Department of Mathematics Colloquia and Seminars

The following table gives the information for all colloquium, applied and computational math seminars, graduate student seminars, and topology, analysis, and geometry seminars at Central Michigan University.

Please scroll down to read the abstract of each event.

Colloquia abstracts

You’ve got a friend in me! Or do you? Using Backbones of Bipartite Projections to Unveil Social Networks

Date: 4/20/2023
Speaker: Rachel Domagalski, Advanced Analytics Center of Expertise, General Motors
Abstract: Signed networks are difficult to collect. Individuals often don’t want to report their negative relationships, and occasionally researchers are prohibited from asking about them. One solution is to infer true relationships from bipartite data. Bipartite projections are used in a wide range of network contexts including politics (bill co-sponsorship), geography (firm co-location), genetics (gene co-expression), economics (executive board co-membership), and innovation (patent co-authorship). However, because bipartite projections are always weighted graphs, which are inherently challenging to analyze and visualize, it is often useful to examine the ‘backbone’, an unweighted subgraph containing only the most significant edges. We’ll discuss different methods of backbone extraction, the distributions behind these methods, and their applications to real-world scenarios.

A Diffeological Approach to Integrating Infinite Dimensional Lie Algebras

Date: 4/18/2023
Title: Joel Villatoro, Washington University in St. Louis
Abstract: Lie groups and Lie algebras are mathematical objects commonly used as models for studying symmetry in geometry. Given a Lie group, one can always construct an associated Lie algebra through a process known as “differentiation.” The reverse procedure is called integration. For finite-dimensional cases, it is well-established that every Lie algebra arises as the derivative of some Lie group, meaning that every finite-dimensional Lie algebra is integrable. However, this is not always true for infinite-dimensional cases. One particularly interesting class of infinite-dimensional Lie algebras comes from Lie algebroids. In this talk, we will provide an overview of the history of integrating Lie algebroids and the challenges encountered. We will then explore how diffeologies might help us overcome some of these issues.

Mathematical Conversations With ChatGPT

Date: 4/6/23
Speaker: Debraj Chakrabarti (CMU)
Abstract: ChatGPT, the recently developed "large language model", has been making headlines. ChatGPT responds to questions posed by users in a startlingly human-like manner, based on a statistical algorithm. Among its latest exploits is to pass the US Uniform Bar Examination with a score in the 90th percentile. There are legitimate fears about the possible impact of these technologies, for example on office jobs that can be potentially replaced by a chatbot. What will be the impact of AI technologies on mathematics and the mathematical profession? It is too early to reach a definite conclusion, but not too early to be worried. In this open-ended presentation, we will ask ChatGPT simple mathematical and logical questions to try to gauge its mathematical capabilities and try to see if there is any reason to worry about our jobs.

Edge Covers of Graphs: The Story of Building an Accessible and Inclusive Undergraduate Research Project

Date: 3/2/2023
Speaker: Feryal Alayont
Abstract: An edge cover of a graph is a subset of the edges so that each vertex is an endpoint for at least one edge. Edge covers provide new combinatorial interpretations to certain number sequences. For example, the number of edge covers of path graphs form the Fibonacci sequence. In summer 2022, five students and I investigated similar sequence patterns in other graph families. I will describe our mathematical results about edge covers of various graph families along with the unusual logistical set-up of this project that allowed for a diverse group of students to participate in undergraduate research. Successes and challenges of this approach will be discussed.

Topology and Order

Date: 11/3/2022
Speaker: Tom Richmond
Abstract: Several connections between topology and order will be presented. We show that every topology on a finite set X is characterized by an equivalence relation on X and a partial order on the equivalence classes. After several examples, we use this link to count the number of nested topologies on a finite set. We briefly introduce complementation in the lattice of topologies on a finite set and give an algorithm to generate complements for a totally ordered convex topological space.

Fall23--Spring24 Graduate Courses Exhibition

Date: 10/13/2022
Speaker: Faculty of Mathematics
Abstract: In this exhibition, some of the Faculty members who plan to teach elective graduate courses ​including MTH 534, MTH 633, MTH 634, MTH 643, MTH 725 and MTH 737 will give a short presentation about the topics to be covered in the course. It is an opportunity for students to sample the course materials and ask questions to the instructors before they submit Graduate Student Survey Form on Friday, Oct 21, 2022. Pizza will be provided during the presentation. All graduate students and faculty are welcome!

Tropical Jacobians and Prym varieties

Date: 10/4/2022
Speaker: Dmitry Zakharov
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.

Homotopy Theory for Graphs

Date: 9/29/2022
Speaker: Laura Scull
Abstract: Homotopy theory studies deformations of geometric objects, an inherently continuous concept. In this talk, we will explore how this translates to the discrete category of graphs. We will see how we can use a little category theory to define a notion of homotopy for graphs, creating an analogous concept of discrete deformations. We will explore the properties of these deformations by looking at the 'fundamental group' of the graph created from homotopy classes of walks. No prior knowledge of either homotopy theory or graph theory will be needed, and many examples and pictures will be given. Parts of this work are joint with Dr. Tien Chih, and parts were developed in collaboration with Fort Lewis College undergraduate students as part of a CURM funded research project.

Curricular Training

Date: 9/22/2022
Speaker: Kristi Wood
Abstract: Kristi Wood, Associate Director/Curricular Process and Systems in CIS, will give our department training on the new Curriculm Strategy system for the curricular workflow. (That is, the system that replaces the old green/blue/pink forms.) This system is needed to do MCS updates/changes, etc.

Graduate student seminar abstracts

Edge-Colored Graphs and Nilpotent Lie Algebras

Speaker: Meera Mainkar
Abstract: In this talk we discuss the symmetries of various combinatorial and algebraic structures. We study the interesting symmetries of edge-colored graphs which give rise to the automorphisms of the associated nilpotent Lie algebras. Nilpotent Lie algebras are algebraic structures useful in geometry and dynamics. This is joint work with Debraj Chakrabarti and undergraduate student Savannah Swiatlowski. 

Agent-Based Models

Speaker: Karleigh Pine
Abstract: Agent-based models (ABMs) simulate the formation and evolution of social processes at a fundamental level by decoupling agent behavior from global observations. Social networks provide a natural set of tools for understanding the emergent relationships of these systems. In this talk I will explore the effect of agent behavior on the associated social graphs in an ABM using methods from network science. I will also discuss how observations from network science about real-world networks can be used for validation of an ABM. In addition to this research, I will highlight a few areas of math that have been useful in my work as a mathematician in the defense industry."

Stability analysis of some numerical schemes for Navier-Stokes equations 

Speaker: Xiaoming Zheng
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.

Sequences and Series in topological vector spaces 

Speaker: Thomas Gilsdorf 
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.

A deterritorialization of mathematics education 

Speaker: Ana Dias 
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.

A brief history of curriculum studies and mathematics curricula 

Speaker: Ana Dias 
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

Preserve one, preserve all

Speaker:  Meera Mainkar
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.

Q-ball Imaging

Speaker: Hiruni Pallage 
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. 

Smooth functions on convex sets

Speaker: Jordan Watts
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.

From real line to measure space

Speaker: Isaac Cinzori
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.

"Oh, the thinks you can think" on y²=x³

Speaker: C-Y. Jean Chan
Abstract: In the xy-plane, the zero set of y²−x³is a curve that can be parametrized by (t²,t³). 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!

Edge Covers of Graphs

Speaker:   Evan Henning
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.

Applied and computational math seminar abstracts

Scheduling and Planning High Uncertainty Seasonal Products at Dow

Speaker: Kyle Harshbarger, Senior Innovation Manager, Dow Inc., Midland, Michigan
Abstract: Dow manufactures sells products with a joint problem of high seasonality and high uncertainty. Proper framing and modeling of the problem allows scheduling and planning processes to be brought under control. Planning for the whole season requires changing buffers with backwards scheduling. Scheduling requires balancing immediate high service level needs to avoid stockouts and rest-of-season analysis to minimize excess inventory. Model fitting of sales history identified a Gamma distribution as best fit to predict future demand in future periods. Monte Carlo approaches are used to accumulate periodic demand for operational requirements.

Machine Learning Glassy Dynamics

Speaker: Andrea Liu
Abstract: The three-dimensional glass transition is an infamous example of an emergent collective phenomenon in many-body systems that is stubbornly resistant to microscopic understanding using traditional mathematical statistical physics approaches. Establishing the connection between microscopic properties and the glass transition requires reducing vast quantities of microscopic information to a few relevant microscopic variables and their distributions. I will demonstrate how machine learning, designed for dimensional analysis reduction, can provide a natural way forward when standard statistical physics tools fail. We have harnessed machine learning to identify a useful microscopic structural quantity for the glass transition, have applied it to simulation and experimental data, and have used it to build a new mathematical model for glassy dynamics.

Data-Driven Fractional Modeling, Analysis, and Simulation of Anomalous Transport and Materials

Date: 1/27/2023
Speaker: Mohsen Zayernouri (Michigan State University)
Abstract: The classical calculus and integer-order differential and integral models, due to their inherently local characters in space-time, cannot fully describe/predict the realistic nonlocal and complex nature of the anomalous transport phenomena. Nature is abundant with such processes, in which for instance a cloud of particles spreads in a different manner than traditional diffusion. This emerging class of physical phenomena refers to fascinating processes that exhibit non-Markovian (long- range memory) effects, non-Fickian (nonlocal in space) interactions, non- ergodic statistics, and non-equilibrium dynamics. The phenomena of anomalous transport have been observed in a wide variety of complex, multi-scale, and multi-physics systems such as: sub-/super-diffusion in subsurface transport, kinetic plasma turbulence, aging polymers, glassy materials, in addition to amorphous semiconductors, biological cells, heterogeneous tissues, and fractal disordered media. In this talk, we present a series of recent spectral theories and global spectral methods for efficient numerical treatment of fractional ODEs/PDEs. Finally, a number of applications including fluid turbulence, power-law rheology, and material failure processes will be also presented, in which fractional modelling emerge as a natural language for high-fidelity modelling and prediction.

Topology, analysis and geometry seminar abstracts

On Rational Convexity of Totally Real Sets

Speaker: Blake Boudreaux
Abstract: A compact set X$\mathrm{<br>}$ in Cⁿ is said to be rationally convex if for every point z not in X$\mathrm{<br>}$ there is a polynomial P$\mathrm{<br>}$ so that P(z)=0 but whose zero avoids X$\mathrm{<br>}$. In view of the Oka-Weil theorem, any function holomorphic on a rationally convex compact X$\mathrm{<br>}$ can be approximated uniformly on X$\mathrm{<br>}$ by rational functions with poles off X. A totally real manifold M is one whose tangent space has no complex structure (i.e., multiplication of tangent vectors by 'i$\mathrm{<br>}$' ejects them from the tangent space). By a classical result of Duval-Sibony, a totally real manifold M in Cⁿ${\mathrm{<br>}}^{}$ is rationally convex if and only if there exists a Kähler form dd^c(u) for which M$\mathrm{<br>}$ is isotropic. Under a mild technical assumption, we generalize this necessary and sufficient condition to the setting of totally real sets (zero sets of strictly plurisubharmonic functions).

Introduction to Diffeological Spaces

Speaker: Emilio Minichiello
Abstract: A diffeological space consists of a set X together with a collection D of set functions U -> X where U is a Euclidean space, that satisfy three simple axioms. In this talk we will describe how this simple definition provides a new, powerful framework for differential geometry. Namely, every finite dimensional smooth manifold is a diffeological space, as are many infinite dimensional ones, orbifolds, and many other objects of interest in differential geometry. Further, the category of diffeological spaces is much better behaved than the category of finite dimensional smooth manifolds, in a way that we will make precise. Despite the fact that diffeological spaces are much more general than manifolds, many classical constructions in differential geometry still make sense for them, such as tangent spaces, differential forms, homotopy theory and fiber bundles. However, recent results show that many of the cherished and basic theorems of smooth manifold theory fail for general diffeological spaces, but this failure opens up worlds of interesting possibilities. We will review two such results. One being the difference between the internal and external tangent space of a diffeological space, and the obstruction between Cech cohomology and deRham cohomology. If time permits, I will discuss the recent work of my preprint "Diffeological Principal Bundles and Principal Infinity Bundles".