# MathCS Seminar 2010

## Fall 2010

The seminar talks are in Von Neumann Hall 116 (545 W Palm Ave corner of W Palm Ave and railroad, Orange, CA 92866)

The talks usually start at 4 pm. From 3:30 pm to 4:00 pm is time for refreshments and interesting conversations with the speaker, so consider coming a bit early.

### Thursday, December 9, at 3:30 pm in VN 116

*Speaker:* Prof. M. Andrew Moshier (Chapman University)

*Title:* A Bitopological Point-free Approach to Compactifications

*Abstract:* D-frames, consist of a pair of frames, together with two relations which serve as abstractions of disjointness and covering, are a suitable point-free counterpart to bitopological spaces. In particular, the "topological" separation axioms of regularity and normality have natural analogues in d-frames. We develop a bitopological point-free notion of complete regularity and characterise all compactifications of completely regular d-frames. Given that normality of topological spaces does not behave well with respect to products and subspaces, probably the most surprising result is this: The compact regular coreflection of completely regular d-frames (Stone-C(ech compactification) factors through the subcategory of normal d-frames. Moreover, any compactification can be obtained by first producing a regular normal d-frame and then applying the Stone-C(ech compactification to it. Our point-free bitopological compactification subsumes all classical compactifications of frames as well as Smyth?s stable compactification.

This is joint work with Olaf Klinke and Achim Jung (Univ. of Birmingham, UK).

### Thursday, November 18, at 4 pm in VN 116

*Speaker:* Prof. John Harding (New Mexico State University)

*Title:* Subalgebras of orthomodular lattices

*Abstract:* Recent approaches to the foundations of quantum mechanics initiated by Isham and others make use of the topology determined by the abelian subalgebras of a von Neumann algebra to treat various quantum mechanical concepts as sheafs of structures that locally behave as the classical counterparts. This leads to our investigation of properties of the poset of abelian subalgebras of a von Neumann algebra, or more generally, of the poset of Boolean subalgebras of an orthomodular lattice.

### Tuesday, November 9, at 4 pm in VN 116

*Speaker:* Fabrizio Colombo, Politecnico of Milano

*Title:* An introduction to the F-functional calculus

*Abstract:* In this talk, I will recall the classical Fueter mapping theorem and I will show how to provide a new version of the result written in integral form.
I will use this integral representation to construct a functional calculus for n-tuples commuting operators using a subclass of monogenic functions.

*Speaker:* Irene Sabadini, Politecnico of Milano

*Title:* Some new results on the Dirac complex

*Abstract:* The complex of several Cauchy-Fueter operators is nowadays quite well understood and its knowledge implies several analytical consequences on the space of regular functions with quaternionic values. The higher dimensional analogue, i.e. the complex of several Dirac operators, is known in the case of three operators in the so-called "radial case". We will show how to prove that the radial case is in fact the general case, using commutative algebra techniques.

### Thursday, September 30, at 4 pm in VN 116

*Speaker:* Prof. Daniele Struppa (Chapman University)

*Title:* The mathematical legacy of Leon Ehrenpreis: 1930 - 2010

*Abstract:* Leon Ehrenpreis was one of the greatest Fourier analysts
of the twentieth century. In this talk I will discuss his celebrated Fundamental Principle, and its implications for contemporary research, including some recent results of mine on approximate solutions of differential equations.

### Thursday, September 16, at 4 pm in VN 116

*Speaker:* Dr. Peter Jipsen (Chapman University)

*Title:* Relation algebras as residuated lattices expanded with a De Morgan negation

*Abstract:* In this talk we consider the variety DmRL' of residuated lattices with a unary De Morgan operation '. It has the same signature as relation algebras and includes a subvariety qRA of quasi-relation algebras defined by adding equations that ensure ' is a homomorphism from the residuated lattice to its dual. We show how relation algebras are a natural subvariety of qRA, but that qRA has the distinct advantage of having a decidable equational theory. This type of result is only possible if the order structure of relation algebras is generalized to De Morgan lattices, since Kuruczs, Nemeti, Sain and Simon proved in 1993 that any `large enough' variety with Boolean algebra reducts and with an associative operator has an undecidable equational theory.
We also extend a result of Jonsson and Tsinakis [1993], where relation algebras are shown to be term-equivalent to a subvariety of residuated Boolean monoids, to the more general setting of quasi relation algebras. Much of the work reported here was done in collaboration with Nikolaos Galatos.

### Thursday, September 9, at 4 pm in VN 116

*Speaker:* Dr. Mihaela Vajiac (Chapman University)

*Title:* The notion of Holomorphicity in the context of Multicomplex Spaces

*Abstract:* In this presentation we study regularity notions and properties of functions defined on multicomplex spaces BC_n. Our results constitute a generalization of our previous research in this field (joint work with D.C. Struppa and A. Vajiac).

## Summer 2010

### Thursday, July 22nd, at 10:00am in VN116

*Speaker:* Dr. Dan Zaffran, KEIST (South Korea)

*Title:* Euler's formula and (much) more

* Abstract:* A cube has F=6 faces, E=12 edges and V=8 vertices. A pyramid with a square base has F=5 faces, E=8 edges and V=5 vertices. Euler discovered in 1750 that for these two cases, or for any other polyhedron, F-E+V=2. He published the result, but he confessed that he was not able to prove it. This celebrated "Euler's formula" is the starting point of many results and conjectures in higher dimensions. I will explain some of them, and focus on the surprising methods that have been used to solve these problems: topological manifolds and their algebraic topology, algebraic geometry...

## Spring 2010

The seminar talks are usually in Von Neumann Hall 116 (545 W Palm Ave corner of W Palm Ave and railroad, Orange, CA 92866)

### Tuesday, May 11th, at 4:30pm in VN116

*Speaker:* Dr. Ali Nayeri,, Chapman University

*Title: * The State of the Universe: Certain Present, Dark Future and Shady Past

* Abstract:* The past 20 years have seen dramatic advances in our understanding about the universe. We seem to have established the basic parameters describing the behavior of our expanding Universe, thereby putting cosmology on a firm empirical footing. But the emerging standard? model of the universe leaves many details to be worked out, and new ideas are emerging that challenge the theoretical framework on which the structure of the Big Bang is based. There is still a great deal left to explore in cosmology.

In this talk I will first review the status of the universe as we (think) we understand now. After reviewing the standard model of cosmology and the problems within I will present the new idea of structure formation and avoiding the initial singularity based on string theory which was developed in Harvard recently by my colleagues and I.

### Thursday, May 6th, at 4:00pm in VN116

*Speaker:* Dr. Boyana Norris, Argonne National Laboratory, Mathematics and Computer Science Division

*Title: * Annotation-Based Empirical Performance Tuning of Scientific Applications

* Abstract:* In many high-performance scientific applications, computational scientists spend significant time tuning codes for a particular high-performance architecture. Tuning approaches range from the relatively nonintrusive (e.g., by using compiler options) to extensive code modifications that attempt to exploit specific architecture features. Intrusive techniques often result in code changes that are not easily reversible, which can negatively impact readability, maintainability, and performance on different architectures. We describe an extensible annotation-based empirical tuning system called Orio, which is aimed at improving both performance and productivity by enabling scientific software developers to insert annotations in the form of structured comments that trigger a number of low-level performance optimizations on a specified code fragment. To maximize the performance tuning opportunities, we have designed the Orio framework to support both architecture-independent and architecture-specific code optimizations. Given the annotated code as input, Orio generates many tuned versions of the same operation and empirically evaluates the versions to select the best performing one for production use. We have also integrated other, independently developed, transformation approaches into Orio, such as the Pluto automatic parallelization tool for generation of efficient OpenMP-based parallel code and the PrimeTile tool for parametrized tiling of imperfect loop nests. We describe experimental results involving a number of computational kernels, including dense array and sparse matrix operations.

### Thursday, April 22nd, at 3:00pm in VN116

*Speaker:* Dr. Zair Ibragimov, Cal State Fullerton

*Title: * Hyperbolization of metric spaces

* Abstract:* Suppose Z is a locally compact noncomplete metric space. We introduce a metric u_Z on Z, which induces the topology of Z and turns Z into a hyperbolic metric (in the sense of Gromov). The metric u_Z can be thought of as a canonical Gromov hyperbolic metric of Z since it appears that many Gromov hyperbolic metrics introduced in geometric function theory (such as: the Poincare metric, Barbilian metric, the hyperbolic cone metric, the j-metric and the hyperbolic metric of the hyperspaces) are special cases of u_Z (up to a quasiisometry). In particular, this in combination with the Gromov hyperbolicity of u_Z gives alternative proofs of the Gromov hyperbolicity of these metrics.

### Thursday, April 15th, at 3:00pm in VN116

*Speaker:* Dr. David Porter, Professor of Economics and Mathematics, Chapman University

*Title: * Combinatorial Auctions (with Stephen Rassenti)

* Abstract:* A combinatorial auction is a resource allocation process that can be
implemented when multiple resources must be simultaneously allocated
amongst competing users, and information concerning the values of the
various possible uses and constraints affecting those uses is
decentralized. We will go through the design and implementation of
various methods for solving this problem and the computational and
incentive issues involved in such designs.

### Tuesday, April 6th, at 3:00pm in VN116

*Speaker:* Prof. Le Hung Son, Hanoi University of Technology, Vietnam

*Title: *Additive Cousin Problem and Hartogs Extension Theorem in Clifford and Quaternion Analysis

* Abstract:* It is known that the Additive Cousin Problem and Hartogs Extension Theorem are the fundamental events in Complex Analysis of several variables. The first one allows the construction of the global meromorphic functions by given local singularities and the second one states that holomorphic functions of several complex variables possess non isolated singularities. This talk deals with some remarks and comments concerning the same problems in Clifford Analysis, Quaternion Analysis and the theory of functions taking value in a Matrix Algebra.
Furthermore in this talk we will try to reflect some new ideas and results on Clifford and Quaternion Analysis of higher dimensions and their applications to Partial Differential Equations, Mathematical Physics and Engineering Problems.

### Thursday, March 25th, at 3:00pm in VN116

*Speaker:* Dr. Hendrik De Bie, Belgium and Oregon State

*Title: *The Clifford-Fourier transform

* Abstract:* The Clifford-Fourier transform has been introduced a couple of years
ago by Brackx et al. So far, it is probably the most interesting
attempt at introducing a Fourier transform in the setting of Clifford
analysis, because it is defined by a similar exponential operator as
the classical Fourier transform. However, several questions regarding
this transform are still open: the integral kernel is not known, it is
not known on which function spaces the transform is defined, there is
only a formal proof of the inversion formula, etc.

In recent work with Yuan Xu, I have succeeded in answering most of these questions. We have found explicit expressions for the kernel in all even dimensions, leading to estimates of the transform. We have introduced a generalized translation operator and a related convolution structure, which can be used to prove the inverse of the transform.

In this talk, I first give an overview of the classical Fourier transform and how it can be generalized to the Clifford-Fourier transform (stressing the Lie algebraic background). Next, I will explain our recent results.

[No special knowledge of Clifford analysis is necessary to follow the talk]

### Thursday, March 18th, at 3:00pm in VN116

*Speaker:* Dr. Peter Jipsen, Chapman University

*Title: * Periodic lattice-ordered pre-groups are distributive

* Abstract:* \emph{Pregroups} are algebras of the form
$(A,\cdot,1,^l,^r,\le)$
such that $(A,\le)$ is a poset, $(A,\cdot,1)$ is a monoid, $x\le
y\Rightarrow uxv\le uyv$,
and $^l,^r$ are two unary operations, called the \emph{left and
right adjoint} that satisfy
$$
x^l\cdot x\le1\le x\cdot x^l\qquad x\cdot x^r\le1\le x^r\cdot x.
$$
They were defined by J.~Lambek in 1999 in a paper that provides a
new approach to categorial grammars, and they have been studied in
a series of recent papers by W.~Buszkowski from an algebraic and
proof theoretic perspective. Pregroups give an abstract setting for
studying maps on a poset that have all iterated left and right adjoints
(i.e. all residuals and dual residuals). E.g. partially ordered groups,
and hence groups, are pregroups. A \emph{lattice-ordered pregroup}
or $\ell$-pregroup is a pregroup expanded with $\wedge,\vee$ such
that the underlying poset is a lattice with respect to these operations.
The variety of $\ell$-pregroups contains all $\ell$-groups and is
itself contained in the variety of involutive residuated lattices.
An $\ell$-pregroup is \emph{periodic} if it satisfies the identity
$x^{ll\ldots l}=x$ for some finite iteration of left adjoints. In
this talk it is proved that all such $\ell$-pregroups have distributive
lattice reducts. The proof was found with assistance from the equational
theorem prover Waldmeister. It is still an open problem whether the
same result holds for all $\ell$-pregroups.

### Thursday, February 26th, at 4:00pm in VN116

*Speaker:* Dr. Marina Borovikova, CSUF

*Title: * Symmetric products of lines and circles.

* Abstract:* The notion of symmetric product of topological spaces was introduced in 1931 by two Polish mathematicians, Karol Borsuk and Stanislaw Ulam. Since then it has been studied by many authors, mostly in topology. Recently, Z. Ibragimov has initiated the study of the symmetric product of metric spaces from a point of view of geometric function theory. Given a metric space $X$ and $n\geq 2$, the $n^{th}$ symmetric product $X$ is the set of all subsets of $X$ of cardinality less or equal to $n$, equipped with the Hausdorff metric. In this talk, I will discuss some results of K. Borsuk, S. Ulam and R. Bott on the $3^{rd}$ symmetric product of the real line $\mathbb R$ and the unit circle $\mathbb S^1$, and present some new results. (Joint work with Z. Ibragimov).

### Thursday, February 18th, at 4:00pm in VN116

*Speaker:* Dr. Oscar Villareal, UCI

*Title: * Rational points on commutative algebraic groups and the Zilber-Pink conjecture.

* Abstract:* Let $A$ be an semiabelian variety and $X$ a subvariety.
The Zilber-Pink conjecture states that the intersection of $X$ with
all of the semiabelian subvarieties of $A$ of codimension greater than
the dimension of $X$ is not Zariski dense. We give some of the
motivation behind this conjecture along with known results.

### Thursday, February 11th, at 4:00pm in VN116

*Speaker:* Oepomo, Tedja (WLAC)

*Title: * Survey of power, QR, and Oepomo's iterative methods for solution of largest eigenvalue of essentially positive matrices.

* Abstract:* Many of the popular methods for the solution of largest eigenvalue of essentially positive irreducible matrices are surveyed with the hope of finding an efficient method suitable for electromagnetic engineering, radiation problems, system identification problems, and solid mechanics. Eigenvalue computations are both fundamental and ubiquitous in computational science and its fast application areas. Some comparisons between several known algorithms, i.e. Power and QR methods, and earlier theory of Oepomo iterative techniques for solving largest eigenvalue of nonnegative irreducible matrices are presented since there is a continuing demand for new algorithm and library software that efficiently utilize and adapt to new applications.

### Thursday, January 28th, at 4:00pm in VN116

*Speaker:* Prof. Jaroslaw Harezlak, Indiana University School of Medicine

*Title: * Impact of the design matrix structure on the performance of LASSO: An empirical study

* Abstract:* High-throughput technologies in medical research provided statisticians with an ever increasing amounts of data. One of the methodological and practical challenges in the analysis of such data is variable selection in regression models. The past 15 years brought a formidable number of methods dealing with the variable selection in the case when the number of covariates is much larger than the number of observations (p >> n). Majority of the methods fall under the category of penalized likelihood which includes ridge regression, LASSO and its variations, SCAD and Dantzig selector.

In our work, we provide simulation results on the performance of LASSO in the case of strong dependence between the columns of the design matrix X. We consider the estimation error, prediction error and a measure of concordance between the true and selected variables. We study the dependence of the results on the design matrix specification, “irrepresentability condition” of Zhao and Yu (2006) and "phase transition" of Donoho and Stodden (2006). We also compare these results with the more common situation of orthogonality of columns of X.

In the compound symmetry case, we find that the increased dependence between the columns of X results in larger estimation error, but decreased prediction error. In the anisotropic correlation case, both estimation and prediction errors are the largest when the covariates exhibit both positive and negative correlations.

### Monday, January 4th to Saturday, January 15th in VN116

*Workshop:* OCCTAL WORKSHOP ON CONTEMPORARY TOPOLOGY

*Speaker:* Prof. Bernhard Banaschewski, McMaster University

*Title: * Essential completions in categories of Archimedean l-groups with order unit

*Speaker:* Prof. Bernhard Banaschewski, McMaster University

*Title: * On the ccc characterization of the function rings in pointfree topology.

*Speakers:* Prof. Rick Ball, University of Denver, and Dr. Joanne Walters-Wayland, OCCTAL

*Title: * The K-Briar Patch

*Speaker:* Dr. Peter Jipsen, Chapman University

*Title: * Some open problems regarding lattice-ordered pre-groups

*Speaker:* Prof. M. Andrew Moshier, Chapman University

*Title: * Locales and Quantales Concretely