A foundational result in equivariant commutative algebra is Cohen's theorem that the infinite variable polynomial ring \(R=\mathbb{C}[x_1,x_2,\ldots]\) is Noetherian up to the action of the infinite symmetric group. This has been applied to prove uniformity results for finite-dimensional structures in algebraic geometry, statistics, and algebraic topology, and motivates the study of other aspects of the equivariant commutative algebra of \(R\). In this talk, we explain an approach to developing the local theory of \(R\)-modules in this equivariant setting by studying a generalization of FI-modules to a "weighted" setting. We introduce these weighted FI-modules, and discuss how they too can be studied from the perspective of commutative algebra up to the action of parabolic subgroups of the infinite general linear group.

We prove an analog of the disintegration theorem for tracial von Neumann algebras in the setting of elementary equivalence rather than isomorphism, showing that elementary equivalence of two direct integrals implies fiberwise elementary equivalence under mild, and necessary, hypotheses. This verifies a conjecture of Farah and Ghasemi. Our argument uses a continuous analog of ultraproducts where an ultrafilter on a discrete index set is replaced by a character on a commutative von Neumann algebra, which is closely related to Keisler randomizations of metric structures. We extend several essential results on ultraproducts, such as Łoś's theorem and countable saturation, to this more general setting.

Real-world networks frequently exhibit a clustering phenomenon where the friends of friends are likely to be friends. We show that the clustering effect is highly correlated with Ricci curvatures of a graph for random clustering graphs with given degree distributions. In particular, we show that for a random clustering graph with certain power-law degree distributions the Ricci curvature (in the sense of Lin, Lu, and Yau) is concentrated around the clustering coefficient.

Based on joint work with Fan Chung (UCSD), Michael Rawson (PNNL), Zhaiming Shen (University of Georgia), and Murong Xu (University of Scranton).

Results on the regularity of operators on -spaces are often proved by means of interpolation operators applied to estimates at the endpoints. A classical example is that of the Hibert transform on the real line, the -behavior of which can be deduced from a weak  type (1,1) estimate and the Marcinkiewicz interpolation theorem. Attempts to apply this idea to the Bergman projection on certain domains such as the Hartogs triangle  in  lead to some unexpected endpoint behavior. In particular, we show that for the Hartogs triangle, at the left endpoint  of the interval of -boundedness, the Bergman projection  on this domain is of restricted strong type  in the sense of Stein-Weiss, that is, for a characteristic function  of a measurable subset , we have

for a constant  independent of . This now determines the -behavior of the Bergman projection via classical interpolation results. We discuss several generalizations of this result to other domains. This is ongoing joint work with Zhenghui Huo of Duke Kunshan University, China.

Crystalline deformation rings play an important role in Kisin's proof of the Fontaine-Mazur conjecture for GL2 in most cases. One crucial step in the proof is to prove the Breuil-Mezard conjecture on the Hilbert-Samuel multiplicity of the special fiber of the crystalline deformation ring. In pursuit of formulating a horizontal version of the Breuil-Mezard conjecture, we develop an algorithm to compute arbitrarily close approximations of crystalline deformation rings. Our approach, based on reverse-engineering the Taylor-Wiles-Kisin patching method, aims to provide detailed insights into these rings and their structural properties, at least conjecturally.

[pre-talk at 3:00PM]

We show that the random degree constrained process (a time-evolving random graph model with degree constraints) has a local weak limit, provided that the underlying host graphs are high degree almost regular. We, moreover, identify the limit object as a multi-type branching process, by combining coupling arguments with the analysis of a certain recursive tree process. Using a spectral characterization, we also give an asymptotic expansion of the critical time when the giant component emerges in the so-called random $d$-process, resolving a problem of Warnke and Wormald for large $d$.

Based on joint work with Balázs Ráth and Lutz Warnke; see arXiv:2409.11747

Entire critical points of the abelian Higgs functional are known to blow down to generalized minimal submanifolds (of codimension 2). In this talk we prove an Allard type large-scale regularity result for the zero set of solutions. In the "multiplicity one" regime, we show the uniqueness of blow-downs and classify entire solutions in low dimensions and minimizers in all dimensions; thus obtaining an analogue of Savin's theorem in codimension two. This is based on a joint work with Guido de Philippis and Alessandro Pigati.

Embryogenesis--generation of functional forms--entails coordinated cell motion (morphogenesis), intercellular communications via morphogen patterns, and cell fate decisions. Morphogenesis and patterning have traditionally been studied separately, and how cell movement affects cell fates remains unclear. Traditional models of pattern formation deal mostly with static tissues, preventing the rationalization of increasingly available spatiotemporal data of morphogens and flows in remodeling tissues. We present a theoretical framework for pattern formation in motile cell environments by describing the dynamics of morphogen exposure felt by moving cells (Lagrangian frame) rather than at fixed laboratory coordinates (traditional Eulerian frame). This cell frame description reveals how morphogenetic motifs such as multicellular attractors and repellers (i.e., the Dynamic Morphoskeleton) and convergent extension flows act as barriers and enhancers to diffusive morphogen transport, revealing a robust synergy between morphogenesis and intercellular signaling. We apply our framework to standard models for dynamic cell fate bifurcations and induction and to experimental data from avian gastrulation flows.

Inverse problem is the study of the recovery of parameters or the governing equations of a system based on given observational data. In this talk I will focus on the techniques used in inverse scattering problems for hyperbolic type equations. Scalar wave equation will be used as an example for showing how one can use higher order linearization and microlocal analysis to retrieve information about the unknown nonlinearity. I will also explain the main ideas used in proving that the bounded time scattering map uniquely determines the nonlinearity in the semilinear 4 by 4 Dirac system, with some mild assumptions.

In this talk I will show you how to maximize your a-mean-ability. More precisely, I will survey some results about maximal amenable subgroups and subalgebras and share a new result which states that it is possible for the space of maximal amenable extensions to be any k-simplex (such as a triangle!). While this talk will technically be about operator algebras, it will be accessible to anyone who knows a bit of group theory (semi-direct products, free products, and the left regular representation).

I will discuss recent results describing the Chow rings and the tautological classes of the moduli space of quasi-polarized K3 surfaces of degree 2. This is based on joint work with Samir Canning and Rahul Pandharipande.

SDSU Climate Informatics Lab has developed a suite of computer code and Apps for visualizing and delivering real climate data for the general public, such as school classrooms. This presentation will specifically demonstrate the following tools:

1.  4-dimensional visual delivery (4DVD) of big climate data: www.4dvd.org.

2. Statistics, machine learning, and data visualization for climate science with R and Python: www.climatestatistics.org

3. Climate mathematics with R and Python:  www.climatemathematics.org

4. 4DVD Rural Heat Island for a California Climate Action project: www.4dvdrhi.sdsu.edu

We will also discuss our proprietary database optimization algorithms for fast queries. Using cutting-edge database technologies and 3D video games, we will outline our product development for the NSF program of AI Institutes and NOAA National Centers for Environmental Information.