Analysis, Geometry and Probability Seminar: Fall 2026 (Carver Hall 282)
September 9th: Rafiq Islam (ISU)
Title: Scalable Sampling Algorithms via Langevin Dynamics
Abstract: Langevin algorithms are popular Markov chain Monte Carlo (MCMC) methods for large-scale sampling
from distributions with probability density $\pi(x)\propto e^{-f(x)}$. In this talk, we present three Langevin
sampling algorithms under strong convexity and smoothness assumptions on $f$. We first study the decentralized
proximal stochastic gradient Langevin dynamics (DE-PSGLD) algorithm for sampling from a convex body
$\mathcal{K}\subset\mathbb{R}^d$ when the data is distributed among $N$ computational agents. Constraints are
handled through a shared proximal regularization based on the Moreau–Yosida envelope, allowing unconstrained
updates to approximate the constrained target distribution. Next, we study unconstrained sampling in a decentralized
setting, where agents perform Bayesian learning collaboratively without sharing individual data. Existing
decentralized SGLD algorithms introduce network-induced bias that persists even when using full batches. To address
this bias, we propose the generalized EXTRA stochastic gradient Langevin dynamics. Finally, we propose the $P$-th
order Langevin dynamics for any $P\geq 3$, with improved dependence on dimension and sampling accuracy. For all
three algorithms, we obtain convergence guarantees and iteration complexities in the 2-Wasserstein distance. We
also validate our algorithms through numerical experiments in Bayesian regression, classification, and deep learning.
September 16th: No speaker. Miller Family Endowed Mathematics Lecture Series.
September 23rd: No speaker.
September 30th: Micah Coats (ISU)
Title: Symmetrization on Hyperbolic Spaces
Abstract:Schwarz symmetrization on a function is a rearrangement of such that measures of level sets are
unchanged and where the rearrangement is radially symmetric and decreasing. It is useful as a tool for optimization
problems and was classically important for proving the isoperimetric inequality. Many of the results from carry
over to , the hyperbolic space, and in particular, Talenti’s theorem for rearrangements also hold in . We follow the
elegant technique outlined by Baernstein and verify that it holds in . Some possible future directions for the technique
will also be discussed.
October 7th: Jonathan Wood (ISU)
Title: From Sightlines to Envelopes: A Differential Geometry Approach to Roadway Visibility
Abstract: Providing adequate stopping sight distance on a horizontal roadway curve requires
that a driver be able to see an object far enough ahead to perceive, react to, and stop before reaching
it. Obstructions on the inside of the curve can reduce the available sight distance below this
requirement. Consequently, a region along the interior of the curve must remain free of visual obstructions.
As the driver approaches, enters, and traverses the curve, the sightlines connecting the driver to an object
located a prescribed distance ahead along the roadway generate a family of line segments whose envelope defines
the required clearance boundary. This seminar develops that boundary using the classical interpretation of an
envelope as the limiting intersections of neighboring members of a family of curves. The sightlines are represented
by a vector-valued mapping, and differential geometry is used to obtain the envelope from the condition that its
Jacobian determinant vanishes. Closed-form expressions are developed for a circular arc joined to tangent segments,
accounting for changes in geometry as the driver and object move along the roadway. The formulation determines
both the required lateral offset and its location along the roadway alignment. It also reduces to the conventional
middle-ordinate equation when the driver and object are both within the circular curve. The presentation
illustrates how a visibility requirement can be expressed and solved as a geometric envelope problem.
October 14th: Jessica Lin (McGill)
Title: Homogenization and Invariant Measures for Nonreversible Diffusions in Random Media
Abstract: In this talk, I will first give an overview of stochastic homogenization for nondivergence form
elliptic equations (from the PDE perspective) and quenched invariance principles for nonreversible diffusion processes
(from the probability perspective). I will then present new quantitative homogenization results for the fundamental
solution and for the unique ergodic invariant measure. This invariant measure is a solution of the adjoint equation in
doubly divergence form, satisfying certain integrability conditions. I will discuss the implications of these homogenization
results, such as heat kernel bounds on the parabolic Green function and quantitative ergodicity for the environmental
process. This talk is based on joint work with Scott Armstrong (NYU/Sorbonne) and Benjamin Fehrman (LSU).
October 21th: Chase Giles (ISU)
Title: Regularity for Fractional Laplacian Transmission Problems
Abstract: Most physical problems that involve the behavior of a quantity
or solution changing across a barrier can be formulated as a transmission problem (PDE or otherwise). Taking
inspiration from the recent literature, we investigate a transmission problem for the fractional Laplacian.
Consequently, we prove an embedding regularity result. Motivated by the fraction Laplacian, we also investigate
fractal interfaces and the regularity of solutions transitioning across them.
October 28th: Joe Groszkiewicz (ISU)
Title:
Abstract:
November 4th: Suraj Kothari (ISU)
Title:
Abstract:
December 2nd: Rob Neel (Lehigh University)
Title:
Abstract: