Georgia Institute of Technology College of Sciences

In the second talk of this seminar series, we continue to follow the manuscript of Carl Miller and building up concepts from algebraic topology. In particular, we will introduce chain complexes to define homology groups and provide some of the standard theory for them. 

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