The reconstruction of flexible proteins is one of the most critical challenges in structural biology, allowing us to gain insights into the functions and mechanisms of biomolecules by observing their motion. Cryogenic electron microscopy (cryo-EM) stands out as an ideal technique for studying the dynamic conformational landscape (i.e., range of motions) as it can capture a snapshot of the entire conformational ensemble. However, this reconstruction task comes with notable mathematical and computational challenges due to massive datasets, sometimes exceeding terabytes, high dimensionality, and substantial noise.

After delving into the basics of the cryo-EM reconstruction problem, I will present a framework for reconstructing the protein distribution in a dataset by representing it in a linear subspace. The initial step can be viewed as a linear algebra problem: how can one compute a basis for proteins (3D volumes) from only incomplete and noisy measurements (2D images)? I will propose a method based on a Nyström extension of a regularized estimator of the covariance of the volumes. In subsequent steps, we will use this low-dimensional basis to reconstruct individual volumes using standard statistical methods and infer motions using elements of optimal control. I will conclude by discussing remaining challenges and open problems.

A relaxed-pace seminar on impromptu subjects related to the interests of the audience.

Everyone is welcome.

The subjects are geometry, probability, combinatorics, dynamics, and more!

Many modern electronic/optical devices rely on waves such as solar cells, antennae, radar, and lasers. These devices are mostly built on a patterned layered structure. For optimizing and characterizing these devices, numerical simulations play a crucial role. Two integral equation methods will be presented. Firstly, we developed a robust and fast computational method based on boundary integral equations for the Helmholtz equation in periodically patterned multilayered media. This method uses near- and far-field decomposition to avoid using the quasi-periodic Green’s function. By construction, far-field contribution can be compressed using Schur complement with minimal computational cost. Both the 2-D and 3-D numerical results will be presented. Secondly, two of the main challenges in wave simulation in layered media using layered media Green’s function will be discussed. The layered media Green’s function is represented by Sommerfeld integrals and evaluation of the layered media Green’s function for a given density function is accelerated by adapting the free-space fast multipole method by compressing the Sommerfeld integrals with multipole expansion with transformed basis. In the low-frequency regime, the convergence of multipole and local expansion follows the free-space method results in O(N) algorithm. Additionally, a slow convergence issue of Sommerfeld integral representation of layered media Green’s function when the target and source points are close to each other is overcome with a mathematically equivalent directional representation of Sommerfeld integrals.

A relaxed-pace seminar on impromptu subjects related to the interests of the audience.

Everyone is welcome.

The subjects are geometry, probability, combinatorics, dynamics, and more!

The Bateman-Horn conjecture gives a prediction for how often an irreducible polynomial takes on prime values. In this talk, I will discuss the proof of Bateman-Horn for two new polynomials – the determinant polynomial on nxn matrices and the determinant polynomial on nxn symmetric matrices. A key tool in the proof is the input of homogeneous dynamics to count the number of integral points on level sets. This talk is based on joint work with Giorgos Kotsovolis.

A relaxed-pace seminar on impromptu subjects related to the interests of the audience.

Everyone is welcome.

The subjects are geometry, probability, combinatorics, dynamics, and more!

Anosov groups constitute a rich class of discrete subgroups of Lie groups, offering both geometric and dynamical intricacies. This class of discrete groups also have deep connections with several current developments in mathematics, such as higher Teichmüller theory and thin groups. For a semisimple Lie group G, each conjugacy class of parabolic subgroups P of G gives rise to a family of Anosov subgroups known as P-Anosov. A natural inquiry is: Which abstract groups can arise as P-Anosov subgroups of G? In this talk, we will discuss some results that fully address this question for many specific pairs of G and P. This talk will be partly based on joint work with Z. Greenberg and J.M. Riestenberg.

The insides of cells are geometrically complex, heterogeneous, and dynamic.  Flows inside of cells can reflect the motion of internal structures, and so can be signatures of how forces transduce to motion, or can be an intrinsic part of a self-organizing process involving other moving parts like biopolymers and molecular motors. I’ll show examples of each, discuss the mathematical models that we’ve developed to describe them, and outline the numerical methods we use to simulate them.