Friday, April 17, 2026

Title: Weighted Inequalities for the Bergman Projection on Low Regularity Domains
Speaker: Nathan Wagner, George Mason University
Time: 3:00pm–4:00pm
Place: CMC 130
Sponsor: C. Felder

Abstract

The Bergman space is a fundamental space of analytic functions in complex analysis, and the Bergman projection is the Hilbert space orthogonal projection onto the Bergman space. As such, different mapping properties of the Bergman projection have been studied extensively and have meaningful connections to various problems in analysis. Motivated by analogy with real-variable singular integral theory, weighted \(L^p\) inequalities for the Bergman projection have been studied since the late 1970s. Much previous work has focused on generalizing unweighted Lebesgue space estimates on general classes of domains in several complex variables, but comparatively little work has been done to extend weighted inequalities until recently.

Of particular interest (even in the unweighted setting) is the case where low regularity assumptions are imposed on the boundary of the underlying domain. In this talk, we will take a broad tour through the history and mathematical context of this subject, highlighting the boundary smoothness hypotheses throughout. We will discuss joint work with Brett D. Wick and Cody B. Stockdale on the strongly pseudoconvex case with near-minimal \((C^4)\) boundary smoothness. We will also highlight a recent contribution with A. Walton Green on this topic in the special case of planar domains. Time permitting, we will include a cursory overview of some areas of ongoing investigation.

Friday, April 10, 2026

Title: Equilateral Triangulations of Riemann Surfaces
Speaker: Christopher Bishop, Distinguished Professor, Stony Brook University
Time: 3:00pm–4:00pm
Place: CMC 130
Sponsor: Nagle Lecture Series Committee

Abstract

After defining harmonic measure on a planar domain, I will discuss “true trees”. These are trees drawn in the plane so that every edge has equal harmonic measure, and so that the measure restricted to each edge is the same from either side of the edge. Such trees correspond in a precise way to polynomials that have exactly two critical values, and they form a special case of a dessin in Grothendieck's theory of dessins d'enfant, where a graph on a topological surface induces a conformal structure on that surface. I will explain the connection between dessins, equilateral triangulations of Riemann surfaces, and branched coverings of the 2-sphere (Belyi's theorem). Although it is easy to see that only countably many compact Riemann surfaces have any equilateral triangulation, Lasse Rempe and I recently proved that every non-compact surface has uncountably many. In particular, this implies that every Riemann surface is a holomorphic branched cover of the 2-sphere, branched over finitely many points. If time permits, I will briefly discuss “infinite true trees” that correspond to entire functions with exactly two critical values. A new, flexible construction of such functions, called “quasiconformal folding”, has had numerous applications in complex analysis and holomorphic dynamics.

Friday, April 3, 2026

Title: Improving the Power of Controlled Variable Selection with Prior Information
Speaker: Xiangyu Zhang, Ph.D. Candidate, Yale University
Time: 3:00pm–4:00pm
Place: CMC 130
Sponsor: J. Gu

Abstract

Genome-wide association studies (GWAS) identify genetic variants associated with traits within genomic regions. Functional annotations provide valuable prior information to prioritize variants and enhance power. However, substantial correlations among variants make it challenging to rigorously control the false discovery rate (FDR) while leveraging prior knowledge. We introduce annotation-informed knockoffs (AnnoKn), a knockoff-based method that performs annotation-informed variable selection with strict control of the false discovery rate. AnnoKn integrates the knockoff procedure with adaptive Lasso within a unified Bayesian framework to incorporate functional annotations. To facilitate applications where individual-level data are inaccessible, we extend AnnoKn to operate on summary statistics. Through simulations and real-world applications, AnnoKn demonstrates superior power in detecting causal variants compared with existing methods while maintaining valid control over false discoveries.

Friday, March 13, 2026

Title: Exploring Functional Clustering through Bayesian Nonparametric Models
Speaker: Wenyu Gao, University of North Carolina at Charlotte
Time: 3:00pm–4:00pm
Place: CMC 130
Sponsor: J. Min

Abstract

Clustering functional data poses significant statistical challenges, especially when data exhibits complex dependencies and the number of latent groups is unknown. Nonparametric Bayesian methods, such as Dirichlet Process Mixture (DPM) models, provide a principled mechanism for inferring cluster structures without pre-specifying the number of clusters. However, traditional DPM frameworks often fail to fully exploit the rich covariate information commonly available in modern functional datasets. To address this limitation, the Weighted Dirichlet Process Mixture (WDPM) model introduces a flexible weighting mechanism that allows external covariates to directly influence the clustering process. Compared with extensions such as the Dependent Dirichlet Process (DDP), the WDPM framework offers interpretation simplicity while retaining the capacity to capture covariate-dependent structure.

This presentation demonstrates how WDPM models can be effectively applied to functional data by integrating subject-level covariates, such as spatial characteristics or demographic and clinical factors, into the clustering process. We illustrate the methodology using functional magnetic resonance imaging (fMRI) data from individuals with autism spectrum disorder (ASD). By embedding relevant covariate information within the weight function, the WDPM model jointly partitions functional connectivity profiles while accounting for known biological and clinical heterogeneity. The results highlight the model’s potential to enhance clustering accuracy and yield more interpretable, scientifically meaningful insights into functional variability within the ASD population.

Friday, February 20, 2026

Title: Biochemical computing for noise-robust neural network implementation
Speaker: Jinsu Kim, POSTECH, Korean Republic
Time: 3:00pm–4:00pm
Place: CMC 130
Sponsor: N. Jonoska

Abstract

Artificial neural networks can be implemented using chemical reaction networks, where species concentrations serve as inputs and outputs. Previous studies have primarily focused on feed-forward architectures and have employed ReLU or discrete activation functions because of their structural simplicity in chemical reaction implementations. However, chemical reactions are inherently stochastic, and the discontinuities of such activation functions may increase sensitivity to noise, particularly during backpropagation. Consequently, achieving noise-robust computation is an important challenge in chemical reaction-based neural network design.

In this work, we propose a chemical reaction framework that performs both feed-forward computation and training using smooth activation functions, thereby avoiding discontinuities in derivatives. All reactions occur within a single well-mixed system and rely only on bimolecular reactions for the training dynamics. We show that the resulting chemical computing scheme maintains accuracy under perturbations in reaction rates.

Friday, February 6, 2026

Title: A guided tour of topological data analysis
Speaker: Peter Bubenik, University of Florida
Time: 3:00pm–4:00pm
Place: CMC 130
Sponsor: M. Elhamdadi

Abstract

Topological data analysis develops methods based on algebraic topology to provide insight into the structure of scientific data. Often these tools come with theorems guaranteeing that their output is stable with respect perturbations of the input. Furthermore, they may be easily combined with machine learning. I will give a survey of some aspects of this research area, including a few of its connections to old and new mathematics, and an application to video data.