Coming Next:
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Knots with Boring Khovanov Homology |
Dean Spyropoulos, NMSU |
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Khovanov homology is a relatively new yet highly influential invariant of links in the 3-sphere. Since its discovery, it has been used to prove previously intractable results in knot theory and 4-manifold topology, all while being combinatorial in design. However, it is also interesting to ask when Khovanov homology is not interesting. After telling you some highlights of the theory, we'll discuss the problem of classifying knots with "boring" (i.e., thin) Khovanov homology. This problem has motivated budding joint work with Keegan Boyle, in which we study Khovanov-thinness when the ambient manifold is replaced with real projective space. |
Science Hall 107 and Zoom12:00PM refreshments | 12:30PM - 1:30PM |
Zoom: Link |
Fall 2025
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Jianjun Paul Tian Evolution Coalgebra I defined a new type of coalgebra recently although I have not carried out detailed study yet. It is not coassociative or cocommutative. This coalgebra has some good properties and structures. It may have interpretation and applications in dynamical systems or stochastic processes. To give a picture, I will explain it from basic algebras, coalgebras, bialgebras, Hopf algebras, evolution. |
Previous Colloquia
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Spring 2025
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Chun Liu, Illinois Institute of Technology Complex Fluids and Molecular Biology: General Diffusion in Biological Environment In this talk, I will discuss topics related to biological fluids, especially charge transport in solutions and proteins (ion channels). One of the key ingredients in these studies is the understanding of diffusion and its relations to other effects, such as hydrodynamics, electrostatics and other particle-particle interactions. I will employ the general framework of energetic variational approaches, especially Onsager's Maximum Dissipation Principles to these generalized diffusion problems. We will also discuss specific analytical issues arising from these studies. |
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Ziwei Ma, University of Tennessee at Chattanooga The Development and Applications of Skew F Distribution The $F$ distribution is central to many classical inferential statistical methods that rely on the assumption of normality. However, when data exhibit multimodality or skewness, this assumption can lead to a lack of robustness, resulting in invalid or inefficient statistical inferences. Consequently, there is considerable interest in relaxing the normality assumption to improve inference about unknown parameters. In this talk, I will first introduce the development of a generalized $F$ distribution based on the skew-normal family of distributions. Following that, I will discuss its application in estimation and hypothesis testing within the framework of skew-normal distributions. Finally, I will highlight some potential research directions in this area. |
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Ziyuan Wang, The University of Wisconsin Oshkosh Unlocking Uncertainty: How Gain-Probability Analysis Empowers Decision-Making with Gamma Distributions Gain-Probability (G-P) analysis provides a powerful, intuitive framework for comparing distributions by estimating the probability that a randomly selected observation from one group exceeds another. While G-P analysis has been applied to various distributions, its extension to gamma distributions—commonly used to model positively skewed data such as product lifetimes, wait times, and streamflow—remains underexplored. This talk presents a comprehensive investigation of G-P analysis for gamma distributions, covering both independent and dependent (bivariate) cases. We derive the mathematical foundations for G-P analysis in the gamma family, develop computational algorithms, and validate the methodology using real-world datasets from healthcare and hydrology. Additionally, we provide open-source tools for computing gain probabilities, enabling researchers and practitioners to apply this framework in their work. The talk concludes with a discussion of future directions, including extensions to other distributions and more complex dependency structures. |
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Robert Smits, NMSU Behavior of Absorbing and Generating $p$-Robin Eigenvalues in Bounded and Exterior Domains We establish rigorous quantitative inequalities for the first eigenvalue of the generalized $p$-Robin problem, for both the classical diffusion absorption case, where the Robin boundary parameter $\alpha$ is positive, and the superconducting \textcolor{blue}{generation regime} (\(\alpha<0\)), where the boundary acts as a source. In bounded domains, we use a unified approach to derive a precise asymptotic behavior for all $p$ and all small real $\alpha$, improving existing results in various directions, including requiring weaker boundary regularity for the case of the classical 2-Robin problem, studied in the fundamental work by R. Sperb. In exterior domains, we characterize the existence of eigenvalues, establish general inequalities and asymptotics as \(\alpha\to 0\) for the first eigenvalue of the exterior of a ball, and obtain some sharp geometric inequalities for convex domains in two dimensions. |
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Cong Wang, University of Nebraska Omaha The A Priori Procedure for Determining Necessary Sample Sizes in Parameter Estimation The A Priori Procedure (APP) aims to determine necessary sample sizes to ensure that sample statistics reliably estimate corresponding population parameters. As data collection in certain fields faces limitations and periodicity, the study of a priori confidence intervals that do not rely on actual data has become increasingly important. A priori confidence intervals enable researchers to identify a sample size 𝓃 such that the deviation and corresponding probability between the sample statistic and the population parameter fall within a specified range. This approach allows researchers to plan sample sizes before data collection, enhancing the reliability and efficiency of their research. This talk will focus on deriving APP equations for determining required sample sizes and constructing confidence intervals for parameters within the family of skewnormal distributions. Applications using real data sets will be presented to illustrate the main results. |
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Yimin Xiao, Michigan State University Thermal Capacity and Hausdorff Dimension Results for Brownian Motion and Related Processes Let W = {W(t), t ∈ IR+} denote d-dimensional Brownian motion. For arbitrary nonrandom compact sets E ⊂ (0 , ∞) and F ⊂ IR°d, we provide an explicit formula for the essential supremum of Hausdorff dimension of W(E) ∩ F. Our formula is related intimately to the thermal capacity of Watson (1978). We also prove that when d ≥ 2, our formula can be described in terms of the Hausdorff dimension of E × F under the parabolic metric in (0 , ∞) × IR°d. This part is joint work with Davar Khoshnevisan. We also study the analogous problems for the Brownian sheet (with Cheuk Yin Lee) and certain Markov processes. |
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March 19 |
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Dionysus Birnbaum, Oregon State University Flavors of Asphericity: DR and CLA In 1941, J.H.C. Whitehead conjectured that every connected subcomplex of a two-dimensional aspherical CW complex was also aspherical. Despite considerable work on this problem, it remains open. In the attempts to resolve this problem, a number of different types of hereditary asphericity have been developed, which seem to fit into a hierarchy of strength. We will examine the relation between two types of asphericity in particular: diagrammatic reducibility (DR) and Cohen-Lyndon asphericity (CLA). |
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Keegan Boyle, NMSU Polynomials Are Knot Difficult We will discuss some open conjectures about polynomials: one classical, and one modern. Both conjectures have a topological formulation in knot theory and produce some interesting pictures. Some of this talk will discuss joint work with Nicholas Rouse. |
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Eva Belmont, Case Western Reserve University Machine Computation Of Unstable Homotopy Groups Of Spheres Computer calculations have driven several recent breakthroughs involving the stable homotopy groups of spheres, including the 2024 solution due to Lin, Wang, and Xu of the remaining case of the Kervaire invariant problem. I will give an overview of the new results and techniques this area, and discuss work in progress (joint with Francis Baer, William Balderrama, and Dan Isaksen) to employ some of the same methods to studying the unstable homotopy groups of spheres. |
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Andre Kornell, New Mexico State University Entropy In Multimatrix Algebras A density matrix is a positive semidefinite complex matrix whose trace is one. We will discuss the theorem that a linear map of complex matrices is a homomorphism of unital *-algebras if and only if its adjoint maps density matrices to density matrices of equal or smaller entropy and does so completely. The relevant notion of entropy is a variant of von Neumann entropy. This talk will emphasize the motivation for this theorem in quantum information theory and noncommutative geometry. |
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Dustin Ross, San Francisco State University A Geometer’s Guide to Log-concavity Chromatic polynomials count the number of ways to color a graph’s vertices so that no two adjacent vertices have the same color. One of the great combinatorial conjectures of the 20th century claimed that the chromatic polynomial of any graph is log-concave, meaning that the square of each of its interior coefficients is at least as big as the product of its neighbors. This conjecture remained unresolved for over 50 years until, in a major breakthrough, June Huh finally resolved it in 2012. In this talk, we’ll explore chromatic polynomials, log-concavity, and a recently-discovered method by which we can view Huh’s result through the lens of classical ideas in geometry. |
Fall 2024
November 22 |
Speaker Title Abstract |
Wei Ning, Bowling Green State University Confidence Distributions for Skew Normal Change‑PointModel Based on Modified Information Criterion In this talk, we will introduce a modified max-cumulative sum (CUSUM) procedure for detecting changes in parameters of skew normal distribution. The corresponding false alarms frequency and the post change detection delay are investigated. Asymptotic behaviors of detection delay and theoretical optimality of the detection procedure have been established. Simulations have been conducted to show the performance of the proposed method and compare it to the other existing methods including CUSUM. Real data are given to illustrate the detection procedure. |
November 15 |
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In this talk I will survey Arveson's non-commutative boundary theory, leading up to his last major open conjecture known as Arveson's hyperrigidity conjecture. This conjecture roughly states that if the non-commutative Choquet boundary coincides with the whole spectrum of the generated C*-algebra, then nets of unital completely positive maps which converge to the identity on generators must converge to the identity on the whole generated C*-algebra. We will showcase a counterexample with a separable type I C*-algebra. |
November 08 |
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Katherine Kosaian, University of Iowa Formalizing Mathematics in Isabelle/HOL Many mathematical algorithms are used in safety-critical contexts. Correctness of these algorithms, and the mathematical results underlying them, is crucial. In formal methods, a piece of software called a theorem prover can be used to formally verify algorithms. In this approach, code for an algorithm is accompanied by a rigorous proof of correctness that only depends on the logical foundations of the theorem prover. Algorithms that have been verified in this way are highly trustworthy and thus safe for use in safety-critical applications. The theorem prover Isabelle/HOL is well-suited for formalizing mathematics. This talk will motivate formalized mathematics, exhibit how mathematics is formalized in Isabelle/HOL, and discuss the challenges that may arise, with a focus on three different use cases: 1) verifying algorithms for real quantifier elimination, 2) verifying Coppersmith’s method, 3) verifying Pick’s theorem. |
October 25 |
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In this talk we give the basic definitions and construction of chains of evolution algebras. We provide a brief review of known chains of evolution algebras (CEAs), construct new finite-dimensional real CEAs, then we study property transitions and time depending dynamics of constructed CEAs. |
October 18 |
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Lance E. Miller, University of Arkansas Arithmetic and Geometry of the Frobenius Lying at the core of nearly all intersections between algebra, arithmetic, and geometry is a particularly map named after G. Frobenius. In this talk, we will see how this map, or its lifts, connect to millennium problems and algebraic geometry. The main role will be played by a theory similar to that of differential equations, but where differentiation is replaced by a Fermat quotient. In addition to classic, and new applications, we will introduce a new emerging field lying at the intersection of differential geometry, number theory, and physics. All new work presented is joint with A. Buium. |
October 11 |
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We study the expressive power of SCR and show that they are capable of universal approximation of any unrestricted linear reservoir system (with continuous readout) and hence any time-invariant fading memory filter over uniformly bounded input streams. Surprisingly, the main technique in our study comes from finite dimensional dilation techniques in operator theory. I will briefly introduce backgrounds on reservoir computing and explain how dilation theory technique are applied in this setting. This is a joint work with Robert Simon Fong and Peter Tino. |
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Mark Allen, Brigham Young University Isoperimetric and Faber-Krahn Inequalities In this talk we review the isoperimetric inequality and how it leads to the Faber-Krahn inequality. We will then discuss how to establish quantitative forms of these inequalities. This entails measuring how close a set is to a ball if the perimeter (or first eigenvalue of the Laplacaian) is close to that of the ball. We will conclude with a recent result on a quantitative resolvent estimate. |
