Seminars, Colloquia, and Conferences
Colloquium
The colloquium meets on Fridays at 4:00pm in
BromfieldPearson 101, unless otherwise indicated.
Spring 2016
March 11, 2016
Elliptic Curves, Triangles and Lfunctions
David Roe (University of Pittsburgh)
Abstract:
Given a positive integer n, is n the area of a right triangle with rational side lengths?
Called the congruent number problem, this question has a rich history culminating in a theorem
of Tunnell (1983) that relates it to elliptic curves of the form y^2 = x^3  n^2x. But the story
does not stop there, since a full resolution of the problem requires the Birch and SwinnertonDyer
conjecture, one of the million dollar Clay Math Prize problems.
I will describe the connection between the *rank* of an elliptic curve, the congruent number
problem and the Birch and SwinnertonDyer conjecture, giving many examples. If time permits,
I will also explore the rank of elliptic curves further by describing the search for curves
of high rank and some recent heuristics that suggest that the rank of an elliptic curve over*Q* may
be absolutely bounded.
March 18, 2016
The Theory of Chaos: Yesterday, Today and Tomorrow
Yakov Pesin (Pennsylvania State University)
Abstract:
The talk is a brief historical account of the development of the theory that
deals with the phenomenon widely known as "deterministic chaos"  the appearance
of irregular chaotic motions in purely deterministic dynamical systems on compact
phase spaces.
The hyperbolic theory of dynamical systems provides a mathematical foundation for
this paradigm and thus serves as a basis for the theory of chaos. The hyperbolic
behavior can be interpreted in various ways and the weakest one is associated with
dynamical systems with nonzero Lyapunov exponents.
I will describe various types of hyperbolicity, outline some examples of systems
with hyperbolic behavior and discuss the stillopen problem of whether chaotic
dynamical systems are generic. This genericity problem is closely related to two
other important problems in dynamics on whether systems with nonzero Lyapunov
exponents exist on any compact phase space and whether chaotic behavior can coexist
with a regular (nonchaotic) one in a robust way.
April 1, 2016
Markedlength spectrum rigidity for unfriendly spaces
David Constantine (Wesleyan University)
Abstract:
The markedlength spectrum rigidity problem is the following:
Given a geometric space, if we know the lengths of the shortest
curves representing each element of the fundamental group, can we
recover the full geometry of the space. This problem has been most
fruitfully studied in negative curvature, where closed geodesics are
dense. The answer is yes for closed, negatively curved Riemannian
surfaces, and for compact, negatively curved locally symmetric
spaces of all dimensions. In this talk I'll give a general overview
of this problem, and discuss a few recent results on extending MLS
rigidity away from the Riemannian setting, to metrics which have
some nonsmooth structure, and which even allow some zero curvature.
Portions of this work are joint with JeanFrancois Lafont.
April 8, 2016
Norbert Wiener Lecture
Noam Elkies (Harvard University)
April 15, 2016
Martin Guterman Lecture
John Urschel (MIT)
April 22, 2016
Structure of zero sets
of random waves on a manifold
Yaiza Canzani (Harvard University)
Abstract:
There are several questions about the zero set of Laplace eigenfunctions
that have proved to be extremely hard to deal with and remain unsolved.
Among these are the study of the size of the zero set, the study of the
number of connected components, and the study of the topology of such
components. A natural approach is to randomize the problem and ask the
same questions for the zero sets of random linear combinations of eigenfunctions.
In this talk I will present some recent results in this direction.
Past Colloquia >
