Let $D$ be a relatively compact $C^2$ domain in a complex manifold $X$ of dimension $n$. Assume that $H^1(D,\Theta)$ vanishes, where $\Theta$ is the sheaf of germs of holomorphic tangent fields of $D$. Suppose that the Levi-form of the boundary $b D$ has at least $3$ negative eigenvalues or at least $n-1$ positive eigenvalues pointwise. We will show that if a formally integrable  almost complex structure $H$ of the Holder class $C^r$ with $r>5/2$ on $D$ is sufficiently close to the complex structure on $D$, there is a embedding $F$ from $D$ into $X$ that transforms the almost complex structure into the complex structure on $F(D)$, where  $F $ has class $C^s$ for all $s<r+1/2$. This result was due to R. Hamilton in the 1970s when both $b D$ and $H$ are of class $C^\infty$.

Let $X$ be a normal variety with a projective contraction $f:X\to S$. Assume $U$ is an open subset of $S$ and $L_U$ is a Cartier divisor on $X_U:=X\times_S U$ such that $L_U$ is numerical trivial over $U$.

We will discuss about when it is possible and how to extend $L_U$ to a global Cartier divisor $L$ on $X$ such that $L\equiv_f 0$.

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.

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.

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.

Universal enveloping algebras of finite-dimensional Lie algebras are fundamental examples of well-behaved noncommutative rings. On the other hand, enveloping algebras of infinite-dimensional Lie algebras remain mysterious. For example, it is widely believed that they are never noetherian, but there are very few examples whose noetherianity is known. In this talk, I will summarize what is known about the noetherianity of enveloping algebras, with a focus on Lie algebras of derivations of associative algebras.

The canonical van der Waerden theorem states that, for large enough $n$, any colouring of $[n]$ gives rise to monochromatic or rainbow $k$-APs. In joint work with Alvarado, Kohayakawa, Morris and Mota, we study sparse random versions of this result. More concretely, we determine the threshold at which the binomial random set $[n]_p$ inherits the canonical van der Waerden properties of $[n]$, using the container method.

We describe a new and elementary proof of the fact that many groups of piecewise-projective transformation of the line are non-amenable by constructing an explicit action without fixed points. One the one hand, such groups provide explicit counter-examples to the Day-von Neumann problem. On the other hand, they illustrate that we can distinguish many "layers" of relative non-amenability between nested subgroups.

We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus's (2010) algorithm used the HOSVD to compute normal forms of almost all n-qubit pure states under the action of the local unitary group. Taking advantage of the double cover SL2(C)×SL2(C)→SO4(C), we produce similar algorithms (distinguished by the parity of n) that compute normal forms for almost all n-qubit pure states under the action of the SLOCC group.

Classical extremal results in graph theory (such as Turán's theorem) concern the maximal size of of a graph of given order and without certain subgraphs. Bollobás, Erdős, and Szemerédi in 1975 studied extremal problems in multipartite graphs. One of their problems (in its complementary form) was determining the maximal degree of a multipartite graph without an independent transversal. This problem has received considerable attention and was settle in 2006 (Szabó--Tardos and Haxell--Szabó). Other questions asked by Bollobás, Erdős, and Szemerédi remain open, such as determining:

(1) the maximum degree in a multipartite graph without a partial independent transversal, and;
(2) the minimum degree that forces an octahedral graph in balanced tripartite graphs.

In this talk I will survey recent progress on these and other related problems.