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Title: Spherical functions: preimage-resistant and collision-free Speaker: Alexander Ushakov, Stevens Institute of Technology Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: D. Savchuk

The main goal of this work is to create bridges between problems of computational group theory (namely the problem of finding a solution for a spherical equation) and foundations of lattice-based cryptography. We introduce a family of spherical functions and demonstrate that it is preimage-resistant and collision-free, assuming that certain lattice approximation problems are hard in the worst case.

Title: Singular fibers in algebraic fibrations of genus two and their monodromy factorizations Speaker: Sumeyra Sakalli Time: 2:00pm–3:00pm Place: CMC 130 Sponsor: D. Savchuk

Kodaira classified all singular fibers that can arise in algebraic, elliptic (i.e., genus one) fibrations. Later, Harer, Kas and Kirby gave the monodromy factorizations of Kodaira's fibrations that were used extensively by topologists to construct interesting 4-manifolds. After Kodaira, Ogg, Iitaka and then Namikawa and Ueno gave a classification of singular fibers in genus two fibrations. In this work, joint with J. Van Horn-Morris, we split these algebraic genus two fibrations into Lefschetz fibrations and determine the monodromy factorizations. More specifically, we look at four families of hypersurface singularities in \(C^3\). Each hypersurface comes equipped with a fibration by genus 2 algebraic curves which degenerate into a single singular fiber. We determine the resolution of each of the singularities in the family and find a flat deformation of the resolution into simpler pieces, resulting in a fibration of Lefschetz type. We then record the description of the Lefschetz fibration as a positive factorization in Dehn twists. This gives us a dictionary between configurations of curves and monodromy factorizations for some singularities of genus 2 fibrations.

Title: Which links matter most? Sparsifying networks with effective resistance Speaker: Cris Moore, Santa Fe Institute Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: Nagle Lecture Series Committee

Network science has increasingly become central to the field of epidemiology. However, many networks derived from modern datasets are not just large, but dense, with a high average degree. One way to reduce the computational cost of simulating epidemics on these networks is sparsification, where a small number of edges are selected and reweighted based on some measure of their importance. Following recent work in computer science, we find that the most accurate approach uses the effective resistances of edges, which can be computed from the graph Laplacian. The resulting sparse network preserves both the local and global behavior of the SIR epidemic model, including the probability each node becomes infected and its distribution of arrival times. This holds even when the sparse network preserves less than 10% of the edges of a mobility network from the United States. Our work helps illuminate which links of a network are most important to disease spread. Defining edge importance using purely topological methods, or by thresholding edge weights, does not perform nearly as well. We believe this same approach may be useful in other biological, social, and physical network models as well.

This is joint work with Alexander Mercier (Harvard School of Public Health) and Sam Scarpino (Northeastern).

Title: Quasisymmetric Koebe Uniformization Speaker: Hrant Hakobyan, Kansas State University Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: A. Danielyan

A deformation of a metric space is said to be quasisymmetric if it roughly preserves the shapes (or roundedness) of its subsets. We study the question of when a metric space that is homeomorphic to a domain in \(\mathbb{S}^2\) can be mapped onto such a domain by a quasisymmetry. For this, we introduce and study spaces satisfying what we call a Transboundary Loewner Property (or TLP). This property is analogous to Heinonen-Koskela’s Loewner property and is defined in terms of Schramm’s transboundary extremal length. Our main result states that under some mild conditions, a space \(X\) is quasisymmetric to a countably connected circle domain in \(\mathbb{S}^2\) if and only if it satisfies the TLP. The talk is partly based on joint works with Wen-Bo Li and Jonathan Rehmert.

Title: Reliability Analysis of Artificial Intelligence Systems Using Recurrent Events Data from Autonomous Vehicles Speaker: Jie Min Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: D. Savchuk

Artificial intelligence (AI) systems have become increasingly common, and the trend will continue. Examples of AI systems include autonomous vehicles (AV), computer vision, natural language processing, and AI medical experts. To allow for safe and effective deployment of AI systems, the reliability of such systems needs to be assessed. Traditionally, reliability assessment is based on reliability test data and the subsequent statistical modeling and analysis. The availability of reliability data for AI systems, however, is limited because such data are typically sensitive and proprietary. The California Department of Motor Vehicles (DMV) oversees and regulates an AV testing program, in which many AV manufacturers are conducting AV road tests. Manufacturers participating in the program are required to report recurrent disengagement events to California DMV. This information is being made available to the public. In this talk, we use recurrent disengagement events as a representation of the reliability of the AI system in AV, and propose a statistical framework for modeling and analyzing the recurrent events data from AV driving tests. We use traditional parametric models in software reliability and propose a new nonparametric model based on monotonic splines to describe the event process and to estimate the cumulative baseline intensity function of the event process. We develop inference procedures for selecting the best models, quantifying uncertainty, and testing heterogeneity in the event process. We then analyze the recurrent events data from four AV manufacturers, and make inferences on the reliability of the AI systems in AV.

Title: Some exact solutions associated with a modified Benjamin-Ono equation with variable coefficients Speaker: Solomon Manukure, Florida A&M University Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: W. Ma

Research in mathematical physics has long been centered on nonlinear integrable systems. In recent times, there has been a heightened interest in integrable equations involving variable coefficients. While such equations are more complex, they provide a more precise depiction of various real-life physical phenomena, particularly when considering the diverse nature of wave propagation media.

In this presentation, we will discuss a modified nonlinear equation derived from the well-known Benjamin-Ono equation, incorporating variable coefficients. The discussion will specifically revolve around various exact solutions, including solitons, lump solutions, rogue waves, breather solutions, and interaction solutions, most of which have been observed in oceanic phenomena.

Title: Tackling Posterior Drift via Linear Adjustments and Exponential Tilts Speaker: Subha Maity, University of Michigan Time: 2:00pm–3:00pm Place: NES 102 Sponsor: L. Lu

I will speak on some of my recent work on transfer learning from a source to a target population in the presence of 'posterior drift': i.e. the regression function or the Bayes classifier in the target population is different from that in the source. In the situation where labeled samples from the target domain are available, by modeling the posterior drift through a linear adjustment (on an appropriately transformed scale), we are able to learn the nature of the posterior drift using relatively few samples from the target population as compared to the source population, which provides an abundance of samples. The other (semi-supervised) case, where labels from the target are unavailable, is addressed by connecting the probability distribution in the target domain to that in the source domain via a formulation involving the exponential family and learning the corresponding parameters. Both approaches are motivated by ideas originating in classical statistics. I will present theoretical guarantees for these procedures as well as applications to real data from the UK Biobank study (mortality prediction) and the Waterbirds dataset (image classification).

Title: Statistical Shape Analysis of Brain Structure Surfaces Speaker: Yuexuan Wu, University of Washington Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: L. Lu

Over the past 30 years, magnetic resonance imaging (MRI) has become a ubiquitous tool for visualizing brain structures. Understanding the structural characteristics of the brain is essential for disease diagnosis and treatment. However, accurate quantification of complex brain structures remains challenging due to issues with shape extraction, representation, and modeling. This talk will introduce efficient frameworks of elastic shape analysis to model both cross-sectional and longitudinal shape changes in 3D brain subcortical structure surfaces. We develop a set of tools to systematically quantify differences in surface shapes from raw structural MRI data. We apply the developed approaches to the Grady Trauma Project (GTP) and three longitudinal neuroimaging data sets (ADNI, HCP, and OpenPain). In the GTP data set, we integrate accurate morphological features and other clinical covariates to model post-traumatic stress disorder (PTSD) outcomes, which provides a vital tool to visualize localized deformations in brain anatomy and predict PTSD severity. With the longitudinal neuroimaging data sets, we showcase the wide applications of our framework in estimating continuous spatiotemporal shape changes from sparse longitudinal data, building life-span growth patterns, and comparing shape development differences among different groups for Alzheimer's disease and aging. This talk will also briefly introduce other ongoing projects on statistical analysis of brain shape.

Title: Sparse Causal Learning: Challenges and Opportunities Speaker: Dingke Tang, University of Toronto Time: 1:00pm–2:00pm Place: NES 102 Sponsor: L. Lu

There has been a recent surge in attention towards trustworthy AI, especially as it starts playing a pivotal role in high-stakes domains such as healthcare, the justice system, and finance. Causal inference emerges as a promising path toward building AI systems that are stable, fair, and explainable. However, it often hinges on precise and strong assumptions. In this talk, I introduce sparse causal learning as a common ground between trustworthy AI and robust causal inference. Here, sparsity plays a dual role in enhancing explainability and ensuring the robust identification of causal effects. Specifically, I reconsider the supervised learning problem of predicting an outcome using multiple predictors through the lens of causality. I demonstrate that it is possible to remove spurious correlations caused by unmeasured confounding by leveraging low-dimensional structures in the predictors. I study its identifiability using an expert voting approach and show that sparsity provides a promising path to transforming exact causal inference methods into multiply robust identification frameworks. Furthermore, I introduce the synthetic instrument, a novel tool for constructing instrumental variables and estimating causal effects. This new approach leads to algorithms that are theoretically justifiable, computationally feasible, and statistically sound.

Title: Syzygies, Curves, and Combinatorics Speaker: Aleksandra Sobieska, University of Wisconsin-Madison Time: 1:30pm–2:30pm Place: CMC 108 Sponsor: T. Bieske

The interplay between algebraic varieties and minimal free resolutions has long been a fruitful field of study, where the algebra inherent to an ideal reveals information about the geometry of the variety. Recently, interest has arisen in this same connection in nonstandard graded settings, but the fracturing of classical definitions and failure of certain classical results mean that the connection is not yet fully understood. In this talk, I will share some results on syzygies of weighted rational (joint with Davis) and affine monomial curves (joint with Braun, Gomes, Miller, and O'Neill), where combinatorial methods make the problem more tractable.

Title: A Spatially Correlated Competing Risks Time-to-Event Model for Supercomputer GPU Failure Data Speaker: Jie Min, Virginia Tech Time: 2:00pm–3:00pm Place: NES 102 Sponsor: L. Lu

Graphics processing units (GPUs) are widely used in many high-performance computing (HPC) applications, such as imaging/video processing and training deep-learning models in artificial intelligence. GPUs installed in HPC systems are often heavily used, and GPU failures occur during HPC system operations. Thus, the reliability of GPUs is of interest for the overall reliability of HPC systems. In this talk, I will introduce in-depth statistical modeling of GPU failure times to study the effect of GPU cabinet's connectivity location in supercomputers on GPU failures under competing risks with covariates and spatially correlated random effects. In particular, two major failure types of GPUs are considered. The connectivity locations of cabinets are modeled as spatially correlated random effects, and the positions of GPUs inside each cabinet are treated as covariates. A Bayesian framework is used for statistical inference. The proposed model is capable of being applied to more general applications.

Title: L-spaces, orderability and taut foliations Speaker: Idrissa Ba, Collège de Bois-de-Boulogne Time: 2:00pm–3:00pm Place: CMC 108 Sponsor: T. Bieske

\(L\)-spaces, orderability and taut foliations relate three distinct areas of mathematics: topology, algebra and analysis. For an irreducible, closed, orientable and connected 3-manifold \(M\), consider the following three statements:

In 2013 Boyer-Gordon-Watson conjectured that b) and c) are equivalent, and in 2014 Juhasz conjectured that a) and c) are equivalent. In this talk, in addition to explaining these conjectures we will discuss our work in relation to these conjectures.

Title: E-values, Multiple Testing and Beyond Speaker: Guanxun Li, Texas A&M Time: 2:00pm–3:00pm Place: CMC 130 Sponsor: L. Lu

We discover a connection between the Benjamini-Hochberg (BH) procedure and the recently proposed e-BH procedure with a suitably defined set of e-values. This insight extends to a generalized version of the BH procedure and the model-free multiple testing procedure (BC) with a general form of rejection rules. The connection provides an effective way of developing new multiple testing procedures by aggregating or assembling e-values resulting from the BH and BC procedures and their use in different subsets of the data. In particular, we propose new multiple testing methodologies in three applications, including a hybrid approach that integrates the BH and BC procedures, a multiple testing procedure aimed at ensuring a new notion of fairness by controlling both the group-wise and overall false discovery rates (FDR), and a structure adaptive multiple testing procedure that can incorporate external covariate information to boost detection power. One notable feature of the proposed methods is that we use a data-dependent approach for assigning weights to e-values, significantly enhancing the efficiency of the resulting e-BH procedure. The construction of the weights is non-trivial and is motivated by the leave-one-out analysis for the BH and BC procedures. In theory, we prove that the proposed e-BH procedures with data-dependent weights in the three applications ensure finite sample FDR control. Furthermore, we demonstrate the efficiency of the proposed methods through numerical studies in the three applications.

Title: Multiple zeta values: from arithmetic to geometry and physics Speaker: Cezar Lupu, Yanqi Lake Beijing Institute of Mathematical Sciences and Applications/Yan Mathematical Sciences Center, Tsinghua University Time: 2:00pm–3:00pm Place: CMC 108 Sponsor: T. Bieske

In this talk, we give a survey on multiple zeta values with an emphasis on their applications ranging from arithmetic properties of odd zeta values to Dirichlet eigenvalues in a regular polygon and periods of zig-zag graphs in quantum field theory. We explore various identities involving different families of multiple zeta values and we emphasize their importance in proving some very important conjectures.

Title: Exotic 4-manifolds and complex surfaces, algebraic fibrations, singularities and knots Speaker: Sümeyra Sakalli, University of Arkansas Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: T. Bieske

Exotic manifolds are smooth manifolds which are homeomorphic but not diffeomorphic to each other. After Donaldson showed that Dolgachev’s surface is an exotic complex projective plane blown-up at 9 points, constructing exotic 4-manifolds has been an active research area in symplectic and low dimensional topology over the last 35 years, with a lot of progress as well as many open problems.

In this talk, I will describe my research on exotic 4-manifolds in relation to complex surfaces, algebraic fibrations, singularities and also knots. In particular, I will present our constructions of smallest exotic 4-manifolds with nonnegative signatures; deformation and splitting singularities in algebraic fibrations; and generalized chain surgeries, which are symplectic operations we defined, and are useful for building exotic 4-manifolds. I will also discuss slice knots in definite 4-manifolds. Some of these works are joint with Akhmedov, Karakurt, Kjuchukova-Miller-Ray, Park, Van Horn-Morris and Yeung.

Title: Analyzing structure data using polynomial invariants Speaker: Pengyu Liu, University of California, Davis Time: 2:00pm–3:00pm Place: CMC 109 Sponsor: T. Bieske

Advancements in innovative technology such as high-throughput sequencing, cryogenic electron microscopy and artificial intelligence have enabled production of myriad structure data in life sciences. These data include molecular structures of nucleic acids and proteins, as well as phylogenetic trees of pathogens. Analyzing extensive structure data is challenging without proper representations, primarily due to structure polymorphism. Polynomial invariants, such as the Tutte polynomial for graphs and the Jones polynomial for knots, are essential mathematical objects in algebraic combinatorics and algebraic topology. They encode structural information and are compatible with modern data analytic tools. In this talk, we introduce a computationally efficient, interpretable and complete polynomial invariant for rooted and unrooted trees. We develop methods based on the polynomial invariant to study pathogen diversification and nucleic acid structures. With this approach, we reveal distinct diversification patterns of seasonal and tropical human influenza virus A H\(_3\)N\(_2\), as well as a strong correlation between the nascent RNA secondary structure and R-loop formation.

Title: Stable polynomials — some properties and applications Speaker: Alan Sola, Stockholm University, Sweden Time: 4:00pm–5:00pm Place: CMC 130 Sponsor: D. Khavinson

A polynomial \(p\) in \(d\) complex variables is said to be stable with respect to a domain \(D\) if \(p\) has no zeros in \(D\). Such polynomials play an important role in several areas of mathematics, and appear in some unexpected situations. I will survey some basic facts about stable polynomials in one and two variables, and will then describe recent joint work with Bickel, Knese, and Pascoe that relies on properties of stable polynomials.