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Title: Stability, hyperlinearity and soficity of amenable groups Speaker: Kate Juschenko, University of Texas at Austin Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: D. Savchuk
The concepts now known as classical notions of sofic and hyperlinear groups were introduced by Gromov. There is no known example of a non-sofic group. In this talk, I will discuss the field and explore possible approaches in the search for a non-sofic group, rooted in certain stability properties of sofic approximations .
Title: Integral Transformation of KZ-Type Equations and the Construction of Braid Group Representations Speaker: Haru Negami, Chiba University, Japan Time: 2:00pm–3:00pm Place: CMC 108 Sponsor: W. Ma
This work establishes a correspondence between the multiplicative middle convolution of Knizhnik-Zamolodchikov (KZ)-type equations [2], an integral transformation that reproduces the KZ-type equation, and the Katz-Long-Moody construction [3], an algebraic method for constructing infinitely many representations of the braid group \(B_n\).
Braid group representations play a pivotal role in mathematics, with applications in topology, representation theory, and mathematical physics [1]. The KZ equation [4], central to conformal field theory, is connected to Appell-Lauricella hypergeometric series, Selberg integrals, and other areas. The KZ-type equation is a certain generalization of the KZ equation. The fundamental group of the domain of the \(n\)-variable KZ-type equation corresponds to the pure braid group \(P_n\). Thus, the monodromy representation of the KZ-type equation is the anti-representation of \(P_n\). Haraoka’s convolution method can be interpreted as constructing anti-representations of \(P_n\) from existing ones.
Katz’s middle convolution technique, extended by Dettweiler and Reiter, enhances the analytic framework by constructing differential equations with specified monodromy representations. On the algebraic side, the Katz-Long-Moody construction systematically generates representations of \(B_n\) from those of \(F_n B_n\), extending Long’s foundational methods [5]. This unified approach combines algebraic and geometric perspectives to study the braid group.
Finally, the potential applications of this method to other fields, such as knot theory and quantum computing, are also discussed.
References
Title: Hamiltonian Circuits on a Torus and Hyperbolic Escher-like images Speaker: Joseph A. Gallian, University of Minnesota Duluth Time: 2:00pm–3:00pm Place: CMC 130 Sponsor: Nagle Lecture Series Committee
This talk concerns the existence of Hamiltonian circuits in an \(m\times n\) directed grid embedded on a torus. We also consider variations and generalizations on this theme. We conclude by showing how to create Escher-like images using Hamiltonian paths in hyperbolic symmetry groups and how to create other Escheresque images using exponential functions, logarithms, and factor groups.
Title: Abelian dynamical Galois groups over function fields Speaker: Andrea Ferraguti, Università di Torino Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: G. Micheli
Dynamical Galois groups are constructed by iterating a rational function over a field \(K\) and looking at the tower of preimages of a fixed point of \(P^1\). A couple of years ago, Andrews and Petsche conjectured that when \(K\) is a number field and the function is a polynomial, such groups can only be abelian in trivial cases. This has only been proven to be true in a handful of cases, e.g. when \(K\) is the field of rationals. In this talk, we will consider the same question when \(K\) is a global function field. In this talk I will show how, combining a series of reduction steps that involve the construction of Bottcher coordinates in positive characteristic, it is possible to prove that if the rational function \(f\) has a superattracting cycle the dynamical Galois group of \(f\) with basepoint \(\alpha\) can be abelian only if the pair \((f,\alpha)\) is defined over the constant field of \(K\). This is joint work with P. Ingram and C. Pagano.
Title: Stable homotopy theory and cobordism Speaker: Hassan Abdallah, Wayne State University Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: M. Elhamdadi
Two manifolds of the same dimension are said to be cobordant if their disjoint union is the boundary of a compact manifold one dimension higher. Cobordism is an equivalence relation that endows manifolds with a group structure via disjoint union and a graded ring structure via cartesian product. By the pioneering work of Lev Pontryagin and Renee Thom in the early-to-mid 20th century, stable homotopy theory provides powerful tools for classifying manifolds up to cobordism and describing their associated algebraic structure. In the years since, cobordism theory has been found to sit at multiple interesting intersections in mathematics. I will trace some of the history of cobordism theory, discuss various applications, and touch on some recent results in the area.
Title: Bent Functions on Finite Groups: Existence, Constructions, and Relations to Relative Difference Sets Speaker: Bangteng Xu, Eastern Kentucky University Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: X. Hou
Boolean bent functions were introduced by Rothaus in the 1970s as a class of highly nonlinear Boolean functions, and since then, various generalizations have been studied. Among these generalizations are bent functions on finite groups, including both finite abelian and nonabelian groups. Bent functions on finite abelian groups were introduced by Logachev et al. in the 1990s, and in the 2010s, Poinsot extended the concept to finite nonabelian groups. The existence and construction of bent functions are central topics in current research due to their deep connections to combinatorics, coding theory, and cryptography.
Bent functions are closely related to perfect nonlinear (PN) functions between finite groups. While the graphs of PN functions are known to form relative difference sets, the graphs of bent functions do not necessarily exhibit this property in general. This distinction leads to interesting structural questions about bent functions.
In this talk, we will begin by recalling the definition of bent functions and discussing their basic properties. Next, we will discuss the existence and constructions of bent functions, focusing on both abelian and nonabelian settings. Finally, we will investigate the necessary and sufficient conditions under which the graphs of bent functions are relative difference sets, highlighting the implications for the study of nonlinearity and combinatorial structures.
Title: Geometry of the doubly periodic Aztec dimer model Speaker: Tomas Breggren, Royal Institute of Technology (KTH), Sweden Time: 2:00pm–3:00pm Place: CMC 120 Sponsor: H. Pendharkar
Random dimer models (or equivalently random tiling models) have been a subject of extensive research in mathematics and physics for several decades. In this talk, we will discuss the doubly periodic Aztec diamond dimer model of growing size. In this limit, we see three types of macroscopic regions — known as rough, smooth, and frozen regions. We will discuss how the geometry of the arctic curves, the boundary of these regions, can be described in terms of an associated amoeba. In particular, we determine the number of frozen and smooth regions and the number of cusps on the arctic curves.
Title: Schur/Hadamard Multipliers on Large Matrices Speaker: Chiang Chuah Time: 2:00pm–3:00pm Place: NES 103 Sponsor: H. Pendharkar
The boundedness of Schur/Hadamard Products plays an important role in engineering and data science. In particular, the storage of information in large data is closely related to the study of the boundedness of the Schatten-\(p\) norm of matrices in non-commutative harmonic analysis. In this talk, we provide a Marcinkiewicz type multiplier theory for the Schur multipliers on the Schatten \(p\)-classes. This generalizes a previous result of Bourgain for Toeplitz type Schur multipliers and complements a recent result by Conde-Alonso, Gonzalez-Perez, Parcet and Tablate. As a corollary, we obtain a new unconditional decomposition for the Schatten \(p\)-classes for \(p>1\). Similar results can also be extended to the case of \(R^d\) and \(Z^d\), where \(d\ge2\).
Title: Nonlinear steepest descent on a torus: A case study of the Landau-Lifshitz equation Speaker: Andrei Prokhorov, University of Chicago Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: H. Pendharkar
The Landau-Lifshitz equation describes the time evolution of spins in an anisotropic continuous spin chain. It is an integrable nonlinear wave equation that follows the inverse scattering procedure established by Sklyanin. We consider fast-decaying initial conditions without solitons and apply the Riemann-Hilbert method, set on a torus and developed by Rodin, for large-time asymptotic analysis.
First, we determine the class of scattering data that ensures the initial conditions belong to the Schwartz class. Then, using the solvability of the corresponding Riemann-Hilbert problem established by Rodin, we construct a singular integral equation representation for the solution using a scalar Cauchy kernel defined on the torus. We proceed with the standard nonlinear steepest descent method, employing a parabolic cylinder parametrix. Finally, the asymptotics of the solution is obtained through the iterative analysis of a small-norm singular integral equation. Our result provides more rigorous derivation of formulas obtained Bikbaev, Bobenko, and Its. This is joint work with Alexander Its and Harini Desiraju.
Title: Mackey Deformation Space in Unitary Representations of Reductive Lie Groups and Their Associated \(C^*\)-Algebras Speaker: Angel Roman, Penn State Time: 2:00pm–3:00pm Place: CMC 109 Sponsor: H. Pendharkar
A reductive Lie group has an associated semidirect product group called the Cartan motion group. Their algebraic structures are quite different, however, in 1975 George Mackey made a striking observation. He proposed that there could be a correspondence between the set of equivalence classes of irreducible unitary representations of a reductive group and the set of equivalence classes of irreducible unitary representations of its associated Cartan motion group.
The study of the Mackey analogy can be carried out by studying the group \(C^*\)-algebras of both the reductive group and the Cartan motion group. In a joint work with Nigel Higson and Pierre Clare, we constructed an embedding between the two \(C^*\)-algebras. In order to carry out the construction, I shall introduce the so-called “Mackey” deformation space, which bundles the reductive group and the motion group into a single smooth manifold. By studying this deformation space, and by extension, the new embedding, we can characterize the Mackey Analogy beyond that of a mere coincidence of parameters.
If time permits, I will also present other work in which the Mackey deformation space plays a central role. Among them is a joint work with Yanli Song and Xiang Tang where we apply the deformation to orbital integrals of reductive groups.
Title: The Tait conjectures for classical, virtual, and welded knots Speaker: Hans U. Boden, McMaster University Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: M. Elhamdadi
The Tait conjectures were originally posed in the 1900s but remained open for nearly 100 years. They were ultimately resolved in the late 1980s using the new (at the time) technology from quantum topology, namely the Jones polynomial. In a more recent development, Josh Greene found a geometric characterization of alternating knots as those that simultaneously bound positive and negative definite spanning surfaces, and he used this to give a new proof of the Tait conjectures. In this talk, I will survey the results obtained by attempts to adapt these methods to knots in thickened surfaces and virtual knots.
Title: Painlevé Equations and Orthogonal Polynomials: A Century of Unexpected Connections Speaker: Ahmad Barhoumi, KTH Royal Institute of Technology, Stockholm, Sweden Time: 4:00pm–5:00pm Place: CMC 130 Sponsor: L. Skrzypek
More than a hundred years ago, Painlevé sought to classify certain second-order differential equations, and what began as a purely mathematical question blossomed into a flurry of connections with mathematical physics, combinatorics, and many more disperate areas. In this talk, I will highlight the deep connections Painlevé equations have to some of these areas and how the path connecting the two seems to always go through orthogonal polynomials.
Title: The geometry of the deep linear network Speaker: Govind Menon, Brown University Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: R. Teodorescu
The deep linear network (DLN) is a phenomenological model of deep learning introduced by computer scientists. This talk describes a rigorous study of training dynamics in the DLN from the viewpoint of the geometric theory of dynamical systems. We identify several surprising connections between the DLN and other areas of mathematics, especially the theory of minimal surfaces and random matrix theory. This allows us to establish a thermodynamic framework for the analysis of the DLN.
The talk will also include some speculation on the relationship between the DLN and deep learning. The unifying theme is the use of Riemannian geometry.
This is joint work with several students and colleagues (Nadav Cohen, Tejas Kotwal, Lulabel Seitz, Zsolt Veraszto and Tianmin Yu).
Title: Nonlinear Waves, Differential Operators, and Mathematical Physics Speaker: Xudan Luo, Chinese Academy of Sciences, Beijing Time: 2:00pm–3:00pm Place: NES 104 Sponsor: L. Skrzypek
Theoretical and applicable aspects of nonlinear waves are relevant to subjects as diverse as relativity, plasma physics, materials science, nonlinear optics, random media, atmosphere and ocean dynamics, and biology. Relevant predictions are often tested against physical experiments and open avenues for collaborations and interactions that transcend traditional disciplinary boundaries in many of these fields. In this talk, we discuss connections between nonlinear wave motion and differential operators. In addition, two of recent projects will be mentioned, including periodic wave motion in deep water and two-dimensional atmospheric internal waves, which are connected to a non-self-adjoint Dirac operator and a fractional Laplace operator, respectively.
Title: Existence of extremal subspaces for some restricted \(n\)-width problems in Hilbert spaces Speaker: Christopher Felder, Zorn Postdoctoral Fellow, Indiana University, Bloomington Time: 4:00pm–5:00pm Place: CMC 130 Sponsor: L. Skrzypek
This talk will be concerned with the existence of solutions to the minimization problem \(\inf\{\inf{|x-h|_H:x\text{ in }X_n\}: X_n}\), where \(X_n\) is varied over all \(n\)-dimensional subspaces of a subset of a Hilbert space \(H\). This is a restricted version of the well-known \(n\)-width minimization problem introduced by Kolmogorov in the 1930s. A subspace for which the infimum is achieved is called extremal. We will present a user-friendly theorem which gives sufficient conditions for the existence of an extremal subspace, and discuss some applications to approximations using reproducing kernels in reproducing kernel Hilbert spaces. Time permitting, we will also mention some connections to machine learning.
Title: A topologist's view of algebraic singularities Speaker: Jeremy Van Horn-Morris, University of Arkansas Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: S. Sakallı
Most of the tools that algebraists, as well as high dimensional topologists, use to study singularities are, unsurprisingly, algebraic, but you can still achieve significant intuition by examining singularities directly. We'll take a visual tour of some low dimensional singularities and see how this can enhance our algebraic understanding.
Title: Sparse Bayesian Variable Selection in High Dimensional Regression Models Speaker: Zhuanzhuan Ma, University of Texas Rio Grande Valley Time: 1:00pm–2:00pm Place: CMC 120 Sponsor: L. Lu
With the rapid development of computational technology, Bayesian methods have received much attention in the model selection for high-dimensional data, which are frequently collected in various research and industrial areas. This presentation consists of two distinct but related research projects. In the first project, we constructed a Bayesian hierarchical modeling framework for the problems of variable selection in logistic regression of high-dimensional settings, in which the model dimension exceeds the sample size. We assign a modified global-local shrinkage prior including horseshoe and normal-gamma priors on the regression coefficients and propose an efficient sampling algorithm based on the Gibbs sampler to draw samples from the full conditional posterior distributions to make the posterior inference. Simulation studies and two real-data applications are conducted to compare the performance of the proposed Bayesian method and existing ones in the literature. Given the potential ability of the proposed Bayesian method to be extended to other situations, in the second project, we actively investigate how to generalize our computational framework for parameter estimation and variable selection in Bayesian binary quantile regression. Generalizing this method for high-dimensional multi-response problems (e.g., quantitative and qualitative responses) will be an area of future investigation.
Title: Discrete Autoregressive Switching Processes in Sparse Graphical Modeling of Multivariate Time Series Data Speaker: Beniamina Hadj-Amar, Rice University Time: 1:00pm–2:00pm Place: NES 108 Sponsor: L. Lu
We propose a flexible Bayesian approach for sparse Gaussian graphical modeling of multivariate time series. We account for temporal correlation in the data by assuming that observations are characterized by an underlying and unobserved hidden discrete autoregressive process. We assume multivariate Gaussian emission distributions and capture spatial dependencies by modeling the state-specific precision matrices via graphical horseshoe priors. We characterize the mixing probabilities of the hidden process via a cumulative shrinkage prior that accommodates zero-inflated parameters for non-active components, and further incorporate a sparsity-inducing Dirichlet prior to estimate the effective number of states from the data. For posterior inference, we develop a sampling procedure that allows estimation of the number of discrete autoregressive lags and the number of states, and that cleverly avoids having to deal with the changing dimensions of the parameter space. We thoroughly investigate performance of our proposed methodology through several simulation studies. We further illustrate the use of our approach for the estimation of dynamic brain connectivity based on fMRI data collected on a subject performing a task-based experiment on latent learning.
Title: TBA Speaker: Andrew Green, Washington University in St. Louis Time: 2:00pm–3:00pm Place: CMC 109 Sponsor: L. Skrzypek
TBA
Title: Methods for Dependent Functional Data in Biomedical and Industrial Applications Speaker: Chi-Kuang Yeh, McGill University Time: 1:00pm–2:00pm Place: NES 108 Sponsor: L. Lu
Functional data analysis has gained prominence with the increasing availability of complex, high-dimensional data observed continuously over time. Sequentially observed functional data (FD), referred to as functional time series (FTS), pose unique challenges in capturing and modeling serial dependencies, particularly in the presence of outliers or irregular data patterns. In this talk, I present spherical autocorrelation, a method for measuring serial dependence in FTS that examines angles between lagged functional pairs projected onto a unit sphere. By capturing both the direction and magnitude of dependence, this approach addresses limitations of traditional autocorrelation measures while maintaining robustness to atypical curves in the data. The asymptotic properties of the proposed estimators are established, enabling the construction of confidence intervals and portmanteau tests for white noise. Simulation studies validate the method’s effectiveness, and applications to model selection for daily electricity price curves and volatility measurement in densely observed asset prices demonstrate its versatility. The talk concludes with a discussion of potential extensions, including applications to multivariate FD, as well as future directions for other ongoing projects inspired by real-world challenges.
Title: Dimension Reduction for High-Dimensional Multivariate Heteroskedastic Time Series via Envelope Methods Speaker: Toktam Valizadeh, Southern Illinois University Carbondale Time: 1:00pm–2:00pm Place: CMC 120 Sponsor: L. Lu
Multivariate time series data, prevalent in fields such as finance, economics, and neuroscience, often exhibit heteroskedasticity—an essential feature that, if ignored, can result in inefficient and biased estimation. Standard models like vector autoregressive (VAR) and matrix autoregressive (MAR) models are widely used but fail to fully address this critical feature. While conditional heteroskedasticity models offer partial solutions, they often rely on oversimplified assumptions, such as constant unconditional covariance, and are often overparameterized, making them challenging to work with in practice. To bridge these gaps, we introduce the Unconditional heteroskedastic Envelope VAR (HEVAR) and heteroskedastic Envelope MAR (HEMAR) models. These models improve estimation efficiency by constructing a minimal common reducing subspace that links the mean function to heteroskedastic covariance structures. By incorporating common-heteroskedasticity constraints, the approach reduces parameter dimensionality, stabilizes estimation, and enhances inference accuracy. We also develop quasi-maximum likelihood (QML) methods for parameter estimation, establish their asymptotic properties, and evaluate their performance through simulations and theoretical analysis. We applied our proposed models to real-world economic datasets, demonstrating their ability to capture complex time-varying structures and uncover meaningful insights into high-dimensional, heteroskedastic systems.
Title: Identifying hidden heterogeneous effect of biological molecules via local feature selection using knockoffs Speaker: Jiaqi Gu, Stanford University Time: 1:00pm–2:00pm Place: NES 108 Sponsor: L. Lu
Interpreting individual heterogeneity within diverse populations is essential for optimizing healthcare quality in precision medicine practice. Despite the availability of vast and complex health-related datasets, most existing methods focus on the inference of a particular population or a few predefined subpopulations. For more precise inference of disease mechanisms and the development of individualized optimal therapies, we introduce a novel framework of local feature selection, which identifies important features associated with the outcome of interest tailored to individual profiles. Leveraging the recently developed knockoffs methodology, our framework integrates seamlessly with modern machine learning models while maintaining rigorous error rate control. We validate our method’s statistical properties via extensive simulations and its ability to uncover hidden heterogeneity of Alzheimer’s disease at the cellular level via an application on single-cell RNA sequencing data. We envision our framework as a crucial step toward addressing key challenges in precision medicine by generating novel and interesting hypotheses for confirmatory biological experiments.
Title: Knotoid polynomials Speaker: Wout Moltmaker, University of Amsterdam Time: 3:00pm–4:00pm Place: CMC 130 Sponsor: N. Jonoska
Knotoids are a simple generalization of knot diagrams in which we allow for open ended diagram components. The difference with e.g. braids is that the endpoints may lie anywhere in the diagram, also in interior regions. In this talk we discuss polynomial invariants of knotoids. Specifically we discuss the mock Alexander polynomial and arrow bracket polynomial, which are generalizations of the classical Alexander and Jones polynomials for knots respectively. We show a symmetry in the mock Alexander polynomial for spherical knotoids that was conjectured by L. Kauffman, and give an arrow version of the Thistlethwaite theorem for knotoids. Namely that the arrow bracket polynomial of a knotoid can be obtained as an evaluation of the Bollobás-Riordan polynomial of a ‘marked’ ribbon graph associated to the knotoid. Here a marked ribbon graph is a knotoidal analogue of ribbon graphs allowing for linearized vertices.