Mathematical Conversations

Oct 09 2019

Finite fields and the Ax-Grothendieck theorem

Speaker: Remy van Dobben de Bruyn
6:00pm | Dilworth Room
The Ax-Grothendieck theorem from the 1960s says that an injective polynomial $f : \mathbb C^n \to \mathbb C^n$ is also surjective. It is one of the first examples of the powerful technique in algebraic geometry of using finite fields to prove results over the complex numbers...
Apr 10 2019

How do computers do arithmetic, and should we believe the answers?

Speaker: Scott Tremaine
6:00pm | Dilworth Room
When designing the first computer built at IAS, von Neumann rejected floating-point arithmetic as neither necessary nor convenient. In 1997 William Kahan at Berkeley, who designed the famously accurate algorithms on Hewlett-Packard calculators, said that "floating–point...
Apr 03 2019

A glamorous movie star, the "bad boy" of music, and the development of spread spectrum communications

Speaker: Mark Goresky
6:00pm | Dilworth Room
An unlikely couple devised one of the first spread spectrum communication systems. Today these systems use sophisticated mathematics and are ubiquitous. This is a verbatim repeat (by popular demand) of a talk I gave about 6 years ago.
Mar 27 2019

A curious family of curves

Speaker: Amie Wilkinson
6:00pm | Dilworth Room
I will construct a family of curves in the square that illustrates the interplay between hyperbolic dynamics and pathology.
Mar 20 2019

from dynamics to contact topology and back

Speaker: Jo Nelson
6:00pm | White Levy Room
This is a light survey of the origins of contact topology and its applications to dynamics. We will use anecdotes and images to illustrate ideas.
Mar 13 2019

Wiggling and wrinkling

Speaker: Daniel Álvarez-Gavela
6:00pm | Dilworth Room
The idea of corrugation goes back to Whitney, who proved that homotopy classes of immersed curves in the plane are classified by their rotation number. Generalizing this result, Smale and Hirsch proved that the space of immersions of a manifold X into a manifold Y is (weakly...
Mar 06 2019

From Celestial Mechanics to the Arnold Conjectures

Speaker: Umberto Hryniewicz
6:00pm | Dilworth Room
The study of the planar-circular-restricted 3-body problem led to Poincaré's "last geometric theorem", nowadays known as the Poincaré-Birkhoff theorem. It is a fixed point theorem for certain area-preserving annulus homeomorphisms. Birkhoff's proof did not allow for...
Feb 27 2019

Hodge theory: matrices and differential equations

Speaker: Simion Filip
6:00pm | Dilworth Room
Solutions to some differential equations are related to geometric structures on the underlying manifold. For instance certain hypergeometric equations are related to the uniformization of Riemann surfaces. I will start by recalling some classical theorems from the 19th...
Feb 20 2019

Finite fields and the Ax–Grothendieck theorem

Speaker: Remy van Dobben de Bruyn
6:00pm | Dilworth Room
The Ax–Grothendieck theorem from the 1960s says that an injective polynomial $f \colon \mathbb{C}^n \rightarrow \mathbb{C}^n$ is also surjective. It is one of the first examples of the powerful technique in algebraic geometry of using finite fields to prove results over the...
Feb 13 2019

Harmonic measure and boundary regularity

Speaker: Zihui Zhao
6:00pm | Dilworth Room
Given a domain, the harmonic measure is a measure that relates any boundary function to its harmonic extension; it is also the hitting probability of the boundary for a Brownian motion moving inside the domain. We will talk about the relationship between the harmonic measure...

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