TBD

Wavelets provide a versatile framework for signal representation and analysis, integrating ideas from harmonic analysis, approximation theory, and practical algorithm design. In this talk, we introduce foundational concepts in wavelet theory, focusing on classical results regarding wavelet expansions and approximations. Building on these basics, we explore modern developments and discuss how these approaches can balance theoretical rigor with practical convenience. The presentation aims to offer both a solid introduction to classical wavelet theory and a glimpse into current and future research directions. Part of the talk is based on joint work with Hyojae Lim.

The Langlands program, introduced by Robert Langlands, is a set of conjectures that attempt to build bridges between two different areas: Number Theory and Representation Theory (Automorphic forms). The program is also known as a generalization of a well-known theorem called Fermat’s Last Theorem. More precisely, when Andrew Wiles proved Fermat’s Last Theorem, he proved a special case of so-called Taniyama-Shimura-Weil Conjecture, which states that every elliptic curve is modular. And as a corollary, he was able to prove Fermat’s Last Theorem since Taniyama-Shimura-Weil Conjecture implies that certain elliptic curves associated with Fermat-type equations must be modular, leading to a contradiction. Note that the Langlands program is a generalization of the Taniyama-Shimura-Weil conjecture. In the first part of the colloquium, we briefly go over the following subjects: (1) Fermat’s Last Theorem (2) Taniyama-Shimura-Weil conuecture And then, in the remaining of the talk, we start to explain a bit of the Langlands program (3) Langlands program and L-functions (4) (If time permits) Recent progress This colloquium will be accessible to graduate students in other fields of mathematics (and undergraduate students who are interested in Number theory) at least in the first part.

TBD

We study stochastic motion of objects in micrometer-scale living systems: tracer particles in living cells, pathogens in mucus, and single cells foraging for food. We use stochastic models and state space models to track objects through time and infer properties of objects and their surroundings. For example, we can calculate the distribution of first passage times for a pathogen to cross a mucus barrier, or we can spatially resolve the fluid properties of the cytoplasm in a living cell. Recently developed computational tools, particularly in the area of Markov Chain Monte Carlo, are creating new opportunities to improve multiple object tracking. The primary remaining challenge, called the data association problem, involves mapping measurement data (e.g., positions of objects in a video) to objects through time. I will discuss new developments in the field and ongoing efforts in my lab to implement them. I will motivate these techniques with specific examples that include tracking salmonella in GI mucus, genetically expressed proteins in the cell cytoplasm, active transport of nuclei in multinucleate fungal cells, and raphid diatoms in seawater surface interfaces.