Colloquium Series | New Mexico State University

Coming Next:

04/25

The Development and Applications of Skew F Distribution, Ziwei Ma, University of Tennessee at Chattanooga

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.

Science Hall 107 and Zoom

12:00PM refreshments | 12:30PM - 1:20PM

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Spring 2025

May 2	Speaker Title Abstract	Chun Liu, Illinois Institute of Technology TBA TBA
April 25	Speaker Title Abstract	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.
April 11	Speaker Title Abstract	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.
April 4	Speaker Title Abstract	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.
March 28	Speaker Title Abstract	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.
March 28	Speaker Title Abstract	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.
Special Colloquium March 19	Speaker Title Abstract	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).
February 28	Speaker Title Abstract	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.
February 14	Speaker Title Abstract	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.
January 31	Speaker Title Abstract	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.
January 24	Speaker Title Abstract	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.

Previous Colloquia

Spring 2024

May 03	Speaker Title Abstract	Ishan Levy, Massachusetts Institute of Technology Stable Homotopy Groups of Spheres, Telescopes, and Algebraic K-Theory The stable homotopy groups of spheres are a mathematical object which play a fundamental role in many areas such as geometric topology. These groups are incredibly complicated and have a rich structure, which is captured via the study of spectra. Spectra can be viewed as analogs of abelian groups in the realm of homotopy theory, and chromatic homotopy theory provides an approach to studying spectra by decomposing them into 'monochromatic' pieces. Each monochromatic piece of a spectrum is built out of objects called telescopes. I will explain how the stable homotopy groups of telescopes can be used to produce infinite families of elements in the stable homotopy groups of spheres. Ravenel's long-standing telescope conjecture sought to describe telescopes in terms of spectra whose homotopy groups are more understandable. I will explain recent advances which use algebraic K-theory as a new tool to probe the homotopy groups of telescopes and in particular disprove the telescope conjecture. As a consequence, we are able to produce new lower bounds on the asymptotic behavior of the stable homotopy groups of spheres.
April 19	Speaker Title Abstract	Jianguo (James) Liu, Colorado State University Numerical Simulations for Subdiffusive Transport in Poroelastic Media Many biological and geological problems can be modeled as transport in porous media, e.g., gas and oil extraction from petroleum reservoirs and drug delivery to cancer sites. Moreover, the media could be poroelastic and transport is subdiffusive. In this talk, we present results from our on-going efforts for development of efficient and robust numerical solvers for time-fractional convection-diffusion problems and (linear and nonlinear) poroelasticity problems. We pay special attention to positivity-preserving, local mass conservation, and free of Poisson-locking. This talk is based on a couple of collaborative projects with researchers at Jilin University (China), University of Kansas, and Colo State.
April 12	Speaker Title Abstract	Tài Hà, Tulane University Algebraic Perspectives Of Polynomial Interpolations In Several Variables The polynomial interpolation problem considers the construction and the number of polynomials of a fixed degree that vanishes at a given set of point with prescribed multiplicities. In this talk, we will examine a few open problems for polynomial interpolations in several variables. We will discuss algebraic approaches to these problems, particularly, one that investigates containment between powers of ideals.
April 05	Speaker Title Abstract	Arnab , Heriot-Watt University Nowcasting Growth Using Google Trends Data: A Bayesian Structural Time Series Model This paper investigates the benefits of internet search data in the form of Google Trends for nowcasting real U.S. GDP growth in real time through the lens of mixed frequency Bayesian Structural Time Series (BSTS) models. We augment and enhance both model and methodology to make these better amenable to nowcasting with large number of potential covariates. Specifically, we allow shrinking state variances towards zero to avoid overfitting, extend the SSVS (spike and slab variable selection) prior to the more flexible normal-inverse-gamma prior which stays agnostic about the underlying model size, as well as adapt the horseshoe prior to the BSTS. The application to nowcasting GDP growth as well as a simulation study demonstrate that the horseshoe prior BSTS improves markedly upon the SSVS and the original BSTS model with the largest gains in dense data-generating-processes. Our application also shows that a large dimensional set of search terms is able to improve nowcasts early in a specific quarter before other macroeconomic data become available. Search terms with high inclusion probability have good economic interpretation, reflecting leading signals of economic anxiety and wealth effects.
March 22	Speaker Title Abstract	Bella Tobin, Agnes Scott College Stochastic Dynamical Systems and Their Julia Sets In arithmetic dynamics we often consider the behavior of points under iteration of a rational function $f$ defined over a field of number theoretic interest. Every rational map has a Julia set which is often considered the locus of chaos. The Julia set is defined as the complement of the set of points on which $f$ is equicontinuous and it has many interesting properties. A stochastic dynamical system is a set of maps $S$ together with a probability measure $\nu$ on $S$. Points are iterated under elements of $S$ according to $\nu$. For such a system we can define a stochastic Julia set in both the complex and p-adic setting. In this talk we will discuss stochastic dynamical systems, how we build their Julia sets, and some of the interesting properties of their Julia sets. While most of this work is number theoretic in nature it will be discussed in a general and broad setting. No number theory background will be necessary to enjoy the beauty of stochastic dynamical systems.
January 26	Speaker Title Abstract	Sumeyra Sakalli, University of Arkansas Exotic 4-Manifolds and Complex Surfaces, Algebraic Fibrations, Singularities and Knots 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 the 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.
January 22	Speaker Title Abstract	Keegan Boyle, University of British Columbia Alexander Polynomials and Symmetric Knots The Alexander polynomial is an invariant of a knot in the 3-sphere, defined in terms of the homology of a particular covering space of the knot complement. It is easy to compute and contains significant geometric information about the knot. Consequently, it has received a lot of attention in the literature over the last century. In this talk, I will discuss several of my recent theorems about the Alexander polynomial. These theorems indicate how new geometric information about the knot can be extracted from the polynomial in the presence of a symmetry, and have applications to 4-manifold topology. The proofs of these theorems include a mix of classical tools and methods from algebraic and geometric topology, and modern inventions such as Heegaard Floer homology.
January 19	Speaker Title Abstract	William Balderrama, University of Virginia Equivariant Maps Between Spheres A fundamental problem in topology is to classify all continuous maps between two spheres up to continuous deformation, that is, up to homotopy. This problem becomes even richer when considering maps that are required to respect additional symmetries that may be present. Perhaps the first major result along these lines is the Borsuk--Ulam theorem, which asserts that there exists no continuous map from a sphere to a sphere of lower dimension that is moreover an odd function (in the usual calculus sense). In this talk, I will discuss how these ideas led to some of the early development of equivariant stable homotopy theory, and describe some recent work in this area.

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	Speaker Title Abstract	Adam Dor-on, Haifa University, Israel Non-commutative Boundary Theory and Arveson's Hyperrigidity Conjecture In his foundational work, Arveson extended the classical boundary theory for subalgebras of functions to the non-commutative world. His theory provided us with non-commutative analogues of Shilov and Choquet boundaries which, classically, are smallest subsets of the space for which the maximum modulus principle holds. Arveson's theory has had a profound influence in the fields of operator theory and operator algebras, leading up to the resolution of several problems in dilation theory, non-commutative convexity, structure theory of C-algebras and classification theory of operator algebras. 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. All necessary background will be provided throughout the talk, and the construction of the counterexample will be clear (at least) to third-year undergraduate students.
November 08	Speaker Title Abstract	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	Speaker Title Abstract	Sherzod Murodov, Washington State University Dynamics of Chains of Finite-Dimensional Evolution Algebras Historically, mathematical methods have been applied successfully to population genetics for a long time. Investigation of mathematics to population genetics goes to Mendel’s (Gregor Johann Mendel, 1822-1884) law, where he exploited symbols that are quite algebraically suggestive to express his genetic laws. Mendel ﬁrst exploited symbols that are quite algebraically suggestive to express his genetic laws. Thus, mathematicians and geneticists once used non-associative algebras to study Mendelian genetics and it was later termed “Mendelian algebras” by several other authors. Now, non-Mendelian genetics is a basic language of molecular geneticists. Non-Mendelian inheritance plays an important role in several disease processes. Non-Mendelian genetics oﬀers to mathematics new type of genetic algebras, denominated evolution algebras. 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	Speaker Title Abstract	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	Speaker Title Abstract	Boyu Li, New Mexico State University Universality Of Simple Cycle Reservoirs Using Dilation Theory Reservoir computation models form a subclass of recurrent neural networks with fixed non-trainable input and dynamic coupling weights. Reservoir models have been successfully applied in a variety of tasks and were shown to be universal approximators of time-invariant fading memory dynamic filters under various settings. Simple cycle reservoirs (SCR) have been suggested as severely restricted reservoir architecture, with equal weight ring connectivity of the reservoir units and input-to-reservoir weights of binary nature with the same absolute value. Such architectures are well suited for hardware implementations without performance degradation in many practical tasks. 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.
September 13	Speaker Title Abstract	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.

New Mexico State University