Colloquia

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.

The hippocampus is capable of rapidly learning incoming information, even if that information is only observed once.   Further, this information can be replayed in a compressed format during Sharp Wave Ripples (SPW-R).  We leveraged state-of-the-art techniques in training recurrent spiking networks to demonstrate how primarily interneuron networks can: 1) generate internal theta sequences to bind externally elicited spikes in the presence of septal inhibition, 2) compress learned spike sequences in the form of a SPW-R when septal inhibition is removed, 3) generate and r

The incompressible Euler equation of fluid mechanics describes motion of ideal fluid, and goes back to 1755. In two dimensions, global regularity of solutions is known, and double exponential in time upper bound on growth of the derivatives of solution goes back to 1930s. I will describe a construction of example showing sharpness of this upper bound, based on work joint with Vladimir Sverak. The construction has been motivated by a singularity formation scenario proposed by Hou and Luo for the 3D Euler equation.

Let X be a smooth separated geometrically connected variety defined over a characteristic p finite field, f : Y → X a smooth projective morphism, and w a non-negative integer. A celebrated result of Deligne states that the higher direct image Qℓ-sheaf Rwf∗Qℓ is semisimple on X geometrically for all prime ℓ not equal to p. By comparing the invariant dimensions of sufficiently many ℓ-adic and mod ℓ representations arising from the sheaves Rwf∗Qℓ and Rwf∗Fℓ respectively, we prove that the Fℓ-sheaf Rwf∗Fℓ is likewise semisimple on X geometrically if ℓ is sufficiently large.