UCSC Mathematics Colloquia
Winter 1999

COLLOQUIUM CHAIRMAN
ARTHUR E. FISCHER
aef@cats.ucsc.edu

Everything that counts cannot be counted and everything that can be counted does not count.
Sign hanging in Albert Einstein's office in Princeton

The Mathematics Colloquia will be held on Tuesdays at 4:00 pm in Room 372 Applied Sciences. There will be a Mathematics Tea beginning at 3:30 pm in Room 358 Applied Sciences. For further information regarding this series of talks, please visit our Web site or contact the Colloquium Chairman above. To subscribe to a weekly email Colloquium announcement, email joann@cats.ucsc.edu.

January 5

WELCOME BACK COLLOQUIUM TEA

January 12

ILLUMINATING THE SHADOW PROBLEM

MOHAMMAD GHOMI
University of California, Santa Cruz and University of South Carolina at Columbia
ghomi@cats.UCSC.EDU

Abstract: A convex surface, e.g., an egg shell, has the property that, when illuminated from any given direction, the shadow cast on the surface is a simply connected region. The "shadow problem" asks whether the converse of this phenomenon is true as well, i.e., does the simply connectedness of the shadows imply the convexity of the surface? In this talk the speaker proves that the answer is yes, and will also discuss some of its implications and related open problems.

January 19

TOM NODDY'S BUBBLE MAGIC

TOM NODDY
tnoddy@aol.com

Abstract: Bubbles are not just for children any more. The mathematics of bubbles are explored up close and personal in this unique hand's on demonstration of minimal surfaces in action. These demonstrations have appeared in more than 40 countries and have been featured on The Tonight Show, in Der Spiegel, and at the International Congress of Mathematicians in Berlin this past summer. Come see why Johnny Carson said "Tom Noddy's sensational!" Professor Anthony Tromba will provide some additional mathematical comments.

January 26

NORMAL FORMS IN LOCAL ANALYSIS AND WHAT ARE THEY FOR?

MICHAIL ZHITOMIRSKII
University of California, Santa Cruz, and Technion University, Israel
mzhi@cats.ucsc.edu

Abstract: The talk is devoted to local classification problems and applications of normal forms. A number of examples concerning functions, mappings, differential equations, differential forms, foliations, distributions, and control systems will be given in order to discuss the following points: 1. Is it possible to use a list of normal forms? 2. Normal forms in 19th, 20th and 21st centuries. 3. Relation between normal forms and singularity theory. 4. Classification results: the true meaning. 5. Reduction theorems, invariants and covariants. 6. Methods to obtain normal forms. 7. Terminology and "ideology" of singularity theory.

February 2

THE MONSTER LIE ALGEBRA AND MOONSHINE

ELIZABETH JURISICH
University of California, Santa Cruz
jurisich@count.ucsc.edu

Abstract: In this talk I will give an overview of the construction and structure of the "Monster" Lie algebra. The construction of the Monster Lie algebra, along with its denominator identity, are a crucial part of Richard Borcherds' proof of the celebrated "Moonshine Conjectures" of Conway and Norton relating to the Monster simple group. I will present an interesting and useful decomposition of this Lie algebra which simplifies the theory and part of the proof of the Moonshine conjectures.

February 9

SEMISIMPLE QUANTUM GROUPS

SHLOMO GELAKI
University of Southern California
gelaki@math.usc.edu

Recruitment Speaker

February 11

REGULAR REPRESENTATIONS OF VERTEX OPERATOR ALGEBRAS

HAISHENG LI
Rutgers University
hli@crab.rutgers.edu

Recruitment Speaker

February 16

A MODULAR VARIETY AND A CONJECTURE OF BEUKERS

SCOTT AHLGREN
Pennsylvania State University
ahlgren@math.psu.edu

Recruitment Speaker

February 18

AUTOMORPHIC FORMS ON GL(2) AND THE RANK OF CLASS GROUPS

SIMAN WONG
Brown University
siman@math.brown.edu

Recruitment Speaker

February 22

Title to be announced.

MARCIN MAZUR
University of Chicago
mazur@math.uchicago.edu

Recruitment Speaker

February 23

ELLIPTIC COMPLEXES RELATED TO ELECTROMAGNETISM AND GRAVITY

ROBERT BEIG
University of Vienna
beig@pap.univie.ac.at

Abstract: In the Einstein theory of gravitation one is interested in finding initial data for the gravitational field. These comprise "TT-tensors", i.e. symmetric tensors on a complete Riemannian 3-manifold M which have zero trace and zero divergence. When M is (locally) conformally flat, finding such tensors naturally involves an elliptic complex in exactly the same way as solving the "Gauss law constraint" of the Maxwell theory involves the Hodge-deRham complex. In the latter theory, when the second cohomology of M is trivial, there is a well-defined procedure, after deleting some points from M ("punctures"), of "adding charge" to any particular solution. The analogous process for gravity is related to the notion of "boosting or spinning up a black hole".

February 25

BESSEL DISTRIBUTIONS, BESSEL FUNCTIONS AND SPECIAL VALUE OF L-FUNCTIONS

EHUD BARUCH
Weizmann Institute
baruch@narkis.wisdom.weizmann.ac.il

Recruitment Speaker

March 2

CONTINUA OF H-GRAPHS AND THEIR STABILITY

JOHN McCUAN
Mathematical Sciences Research Institute
john@msri.org

Abstract: We will discuss the concavity problem for non-parametric solutions of a certain Dirichlet problem for the equation of constant mean curvature. A general discussion of quasi-concavity and some partial results will be given.

March 9

BI-CLOSED ALGEBRAS OF DIFFERENTIAL OPERATORS

ALEXANDRE KIRILLOV
University of Pennsylvania and Mathematical Sciences Research Institute
kirillov@math.upenn.edu

Abstract: Different kinds of non-commutative operator algebras occur in pure and applied mathematics. Two of them are especially frequently used and thoroughly studied: Von Neumann algebras and algebras of differential operators on manifolds. These two sorts of algebras seem to have nothing in common with each other. Differential operators are usually unbounded in any Hilbert or Banach space and bounded operators never satisfy the canonical commutation relations. Nevertheless, we shall try to use notions and tools of one theory in the another. We define bi-closed algebras of differential operators on an algebraic manifold. We also introduce a notion of an almost bi-closed algebra. The main test of any new definition is the existence of good examples. It seems that many examples of almost bi-closed algebras can be obtained by considering algebraic group actions on smooth affine manifolds. We shall provide some interesting examples in this talk and hope that there are others.

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