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.
We'll introduce the notion of Langlands reciprocity through its connections to counting problems in modular arithmetic. We'll then discuss some modern geometric approaches to these questions using congruences of a different nature.
How do neural circuits in the brain accomplish rapid learning? When foraging for food in a previously unexplored environment, animals store memories of landmarks based on as few as one single view. Also, animals remember landmarks and navigation decisions that eventually lead to food, which requires that the brain associate events with delayed outcomes. I will present evidence that a particular neural circuit structure found in the hippocampus and cortex enables exactly this type of one-shot learning across a delay.
The brain network has an exquisite ability to process sensory information robustly and efficiently. The complex connectivity structure and rich neuronal dynamics in the brain network are believed to underlie such optimal coding capability. In this talk, I will introduce two different approaches investigating how the brain network structure, dynamics, and computational strategies shape each other.
Chemotaxis is the mechanism by which unicellular or multicellular organisms direct their movements in response to a stimulating chemical in the environment. Bacterial chemotaxis was discovered by T. W. Engelmann and W. Pfeffer in 1880s, and over one century's research has illustrated its importance in many physiological processes. In the 1970s, E. Keller and L. Segel proposed a system of two coupled partial differential equations to describe the traveling bands of \textit{E.
This talk will give an introduction to my current project, which aims to write all the theorems and definitions of mathematics in a computer-readable form. By "computer-readable", we mean much more than TeX, Maple, or Sage. We mean that the math is expressed in terms of the rules of logic and foundations of mathematics. This project is expected eventually to encompass all branches of mathematics.