Georgia Institute of Technology College of Sciences

Let $M$ be the underlying smooth $4$-manifold of a degree $d$ del Pezzo surface. In this talk, we will discuss two related results about finite subgroups of the mapping class group $\text{Mod}(M) := \pi_0(\text{Homeo}^+(M))$. A motivating question for both results is the Nielsen realization problem for $M$: which finite subgroups $G$ of $\text{Mod}(M)$ have lifts to $\text{Diff}^+(M) \leq \text{Homeo}^+(M)$ under the quotient map $\pi: \text{Homeo}^+(M) \to \text{Mod}(M)$? For del Pezzo surfaces $M$ of degree $d \geq 7$, we will give a complete classification of such finite subgroups. Furthermore, we will give a classification of, and a structure theorem for, all involutions in $\text{Mod}(M)$ for all del Pezzo surfaces $M$. This yields a positive solution to the Nielsen realization problem for involutions on $M$ and a connection to Bertini's classification of birational involutions of $\mathbb{CP}^2$ (up to conjugation by birational automorphisms of $\mathbb{CP}^2$).

We will review how divisors on abstract algebraic curves are connected with projective embeddings and then see how that language translates to tropical curves and tropicalization. This talk aims to explain some of the connections between tropical curves and algebraic curves that was not discussed during the seminar on tropical Brill-Noether theory.

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We describe a spatial Moran model that captures mechanical interactions and directional growth in spatially extended populations. The model is analytically tractable and completely solvable under a mean-field approximation and can elucidate the mechanisms that drive the formation of population-level patterns. As an example, we model a population of E. coli growing in a rectangular microfluidic trap. We show that spatial patterns can arise because of a tug-of-war between boundary effects and growth rate modulations due to cell-cell interactions: Cells align parallel to the long side of the trap when boundary effects dominate. However, when cell-cell interactions exceed a critical value, cells align orthogonally to the trap’s long side. This modeling approach and analysis can be extended to directionally growing cells in a variety of domains to provide insight into how local and global interactions shape collective behavior. As an example, we discuss how our model reveals how changes to a cell-shape describing parameter may manifest at the population level of the microbial collective. Specifically, we discuss mechanisms revealed by our model on how we may be able to control spatiotemporal patterning by modifying cell shape of a given strain in a multi-strain microbial consortium.

Recording Link: https://bluejeans.com/s/0g6lBzbf0XT

In modern data analysis, the datasets are often represented by large-scale matrices or tensors (the generalization of matrices to higher dimensions). To have a better understanding or extract   values effectively from these data, an important step is to construct a low-dimensional/compressed representation of the data that may be better to analyze and interpret in light of a corpus of field-specific information. To implement the goal, a primary tool is the matrix/tensor decomposition. In this talk, I will talk about novel matrix/tensor decompositions, CUR decompositions, which are memory efficient and computationally cheap. Besides, I will also discuss the applications of CUR decompositions on developing efficient algorithms or models to robust decompositions or data completion problems. Additionally, some simulation results will be provided on real and synthetic datasets.