# MathCS Seminar 2009

## Fall 2009

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

### Thursday, December 3rd, at 4:00pm in VN 116

*Speaker:* Prof. Robert Desharnais (CSULB)

*Title: * Chaos in a Bottle: Experimental Nonlinear Population Dynamics

* Abstract:* Research in ecology is increasingly interdisciplinary. This presentation will describe a long-term collaboration among biologists, mathematicians, statisticians which is focused on understanding how nonlinearity manifests itself in the dynamics of ecological populations.

### Thursday, November 19th, at 5:15pm in VN 116

*Speaker:* Prof. Irene Sabadini, Dipartimento di Matematica, Politecnico di Milano

*Title: * Slice Hyperholomorphy and its Functional Calculus

* Abstract:* TBA

### Thursday, November 5th, at 4:00pm in VN 116

*Speakers:* Prof. Cyril Rakovski and Lisa Brown, Chapman University

*Title: * On the ranking of the disease susceptibility locus in family-based candidate gene studies: a simulation-based analysis

* Abstract:* We carried out an extensive family-based candidate gene simulation study to analyze the position of the true causal single nucleotide polymorphism in the complete list of marker p-values ordered according to their statistical significance. We used the real haplotype structures of 10 genes from the HapMap dataset, various sample sizes that current studies employ and disease models that mimic the characteristics of complex human disorders. We found that the all three factors, gene, disease model and sample size have profound effect on the rank of the causal SNP.

### Thursday, October 22nd, at 4:00pm in VN 116

*Speaker:* Dr. Mihaela Vajiac, Chapman University

*Title: * Bicomplex Hyperfunctions

* Abstract:* In this talk we will develop the foundations for a theory of hyperfunctions as cohomology classes of bicomplex hyperholomorphic functions. The sheaf H of bicomplex hyperholomorphic functions was defined and studied in an earlier paper. We discuss some cohomological properties of H, we compute its flabby dimension and we use the knowledge of its resolution to define a sheaf of hyperfunctions. These hyperfunctions will be objects defined on a codimension three real analytic variety in the space BC of hypercomplex numbers. This is consistent with the fact that the flabby dimension of H is three.

### Thursday, October 8th, at 4:00pm in VN 116

*Speaker:* Dr. Adrian Vajiac, Chapman University

*Title: * Singularities of functions of one and several bicomplex variables

* Abstract:* In this talk we introduce the notion of regularity for functions of one, as well as several bicomplex variables.
Moreover, using computational algebra techniques, we prove that regular functions of one bicomplex variable have the property that their compact singularities can be removed.

### Thursday, September 17th, at 4:00pm in BK207

*Speaker:* Prof. Andrew Moshier, Chapman University

*Title: * Concrete Limits of Locales

* Abstract:* In the category of locales, dual to the concrete category of frames, limits are usually constructed as co-limits of frames. In this talk, we reconsider the limit constructions by viewing locales concretely as certain complete meet semi lattices.

### Thursday, August 27th, at 4:00pm in BK207

*Speaker:* Dr. Lev Vaidman, Tel Aviv University

*Title: * Where is the Quantum Particle between two Measurements?

* Abstract:* Wheeler Delayed Choice experiment, Elitzur-Vaidman Interaction-free Measurement, and Hosten-Kwiat Counterfactual Computation will be discussed to answer Bohr's forbidden question: "Where is a quantum particle while it is inside a Mach-Zehnder Interferometer?" The analysis reveals a paradoxical feature of a pre- and post-selected quantum particle: it can reach a certain location without being on the path that leads to and from this location.

## Spring 2009

### Thursday, March 5th, at 4:00pm in BK207

*Speaker:* Dr. Peter Jipsen, Chapman University

*Title:*
Topological duality, canonical extensions and decidability for lattices with quasioperators.

*Abstract:*
Lattice-ordered monoids, residuated lattices, modal lattices and De Morgan lattices are all examples of lattices with quasioperators, i.e. operations that preserve joins or send meets to joins in each argument or preserve meets or send joins to meets in each argument. In a previous seminar talk Prof. Moshier presented a duality between bounded lattices and a natural subcategory of topological spaces. We will review this duality and extend it to bounded lattices with quasioperators. We consider the connection to earlier dualities of Urquhart, Hartung, and Hartonas and to the canonical extension by Gehrke and Harding, as well as to Galois frames with additional relations.

Within this framework, we then discuss techniques for proving the decidability of the equational theory of various varieties of bounded lattices with quasioperators. We will also consider algorithms for constructing finite models in these varieties, and present an implementation that enumerates all models based on a given finite Galois frame.

### Thursday, February 19th, at 4:00pm in BK207

*Speaker:* Professor Bernhard Banaschewski

*Title:*
Point-free Topology and the Representation of Lattice-ordered Rings

*Abstract:*
The classical representation of archimedian f-rings with unit by continuous functions on topological spaces unavoidably requires the use of extended realvalued functions, with all the inherent complications that involves. By way of contrast, the natural counterpart of the usual notion of a real-valued continuous function in point-free topology provides a setting in which these l-rings can be embedded in the corresponding rings of real-valued continuous functions. This talk will describe that point-free representation in terms of the relevant function ring functor and its left adjoint, and then show how this adjointness may be used to obtain certain results about f-rings from point-free topology.

This seminar is part of CHAPMAN UNIVERSITY's "WORKSHOP ON POINT-FREE TOPOLOGY" to which you are also invited. Week of: February 17–20, 2009

### Week of February 16th-20th, in BK207

*WORKSHOP ON POINT-FREE TOPOLOGY* in honor of Bernhard Banaschewski’s visit to ChapmanvUniversity.

Organized by Dr. Andrew Moshier

*Initial schedule:*
Feb 17

11:00 – 1:00 B. Banaschewski Essential Completion in Point-free Topology, Part I.

4:00 – 5:00 M. A. Moshier Skew frames and Point-free Bitopology.

Feb 18

11:00 – 1:00 B. Banaschewski Essential Completion in Point-free Topology, Part II.

Feb 19

10:00 – 11:15 F. Dashiell Kuratowski Reduction in ?-frames.

4:00 – 5:00 B. Banaschewski Point-free Topology and the Representation of Lattice-ordered Rings.

Feb 20

11:00 – 1:00 B. Banaschewski Essential Completion in Point-free Topology, Part III.

For more information and updates contact: M. Andrew Moshier (714) 997 6628

### Thursday, February 12th, at 4:00pm in BK207

*Speaker:* Dr. Andrew Moshier Chapman University

*Title:*
Topological Duals and Canonical Extensions for Arbitrary Bounded Lattices

*Abstract:*
Is there a subcategory of Top (the category of topological spaces) that is dually equivalent to BLat (the category of bounded lattices)?

This question has only been affirmatively answered either by restricting BLat to special cases, e.g., Boolean or distributice lattices, or by expanding Top to a category of spaces with extra structure. The latter approach indeed yields duality theories for the general case of BLat, but the results are highly technical and have turned out not to be particularly useful to lattice theorists.

We give an affirmative answer to the above question with no qualifications. The result is a significant improvement in the theory of topological duality for lattices. To illustrate the point, we show that the canonical extension of a lattice 'lives' inside its topological dual in a very natural way. Time permitting, as a further illustration, we will also derive a representation theory for orthocomplemented lattices from the general setting.

### Friday, January 12th, at 1:00pm in BK207

*Speaker:* Dr. Rebecca Goldin, GMU, Director of Research at STATS (www.stats.org), a nonprofit affiliated with GMU.

*Title:*
The Two-Sphere Spinning Around an Axis: an Introduction to Symplectic Geometry with Hamiltonian Group Actions

*Abstract:*
I will describe a beautiful relationship between symplectic manifolds with Hamiltonian group actions, and convex polytopes. The essential goal in this "business" is to find a way of describing difficult geometry in terms of combinatorial objects such as polytopes. We will illustrate these ideas through one of the most basic example, the two-sphere with a circle rotating it around an axis. All terms will be defined.