Colloquia

We discuss a dimension-free deformation theory for Alexander maps and its applications.

In 1920, J. W. Alexander proved that every closed orientable PL (piecewise linear) n-manifold can be triangulated so that any two neighboring n-simplices are mapped to 
the upper and the lower hemispheres of Sⁿ, respectively. Such maps are called Alexander maps.   Rickman introduced a powerful 2-dimensional deformation method for Alexander maps, in his celebrated proof (1985) of  the sharpness of the Picard theorem in R³

Abstract:

Abstract:

Abstract:

Abstract:

Galois representations and modular forms are important objects of study in modern algebraic number theory.  To study the relationship between them, it is often fruitful to study congruences between them.  I will give an introduction to this theory, and I will conclude by discussing some recent results and applications.

We will begin with two disparate and highly influential questions in arithmetic. For what odd primes p is it straightforward to prove that the Fermat equation x^p + y^p = z^p has no non-trivial solutions among the rational numbers? And considering all possible elliptic curve equations, one particular example being y^2 + y = x^3 - x^2, what are all of the possibilities for the structure of the rational solutions as an abelian group?

It has been long proposed that the brain should perform computation efficiently to increase the fitness of the organism. However, the validity of this prominent hypothesis remains largely debated. I have investigated how the idea of efficient computation can guide us to understand the operational regimes underlying various functions of the brain.

Past decades of auditory research have identified several acoustic features that influence perceptual organization of sound, in particular, the frequency of tones and the rate of presentation. One class of stimuli that have been intensively studied are sequences of tones that alternate in frequency. They are typically presented in patterns of repeating triplets ABA_ABA_... with tones A and B separated in frequency by several semitones (DF) and followed by a gap of silence "_".

Graphs can be viewed as (non-archimedean) analogues of Riemann surfaces. For example, there is a notion of Jacobians for graphs. More classically, graphs can be viewed as electrical networks.

I will explain the interplay between these points of view, as well as some recent application in arithmetic geometry.