The Strominger-Yau-Zaslow conjecture predicts Calabi-Yau manifolds admit special Lagrangian fibrations and provides a recipe for the mirror construction. In this talk, I will explain the existence of special Lagrangian fibrations in certain log Calabi-Yau surfaces. Moreover, with the suitable mirror map the special Lagrangian fibrations on the mirror pairs are dual to each other. The study of these special Lagrangian fibrations also accidentally proved the Torelli theorems of certain types of gravitational instantons. Moreover, these setup the foundation for the equivalence of certain open Gromov-Witten invariants and log Gromov-Witten invariants.

Given a graph $H$, we say that a graph $G$ is properly rainbow $H$-saturated if there exists some proper edge-coloring of $G$ which has no rainbow copy of $H$, but adding any edge to $G$ makes such an edge-coloring impossible. The proper rainbow saturation number is the minimum number of edges in an $n$-vertex properly rainbow $H$-saturated graph. In this talk, we will discuss new bounds on the proper rainbow saturation number for odd cycles, paths, and cliques.

I will discuss the Nash iterative construction of non-conservative weak solutions to the Euler equations, with a particular focus on the difficulties presented by the two-dimensional case. I will then present a linear decoupling method, which enabled the construction of examples achieving sharp regularity for the 2D Euler equations, as well as for other systems.

The theory of closed H-fields is model complete and axiomatizes the theory of transseries and maximal Hardy fields, as Aschenbrenner, Van den Dries, and Van der Hoeven have shown in a long series of works. To better understand large closed H-fields, such as maximal Hardy fields, I recently extended this model completeness to the theory of tame pairs of closed H-fields. Building on this work, I will explain how to extend differential-algebraic dimension on a closed H-field to tame pairs of closed H-fields so that it is a fibred dimension function in the sense of [L. van den Dries, "Dimension of definable sets, algebraic boundedness and Henselian fields", Ann. Pure Appl. Logic 45.2 (1989), 189–209] and the nonempty dimension zero definable sets are exactly the nonempty discrete definable sets. The model-theoretic notion of coanalyzability will also make an appearance.

Let $X$ be a K3 surface. Consider a finitely supported probability measure $\mu$ on $\operatorname{Aut}(X)$ such that $\Gamma_{\mu} = \langle \operatorname{Supp}(\mu)\rangle < Aut(X)$ is non-elementary. We do not assume that $\Gamma_{\mu}$ contains any parabolic elements. We study and classify hyperbolic ergodic $\mu$-stationary probability measures on $X$.

Undergrads who participated in the DRP will give short presentations on the topic they learned and read about over the course of the semester. Detailed titles and abstracts will be posted soon. This semester twenty+ UIC undergraduates participated in the Directed Reading Program at UIC. In the DRP undergrads are paired with a supportive graduate student mentor and read through some mathematical text of the mentees choosing. One explicit goal of the program is to help undergraduates build mathematical strength, maturity, and independence. Titles and Abstracts can be found here: https://docs.google.com/document/d/1wRRme1QXOUrA9VubACfUIepr4ybkueXqu4OYvYTjfwE/edit?usp=sharing

Undergrads who participated in the DRP will give short presentations on the topic they learned and read about over the course of the semester. Detailed titles and abstracts will be posted soon. This semester twenty+ UIC undergraduates participated in the Directed Reading Program at UIC. In the DRP undergrads are paired with a supportive graduate student mentor and read through some mathematical text of the mentees choosing. One explicit goal of the program is to help undergraduates build mathematical strength, maturity, and independence. Titles and Abstracts can be found here: https://docs.google.com/document/d/1wRRme1QXOUrA9VubACfUIepr4ybkueXqu4OYvYTjfwE/edit?usp=sharing

A mixed identity in group G is an equation W(x)=1 where W is a non-trivial word in the free product G∗⟨x⟩, which is satisfied for all x∈G. Mixed Identity Free (MIF) means that no such identity holds on G. When G has no mixed identities, one wishes to find such x effectively (w.r.t. the word metric). Set f(n)=min { | g | : g∈G , W(g)≠1 for all W∈B(n) } where B(n) is the n-th ball in G∗⟨x⟩. If f is sub-exponential, there are outstanding applications for the reduced C*-algebra of the group, especially when the group also has rapid decay. Recently, Elayavalli and Schafhauser gave a negative answer for the C*-algebraic Tarski problem by studying this property for free groups. More recently, Itamar Vigdorovich extended their work to uniform lattices in SL(n,R). What we proved is: Theorem 1. For a f.g. linear group \Gamma with MIF, the function f is linear (i.e. f(n)