Coming Next:
Spring 2025
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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. |
Previous Colloquia
Spring 2024
May 03 |
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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 |
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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 |
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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 |
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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 |
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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. |
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Fall 2024
November 22 |
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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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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 |
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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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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 |
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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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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. |
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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. |
