Given a variety X over a field, it is generally difficult to understand the structure of vector bundles on X. As a first approximation, we might try to understand vector bundles only up to isomorphism. Classical isomorphism invariants of algebraic vector bundles include Chern classes and Euler classes, so we can study the extent to which these invariants determine a vector bundle. The analogous question in topology is a finite one: given a finite-dimensional manifold M, there are only finitely many isomorphism classes of complex rank r topological vector bundles on M with given topological Chern classes. However, such a finiteness result is not, in general, known in algebraic geometry. In this talk, I will discuss conditions under which algebraic characteristic classes determine algebraic rank 2 vector bundles on a given smooth affine fourfold up to finite choices. As a consequence, I will deduce complete isomorphism classification results for certain examples. This is joint work with Thomas Brazelton and Tariq Syed.

We introduce a positivity-preserving discontinuous Galerkin (DG) scheme for hyperbolic PDEs on unstructured meshes in 2D and 3D. The standard DG spaces are augmented with either polynomial or non-polynomial basis functions. The primary purpose of these augmented basis functions is to ensure that the cell average from the unmodulated DG scheme remains positive. We explicitly obtain suitable basis functions by inspecting the method of characteristics on an auxiliary problem. A key result is proved which demonstrates that the augmented DG scheme will retain a positive cell average, provided that the inflow, source term, and variable coefficients are positive. Standard slope limiters can then be leveraged to produce a high-order conservative positivity-preserving DG scheme. Numerical experiments demonstrate the scheme is able to retain high-order accuracy as well as robustness for variable coefficients and nonlinear problems.

We will discuss a characterization of first-order theories realizing a certain combinatorial tree configuration in terms of special coheirs.

The VIX--volatility index--is a number. It is transmitted to the world every 15 seconds from downtown Chicago, viewed right outside the window from UIC, at Cboe. It features Dispersion, Probability, Fear, and other topics of interest to statistics students. It is not related to the Black Scholes model, its volatility is not Variance, and the related Volatility of Volatility (VVIX) measures the Fear of Fear. An introduction to the VIX accessible to statistics students is presented via the Science Fictional Options Universe (SFOU) wherein, among other things, Probability does not exist: Never Happened. Not subjective, physical, or frequentist; no dice. People understand that the future is uncertain and have a sense about what is more or less likely, but "likely" in the SFOU is an informal concept, you know what I mean. In our universe it is like Hygge. *Disclosure: I am on the board of directors at the Cboe Futures Exchange that produces the VIX.

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

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Hamiltonian systems have long been a cornerstone of classical mechanics, providing a powerful framework to describe and analyze the motion of physical systems. But what happens when these mathematical structures take flight? In this talk, we will explore how the geometric principles of Hamiltonian mechanics play a crucial role in the modeling and control of modern aerial vehicles, including drones. From symplectic structures and variational principles to optimal control and real-world applications, we will uncover the elegant mathematical tools that bridge fundamental physics with cutting-edge drone technology. Whether you're interested in geometry, control theory, or just fascinated by the math behind autonomous systems, this talk will offer a compelling journey from theory to application in the skies.

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