Colloquia

Eisenstein congruences and arithmetic

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?

Efficient Computation in the Brain - Using Computational Principles to Unify Behavior, Representation and Mechanism

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.

Horse Versus Morse – Can You Hear It? Dynamic Features of Bistable Perception in Auditory Streaming

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 and Arithmetic Geometry

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.

Using Top-Down Approaches to Investigate Compressible Sequence Learning in the Hippocampus

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