Central Michigan University Mathematics
Graduate Student Seminar (GSS)

Organizer

Graduate Student Seminar Coordinator: Debraj Chakrabarti
Because there might be a time delay in updating the webpage, please always check with the Coordinator for the available dates.

Meeting Times and Platforms

Regular GSS Talks are on Tuesdays, 4:00–4:50pm, in person in Room 227 of Pearce Hall. Talks at other times which may be attended by students taking the GSS for credit are noted in the schedule are noted in bold.

SCHEDULE 2024–2025

Abstracts

Speaker: Debraj Chakrabarti
Title: The Cross product in higher dimensions
Abstract: The cross product of two vectors in three-dimensional space is discussed in calculus -3, along with the dot product. We do teach students how to generalize the dot product in higher dimensions (as an inner product structure on vector spaces). What is the correct higher dimensional generalization of the cross product? We will answer this question, and go over the history and geometric significance of the exterior product that generalizes the cross product.

Speaker: Xiaoming Zheng
Title: Gaussian Random Fields
Abstract: This talk is a basic introduction of Gaussian Random Field (GRF). In statistics, a Gaussian random field (GRF) is a random field involving Gaussian probability density functions of the variables. GRFs are widely used across multiple scientific, engineering, and data-driven fields, such as temperature, rainfall, soil composition, air pollution, porous materials, composite materials, turbulent flows, and large-scale structure of the universe. We will talk about GRFs, covariance operator (differential form and integral form), and spectrum by Fourier Transform.

Speaker: Meera Mainkar
Title: The Matrix Exponential
Abstract: The exponential of a matrix plays a crucial role in the theory of Lie groups. In this talk, we will see some basic properties of the matrix exponential and matrix logarithm.  

Speaker: Lino Guarjado
Title: Mathematician's Stories: Investigating the Storylines of Mathematicians Entering the Mathematics Professoriate
Abstract: Humanizing proof-based courses has been a point of interest for me in the recent years. Part of humanizing proof-based courses, to me, is highlighting the stories of mathematicians from marginalized communities, offering points of connection to students from the same or similar communities. From an NSF-funded project, working with faculty from various institutions (mathematicians and math educators), we have developed a set of interesting proofs and mathematician stories to go with them. These stories, meant for an introduction to proof audience, showcase the mathematical journeys of diverse mathematicians (men and women of color). In this talk, we will see how these stories counter the dominant storyline that mathematicians' career paths are linear and stem from an uninterrupted interest and success in mathematics. By presenting different ways of engaging with mathematics and entering the professoriate, these stories further disrupt dominant narratives of individualism, innate ability, and dehumanized mathematics. The stories emphasized joy in community and collaboration, renegotiating mathematical identity when encountering struggles, and bridging school and home life. These stories contribute to our understanding of basic narrative arcs and the ways mathematicians present their stories to inspire students.

Speaker: Dmitry Zakharov
Title: Quasideterminants
Abstract: The determinant of a matrix over a field is a polynomial in the entries of the matrix, and is nonzero if and only if the matrix is invertible. We can use the same formula to define the determinant of a matrix over a commutative ring, such as the integers. I will talk about several attempts to generalize determinants to certain classes of non-commutative rings, such as quaternions and quantum rings. The most general such construction, which works over an arbitrary division ring, is called the quasideterminant and was introduced by Gel’fand and Retakh. The key difference with the commutative case is that the quasideterminant is a rational function of the matrix entries, rather than a polynomial.

Speaker: Alessandro Monguzzi
Title: Convergence of ergodic means of Hilbert space shift operators.
Abstract: In this talk, I will give a basic introduction to abstract Hilbert space shift operators and discuss the pointwise convergence of their ergodic means. Unilateral shift operators are key examples of non-surjective isometries and have a rich history in hard analysis and operator theory. I will review some well-known results and illustrate them with a few examples. If time permits, I will also present one of my own results on the speed of convergence of the ergodic means of shift operators. The talk is intended to be accessible to a general mathematical audience.

Speaker: Lisa DeMeyer
Title: Vertex Coloring of the Zero-divisor Graph
Abstract: The zero-divisor graph associated to a commutative ring, $R$, was first introduced by I. Beck in 1988. Beck’s primary aim was to study the vertex coloring of the zero-divisor graph of $R$. A proper coloring of a graph, $G$, is a coloring of the vertices of $G$ so that no two adjacent vertices have the same color. We will discuss Beck’s results, and several results which followed Beck’s 1988 paper. Several generalizations of the zero-divisor graph will be presented, and we will discuss known graph coloring results in each context.

Speaker: Isaac Cinzori
Title: An exposition of the internal tangent space
Abstract: In certain spaces which are not manifolds, it is possible to define a notion of ``tangent space" which extends the conventional definition of tangent space at a point of a manifold. In particular, on spaces where one can define a diffeology (an alternate smooth structure), one can define the ``internal tangent space" at a point in the space. In this talk, I will discuss how to calculate the internal tangent space in the case where the space is the quotient space of a manifold by a proper Lie group action, beginning with the manifold case and proceeding to a nontrivial example and an outline of the general approach. As this talk is aimed at graduate students, some time will be spent developing the notions of manifold and tangent space, in addition to the primary material.

Speaker: Pete Vermeire
Title: The Amazing Synthesis
Abstract: We will discuss what David Mumford called “The Amazing Synthesis”, the categorical equivalence(s) between compact Riemann surfaces, smooth algebraic curves, and field extensions of the complex numbers of transcendence degree one. During the talk we will define what these objects are.

Past Graduate Student Seminars: Spring 2023/Fall 2022