Title: Ribbon disk formulae for the $\alpha_2$- and $\alpha_3$-invariants of ribbon $2$-knots 

Abstract: Finite type invariants for classical knots have been extended to ribbon $2$-knots, and various related studies have been developed. In this talk, we focus on the degree $2$ and $3$ finite type invariants derived from the Alexander polynomial, called the $\alpha_2$- and $\alpha_3$-invariants, and present explicit formulae for these invariants in terms of ribbon disks. We will begin with the Gauss diagram formula for the second coefficient of the Conway polynomial of a classical knot, and then introduce ribbon $2$-knots, ribbon disks, and their Gauss diagrams. Some examples will also be given. 

Title: Yang-Baxter cohomology and the Zamolodchikov tetrahedron equation

Abstract: In this talk, I will discuss some recent applications of Yang-Baxter cohomology to the construction of solutions to the Zamolodchikov tetrahedron equation, and how these can be used in low-dimensional topology. I will present some connections with computational physics and topological quantum field theory, along with the construction of some quantum invariants in dimensions $3$ and $4$. 

Title: An Interesting Class of Nilpotent Lie Algebras

Abstract: We discuss a new class of nilpotent algebras.  We describe connections to problems in linear dating back to the 1970s.  We present examples, state obstructions to existence, describe structure and give geometric applications.

Title: Circuit Decompositions for Triangulations of Surfaces

Abstract: An Euler circuit of a graph is a closed path that visits every edge of the graph exactly once. Euler circuit and circuit decomposition problems can also be formulated for higher dimensional simplicial complexes. An Euler k-circuit in K is a cyclic sequence of vertices v_1...v_n such that every k+1 adjacent terms { v_i,v_{i+1},...,v_{i+k} } (indexed modulo n) form a k-simplex, and every k-simplex of K appears exactly once in the sequence v_1v_2...v_n(v_1v_2...v_k). We investigate the 2-circuit decomposition problem for triangulated closed compact surfaces. We use interior angles of paths to define an obstruction that lives in the first cohomology of the surface. 

Title: Equivalence of Integral Hermitian Forms over Number Fields

Abstract: Given two hermitian matrices with entries in a number ring, one may ask whether or not they are equivalent under an integral change of basis.  Does there exist an algorithm to decide whether or not two given matrices are equivalent? If so, is there an algorithm to effectively produce the transition matrix? It turns out that these questions can be reduced to a problem involving the quantitative behavior of a related group action. Using tools from homogeneous dynamics and ergodic theory, we show that if two hermitian matrices are equivalent, then a transition matrix can be found after a finite search.

Title: Using heights to find rational points on curves

Abstract: A central question in number theory is finding the set of integer or rational number solutions to a given system of polynomial equations. This question, called the Diophantine problem, has more than 1700 years of history and is still an active area of research. In this talk, we will start with finding rational numbers x and y satisfying the equation -139y^2=x^3+10x^2-20x+8. One key ingredient is heights, which measure the arithmetic complexity of such rational solutions. While we explore this question, I will introduce some key players in number theory and their interactions: elliptic curves, modular curves, modular forms, L-functions, and the more general height theory. In particular, we will see a connection between the heights of rational points and the derivatives of L-functions, known as the Gross—Zagier formula. One direction of my work is a generalization of this formula, where a key step involves constructing Green’s functions on graphs. Another direction is concrete computations of rational points on some modular curves when the set of rational points is known to be finite. I will introduce my past, current, and possible future work in these two directions.

Title: Redux: an Online, Interactive Knowledgebase of Computational Logic Problems and Algorithmic Solutions

Abstract: Supported by our recent $300,000 Accelerate Idaho AI Innovation Award, students and faculty in the ISU Computer Science (CS) department have been building a platform called Redux to facilitate a crowdsourced knowledge base of canonical CS problems and algorithms—particularly those that involve discrete mathematics, combinatorial optimization, and graph theory. Aimed at serving pedagogical and practical use cases, Redux now hosts 49 problems and 93 algorithms (and counting), along with an assortment of other features, including step-by-step algorithm visualizations, problem-to-problem mappings, and solution verifiers. Much of the content of the knowledgebase has its foundations in mathematics, and the team eagerly seeks collaboration. Redux is available online at https://redux.isu.edu/.