## Central Michigan University, Department of Mathematics

### Fall 2022 and Spring 2023

The graduate student seminar features talks on topics in mathematics by faculty, students and visitors, at a level accessible to graduate students.

### Organizer

If you would like to give a talk, please email me!

### Regular Time and Place

Tuedays, 4:00 –5:00pm, Pearce Hall, Room 227.
Some talks may be at other times and venues.

 Date Speaker Title

 Oct 4 (COLLOQUIUM) Dmitry Zakharov (Central Michigan) Tropical Jacobians and Prym varieties Oct 11 Xiaoming Zheng (Central Michigan) Stability analysis of some numerical schemes for Navier-Stokes equations Oct 25 Thomas Gilsdorf (Central Michigan) Sequences and series in topological vector spaces Nov 22 Ana Dias (Central Michigan) A deterritorialization of mathematics education Jan 10 Jan 17 RESERVED Jan 24 Ana Dias (Central Michigan) A brief history of curriculum studies and mathematics curricula Jan 31 RESERVED Feb 7 Meera Mainkar (CMU) Preserve One, Preserve All Feb 14 Feb 21 Hiruni Pallage (CMU) Q-ball Imaging Feb 28 Jordan Watts (CMU) Smooth functions on convex sets. Mar 14 Isaac Cinzori (CMU) From real line to measure space Mar 21 C-Y. Jean Chan (CMU) "Oh, the thinks you can think" on $y^2=x^3$ Mar 28 Evan Henning (GVSU) Edge Covers of Graphs Apr 4 Koksal Karakus (CMU) The sine-Gordon equation and solitons Apr 11 Meera Mainkar (CMU) TBA Apr 18 Joel Villatoro (WUSTL) TBA Apr 25  (3:30 pm) Rasha Al-Mughrabi (CMU) Ph.D. Thesis Defense Bergman Kernels of Monomial Polyhedra

### Abstracts

Speaker: Dmitry Zakharov

Title: Tropical Jacobians and Prym varieties

Abstract: The aim of tropical algebraic geometry is to find polyhedral, piecewise-linear analogues of algebro-geometric objects. The tropical analogue of an algebraic curve is a metric graph. The Jacobian of a metric graph is a real torus that records the intersection numbers of the cycles on the graph. I will talk about the tropical version of an important construction in algebraic geometry: the tropical Prym variety of a double cover of metric graphs. This is a real torus that records those cycles on the source graph that vanish on the target graph. I will derive a formula for the volume of the Prym variety. I will also describe the tropical version of a classical algebraic construction, the trigonal construction of Recillas. This is joint work with Yoav Len and Felix Roehrle.

Speaker: Xiaoming Zheng

Title: Stability analysis of some numerical schemes for Navier-Stokes equations

Abstract: This talk presents how the basic inequalities are used in the stability analysis in some numerical schemes of Navier-Stokes equations. The basic inequalities include Cauchy-Schwarz inequality, Young's inequality, one inequality often used in numerical analysis, and a fantastic algebraic identity.

Speaker: Thomas Gilsdorf

Title: Sequences and Series in topological vector spaces

Abstract: In calculus or other contexts in which a metric is present, sequences can be used to determine properties such as closures and completeness, and for proving important results. For spaces in which the topology does not come from a metric, sequences can no longer be used in the same way, and other structures are needed. In this talk, we will see that for topological vector spaces there are types of convergence of sequences and of series that are useful, even when the spaces are nonmetrizable. In particular, the three types we will see are: Mackey convergence of sequences, K convergence of series, and convergence of series in spaces with completing webs. These types of convergence can be used to obtain results related to the closed graph theorem and duality theory.

Speaker: Ana Dias

Title: A deterritorialization of mathematics education

Abstract: What counts as research in mathematics education? Félix Guattari used the term deterritorialization to denote the destruction of social territories, collective identities, and systems of traditional values. Still according to Guattari, when you engage in a project, for example, a new research project, you can internalize a pre-existing model, a consummate object against which one could measure the ends and the means, or you can try to live the field of the possible that is carried along by the assemblages of enunciation. What are the consummate objects that comprise the dominant models of research in mathematics education and what different models are reterritorializing research on the borders? I will do an exposition on a few types of research in mathematics education that are out there in the present scenario, but that are made invisible by the machinic assemblage of research publishing, including the hierarchy of journals, modes of training and transmission, and the fetish of evidence.

Speaker: Ana Dias

Title: A brief history of curriculum studies and mathematics curricula

Abstract: In this presentation I will present a brief history of the field of curriculum studies and trace a parallel to events in the history of mathematics education. The field of American curriculum studies has moved, since the 1970s, from curriculum development to curriculum understanding. The scholarship in the field of curriculum today is quite different from that which grew out of an era in which schools and education were for the first time expanded to the masses, and when keeping the curriculum ordered and organized were the main motives of professional activity. According to William F. Pinar, William M. Reynolds, Patrick Slattery, and Peter Taubman, in their seminal portrait of the field, many degrees of complexity have entered the conceptions of what it means to do curriculum work, to be a curriculum specialist, and to work for curriculum change. Nonetheless, these changes are scarcely known or acknowledged in the field of mathematics education, where curricular work is often equated to deciding the scope and sequence of content, planning and evaluating. For this reason, I think this talk may be not only informative for graduate students in general, but an opportunity to consider conducting research in curriculum within the context of mathematics education

Speaker:  Meera Mainkar

Title: Preserve one, preserve all

Abstract: The classical theorem of Beckman and Quarles states the following: A function from the Euclidean plane to itself that preserves unit distances must preserve all distances. We will discuss some key steps of the proof. We will also briefly discuss our recent result generalizing this theorem. This is joint work with Ben Schmidt.

No prerequisite knowledge is required, and the talk will be accessible to students.

Speaker: Hiruni Pallage

Title: Q-ball Imaging

Abstract: Magnetic Resonance Imaging (MRI) is a technique that allows us to take images of the brain so that we can identify some diseases. The basic types of MRI are proton density images and T2-weighted images. The advanced versions are Diffusion Weighted Imaging (DWI) and Diffusion Tensor Imaging (DTI) in which we utilize the diffusivity of water molecules. To overcome the limitations of the above forms of MRI we focus on Q-Ball Imaging (QBI). Yet these methods demand a higher imaging time which is not preferable in human brain imaging. The existing literature focuses on reducing imaging time by estimating signal values at certain locations of the brain using: the nearest neighboring locations, the similarity of the corresponding signals, and the (radial basis) interpolation. During this presentation, I will talk about the background of MRI imaging, spherical harmonic basis, Funk Hecke theorem, and my work under the supervision of Professor Kim Yeonhyang.

Speaker: Jordan Watts

Title: Smooth functions on convex sets

Abstract: When you take derivatives, you often don't think too hard about what's actually happening behind all of the symbol manipulation that you're doing.  But there can be some non-trivial mathematics, and in fact, some ambiguity if you look closely!

In this talk, we will consider two definitions of a smooth (i.e. "infinitely-differentiable") real-valued function on a convex set in Euclidean space: an "internal" one where differentiability depends on information inside the convex set, and an "external" one where differentiability depends on information in a neighbourhood around the set.  In the literature, there are many different definitions (and often it is not clear which one is being used), but any reasonable one will be equivalent to either the internal or external one.  We will show that if the convex set is locally compact (e.g. it is open or closed), then both "internal" and "external" definitions coincide.  We will then also give a counterexample in the plane where they differ if the convex set is not locally compact.  No knowledge beyond a rigorous single-variable calculus course (such as MTH 532) and elementary multivariable calculus course (such as MTH 233) will be assumed.

Speaker: Isaac Cinzori

Title: From real line to measure space

Abstract: While many introductory courses in real analysis and measure theory develop the subject on the real line, without too much work the results can be extended to general measure spaces (sets equipped with a sigma-algebra and measure). In this talk, I discuss how to go about this extension and some of the interesting consequences which follow from it.

Speaker: C-Y. Jean Chan
Title: "Oh, the thinks you can think" on $y^2=x^3$
Abstract: In the $xy$-plane, the zero set of $y^2-x^3$ is a curve that can be parametrized by $(t^2, t^3)$. We can think about polynomial functions defined on this curve and determine when two such functions are equal. Many algebraic structures such as quotient rings, semigroup rings, toric varieties can be interpreted using such parametrized zero sets. We will explore how these seemingly independent concepts are linked together and how the discrete structure of lattice points benefits us in understanding these topics. And what else we can think up if only we try!

Speaker:
Evan Henning

Title: Edge Covers of Graphs

Abstract: Graphs are mathematical models used to represent relationships between discrete objects, where objects are represented by vertices and any two related objects are connected by an edge. An edge cover of a graph, defined similarly to a vertex cover of a graph, is a subset of the graph edges such that each vertex is an endpoint of at least one edge in this set. The edge covers of graphs provide combinatorial interpretations of sequences, such as the famous number sequences Fibonacci and Lucas numbers being the number of edge covers of path and cycle graphs, respectively. In our work, we have studied edge covers of various graph families, including caterpillars, chorded cycles, ladder graphs, and spider graphs. We will discuss our recursive methods for counting edge covers and the new sequences and combinatorial interpretations of known sequences that we have obtained.

Speaker: Koksal Karakus
Title:
The sine-Gordon equation and solitons
Abstract: The Sine-Gordon equation, clasically given as $u_{xx} - u_{tt} = \sin u$, is a non-linear partial differential equation that initially came up in the study of surfaces with constant negative curvature, and in time has been of interest in a diverse set of disciplines from chemistry to DNA biology to quantum physics. The equation leads to soliton (solitary wave) solutions, which led to a recent revival of interest in the equation. In this talk I will talk about the derivation of some of the soliton solutions of the equation, as well as the Bäcklund transformation which allows to find new solutions from present ones. I will also briefly mention how the equation relates to thin-film dynamics.