Information

Thursday 11:15 AM, online
Organizers: Shannon Golden and Daniel Jonas. Email us if you would like to give a talk!
Seminar attendance sheet: PDF
A useful collection of resources on giving mathematics talks.

Schedule

Date	Speaker	Title	Advisor
August 27	Colin Roberts	A multiscale approach to modeling university impact on municipal COVID-19 dynamics	Clayton Shonkwiler
September 3	Christina Rigsby	Navigation with Machine Learning	Simon Tavener
September 10	Kyle Salois	On the Equality of a Class of Shifted Skew Schur Functions
September 17	Brandon Van Over	An Investigation of Mumford's Formula Over the Moduli Space of Pseudostable Curves
September 24	Canceled
October 1	Danny Long	(Practice Qual Part II) Mechanism-Enabled Population Balance Modeling	Wolfgang Bangerth
October 8	Kirk Bonney	WTF is a Modular Form
October 15	Canceled
October 22	Nate Mankovich	A Search for the Center in the Reals and Beyond	Michael Kirby
October 29	John Bush	Diameter-extremal subsets of spheres	Henry Adams
November 5	Shannon Golden	Coset Enumeration is basically just augmented sudoku anyway	Alexander Hulpke
November 12	Harley Meade	Reframing Frames through Combinatorics	Emily King
November 19	Rachel Tremaine	If You Like it Then You Should've Put a Ring on It: How Ring Learning with Errors Increases Key Exchange Security
November 26	No Greenslopes	Fall Break
December 3	Tatum Rask	SIR Modeling of American Chestnut Blight
December 10	Amaury Miniño	The Point of Pointless Topology: a case study in how mathematicians give different names to the same thing

Abstracts

August 27: Colin Roberts, A multiscale approach to modeling university impact on municipal COVID-19 dynamics.

Epidemiological models and COVID-19 data exist on various spatial scales ranging from intra-host mechanisms to global viral spread. Our goal is to create an assimilation scheme to incorporate various data streams into a multiscale epidemiological model of inter-host dynamics. A multiscale framework will allow the usage of the more accurate, but computationally expensive, agent based models as well as more efficient compartmental models to reflect macroscale dynamics of pathogen spread. We use synthetically generated data from small scale agent based models in conjunction with real data to produce contact values for the macroscale compartmental model. As an application, we use this framework to examine the role of a university in municipal spread of COVID-19 and the impacts of various intervention strategies.

September 3: Christina Rigsby, Navigation with Machine Learning.

We investigate the application of machine learning to GPS-denied navigation, using an established system created in the University of Florida's Research and Engineering Education Facility (REEF), the REEF Estimator. Modular in structure, the REEF Estimator was designed to be incrementally modified with relative ease [REEF]. Our primary goal was to replace the bayesian RGBD Odometry unit with one using machine learning and compare results to those previously seen.

To work our way up to this task, we completed preliminary projects: (1) the Speed Challenge [SpeedChallenge] and (2) visual odometry using data from a ground robot [frei]. For the Speed Challenge, we used dashcam imaging to estimate the car’s speed. In doing this, we learned about custom cost functions, controlling overfitting, and cutting computational cost. We also observed that neural networks simply optimize over the prescribed cost function, not always in a way they are intended to: instead of the network estimating speed by comparing consecutive images and their differences, it was learning the relationship between contextual cues and car speeds. This unintended learning motivated our second preliminary project in which we forced the neural network to compare consecutive images through a custom cost function. Although we have not extensively experimented with visual odometry, we discuss how this is done with machine learning and present ideas on how to analyze its performance.

September 10: Kyle Salois, On the Equality of a Class of Shifted Skew Schur Functions.

A symmetric function \(f\) is a function in the variables \(x_1,...,x_n\) such that permuting the order of the variables does not change the function. There are many interesting bases of \(\Lambda_R\), the ring of symmetric functions with coefficients in a ring \(R\), with the Schur basis \(s_{\lambda}\) among the most interesting due to its proximity to other fields of math. While Schur functions come from semistandard fillings of the Young tableaux for integer partitions, Schur Q functions examine weakly amenable fillings of tableaux corresponding to shifted integer partitions. We explore some properties of Schur Q functions, and make headway on a question related to expressing a skew shifted Schur Q function \(Q_{\lambda / \mu}\) in terms of non-skew Schur Q functions \(Q_{\lambda}\) by examining a class of tableaux inspired by the work of Stephanie van Willigenburg on ribbon tableaux.

September 17: Brandon Van Over, An Investigation of Mumford's Formula Over the Moduli Space of Pseudostable Curves.

A current hot topic in mathematics, the moduli space of curves of a fixed genus and number of marked points \((\mathcal{M}_{g, n})\) is a central object of study in algebraic geometry which has interesting connections to physics, combinatorics, and differential geometry. In this talk we compare the intersection theory of two compactifications of this space, the Deligne-Mumford compactification \((\overline{\mathcal{M}}_{g, n})\), and the Pseudostable compactification \((\widetilde{\mathcal{M}}_{g, n})\). We do so via studying their Chow rings, and in particular investigate how a famous formula due to Mumford proven for Chow classes in the Deligne-Mumford compactification changes for the Pseudostable compactification.

September 24: Canceled

October 1: Danny Long, (Practice Qual Part II) Mechanism-Enabled Population Balance Modeling.

Nanoparticles are useful in many applications within the scientific community. The chemistry community is interested in creating nanoparticles with very consistent sizes. However, there is little known about the chemical processes which spur the formation and subsequent growth of nanoparticles. This knowledge is necessary before experiments can be optimized to favor particular nanoparticle sizes.

We begin by exploring a particular set of chemical reactions that produce nanoparticles. Then the work of Derek Handwerk, a recent CSU grad, will be described, which involves an inverse problem to match a theoretical particle size distribution to data. Finally, we detail new work extending Derek's analysis by taking a Bayesian approach to the inverse problem. In other words, instead of finding a single set of parameters to a differential equation that fits the data well, we compute a probability distribution on the parameter space. We discuss the relative strengths/weaknesses of the two methods as well as the future direction of this research.

October 8: Kirk Bonney, WTF is a Modular Form.

A quote of Eichler goes, "there are five fundamental operations of arithmetic: addition, subtraction, multiplication, division, and modular forms." The goal of this talk is to explore what modular forms are and justify why they are such fundamental objects.

October 15: Canceled

October 22: Nate Mankovich, A Search for the Center in the Reals and Beyond.

Finding the center of a set of data is an essential part of numerous common machine algorithms like K-means and LBG clustering. There are multiple ways to calculate the center of a set of real vectors in n-dimensional space like the mean and the median. The mean is a nice convex optimization problem, whereas the median is a nasty one. The Weiszfeld Algorithm is one solution to approximating the median via iteratively calculating weighted means. We will provide an introduction to this type of iterative optimization in the reals. Then we will move into a similar optimization problem to the median problem in projective space. If there's time at the end, we will generalize further and look at a generalization of this problem on the Grassmannian manifold.

October 29: John Bush, Diameter-extremal subsets of spheres.

Diameter-extremal subsets of spheres, as considered by Katz and Lovasz, satisfy the property that all nearby subsets in Hausdorff distance have strictly larger diameter. These subsets are good candidates for vertex sets of simplices that change the homotopy type of Vietoris-Rips complexes defined on spheres. I will describe some known diameter-extremal subsets and relate them to the persistent homology of spheres and generalizations of the Borsuk-Ulam theorem.

November 5: Shannon Golden, Coset Enumeration is basically just augmented sudoku anyway.

Given a group presentation, a sneakily difficult question to answer is "what group is this?" (followed by "is this finite?" and "is it abelian?"). Attempting to answer this question has generated many different tools to help identify properties of the associated group. In this presentation (*pun intended*), we will give some background as to how group presentations are defined and look at a few common examples. The main topic we will explore is the enumeration technique developed by Todd-Coxeter, which will allow us to tell the index of our original group G given a subgroup H. Further topics will be the Reidemeister-Schreier and modified Todd-Coxeter methods of coset enumeration and their benefits and drawbacks.

November 12: Harley Meade, Reframing Frames through Combinatorics.

Full-spark frames are potentially redundant generalizations of orthonormal bases that are maximally robust to erasures. As such, they are important in fields such as signal processing. Though spark concerns the algebraic spread spread of vectors in a frame, it connects us to combinatorics by way of matroid theory. Results are known about which submatrices of \(n\times n\) discrete Fourier transform matrices for \(n\) a prime power, and we attempt to generalize some of these results for the character tables of non-cyclic groups.

November 19: Rachel Tremaine, If You Like it Then You Should've Put a Ring on It: How Ring Learning with Errors Increases Key Exchange Security.

As quantum computing develops at a rapid pace, there exists an increased need for cryptographic systems that cannot (by any presently known algorithm) be broken utilizing quantum technology. Several such systems have been proposed, many of which are based upon the utilization of lattice structures and associated hard problems to create signatures, key exchanges, encoded messages, and the like. This talk will summarize one such cryptographic system, Ring Learning with Errors (RLWE), and how it reflects a standard Diffe-Hellman key exchange protocol to create a more secure path for arriving at a shared secret between two parties.

November 26: Fall break!

December 3: Tatum Rask, SIR Modeling of American Chestnut Blight.

Since the introduction of fungal blight (Cryphonectria parasitica), the American chestnut (Castanea dentata) has gone from a dominant canopy tree in Appalachian forests to an understory species. The introduction of the hypovirus (Cryphonectria hypovirus) to the blight fungus, which reduces pathogen virulence, has improved the survival of some stands of chestnut trees. However, many populations are still declining. We added biological relevance to existing models of the system by incorporating several important classes and interactions that were not previously captured. These additions provide a more complete and biologically accurate description of the demography of chestnut stands. Our results indicate that the future of the American chestnut tree is less than happy, suggesting that scientists should look for new methods of saving this species.

December 10: Amaury Miniño, The Point of Pointless Topology: a case study in how mathematicians give different names to the same thing.

In studying the structure of topological spaces, it becomes natural to focus more on the open sets than the elements of the underlying set. Point-free topology arises from this study, and leads to interesting connections to lattice theory. We will work through its construction and its intersection into ring theory.

Past Semesters