# MathCS Seminar 2003

*Seminar Organizer:* Mihaela Vajiac, *Webpage maintained by:* Peter Jipsen

## Contents |

## Fall 2003

### Wednesday, November 26, 2003, 10 am

*Speaker:* Viorel Costeanu (MIT)

*Title: * The 2-typical de Rham-Witt complex of the integers

*Abstract:* For any ring $A$ and any prime number $p$ there is a construction called the
2-typical de Rham-Witt complex of $A$. It is related to Milnor's $K$-theory,
Quillen's $K$-theory, and topological cyclic homology. This algebraic
structure was studied and quite well understood for $p$ odd. When $p=2$ there
are a few problems that need to be addressed. I will speak about these and
then describe the 2-typical de Rham-Witt complex of the integers.

### Thursday, November 13, 2003, 2:30 pm

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

*Title:* Quantum Cohomology on Symplectic Manifolds. Properties of the
quantum products.

*Abstract:* The quantum products on a symplectic manifold are deformations of the
cup product. The motivation for the theory of Quantum Cohomology comes from
physics, and the mathematical theory has been built rigurously on the theory
of J-holomorphic curves on symplectic manifolds.
We will explore the properties of the quantum products, together with gauge
theoretic techniques developed to anlayze them.

### Thursday, November 6, 2003, 2:30 pm

*Speaker:* Annika Wille (Vanderbilt University; TU Darmstadt, Germany)

*Title:* A Gentzen System for Involutive Residuated Lattices

*Abstract:* Involutive residuated lattices $\langle L,\wedge,\vee,\cdot,e,'\rangle$ are involutive
lattices with a residuated monoid operation, such that

$a\cdot b \le c\quad$ iff $\quad a \le (b\cdot c')'\quad$ iff $\quad b \le (c'\cdot a)'$,

for all $a,b,c \in L$. We can also see an involutive residuated lattice as a Girard quantale, where the underlying lattice need not be complete.

A cut-free Gentzen system for Girard quantales is already given by J. Y. Girard. We will describe a cut-free Gentzen system for involutive residuated lattices from an algebraic point of view. Finally we conclude decidability.

### Thursday, October 30, 2003, 4 pm

*Speaker:* Dr. Jeff Lawson (Trinity University, Texas; Caltech)

*Title:* Lagrangian reduction by symmetry: Variations on a theme

### Thursday, October 23

Seminar cancelled

### Thursday, October 9, 2003, 4 pm

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

*Title:* Mathematical Background for Topological Quantum Field Theories"

### Thursday, October 2, 2003, 4 pm

*Speaker:* Dr. Andrew Moshier (Chapman University)

*Title:* Measure Theoretic, Domain Theoretic and Logical Views of Labelled Markov Processes

*Abstract:*
Recently, researchers have become increasingly interested in models of
computation in the presence of chance. These models are motivated by
concerns about verification of properties, such as operation within safe
envelopes, when the behavior of the system is influenced by random
events. In one such model, "labelled Markov processes," a system is
modelled as a state machine in which transitions are triggered by
external events, indexed by a set of "labels," resulting in stochastic
internal behavior.

We will discuss three related ways to think about labelled Markov processes. In the first, a classical measure theoretic definition slightly generalizes (discrete time) Markov processes. This yields concepts of bisimulation and simulation of the states of a process. In the second, a domain theoretic definition involves the solution of a certain domain equation. The result can be interpreted as a universal labelled Markov process. In the third, we relate the domain theoretic and classical definitions to a logical one by defining a propositional sequent calculus with semantics definable on any labelled Markov process. The result characterizes simulation of two states of a labelled Markov process by inclusion of propositional theories, and bisimulation by equality of propositional theories. In addition, the very definition of the sequent calculus derives from the domain equation used to construct a universal labelled Markov process.

We will give examples of results that are easily proved using the logic. Time permitting, we will also apply the domain theoretic and logical approaches to models of other kinds of probabilistic processes.

### Thursday, September 25, 2003, 4 pm

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

*Title:* An introduction to generalized basic logic algebras

*Abstract:* Some applications in computing and control theory require a
logic with truthvalues other than the absolute "true" and "false". Many
systems of multi-valued logic have been studied in the past, and quite a
few of them are based on using the interval [0,1] as a set of truth
values. This still leaves much choice as to how truthvalues should be
combined when two statements both hold, but with different degrees of
certainty. The system of Basic Logic was introduced by Petr Hajek in a
series of papers and a book (Metamathematics of Fuzzy Logic, Kluwer,
1998) to give a general framework for a logic that is common to all
multi-valued logics.

I will review the algebraic version of this logic, called BL-algebra, relate it to residuated lattices, and show that some of the strong properties of this logic are also true in various generalized contexts. In particular I will prove that the extension of BL-algebras with the so-called projection, gives a class of discriminator algebras, which explains why the projections have such far reaching consequences for basic logic. A modification of this result also holds for residuated lattices with a least element.

## Summer 2003

**Seminar talks during the summer take place at various times but still in Beckman Hall 402 (corner of One University Drive and N. Glassell, Orange, CA).**

### Tuesday, June 10, 2003, 2 pm, **Colloquium Talk**

*Speaker:* Prof. Dexter Kozen, Cornell University
(joint work with Mark Hopkins, AMS Inc.)

*Title:* Parikh's Theorem in Commutative Kleene Algebra

*Abstract:* Parikh's theorem says that every context-free language
is "letter-equivalent" to a regular set. Formally, two sets
are letter-equivalent if any word in one has an anagram
in the other. For example, the CFL $\{a^n b^n \mid n \ge 0\}$
is letter-equivalent to the regular set $(ab)^*$.

The usual proofs of Parikh's theorem involve an induction on parse trees of context-free grammars. However, this result is a special case of a more general result in commutative Kleene algebra, namely: every commutative Kleene algebra is uniformly algebraically closed. Formally, every system of inequalities

$f_i(x_1,...,x_n) \le x_i,\quad 1 \le i \le n$,

where the $f_i$ are polynomials in $K[x_1,...,x_n]$ over a commutative Kleene algebra $K$, has a unique least solution in $K^n$. Moreover, the components of the solution are given by regular expressions in the coefficients of the $f_i$.

The proof involves the definition of a differential operator $d/dx : K[x] \to K[x]$ on polynomials over a commutative Kleene algebra and a version of Taylor's theorem

$f(x+d) = f(x) + d f'(x+d)$.

### Wednesday, May 21, 2003, 4 pm

*Speaker:* Dr. Marcus Kracht (UCLA)

*Title:* Weakly Transitive and Semisimple Varieties of Modal Algebras

*Abstract:* In polymodal logics one often has to generalize notions from monomodal
logic, eg transitivity and symmetry. It turns out that this can be done
using notions that have originally been invented for monomodal logics.
The notion of weak transitivity (going back to Wim Blok) generalizes the
notion of transitivity in the right way. It shall be shown that many
results in modal logic do not depend on transitivity, they only depend
on weak transitivity. One result is the existence of a deduction theorem
for the global frame consequence, finite equivalentiality, the existence
of many splitting algebras. A variety is cyclic if every operator possesses
(or is included in) a converse. Cyclicity generalizes the notion of
symmetry. It turns out that a variety of modal algebras is semisimple
iff it is both weakly transitive and cyclic iff it is discriminator.

## Spring 2003

**All seminar talks take place Thursday afternoons in Beckman Hall 402 (corner of One University Drive and N. Glassell, Orange, CA) at 4 pm.**

### May 8, 2003

*Speaker:* Prof. John Yules (Chapman University)

*Title:* Design of an Ocean Wave Recorder -- My Sabbatical Project

*Abstract:* I plan first to discuss "an oceanographer's view of waves," which encompasses not only wind-generated waves, but also tsunamis and tides. We'll review a wide variety of techniques for making measurements of waves, and I'll show how my plan of attack fits in. Finally, we'll talk about the phenomena that might present themselves for study and perhaps brainstorm about data analysis.

### May 1, 2003

Discussion about departmental goals and assessment

### April 24, 2003

*Speaker:* Dr. Bogdan Suceava (CSU Fullerton)

*Title:* Fundamental Inequalities and Strongly Minimal Submanifolds

*Abstract:* The classical obstruction to minimal isometric immersions into Euclidean space is $Ric \geq 0.$ A question originated in the work of S.-S.Chern (1968) asked if there are any other curvature Riemannian obstructions to minimal isometric immersion in a space form. The answer has been given by B.-Y. Chen (first form in 1993, general form in 2000). An interesting problem is to see examples of Riemannian manifolds with $Ric<0$ which don't admit any minimal isometric immersion into Euclidean space for any codimension. The proof uses a consequence of Chen's fundamental inequality. The fundamental inequalities have been extended recently in complex space forms, where one of the equality cases bring to our attention a new class of geometric objects: strongly minimal submanifolds. These submanifolds (whose classification is not yet completed) would be the Kaehler submanifolds that best fit in an ambient complex space form.

### April 10, 2003

*Speaker:* Dr. Mohamed Allali (Chapman University)

*Title:* Fourier Transforms and Wavelets (Part 2)

### April 3, 2003

*Speaker:* Dr. Mohamed Allali (Chapman University)

*Title:* Fourier Transforms and Wavelets (Part 1)

*Abstract:* Wavelets are now a driving force that has regrouped a
community of mathematicians and engineers sharing representation
techniques. The two-part talk will introduce Fourier Transforms and
Wavelet theory, in particular Multiresolution Analysis.

### March 27, 2003

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

*Title:* Lie Groups, Loop Groups in Lie Groups and Harmonic Maps (Part 3)

### March 20, 2003

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

*Title:* Lie Groups, Loop Groups in Lie Groups and Harmonic Maps (Part 2)

### March 13, 2003

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

*Title:* Lie Groups, Loop Groups in Lie Groups and Harmonic Maps (Part 1)

### March 6, 2003

### February 27, 2003

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

*Title:* An online database of ordered algebraic structures (Part 2)

### February 20, 2003

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

*Title:* An online database of ordered algebraic structures (Part 1)

*Abstract:* Research in ordered algebraic structures has grown from the classical
areas of ordered groups, rings, and fields to the study of a wide
variety of classes of partially ordered algebras, where the order takes
the form of (join- or meet-) semilattices, lattices, or Boolean
algebras. In addition, subclasses determined by various completeness
properties and residuation properties have received considerable
attention, motivated by studies in algebraic logic, topology, and
theoretical computer science. While researchers are familiar with the
structures in their areas of expertise, it is no small task to be
informed about the terminology, definitions, basic properties and
relationships between the many classes of structures in this growing field.

This talk will present a preliminary version of an online database that
aims to (eventually) survey most classes of ordered algebraic structures
that have appeared in the literature. At minimum, a record for a class
contains its name, (several) definition(s), some of the basic properties
and results about the class, and the position of the class in this
hierarchy relative to its ``nearest* sub- and superclasses. Most of the*
classes are categories in a natural way, and concepts from category
theory are used to express structure preserving relationships between
different classes. Currently the database contains around a hundred
classes from action algebras to weakly representable relation algebras.
Although intended mainly as a reference for researchers, with citations
to the literature for further detail, the database aims to store
definitions in machine-readable form to allow additional processing with
computational tools. It will also provide access to families of examples
and (where feasible) implementations of algorithms for computable
syntactic or semantic properties of a given class. For example, if a
class has a decidable equational theory, an implementation of a decision
procedure may be included in the entry for the class. Tools for
visualizing (parts of) the partial order structure of specific examples
and of the hierarchy of classes will be discussed, as well as recent
significant online developments such as MathML and XML transformation
tools that play a role in our approach.

Much work remains to be done to give this database the breadth and consistency that would ensure it is a useful research tool and is helpful to students entering this area. In fact, in a successful scenario, this project will not be completed at some stage, but will continue to grow in a collaborative style, thereby ensuring that the information in the database remains up-to-date.

### February 13, 2003

*Speaker:* Dr. Andrew Moshier (Chapman University)

*Title:* Stably Compact Spaces Predicatively Understood (Part 2)

### February 6, 2003

*Speaker:* Dr. Drew Moshier (Chapman University)

*Title:* Stably Compact Spaces Predicatively Understood (Part 1)

*Abstract:* Stably compact spaces are suitable for investigating relationships between classical mathematics and computation because they include compact regular spaces and topological spaces that arise in domain theory. Proofs of many theorems involving stably compact spaces, however, depend on the axiom of choice and other non-constructive principles. This presents a problem if we are to understand these spaces computationally. The axiom of choice comes into play when we need to speak about existence of points in a space. Impredicativity (roughly, non-constructive quantification over powersets) comes into play when we need to speak about compactness, and more generally about arbitrary collections of opens.

We briefly motivate the concern about predicativity and constructivity, and then consider two alternatives to stably compact spaces: stably compact locales and stable sequent calculi. The former can be thought of as stably compact spaces "without points." The latter can be thought of representations of compactness information in stably compact locales. We sketch proofs showing that stably compact spaces, stably compact locales and stable sequent calculi form equivalent categories, isolating the use of non-constructive principles in these equivalences. To illustrate the point, we also give an entirely predicative proof of Tychonoff's theorem for stable calculi (a product of stable calculi is a stable calculus).