Department of Mathematics

Date : Thursday, December 1, 2022
Time: 2pm - 3pm
Location : Goldsmith 300

Abstract: The stochastic processes of evolution have generated DNA, RNA, and protein sequences. These sequences determine how these entities chemically interact with themselves and each other, form physical structures, and functionally behave as signals and/or machines within cells. My research involves reconstructing the history of the stochastic processes that led to the sequences we observe today and developing methods to make biological predictions from sequences.

I will describe a new method for arranging related protein or RNA sequences into Multiple Sequence Alignments (MSAs) where each column corresponds to a position in an unobserved ancestral sequence. The patterns of conservation and covariation in an MSA can be leveraged for a variety of downstream tasks, including structure prediction. Generating an MSA is typically treated as a preprocessing step. Instead, we designed a method that learns an MSA jointly with a downstream machine learning task. To do so, we reformulated a classic dynamic programming algorithm so that it outputs a probability distribution over alignments and is differentiable. Using contact prediction as a case study, we show that it is possible to learn MSAs in this manner, and for many protein and RNA families doing so improves the accuracy of the predicted contacts. I will also briefly discuss (i) proper benchmarking of methods involving biological sequence data using graph algorithms, and (ii) predicting how the fitness of yeast emerges as a function of its DNA sequence. 

Date: Monday December 12, 2022
Time: 2pm - 3pm
Location : Goldsmith 300

Abstract:Exponential distributions on unitary conjugation orbits of Hermitian matrices, known as Harish-Chandra-Itzykson-Zuber (HCIZ) distributions, are important in various settings in physics and random matrix theory. However, the basic question of efficient sampling from these distributionshas remained open. In this talk, I will present two efficient algorithms to sample matrices from distributions that are close to an HCIZ distribution. Both algorithms exploit a natural self-reducible structure that arises from symmetries of the underlying group orbit. As a motivating application, I will discuss how these sampling algorithms yield a new state-of-the-art algorithm for differentially private low-rank approximation, and I will also sketch some connections to quantum information and integrable systems.  Based on joint work with Jonathan Leake and Nisheeth Vishnoi.

Date: Tuesday December 13, 2022
Time: 2pm - 3pm
Location: Goldsmith 300

My abstract is: This talk comprises two parts. In the first half of the talk (joint with Dejan Slepcev (CMU)), we revisit the problem of pointwise semi-supervised learning (SSL). Working on random geometric graphs (a.k.a point clouds) with few "labeled points", our task is to propagate these labels to the rest of the point cloud. Algorithms that are based on the graph Laplacian are often found to perform poorly in such pointwise learning tasks since minimizers develop localized spikes near labeled data, being essentially constant elsewhere. In the first half of the talk we introduce a class of graph-based higher order fractional Sobolev spaces (H^s) and study their consistency in the large data limit, along with applications to the SSL problem. The mathematical essence of the question is the continuity of the pointwise-evaluation functional, uniformly in the number of data points. A crucial tool is recent convergence results for the spectrum of the graph Laplacian to that of the continuum.  Obtaining optimal convergence rates for the spectrum of the graph laplacian on point clouds is an open question in stochastic homogenization. In the second part of the talk, we consider such a question in the simpler context of periodic homogenization. For a Schrodinger equation on all of space with a confining potential and periodic coefficients, we obtain optimal convergence rates and high order asymptotic expansions for the eigenvalues and eigenfunctions. Our results are optimal also in how high up in the spectrum we can go. Time permitting, we will discuss ongoing work on estimates on similar estimates in the random setting of a Poisson point process on $R^d$, where we use recent tools from stochastic homogenization in order to tame the large-scale geometry of the associated point cloud. This part is joint work with Scott Armstrong (Courant).

Date: Wednesday December 14, 2022
Time: 2pm - 3pm
Location: Goldsmith 300

Abstract:  The perspectives and opinions of people change and spread through social interactions on a daily basis. In the study of opinion dynamics on social networks, one often models social entities (such as twitter accounts) as nodes and their relationships (such as followship) as edges,and examines how opinions evolve as dynamical processes on networks, including graphs,hypergraphs, multi-layer networks, etc. In the first part of my talk, I will introduce a model of opinion dynamics and derive its mean-field limit as the total number of agents goes to infinity.The mean-field opinion density satisfies a kinetic equation of Kac type. We prove properties of the solution of this equation, including nonnegativity, conservativity, and steady-state convergence. The parameters of such opinion models play a nontrivial role in shaping the dynamics and can also be in the form of functions. In reality, it is often impractical to measure these parameters directly. In the second part of the talk, I will approach the problem from an`inverse’ perspective and present how to infer the parameters from limited partial observations. I will provide sufficient conditions of measurement for two scenarios, such that one is able to identify the parameters uniquely. I will also provide a numerical algorithm of the inference when the data set only has a limited number of data points.

Date: Thursday December 15th, 2022
Time: 2pm - 3pm
Location : Goldsmith 300

Abstract: Central pattern generators (CPGs) are neural networks that are intrinsically capable of producing rhythmic patterns of neural activity and are adaptable to sensory feedback to produce robust motor behaviors such as breathing and swallowing. The first half of the talk will be focused on analysis of bursting dynamics in multiple-time-scale systems, via a focus on respiratory CPG neurons. We use geometric singular perturbation theory, bifurcation analysis and modeling to uncover mechanisms underlying a variety of complex bursting activity patterns depending on distinct combinations of burst-generating conductances. Our analysis also yields predictions about how changes in the balance of two different bursting mechanisms contribute to alterations in respiratory neuron activity during prenatal development. Then I will discuss our work on dissecting robustness for motor control, the ability of a system to maintain its performance despite perturbations. Motor systems show an overall robustness, but because they are highly nonlinear, understanding how they achieve robustness due to their different components such as CPGs, sensory feedback and biomechanics is difficult. I’ll illustrate how our recently developed tools from variational analysis can be applied to study motor robustness by considering a neuromechanical model of motor patterns in the marine mollusk Aplysia californica. Our approaches and the results that we have obtained are likely to provide insights into the function and robustness of other motor systems.