical Society, 45, (2014), 767-775.

We show that the F-pure and log canonical thresholds of determinantal varieties agree.

 

Deformation of F-injectivity and finite-length cohomology

(with Jun Horiuchi, Kazuma Shimomoto )

Indiana University Mathematics Journal, 63, 4, (2014) 1139-1157.

We give a sufficient condition for F-injectivity to deform in terms of local cohomology. This includes an appendix written by Karl Schwede and Anurag Singh.

 

The s-multiplicity function of 2 x 2-determinantal rings

(with William D. Taylor )

Proceedings of the American Mathematical Society, 146, 7, (2018), 2797-2810.

We compute the s-multiplicity of ideals of 2 minors of generic matrices via recursion and Groebner basis techniques.

 

Test ideals in rings with finitely generated anti-canonical algebras

(with Alberto Chiecchio, Florian Enescu , Karl Schwede )

Journal of the Institute of Mathematics of Jussieu, 17, 1, (2018), 171-206.

We generalize many theorems about test ideals holding for Q-Gorenstein pairs to the setting of finitely generated anti-canonical algebras. This includes showing the test ideal can be described via alterations, splinters are strongly F-regular, and extend the Hara-Yoshida correspondence relating multiplier and test ideals under reduction.

 

Number Theory

One theme of my research in number theory concerns 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. The Witt-Burnside rings are more closely related to function rings taking values in the classic Witt vectors. Another direction of my work concerns the de Rham-Witt complex which has connections to singularities.

Witt differentials in the h-topology

(with Veronika Ertl )

Submitted.

We prove descent statements for the de Rham-Witt complex in the h-topology. These mirror results by Huber-Joerder and Huber-Kebekus-Kelly for Kaehler differentials.

 

Witt-Burnside rings and p-adic Lipschitz continuous functions

(with Benjamin Steinhurst )

to appear Journal of Commutative Algebra

We give a concrete interpretation of a quotient of a Witt-Burnside ring in terms of p-adic Lipschitz contnuous functions. We then use this to show that in addition to being non-noetherian rings, a particular class of Witt-Burnside rings are also infinite dimensional.

 

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

International Journal of Number Theory, 9, 3, (2013), 747-757.

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

 

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

Documenta Mathematica, 19, (2014), 1291-1316.

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.

 

Nonexistence of Galois representations for quadratic fields

(with Adam Gamzon )

International Journal of Number Theory, 14, 5, (2018), 1505-1524.

We establish new cases of non-existence of irreducible two dimensional continuous Galois representations modulo a prime p. This builds on previous work of Moon-Taguchi and Sengun and slightly improves estimates of Poitou.

 

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.

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 Newton’s method was used in experiments to determine feasible support for these visualization applications.

 

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. Formal topological guarantees are often lacking and can lead to significant flaws. Here, we introduce a modified approach that integrates topological rigor with a continuous model of time.

 

Geometric Topology and Visualizing 1-Manifolds

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

Topology-based Methods in Visualization, Springer, (2011), 1-13.

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).

Visualization of Concept Lattices using Weight Functions

(with Tim Hannan, Alex Pogel )

Conceptual Structures at Work: Contributions to the 12th ICCS, (2004), 1-15.

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. We propose a method that can use this additional information to help draw the lattice by introducing a weight function.

 

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 rooted level aware breadth first search to partition graph-connected formal contexts and examine some of the consequences for the corresponding concept lattices. We discuss potential uses of the results in data mining applications that employ concept lattices, specifically those involving association rules.

 

Group Gradings in Incidence Algebras

(with Eugene Spiegel )

Communications in Algebra, 38, 3, (2010), 953-963.

If X is a bounded countable locally finite partially ordered set, R an integral domain and G a group having the property that no non-identity element has order a unit of R, then it is shown that any G-grading of the incidence algebra I(X,R) is equivalent to a good grading. Further, an example is given showing that not all group gradings of incidence algebras are equivalent to good gradings.