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.
A fully nonlinear, closed-form mapping from time series of pressure measurements at an arbitrary depth in the fluid column to time series of surface displacement is derived from the full Stokes boundary value problem for water waves. The formula is implicit and nonlocal and must be solved numerically, albeit to machine precision.
Shortly after Szemeredi proved that a set of natural numbers with positive upper density contains arbitrarily long arithmetic progressions, Furstenberg gave a new proof using ergodic theory. This major event gave rise to the field of combinatorial ergodic theory, in which problems motivated by additive combinatorics are addressed with ergodic theory. In this talk we give a brief survey of some of the successes in the field, leading into a description of an interesting, yet still unproved result, which would provide a generalization of many earlier results.
A major challenge in the study of dynamical systems is that of model discovery: turning data into models that are not just predictive, but provide insight into the nature of the underlying dynamical system that generated the data and the best representation of an accompanying coordinate system. This problem is made more difficult by the fact that many systems of interest exhibit parametric dependencies and diverse behaviors across multiple time scales.