** **

**JOSEPH
FELS RITT LECTURES**

**
Karen Uhlenbeck**

University of Texas, Austin

&

Texas Institute for Computational and Applied Mathematics

**
From
Solitary Waves to Ubiquitous
Symmetry**

Abstract:

**One of the surprises of
modern mathematics is the appearance of the Korteweg-de Vries
Equation in the organization of new invariants of symplectic
manifolds X (usually called quantum cohomology). Certain special
differential equations (a subset of those known as integrable) have
appeared in the physics literature on topological conformal field
theory for over a decade. Kontsevich's Fields Medal is at least
partly based on his verification of the formulas for X a point which
are based on algebraic structures used for solving Korteweg-de Vries
equation. The appearance of these equations in quantum cohomology is
further reflected in the well known "Virasoro Conjecture". This
asserts that the quantum cohomological invariants are fixed points of
symmetries consisting of half a Virasoro algebra. These algebras are
known to act on many mathematical structures, in particular on the
solution sets of most integrable equations. There is little
speculation or conjecture as to the reason for this truly amazing and
unlikely mating of two entirely different subjects of integrable
systems and topological invariants. **

**On the other hand, the
Korteweg-de Vries equations themselves, which appeared in the 19th
century to describe solitary water waves, have already played several
roles in mathematics. The first lecture will be devoted to an
elementary description of some of these earlier appearances. The
second lecture will be devoted to the Virasoro symmetries and one
explanation of their appearance in integrable systems. One goal of
the talk is to interest the audience in deeper questions about
connections between these two subjects and topology.
**

**The lectures are intended
for a broad mathematical audience rather than for specialists. A
good part of the first lecture should be comprehensible to anyone who
understands some differential equations. There is a nice
web site at Herriot Watt University in Edinburgh which has
some fun information about solitons. In particular, one of their
history pages gives some information about
John Scott
Russell ,who appears to have been a fascinating
man. I also like the particular
photo from their collection of a
water wave soliton best, although the pictures of the corresponding
to a
Sine Gordon solution from Richard Palais' home page is also fun.
**

**An introductory article The Symmetries
of Solutions by Richard Palais contains more
interesting history,
some of which I will repeat in the talk. An
article written by Chuu-Lian Terng and myself which appeared two
years ago in the Notices of the American Mathematical Society gives
an
introduction to one method of viewing inverse scattering. I
will define and explain the Virasoro actions, which can be
understood as acting through the inverse scattering transform.
This article might be good preparation for my explanation of the
Virasoro actions. **

**The address of the
International Affairs Building:
420 West 118th Street**

**Go to the 6th floor- Plaza Entrance**

**Dag Hammarskjold Lounge
**

**Thursday, April 18 &**** ****
Friday, April 19**

**4:30 p.m.**

*Tea will be served at 3:30
p.m. -
508 Mathematics Building
*

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