Thursday, September 4A. Gogolev, Penn State
2:30pmTheory of Dynamical Systems on service to Number Theory
ABSTRACTTools from the theory of dynamical systems proved to be extremely useful in various problems in number theory. Outstanding examples include Furstenberg's proof of Szemeredi theorem, Green-Tao theorem,
solution of Oppenheim conjecture and progress towards Littlewood conjecture.
We will start with questions of uniform distribution and Diophantine approximation that illustrate helpfulness of dynamical tools. Then we will discuss modern ideas in dynamics on homogeneous spaces and outline the proof of Oppenheim conjecture.
No background in neither theory is needed.
Thursday, September 11M. Ghomi, Georgia Tech
2:30pmThe four vertex property and topology of surfaces with constant curvature
ABSTRACTThe classical four vertex theorem of Kneser states that any simple closed curve in the plane has four vertices, i.e., points where the curvature has a local max or min. This result has had many generalizations over the years. In particular, Pinkall has shown that any closed curve in the plane or the sphere which bounds a a compact immersed surface has four vertices. This result holds in the hyperbolic plane as well. In this talk we give an elementary introduction to these results, and present some further generalizations due to the speaker. In particular we show that the sphere is the only compact surface of constant curvature in which every closed curve which bounds an immersed surface has four vertices. Also, we give a similar characterization for the disk among all compact surfaces with boundary.
Thursday, September 25K. Stolarsky, University of Illinois
2:30pmHigher dimensional solutions to low dimensional problems
ABSTRACTWe show how a wide variety of problems that occur in a space of a certain dimension can be solved by considerations of a related mathematical object having a strictly larger dimension. In fact, when it comes to variables, it is sometimes the case that "the more the merrier". The classic example is Liouville's calculation of the integral of exp[-x^2}. We shall show that there are a large number and variety of other examples.
Thursday, October 2R. Schwartz, Brown University
2:30pmThe Devil's Pentagram
ABSTRACTThe pentagram map is a simple iteration one performs on polygons. In the case of a regular pentagon, the diagram looks like the famous pentagram well-known to practitioners of black magic and other dark arts. In general, the construction gives rise to an interesting dynamical system that is related to
determinants, integrable systems, and the monodromy of ordinary differential equations. I will explain the basic construction and give a partly visual tour of many features of the system.
Thursday, October 16F. Malikov, University of Southern California
2:30pmFundamental solutions of differential operators and D-module theory
ABSTRACTI will explain, following a classic paper by J. Bernstein, how the existence of fundamental solutions of differential operators with constant coefficients follows from elementary representation theory of the Weyl algebra. Bernstein's proof marked the emergence of algebraic D-module theory, which proved to be a powerful tool in representation theory of Lie algebras and, more recently, was instrumental for a mathematical formulation of conformal field theory.
Thursday, October 23V. Retakh, Rutgers University
2:30pmIntroduction to Hypergeometric Functions
ABSTRACTThe theory of hypergeometric functions started more than 300 years ago and continues to bring us more and more surprises. Hypergeometric functions and their close relatives, special functions, appear almost everywhere (in Analysis, Combinatorics, Algebraic Geometry, Number Theory, Theoretical
Physics) and always in the heart of the matter. The classical theory of hypergeometric functions (mostly in one variable) was developed by Euler, Gauss, and other greats. The modern theory of hypergeometric functions of several variables was started around 1985 by B. Dwork, I. Gelfand, M. Sato and their schools. It found striking applications to modern physics including the mirror symmetry. In this talk I will try to present an elementary approach to hypergeometric functions.
Thursday, November 6De Witt Sumners, Florida State University
2:30pmDNA Topology
ABSTRACTCellular DNA is a long, thread-like molecule with remarkably complex topology. Enzymes that manipulate the geometry and topology of cellular DNA perform many vital cellular processes (including segregation of daughter chromosomes, gene regulation, DNA repair, and generation of antibody diversity). Some enzymes pass DNA through itself via enzyme-bridged transient breaks in the DNA; other enzymes break the DNA apart and reconnect it to different ends. In the topological approach to enzymology, circular DNA is incubated with an enzyme, producing an enzyme signature in the form of DNA knots and links. By observing the changes in DNA geometry (supercoiling) and topology (knotting and linking) due to enzyme action, the enzyme binding and mechanism can often be characterized. This talk will discuss the tangle model for site-specific recombination.
Thursday, November 13I. Pak, University of Minnesota
2:30pmThe discrete square peg problem
ABSTRACTThe square peg problem asks whether every Jordan curve in the plane has four points which form a square. The problem has been resolved (positively) for various classes of curves, but remains open in full
generality. I will survey various known results and outline two direct proofs for the case of piecewise linear curves.
Thursday, December 4R. Devaney, Boston University
2:30pmCantor and Sierpinski, Julia and Fatou: Crazy Topology in Complex Dynamics
ABSTRACTIn this talk, we shall describe some of the rich topological structures that arise as Julia sets of certain complex functions including the exponential and rational maps. These objects include Cantor bouquets, indecomposable continua, and Sierpinski curves.
Thursday, October 30Josh Sabloff, Haverford College
2:30pmHow to Tie Your Unicycle in Knots: An Introduction to Legendrian Knot Theory
ABSTRACTYou can describe the configuration of a unicycle on a sidewalk using three coordinates: two
position coordinates x and y for where the wheel comes into contact with the ground and
one angle coordinate t that describes the angle that the direction the wheel makes with
the x axis. At a given point (x,y,t), the instantaneous motions of the unicycle (if we do not
want to scrape the tire by trying to move sideways) are constrained to moving in the
direction the wheel is pointing, turning the wheel without moving forward, or some
combination of the two. As you pedal around, you trace out a path in (x,y,t)-space that
obeys the constraints at every point.

The system of constraints at every point in (x,y,t)-space is an example of a "contact
structure," and a path that obeys the constraints is a "Legendrian curve." If the curve
returns to its starting point, then it is called a "Legendrian knot." A central question in
the theory of Legendrian knots is: how can you tell two Legendrian knots apart? How many
are there? In other words, how many ways are there to parallel park your unicycle?

There will NOT be a practical demonstration.