2003 - 2004
September 19, 2003
Topic: "Examples from elements of theory of
computation"
Mostafa Ghandehari, UTA
Abstract: The study of formal languages
is a central topic in theoretical computer science and engineering.
Results from number theory are used to give examples of
regular and non-regular languages. In particular, Goldbach's
conjecture gives examples of two non-regular languages whose
concatenation is regular.
2002 - 2003
February 7, 2003
Topic: "Logarithmic Derivatives, Logarithmic
Forms, and Non Archimedean Picard Theorems"
William Cherry
Abstract: The equation x^2+y^2=1 has the
well-known rational function solutions: x=(1-t^2)/(1+t^2)
and y = 2t/(1+t^2). However, the equation x^n+y^n=1 has
no such non-constant rational function solutions when n
is at least three. One approach to proving this is using
logarithmic derivatives. If f(z) is a rational function
(or more generally a meromorphic function over the complex
numbers or a quotient of convergent power series over a
non-Archimedean valued field), then the logarithmic derivative
f'/f(z) gets small (or in the case of meromorphic functions
does not "grow" very quickly) as z gets large.
Applying the logarithmic derivative lemma to certain systems
of equations results in so called "Picard theorems"
which state that there are no non-constant function solutions
to certain types of systems of algebraic equations. In my
talk, I will discuss the notion of a "logarithmic form,"
connections to types of singularities of algebraic divisors,
the relationship between logarithmic forms, singularities,
and logarithmic derivative lemma, and finally applications
to Picard like theorems in non-Archimedean analysis. Occasionally,
I will point out some connections with complex analysis,
but the talk itself will focus on the algebraic aspects
of the theory.
March 7, 2003
Topic: "Zero Loci of Holomorphic Forms and
Birational Geometry"
Tie Luo, University of Texas at Arlington
Abstract: The zero locus of a holomorphic
vector field has been a subject of study for a long time
on a compact complex manifold. For example, the residue
theorem of Bott relates the zero locus of a vector field
with the Chern numbers of the manifold (which, in high-brow
terms, explains why a vector field on a sphere has to vanish
at some point). Surprisingly, not much has been done about
the zero locus of a holomorphic form. Through studying various
problems in birational geometry of algebraic varieties of
general type (in dimension 1, theses are Riemann surfaces
with more than one hole), we conjecture that a holomorphic
form has to vanish somewhere on a variety of general type.
In this talk, work done, and work in progress, towards verifying
this conjecture will be presented.
March 28, 2003
Topic: "Lifting Automorphic Forms"
Ze-Li Dou, TCU
Abstract: The purpose of this talk is to
describe the idea of lifting, or correspondence, of automorphic
forms, which has received much emphasis in contemporary
research. The necessary technical concepts will all be introduced
during the talk (concepts such as Hecke characters, Hilbert
modular forms, and $L$-functions...), and hence no existing
expertise on these matters is necessary. Some recent work
on theta-lifts might be discussed as well.
April 25, 2003
Topic: "The topology of algebraic surfaces
and reduction modulo p"
Minhyong Kim, University of Arizona
Abstract: We will discuss some classical
relationships between the mod p arithmetic of varieties
and their topology. Furthermore, we will mention one new
result regarding the homeomorphism type of simply-connected
surfaces.
May 1, 2003
Topic: "On the Classification of the Weight
Modules of Lie Superalgebras"
Dimitar Grantcharov, UC Riverside
Abstract: In the early 20th century Hermann Weyl
classified all finite- imensional representations of the
classical Lie algebras in terms of the so- called character
formula. The problem of generalizing Weyl's character formula
to infinite-dimensional representations turned out to be
more difficult. Recently, Olivier Mathieu obtained an explicit
classification of the class of infinite-dimensional weight
representations using the method of coherent families. In
the present talk, I will briefly sketch Mathieu's idea of
coherent families and some of its numerous applications.
2001
February 19, 2001 UNT
Title: " A prototype of a combined digital
and retrodigitized searchable mathematical journal"
G. Michler, Essen University
Abstract: In this lecture a survey is given
on the computer programs used for the retrodigitization
of 6 volumes of the mathematical journal "Archiv der
Mathematik" that appeared in the period 1993 to 1995.
The result may be considered to be a prototype for a mathematical
text recognition system. Furthermore, methods have been
developed by the author's study group and the IT Center
of Essen University to incorporate the retrodigitized texts
into a digital library database. Thus it is possible to
link the retrodigitized back issues with the recent digital
issues of a mathematical journal. Furthermore, full searchability
within the retrodigititzed texts has been achieved.
In this project also the recognition problem
of mathematical formulas has been addressed. Using the programs
of Professor Okamoto's study group (Nagano) most of the
mathematical formulas contained in the 6 volumes of the
Archiv der Mathematik have been recognized and transferred
into latex form. The linkage problem has been solved by
means of the new programs for the recognition of the layout
of the first page of a scanned article. It produces an XML-file
of the bibliographic data of such an article. These bibliographic
meta-data allow the integration of the MVD format into the
digital library database MILESS of the IT Center of Essen
University. The multivalent document format MVD was developed
by T. Phelps (Berkeley); it is another main ingredient of
the retrodigitization program system.
November 2, 2001
Topic: "Noncommutative Curves"
Kim Retert
Abstract: Noncommutative projective geometry
studies noncommutative graded rings by replacing the variety
by a suitable Grothendieck category. One way of studying
the resulting category is to examine the full subcategories
which behave like curves on a commutative variety. Smith
and Zhang initiated such a study by considering the subcategory
generated by a particular type of module they called a "pure
curve module in good position." In order to extend
the applicability of this approach, the definition of pure
curve modules in good position is generalized to modules
called "multistrand" modules. The categories created
from multistrand modules are described and shown (in general)
to be different from the type of category created from a
pure curve module in good position.
Friday, Oct. 1, UNT
Topic: "Matlis duality and applications to
isolated hypersurface singularities"
Prof. Ruth Michler, University of North Texas
Abstract: The talk will start with a brief
review of Matlis duality, canonical modules and modules
of differentials for affine varieties. In recent work the
speaker has established a duality between the torsion module
of differentials of isolated hypersurface singularities
and the highest nonvanishing exterior power of the module
of differentials, the "canonical module". As an
application of this result, she obtained efficient algorithms
for the computation of the number of generators and the
vector space dimension of the torsion module of differentials.
The vector space dimension of the torsion module of differentials
is an analytical invariant of the singularity, called Tjurina
number.
Friday, Dec. 3
Topic: "The Last Obstruction to a Universal
Theory of Descent"
Dr. Paul Feit, University of Texas - Permian Basin
Friday, January 28
Topic: "Old and New Results on the Arc Structure
of Singular Algebraic Varieties"
Dr. Monique Lejeune-Jalabert, CNRS University Versailles
St-Quentin
Abstract: An arc on an algebraic variety
$ V $ defined over the complex numbers is a mapping from
a sufficiently small neighborhood of the origin in the complex
line into $ V $, given by convergent power series. By Artin's
approximation theorem, for any nonnegative integer $ k $,
the set of k-jets of arcs on $ V $ is a constructible set
(i.e. defined by polynomial equations and inequations).
These constructible sets were first studied by J. Nash in
connection with Hironaka's resolution of singularities.
Further related developments will be reviewed. This lecture
is also sponsored by
the Charn Uswachoke Lecture Series and the Millican Lecture
Series.
Fri., Apr 7: UTA
Topic: "The Points of Quadratic Algebras"
Dr. Michaela Vancliff, University of Texas at Arlington
Abstract: Recent joint work with Brad
Shelton will be presented with emphasis on the following
counter-intuitive result. Let A denote a non-commutative
algebra on four generators with six defining relations (each
homogeneous of degree 2), and let Z denote the locus of
zeros of the defining relations of A. If Z is finite, then
the space of (1,1)-forms that vanish on Z is the span of
the defining relations of A. The result concerns the "points"
of A and has a counterpart involving "lines" of
A. Although the results are non-commutative in nature, the
proofs use only commutative algebra.
