In 1995, Lazebnik and Ustimenko introduced the family of q-regular graphs D(k,q), which is defined for any positive integer k and prime power q. The connected components of the graph D(k, q) have provided the best-known general lower bound on the size of a graph for any given order and girth to this day. Furthermore, Ustimenko conjectured that the second largest eigenvalue of D(k, q) is always less than or equal to 2√q, indicating that the graphs D(k,q) are almost Ramanujan graphs. In this talk, we will discuss some recent progress on this conjecture. This includes the result that the second largest eigenvalue of D(5,q) is less than or equal to 2√q when q is an odd prime power. This is joint work with Vladislav Taranchuk.

Roughly speaking, enumerative geometry is a field whose goal is to count the "typical" number of solutions to certain types of families of polynomial equations, particularly when that number is finite. With a great deal of effort, especially in the wake of the work of Hermann Schubert around the turn of the 20th century, mathematicians made rigorous the notion of a "typical" answer and also made rigorous certain simplifying strategies Schubert had suggested. Indeed, making Schubert's arguments precise was the topic of Hilbert's 15th problem, and the field born from this study is now called Schubert calculus. The simplifications Schubert had suggested entail sliding or deforming the geometric objects to be studied while preserving the total number of whatever it is one wants to count. These strategies are what are now called degeneration techniques. In this talk, we will describe some modern questions in Schubert calculus and explain how these questions are studied via Gröbner degeneration in particular.

The Landau equation in kinetic theory is one of the fundamental kinetic equations that describes the evolution of collisional plasmas. The equation includes a quadratic, non-local term that models the effects of binary collisions mediated by the Coulomb force. This collision term introduces substantial mathematical challenges, leaving many fundamental questions--such as the existence of global-in-time smooth solutions--largely open. In this talk, I will explore recent progress made in understanding a simplified model, the homogeneous Landau equation, which retains the complex collision term. In a recent breakthrough work, Luis Silvestre and Nestor Guillen showed the existence of a new monotone functional---the Fisher information---which is used to construct global-in-time solutions for smooth rapidly decaying initial data. I will discuss joint work with Maria Gualdani and Amelie Loher, where we extend these results to general initial data and obtain new results on global-in-time existence and various forms of uniqueness. I will conclude with a discussion of how these results inform future research of the full model.

We study the connections between the automorphisms of an AC field K and the automorphisms of a structure S definable in it. Our methods are model theoretic and use a minimum of algebra. As an application, we show that the automorphisms of a simple algebraic group G over K acts on a copy L of K, with kernel the geometric automorphisms of G.

A random structure on a vertex set $V$ (in a fixed finite relational language) is exchangeable if its distribution is invariant under permutations of $V$. The Aldous--Hoover Theorem says all such distributions are generated from a collection of i.i.d. variables on $[0,1]$, one for each subset of $V$, using a simple rule that was later called "Euclidean structure" by combinatorialists. As the name suggests, an Euclidean structure resembles a relational structure over $[0,1]$, except for the presence of "higher-order variables". One of the original questions of Hoover was to determine which such distributions admit simpler descriptions that do not depend on certain variables. Very little progress was obtained in this problem until it got revisited under the light of the theories of limits of combinatorial objects and quasirandomness. It turns out that asking for a representation of an exchangeable hypergraph in which the Euclidean structure is a usual (measurable) relational structure over $[0,1]$ (i.e., which does not need any higher-order variables) is equivalent to asking for "tamer" Szemerédi regularity lemmas and was solved using the theory of hypergraphons. The dual problem of determining when there is a representation that does not need any low-order variable is more closely related to quasirandomness, which informally is the property of "lack of correlation with simple structures". In this talk, I will introduce exchangeability and quasirandomness theory and talk about recent progress on the aforementioned dual problem. I will assume familiarity with basic logic/model theory, but no prior knowledge in extremal combinatorics, limit theory or quasirandomness will be necessary. This talk is based on joint works with Alexander Razborov and Henry Towsner.

In oncology clinical trials, subjects prematurely discontinuing from the assigned treatment prior to experiencing an event of interest are often handled by noninformative censoring under censor-at random assumption. Such methods can be challenged with respect to the robustness of the ignorable or noninformative censoring and sensitivity analyses using informative censoring are often required. In a recently published article (Jin and Fang, 2024), reference-based methods (including Jump to Reference and Copy Reference) and tipping point analysis for time-to-event data with possibly informative censoring were proposed. These are novel methods to fit the gap in literature for time-to-event analysis with applications in oncology clinical trials. We will describe and facilitate the implementation of these methods in this presentation. Illustrative examples are provided to demonstrate the reference-based methods and tipping point analysis.

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It is well known that for any constant probability p there exists a function k(n) such that the independence number of the binomial random graph G(n,p) is concentrated on two values (i.e.the independence number of G(n,p) is k(n) or k(n)+1 with high probability). In this talk we discuss the extension of this result to p(n) that tends to 0 with n. In particular, we determine the probability at which two point concentration of the independence number of G(n,p) breaks down. We also discuss the independence number of G(n,m), and show that there is a range of values for m in which the independence number of G(n,m) is concentrated on two values while the independence number of the corresponding G(n,p) is not concentrated on two values. Joint work with Jakob Hofstad.

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Consumers have increasingly spent more time on mobile applications, and companies have also allocated more resources to advertisement in mobile applications and are actively seeking ways to improve the click-through rate of in-app ads. However, there is a lack of research leveraging consumers’ mobile application usage to understand in-app advertisement. In this study, we develop an integrated model of mobile application usage and in-app advertising response. We use a hidden-Markov model (HMM), which allows consumer involvement in mobile activities to drive temporal changes in both consumer mobile application usage and in-app advertising response. Our framework captures three components that are understudied in previous research on in-app advertising responses: 1) contextual mobile app in which consumers are targeted; 2) long-range correlation in preceding periods and 3) multitasking across mobile apps. To address the challenge of long-range correlation in traditional HMM, we further extend HMM by incorporating a long short-term memory (LSTM) autoencoder into the state transition. Using a unique panel dataset, we find salient temporal patterns and persistence of consumers’ underlying involvement that govern both application usage and advertisement response. Interestingly, consumers’ responses to advertisements follow an inverted-U shape where consumers are most likely to respond to advertisements in a medium state of involvement. Consumers’ advertisement responses are also subject to a contextual effect. For example, consumers are more likely to respond to advertisements when they use Entertainment apps compared with other apps. Our simulation indicates that incorporating mobile usage information, such as a temporal state of involvement and contextual effects at the individual level (viewing history), can significantly improve the effectiveness of targeting strategies. For instance, incorporating contextual effect and multitasking can increase performance by as much as 21.2%. This improvement can be further enhanced with the help of the LSTM autoencoder to address the long-range correlations in HMM. We are the first to connect consumers’ mobile application usage with their in-app ad response.

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I'll discuss joint work with Alessio Caminata, Luca Schaffler, and Han-Bom Moon in which we study equations defining (the closure of) the locus of n points in projective space that lie on a rational normal curve and apply these equations to resolve a question of Lior Pachter and David Speyer from 2004 on the tropical geometry of the space of phylogenetic trees.

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