# MathCS Seminar 2007

## Fall 2007

The seminar talks are usually in Beckman Hall 207 (corner of N. Glassell St and University Drive, Orange, CA 92866)

### Thursday, November 15, 2007, 1:30-2:30 pm, BK 207

*Speaker:* Dr. Adrian Vajiac, Chapman University

*Title:* Equivariant Localization in Mathematics and Physics II

### Thursday, November 8, 2007, 1:30-2:30 pm, BK 207

*Speaker:* Dr. Adrian Vajiac, Chapman University

*Title:* Equivariant Localization in Mathematics and Physics

*Abstract:* I will describe localization phenomena in mathematics and physics, especially recent applications of equivariant localization techniques and Weyl's integration formula in Topological Quantum Field Theories (TQFT). This is a summary of the talk I gave this summer at the "Recent Developments in Quantum Field Theory" conference held at the Max Plank Institute, Leipzig. I will conclude with a report on a work in progress regarding the relationship between equivariant localization and computational algebra techniques applied to 4 dimensional TQFTs.

### Thursday, October 25, 2007, 1:30-2:30 pm, BK 207

*Speaker:* Dr. Andrew Moshier, Chapman University

*Title:* Representations of Heyting Algebras in Bitopological Spaces II

*Abstract:* See below.

### Thursday, October 18, 2007, 1:30-2:30 pm, BK 207

*Speaker:* Dr. Andrew Moshier, Chapman University

*Title:* Representations of Heyting Algebras in Bitopological Spaces

*Abstract:* Heyting algebras formalize intuitionistic propositional logic in essentially the same way as Boolean algebras formalize classical propositional logic. While one can regard classical propositions as being characterized by their totality (in any world either "p" or "not p"), in contrast, tertium non datur fails in intuitionistic logic. This simple observation lead Kleene to investigate three-valued logic in which a proposition may be true, false or neither. As part of his investigation, Kleene showed that intuitionistic propositions can be translated into three-valued logic so that a proposition is intuitionistically valid if and only if its translate is valid in three-value logic. Of course, Kleene's program went much deeper, leading to many useful insights into the connection between intuitionistic mathematics and computation (recursion theory). But one important thing missing was a suitable topological representation theory. In this talk, we establish a representation theorem for Heyting algebras in bitopological spaces (spaces equipped with two topologies). In the course of developing this, we will see that the essentials of Kleene's ideas about three-valued logic resurface as a natural way to view a bitopological space. Time permitting we will also discuss connections to Priestley Duality via the work of Esakia.

### Tuesday, October 2, 2007, 1:00-2:00 pm, BK 207

*Speaker:* Prof. Gunduz Caginalp, University of Pittsburgh

*Title:* Toward a mathematical theory of financial market dynamics.

*Abstract:* Methodology for understanding the deterministic aspects of the dynamics of asset markets will be presented. These include differential and difference equations, optimization and statistical time-series methods. The integration of mathematical modeling with economics experiments will also be discussed.

### Thursday, September 27, 2007, 1:30-2:30 pm, BK 207

*Speaker:* Dr. Peter Jipsen, Chapman University

*Title:* The structure of the one-generated free domain semiring *Abstract:* In this talk we consider idempotent semirings with an additional unary operation d that has the properties of a domain operation. Concrete models of such algebras are e.g. reducts of relation algebras (with d(x)=(x;x^{-1})\cap id), as well as reducts of Kleene algebras with domain. Applications of these algebras to the semantic of various programming calculi have been considered by several researchers. Since both relation algebras and Kleene algebras have rich and complex (quasi)equational theories, we will only consider the simpler equational theory of idempotent semirings with domain.

The aim of this talk is to give an explicit construction of the one-generated free domain semiring. In particular it is proved that the elements can be represented uniquely by finite antichains in the poset of finite strictly decreasing sequences of nonnegative integers. It is also shown that this domain semiring can be represented by sets of binary relations with union, composition and relational domain as operations.

### Thursday, September 20, 2007, 1:30-2:30 pm

*Speaker:* Visiting Prof. Alberto Damiano, Chapman University

*Title:* Constructing Free Resolutions of Differential Operators - part II

*Abstract:* I will continue to talk about free resolutions for modules over the polynomial ring and define their cohomology modules. I will then shift to the non commutative case, and introduce the Weyl algebra, the Clifford algebra and the Exterior algebra. Gröbner basis type of algorithms have been studied recently for such algebras, allowing one to compute “effectively” syzygies and free resolutions. I will then show some examples of calculation using Singular, from which it will be clear that such methods are not really “efficient”, mostly because of the lack of a proper graded structure on such algebras.

### Thursday, September 13, 2007, 1:30-2:30 pm

*Speaker:* Visiting Prof. Alberto Damiano, Chapman University

*Title:* Constructing Free Resolutions of Differential Operators - part I

*Abstract:* In this series of two talks, I will illustrate how to construct free resolutions associated to invariant differential operators. Since most of the examples of operators that have been studied within the framework of Algebraic Analysis are constant coefficient operators, this theory possesses mostly an algebraic flavor because an operator is viewed as a matrix with polynomial entries. In this first lecture, I will review some of the basic concepts of commutative and homological algebra such as:

- Syzygies of ideals and modules over the polynomial ring

- Complexes of free modules, free resolutions, “uniqueness” of the minimal free resolution of a graded module

- Ext modules and cohomology

Since all such objects can be effectively constructed thanks to the theory of Gröbner bases, I will show some examples of calculations using the software package CoCoA.

## Spring 2007

### Thursday, May 17, 2007, 12:30-1:50 pm

*Speaker:* Dr. Joanne Walters-Wayland

*Title:* Framing "Mel"

*Abstract:* Unfortunately I was unable to attend a recent conference in Baton Rouge honoring Professor Melvin Henriksen's 80th birthday. I had been allocated the opening slot for a short talk which was to be a tribute to Mel and the influence he has had, albeit mainly inadvertently, on frame theory. I decided to take the opportunity to present the talk I had intended to give in Baton Rouge when I was invited to speak at Chapman University. Here is my original *Abstract:*

"As a tribute to Mel Henriksen and, in thanks for, his ability to inspire and motivate, I would like to give a few snapshots of work he has done over the years, taken through a localic lens. Starting with work presented to the AMS in 1954 on finitely generated ideals, the "raison d'etre" of F-frames, P-frames etc, including his work on prime ideals (1965), quasi-F-covers (1987) and pretty bases (1991), and concluding with some results about cozero complemented spaces (2003)."

I will give a brief introduction to Frame Theory at the beginning of the talk, and will conclude, if time permits, with a sketch of some of our (this is joint work with Rick Ball) recent results and problems in these areas.

### Thursday, May 10, 2007, 12:30-1:50 pm

*Speaker:* Prof. Irene Sabatini

*Title:* An algebraic approach to Hermitian monogenic function

*Abstract:* In this talk we describe the algebraic analysis of the system of differential equations described by the Hermitian Dirac operator, which is a linear first order operator invariant with respect to the action of the unitary group. We show that it is possible to give explicit formulae for the first syzygies of the resolution associated to the system, and we study the removability of compact singularities. We will also discuss the quaternionic version of the hermitian system.

### Thursday, May 3, 2007, 12:30-1:50 pm

*Speaker:* Dr. Andrew Moshier, Chapman University

*Title:* Generalizations of Topology with an Eye on Stone Duality

*Abstract:* One view of point-set topology is that it is the study of concrete representations of frames, i.e., complete lattices in which finite meet distributes over arbitrary join. The utility of this view comes into clear focus when we consider Stone Duality and its close relatives such as Priestley Duality. Recently, we have found a bitopological setting in which Stone and Priestley are unified, but this unification leaves open the question of how to relate topology (as the study of concrete representations of frames) to bitopology (as the study of concrete representations of what?). In this talk, we will make some preliminary steps toward a general approach to topological duality theorems in which topology and bitopology appear as special cases. We will look at possible further applications of this to so-called "quantum logic".

### Thursday, April 26, 2007, 12:30-1:50 pm

*Speaker:* Dr. Ovidiu Munteanu, UCI

*Title:* Curvature invariants in submanifold geometry. New developments"

*Abstract:* In this talk we will discuss the structure at infinity of manifolds that admit a weighted Poincare inequality and have a Ricci curvature lower bound defined by the weight function. We will recall Li-Wang's fundamental results on the subject and discuss some new ideas.

### Thursday, April 19, 2007 in BK 207, 12:30-1:50 pm

*Speaker:* Professor Fabrizio Colombo, Politecnico di Milano

*Title:* Identification of convolution memory kernels

*Abstract:* We investigate some abstract integrodifferential inverse problems that can be applied to the heat equation with memory, to the strongly damped wave equation with memory and to some other models.

### Thursday, April 12, 2007, 12:30-1:50 pm

*Speaker:* Visiting Assistant Professor Nelia Charalambous, UCI

*Title:* The Yang-Mills heat equation on compact manifolds with boundary

*Abstract:* Gauge theory is the study of differential equations for fields over a principal bundle. The case of a principal bundle with a nonabelian group was first introduced by R.L. Mills and C.N. Yang ['54] in order to give a model of the weak and strong interactions in the nucleus of a particle. They wanted to mirror the invariance of physics under an infinite dimensional gauge group, also known as the principle of local invariance.

In this talk we will consider a gauge-theoretic heat equation, the Yang-Mills heat equation. The underlying manifold will be smooth, three-dimensional, with a nonempty boundary. We will prove the existence and uniqueness of solutions to this equation, and consider questions about its convergence at infinite time.

### Thursday, March 29, 2007, 12:30-1:50 pm

*Speaker:* Dr. Bogdan Suceava, Cal State Fullerton

*Title:* Curvature invariants in submanifold geometry. New developments.

*Abstract:* Recently, in 2005, B.-Y. Chen has proved the most general form of a fundamental inequality with curvature invariants. The first version of his geometric inequality has been given in 1993, and it inspired in the last decade many geometers. The geometric motivation of this study was a classical fundamental problem in Riemannian geometry stated originally by S.-S. Chern, in a monograph from 1968: When does a given Riemannian manifold admit (or does not admit) a minimal immersion into a Euclidean space of arbitrary dimension? This study extends the study of the embedding problem, in the spirit of J. F. Nash's embedding theorem, in terms of curvature invariants. In our talk, we will survey a few recent results in this area and we will present future directions of study in this class of geometric inequalities with new curvature invariants.

### Thursday, March 8, 2007, 12:30-1:50 pm

*Speaker:* Dr. Adrian Vajiac

*Title:* Overview of Hamilton-Perelman's proof of Poincare Conjecture"

*Abstract:* I will present some of the main ideas behind the works of Hamilton and Perelman on Ricci flows, and how these prove Poincare Conjecture. The speaker is far from being an expert in this field, so the exposition will be informal.

### Thursday, March 1, 2007, 12:30-1:50 pm

*Speaker:* Professor Atanas Radenski

*Title:* Digital Support for Abductive Learning in Introductory Computing Courses

*Abstract:* Students who grew up browsing the Web are skilled in what is usually referred to as abduction, a reasoning process that starts with a set of specific observations and then generates the best possible explanation of those observations. In order to exploit the abduction skills of contemporary students, I have developed digital CS1/2 study packs that promote and support active learning through abduction, i.e., abductive learning. The study packs integrate a variety of digital resources: online self-guided labs, e-texts, tutorial links, sample programs, quizzes, and slides. These online packs stimulate students to learn abductively by browsing, searching, and performing self-guided lab experiments. In two years of study pack use, the failure rate in the CS1/2 courses at Chapman University has been reduced from 14% to 5%. In surveys conducted at Chapman University in 2005/06, students gave the same high rating of 4.5 (on a 1 to 5 scale) to the CS1 and the CS2 digital study packs. The study packs have been published online at studypack.com and adopted in various institutions.

### Thursday, February 15, 2007, 12:30-1:50 pm

*Speaker:* Dr. Mihaela Vajiac

*Title:* Harmonic Maps, Loop Groups, and Virasoro Actions, Part II

*Abstract:* Continuation of Part I last December: Harmonic maps have been the intense subject in differential geometry for some time now. We will talk about the theory of harmonic maps in Lie groups and the loop groups associated to these spaces of maps. We will then move to describing how this can be thought of as a Riemann-Hilbert Factorization Problem and conclude with the newly found Virasoro action on this space.

### Tuesday, February 6, 2007, 12:30-1:50 pm

*Speaker:* Dr. Alberto Damiano, Eduard Cech Center, Charles University, Prague.

*Title:* Constructing a Dolbeault complex for the Dirac operator in several vector variables.

*Abstract:* In this talk I will present the problem of finding a resolution for the system of partial differential equations associated to the classical Dirac operator in several vector variables. It is now widely accepted that such a system constitutes a proper generalization of the Cauchy-Riemann system which defines holomorphic functions of several complex variables, The analysis of the multivariable Dirac operator corresponds then to studying properties of hyperholomorphic functions. So far, mathematicians have attacked the problem of finding compatibiltiy conditions and global properties for the solutions of the system in essentially three different (but strongly related) ways. The first method (see the work of D.C.Struppa, I. Sabadini, F. Colombo et al.) uses techniques of algebraic analysis and Groebner bases to construct a free resolution of the polynomial module given by the cokernel of the symbol of the operator. The second follows the ideas of Baston for case of quaternionic analysis and aims at constructing the so called BGG graph associated to the system. It is an oriented graph in which nodes are representation spaces of a semisimple Lie Algebra [in this case the sum of sl(k) with so(n), where k is the number of variables and n is the dimension of the space] and the arrows are differential operators, the first one being the Dirac operator. The third method is based on some algebraic relations satisfied by the Dirac operator(s) called radial relations.

Calculations with radial relations lead to the construction of an associative, non commutative algebra called algebra of Megaforms. I will discuss mainly the first two methods and their respective advantages and limitations. In particular, I will show that even when using the representation-theoretic point of view, the aid of a computer algebra system becomes crucial to perform experiments, and in some cases even to prove partial results.

### Thursday, January 18, 2007, 12:30-1:50 pm

*Speaker:* Professor Graziano Gentili, University of Firenze

*Title:* Zeroes and singularities of regular maps of a quaternionic variable.

*Abstract:* A geometrical interpretation of a definition originally given by Cullen allows the construction of a new theory of regular functions over the skew field of quaternions. Unlike what happens with other notions of regularity, including the Fueter-regularity, the zero-set and the singular-set of a Cullen-regular function have very nice properties. In this talk we will present the main features of the theory of Cullen-regular functions and show that the zero-sets and singular-sets of such functions consist of isolated points or isolated 2-spheres in the 4-dimensional space of quaternions.