We have been discussing how the study of model theory is structured by drawing dividing lines based on the presence or absence of various combinatorial configurations. For example, we introduced the order property and the binary tree property to characterize stable theories. In this talk, we will focus on the independence property and introduce NIP theories—those that lack the independence property. We will also explore how NIP theories generalizes different features of stable theories we previously discussed.

A stratum of differentials is the moduli space of curves together with a meromorphic form with prescribed multiplicities of zeroes and poles. The strata are phase spaces of an action of SL(2,R) and thus the central object of study in Teichmueller dynamics. On the other hand, they give natural high codimension subvarieties of the moduli of curves with marked points. The strata are non-compact, and we determine the number of their ends, and discuss a viewpoint towards further homology computations. This uses an algebraic compactification of the strata. Based on a joint work with Ben Dozier.

An independent set in a bipartite graph G = (X, Y, E) is balanced if it contains an equal number of vertices from each partition. A balanced coloring of G is a proper coloring of G such that each color class forms a balanced independent set. In this talk, we will discuss the recent extension of these definitions to multipartite hypergraphs. We establish bounds on the balanced independence number and balanced chromatic number of multipartite hypergraphs that are near-optimal as exhibited by the behavior of these parameters on random instances. This talk is partially based on joint work with Yuzhou Wang.

In this talk, we will give an introduction to big Ramsey degrees of homogeneous structures. Then we will concentrate on computability theoretic aspects and discuss work done in the area by various logicians, including work by Cholak, Dobrinen, McCoy.

The upper density of the symmetric difference between two sets of natural numbers gives a notion of distance that can be used to define a metric on the Turing degrees. By work of Monin, this metric is (0,1/2,1)-valued. A degree a is attractive if almost every degree is at distance 1/2 from a, and dispersive otherwise. I will discuss joint work with Jockusch and Schupp, as well as more recent work of Royer, on the distribution of attractive and dispersive degrees, and their connections with the interplay between effective randomness and genericity.

In this talk I will introduce some ideas to prove exponential decay of correlations for certain dynamical systems, first the angle-doubling map and then cat maps. Then I will give a short introduction to Dolgopyat's method which allows us to treat some systems with a neutral direction, in particular flows and some skew-products. I will talk about the important role of the uniform non-integrability (UNI) condition. If time allows, I will finish by explaining how one might approach this with a microlocal point of view based on ideas of Faure, Tsujii as well as Tsujii-Zhang and Leclerc (joint work with Frédéric Faure).

We discuss a technique called recursive compression for proving incomputability results. The method has developed independently in mathematics and theoretical computer science, and gives a way of showing some set is incomputable by reducing the halting problem to it. Recursive compression was used by Durand, Romashchenko, and Chen in 2008 to give a new proof that the Wang tiling problem is incomputable. In quantum information theory, recursive compression was used by Ji, Natarajan, Vidick, Wright, and Yuen in 2020 to prove the MIP*=RE result that the halting problem is reducible to approximating the quantum value of a nonlocal game. Their result implies a negative answer to the longstanding Connes embedding problem in operator algebras. We formulate a general recursive compression lemma which abstracts the technique used in these applications. A recursive compression f of a set A ⊂ 2ω is a polytime computable function which takes as input a program e computing a string x in exponential time, and outputs a program f(e) computing a string y in polynomial time so that x ∈ A iff y ∈ A. If A has a recursive compression, and A and its complement are nonempty, then A is incomputable. We also show a converse of the recursive compression lemma: the halting problem is polytime reducible to an r.e. set if and only if there is a recursive compression. Finally, we generalize the recursive compression lemma throughout the arithmetical hierarchy, giving a way to show that a language is Σ0n-hard using recursive compression. This is joint work with Seyed Sajjad Nezhadi and Henry Yuen.

Testing statistical software is an extremely difficult task. What is more, for many statistical packages, the developer and test engineer are one and the same, may not have formal training in software testing techniques, and may have limited time for testing. This makes it imperative that the adopted testing approach is both efficient and effective and, at the same time, it should be based on principles that are readily understood by the developer. As it turns out, the construction of test cases can be thought of as a designed experiment (DOE). This talk provides a treatment of DOE principles applied to testing statistical software and includes other considerations that may be less familiar to those developing and testing statistical packages.

Given an family of polarized abelian varieties, let $\mathcal{L}$ be a relatively ample line bundle, consider the family effective divisors on the fibers or equivalently the space of theta-functions induced by $\mathcal{L}$. Mumford-Welters work endowed this family with a flat projective connection which is realized via the heat equations. Hitchin's paper on {\em Flat connections and geometric quantizations, 1990} generalizes the above construction. Namely, for a complex simply connected Lie group $G$ and a surface $\Sigma$, the Hitchin connection is a flat projective connection on a vector bundle over $\mathcal{M}_g$, whose fiber over a compact Riemann surface $C$ ($\Sigma$ and a complex structure) is the space of holomorphic sections of natural line bundle $\mathcal{L}$ on the moduli space of principal $G$ bundles on $C$. Hitchin's work can be realized as an initial step in the direction of Witten's proposal of quantization of Chern-Simons theory via the method of geometric quantization. In this talk, we will discuss the construction of a Hitchin-type connection for the moduli of parabolic bundles. We will start with a more general framework of constructing connections from heat operators with a given symbol map due to van Geeman-de Jong. We will also identify our connection with the Wess-Zumino-Witten/Tsuchiya-Ueno-Yamada connection in conformal field theory generalizing a result of Laszlo. If time permits, we will make further comments on the abelian case. This is a joint work with Indranil Biswas and Richard Wentworth.

BPS cohomology is a cohomology theory that "categorifies" refined BPS invariants associated to 3-Calabi-Yau categories, in the sense that this cohomology recovers these invariants after passing to the (virtual) Poincaré polynomial of the BPS cohomology. As well as categorifying refined BPS state counts, this cohomology turns out to have a rich algebraic structure, with links to various constructions and central objects in quantum algebra, cluster algebras and geometric representation theory. I will survey the construction of BPS cohomology, as well as applications in the above areas and beyond.

In the past decade, there has been a significant shift in the types of mathematical objects under investigation, moving from vectors and matrices in Euclidean spaces, to functions residing in Hilbert or Banach spaces, and ultimately extending to probability measures within the probability measure space. Many questions that were originally posed in the context of linear function spaces are now being revisited in the realm of probability measures. One such question is to efficiently find a probability measure that minimizes a given objective functional. In Euclidean space, we devised optimization techniques like gradient descent and introduced momentum-based methods to accelerate the convergence. Now, the question arises: Can we employ analogous strategies to expedite convergence within the probability measure space? We provide an affirmative answer to this question and show that momentum-based acceleration for Euclidean optimization now translates to Hamiltonian flows, and it can achieve arbitrary high-order of convergence. This opens the door of developing methods beyond standard gradient flow.

As machine learning models become increasingly integrated into distributed and language-intensive applications, ensuring their integrity against backdoor attacks is paramount. This talk presents two defense strategies that target vulnerabilities in federated learning and large language models (LLMs). The first part introduces Trusted Aggregation (TAG), a robust defense mechanism for federated learning that leverages a small validation set to estimate permissible updates and filter out malicious contributions. TAG effectively mitigates backdoor risks while preserving task accuracy, even when up to 40% of client updates are adversarial. The second part addresses the threat of syntactic textual backdoor attacks in LLMs. We propose a novel token substitution strategy that alters semantic content while preserving syntactic structures, enabling the detection of both syntax-based and token-based triggers.

While the general surface of degree at least 4 in projective 3-space contains no lines, the maximum possible number of lines on any surface of degree at least 4 over a field k is a classical question dating back to at least Clebsch's work in 1861. When the degree is less than the (positive) characteristic and always in characteristic 0, the number of lines has an upper bound which is quadratic in the degree when the degree is at least 4. In contrast, once the degree is at least one more than the characteristic, it has long been known that there are surfaces with vastly more lines. In this talk, we answer this classical question over an arbitrary field. In particular, we prove that the maximum number of lines on any smooth surface of degree d over any field k is $d^4-3d^3+3d^2$ and show that, up to projective equivalence, a unique surface obtains this sharp upper bound in the infinitely many degrees and characteristics where it is obtained.

Poroelasticity equations arise from many applications in geophysics and biomechanics, so numerical simulations of poroelasticity equations are of great interest. In this talk I discuss advanced finite element methods for poroelasticity and related problems. In the first part, I introduce parameter-robust discretization of poroelasticity and explain that efficient preconditioners can be obtained by the operator preconditioning approach. In the second part, I present hybridizable discontinuous Galerkin (HDG) methods for the problems that Stokes/Navier-Stokes equations and porous/poroelastic equations are coupled with interfaces. The talk is based on joint works with K.-A. Mardal (University of Oslo), M. E. Rognes (Simula Research Laboratory), A. Cesmelioglu (Oakland University) S. Rhebergen (University of Waterloo), and other collaborators.

We prove an "arithmetic" version of Tao's algebraic regularity lemma about graphs uniformly definable in finite fields. Namely with uniformly definable pairs $(G,A)$ ($G$ group, $A$ subset) in place of a graph. We make connections with Green's arithmetic regularity lemma for finite dimensional vector spaces over $F_p$, and results of Gowers on quasirandom groups. (Joint with Atticus Stonestrom.)

Spring 2025 Symposium at Oakton College Friday, April 18, 2025 Oakton College, Skokie Campus 10:00 - 4:30 Sharona Krinsky Executive Director and Director of Programming, Higher Education Center for Grading Reform, California State University, Los Angeles Authentic Assessment and Alternative Grading in the Age of AI Lew Ludwig Director, Center for Learning and Teaching Professor of Mathematics and Computer Science, Denison University Navigating the AI Storm: Strategies for Educators

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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.

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.

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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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