Seminar

(joint with Computational Math Seminar)

Arithmetic statistics is an area devoted to counting a wide range of objects of algebraic interest, such as polynomials, fields, and elliptic curves.  Fueled by the interplay of analysis and number theory, we'll count polynomials and number fields, which though basic objects of study in number theory, are quite difficult to actually count.  How often does a random polynomial fail to have full Galois group?  How many number fields are there?  We will address both of these questions today.

TY 427

The purpose of this series of talks is to introduce Schwarzschild universe and a non-commutative generalization.  This first talk focuses on the Schwarzschild universe, by which we mean a maximal conformal analytic extension of the static, spherically symmetric space-time vacuum.  We shall discuss the structure of its null geodesics (they are elliptic curves), null geodesic deviation, and the theorem proven jointly by the speaker and George Sparling that every null geodesic in Schwarzschild "feels" the temperature of the singularity (a la Gibbons and Hawking).

Abstract. This is a report on joint work with Dan Ciubotaru.  We consider a cuspidal automorphic representation π of a semisimple group G over a function field K.  When G = PGL(n), the Ramanujan conjecture, which asserts that every local component of π is tempered, was proved in connection with the global Langlands correspondence by L.

Abstract: A basic problem in extremal graph theory is to find large graphs
that contain no copy of a specific subgraph. Surprisingly, in
essentially all the known instances, the best constructions for this
problem come from appropriate varieties. I will present several such
constructions, and give a survey of the area.