# Information

- Thursday 11:00 AM, Online and Room: Weber 223
- Organizers: Emily Heavner and Brittany Story. 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 26 | Colin Roberts | Lorentzian Geometry and Topological Electromagnetism | Clayton Shonkwiler |

September 2 | Nate Mankovich | FRD-NNs and Lessons Learned at PNNL | Michael Kirby |

September 9 | Lander Ver Hoef | A Short Introduction to Topological Data Analysis for Remote Sensing | Adams and King |

September 16 | Michael Moy | The Topology of Vietoris-Rips Complexes and Metric Thickenings | Henry Adams |

September 23 | Carter Lyons | Weighted Ensemble for Rare Event Simulations | David Aristoff |

September 30 | Lara Kassab | Detecting Short-lasting Topics Using Nonnegative Tensor Decomposition | Henry Adams |

October 7 | Wei-Yu Hsu | Introduction to anthocyanins and patterns in Plant Pigmentation | Patrick Shipman |

October 14 | Naomi Fahrner | Synthetic Aperture Radar Flight Path Optimization for Resolution and Coverage | Margaret Cheney |

October 21 | Danny Long | Studying nanoparticles with mathematics | Wolfgang Bangerth |

October 28 | Kyle Salois | Symmetric Functions, Shifted Tableaux, and a Class of Distinct Schur $Q$-Functions (A Practice Master's Defense) | Maria Gillespie |

November 4 | Elliot Krause | An introduction to elliptic curves with a view towards point counts | Jeff Achter |

November 11 | Brittany Story | What in the world is sublevelset persistence and how does it relate to sandstone, chemistry, and space networks? | Henry Adams |

November 18 | Justin O'Connor | The Shroud of Turin: Jesus, Physics, and Pixar | Wolfgang Bangerth |

November 25 | No Greenslopes | Fall Break | |

December 2 | Christina Rigsby | An Analysis of Domain Decomposition Methods | Simon Tavener |

December 9 | Tea Time! |

# Abstracts

## August 26: Colin Roberts, *Lorentzian Geometry and Topological Electromagnetism*.

We live in a Lorentzian spacetime, but we tend to think with our Euclidean brains. In this talk, I want to explore spacetime and discuss an algebraic way to describe the Lorentzian geometry. Then, I will talk about the rigid symmetries of spacetime given by the Poincaré group and build its Lie algebra. Finally, I will introduce the de Rham (co)homology and use both geometry and topology together to study electromagnetism. I will also like to chat briefly about internships in general and my experiences over the years working with both MCRN and NASA.

## September 2: Nate Mankovich, *FRD-NNs and Lessons Learned at PNNL*.

In the past few decodes, neural networks (NNs) have become a prolific model for data classification in research and in industry. These models have been able to produce high test accuracies on simple datasets like MNIST and more complex datasets like ImageNet. With some datasets, it is essential to have a NN that is able to correctly classify an image regardless of its rotation. We call a model with this property, rotationally invariant. NNs struggle with this seemingly simple problem. Strategies like training a vanilla NN with data augmentation and using modified NNs like the E(2)-CNN have moved researchers closer to a rationally invariant model. However, these approaches are not perfectly rotationally invariant and are slow because they require generating many rotations of one image and take a long time to train. In this talk, we present a new, simple, approximately rotationally invariant model inspired by work done by Tegan Emerson at the Pacific Northwest National Labs (PNNL). This model is called the Fourier Ring Descriptor - Neural Network (FRD-NN). After introducing FRD-NNs, will benchmark them against a vanilla NN on the rotated MNIST digits dataset. If there’s time at the end, we’ll take a step back and talk about what it’s like to work as a remote intern at PNNL.

## September 9: Lander Ver Hoef, * A Short Introduction to Topological Data Analysis for Remote Sensing*.

In analyzing data from remote sensing instruments, current methods focus primarily on pixel-based or accumulation measures. However, there is a wealth of information contained in the textural and spatial arrangement of the data. While convolutional neural networks have shown some success in leveraging such textural and spatial information, they face challenges in adoption, due to their lack of transparency.

This work presents an introduction to a tool from the mathematical field of topological data analysis, called sublevelset persistent homology, which allows the quantification of the shape and textural information present in an image. Sublevelset persistent homology has several beneficial properties, in particular that it is rotation-, translation-, and scale-invariant. Topological data analysis is particularly useful as a preprocessing step before using other machine learning algorithms.

## September 16: Michael Moy, *The Topology of Vietoris-Rips Complexes and Metric Thickenings
*.

Vietoris-Rips complexes are specific types of simplicial complexes that are used by the applied topology community to study the shapes of datasets. In the first part of the talk, we will look at some specific examples of Vietoris-Rips complexes, identifying topological spaces to which they are homeomorphic or homotopy equivalent. In the second part of the talk, I'll describe how persistent homology makes use of Vietoris-Rips complexes and related measure-theoretic constructions called metric thickenings. This will include the main result from my master's thesis (the stability of persistent homology for Vietoris-Rips metric thickenings), as well as ongoing work.

## September 23: Carter Lyons, *Weighted Ensemble for Rare Event Simulations*.

In the field of molecular dynamics many events of interest, for example the probability of reaching a particular region of state space, are uncommon. So, using naïve Monte Carlo techniques to estimate these uncommon events often requires a computationally infeasible number of independent trials. To reduce the computational burden Weighted Ensemble (WE), an importance sampling/interacting particle Markov Chain Monte Carlo (MCMC) method, is often employed in practice with molecular simulations. By increasing the amount a Markov Chain explores uncommon areas of the state space, WE not only makes it feasible to estimate uncommon events but also does so with a reduction in the variance of these estimates. In this talk we detail Markov Chains, MCMC methods, and the WE algorithm. We also, show how molecular dynamics are simulated and discuss some open questions currently being explored in WE.

## September 30: Lara Kassab, *Detecting Short-lasting Topics Using Nonnegative Tensor Decomposition
*.

Temporal text data such as, news articles or Twitter feeds, often consists of a mixture of long-lasting trends and popular but short-lasting topics of interest. A truly successful topic modeling strategy should be able to detect both types of topics and clearly locate them in time. We show that nonnegative CANDECOMP/PARAFAC tensor decomposition (NCPD) successfully detects such short-lasting topics that other popular methods such as Latent Dirichlet Allocation (LDA) and Nonnegative Matrix Factorization (NMF) fail to fully detect. We demonstrate the ability of NCPD to discover short and long-lasting temporal topics in semi-synthetic and real-world data including news headlines and COVID-19 related tweets.

## October 7: Wei-Yu Hsu, *Introduction to anthocyanins and patterns in Plant Pigmentation *.

Red, blue, and purple colors in plants are typically due to plant pigments called anthocyanins. In a plant cell, an equilibrium is established between anionic and cationic forms of anthocyanins as well electrically neutral colorless forms called hemiketals. In typical pH ranges in cells, the colorless hemiketal would be expected to be the dominant form. Why then, do plants, in fact, display colors? We propose that this is part due to aggregation of the colored forms of anthocyanins. In this talk, we start with the analysis of the steady-state in different pH values from the basic scheme, and then to the aggregation scheme, and the square scheme. Beside the results of the steady-state, we also observe some dynamical results when changing the total concentrations. Lastly, apply the activator-inhibitor system to varying anthocyanin concentration in plants to simulate the colorful spotted pattern formation of plants.

## October 14: Naomi Fahrner, *Synthetic Aperture Radar Flight Path Optimization for Resolution and Coverage*.

Two key components in Synthetic Aperture Radar(SAR) are image resolution and scene coverage. To improve SARimage resolution, one may increase the bandwidth of the system (frequency diversity) or increase the range of aspect angles toview the target (geometric diversity). In most applications, the radar must operate within a fixed and specified frequency band,so increasing the frequency diversity is not feasible. To exploit geometric diversity and maximize scene coverage, one may flyaround the target scene of interest in a non-straight path and steer the radar antenna towards the desired targets. This leads toa hybrid SAR mode that combines stripmap SAR and spotlight SAR modes. To investigate the trade-off between resolutionand coverage, we develop a mathematical tool to quantify the attainable resolution of the target scene by incorporating thedata-collection manifold (DCM). We use the DCM to build an objective function that is used as a means for determining optimalflight paths for desired coverage and resolution objectives by varying a SAR vehicle’s heading, pitch, and antenna steering angles.

## October 21: Danny Long, *Studying nanoparticles with mathematics*.

Nanoparticles are widely useful in applications but are difficult to monitor with traditional measurement techniques. Fortunately, we can use mathematical techniques to learn about the reactions that create nanoparticles. I will describe how I use Bayesian Inversion techniques to understand how these chemical reactions take place. I will then discuss how I can further use the results of the Bayesian Inversion analysis to answer some other questions that are interesting. For example, (i) how can we collect data in such a way to get more reliable results from Bayesian Inversion?, and (ii) how can we set up the experiment to target a certain particle size as the outcome of the chemical reaction? Finally, I will discuss what happens when a nanoparticle system requires the solution to millions of differential equations. Namely, how to approximate the ODEs with a PDE in order to have a computationally feasible set of equations to solve.

## October 28: Kyle Salois, * Symmetric Functions, Shifted Tableaux, and a Class of Distinct Schur $Q$-Functions (A Practice Master's Defense)*.

The Schur $Q$-functions form a basis of the algebra $\Omega$ of symmetric functions generated by the odd-degree power sum basis $p_{d}$, and have ramifications in the projective representations of the symmetric group. So, as with ordinary Schur functions, it is relevant to consider the equality of skew Schur $Q$-functions $Q_{\lambda/\mu}$. This has been studied in 2008 by Barekat and van Willigenburg in the case when the shifted skew shape $\lambda/\mu$ is a ribbon. Building on this premise, we examine the case of near-ribbon shapes, formed by adding one box to a ribbon skew shape. We particularly consider frayed ribbons, that is, the near-ribbons whose shifted skew shape is not an ordinary skew shape. We conjecture with evidence that all Schur $Q$ functions for frayed ribbon shapes are distinct up to antipodal reflection. We prove this conjecture for several infinite families of frayed ribbons, using a new approach via the ``lattice walks'' version of the shifted Littlewood-Richardson rule, discovered in 2018 by Gillespie, Levinson, and Purbhoo.

## November 4: Elliot Krause, *An introduction to elliptic curves with a view towards point counts* .

This is a practice job talk targeted at a general math audience, specifically at those that are not number theorists. I'd appreciate feedback from any and all. This talk will begin with an introduction to elliptic curves with a few historical notes. Elliptic curves are naturally geometric objects, but also have an arithmetical aspect. Diophantus likely wrote down the first elliptic curve circa 300 AD, with contributions to the theory following from Fermat, Newton, Jacobi, Eisenstein, Weierstrass and Poincaré. Elliptic curves have found recent use in the proof of Fermat's last theorem and elliptic curve cryptography. Finding points and counting points (in particular, integer points) on elliptic curves has been a central topic dating back to Diophantus and Fermat. We'll overview a few tools given by the theory of equidistributed sequences and close by giving a few original results on the distributions of points counts for elliptic curves over finite fields.

## November 11: Brittany Story, * What in the world is sublevelset persistence and how does it relate to sandstone, chemistry, and space networks?*.

This talk will consist of an overview of sublevelset persistent homology and three different applications of it. We'll start with the definition of a topological space and work up to more complicated objects. After we build some structure, we'll move on to applications of persistent homology that involve sandstone, chemistry, and space networks and some of the Python that went with each of the projects. Note, this talk is designed to be an introduction, so no prior knowledge will be necessary and questions are encouraged.

## November 18: Justin O'Connor, * The Shroud of Turin: Jesus, Physics, and Pixar*.

This talk will consist of a brief history of the scientific study of the Shroud of Turin - believed by some to be the burial cloth of Jesus Christ. We will look at historical and physical evidence both for and against this possibility - including radiocarbon dating, vacuum UV radiation experiments, pollen analysis, ancient art, and more! before settling into a discussion of the mathematical difficulties of simulating physics on a cloth. We will look both at methods that are fast and pretty - widely used in computer graphics and animation - and then at methods that are physically accurate, and discuss why they are rarely, if ever, used.

## November 25: Fall break!

## December 2: Christina Rigsby, *An Analysis of Domain Decomposition Methods*.

Iterative solvers have attracted significant attention since the mid-20th century as the computational problems of interest have grown to a size beyond which direct methods are viable. Projection methods, and the two classical iterative schemes, Jacobi and Gauss-Seidel, provide a framework in which many other methods may be understood. Parallel methods, or Jacobi-like methods, are particularly attractive as Moore's Law and computer architectures transition towards multiple cores on a chip. We explore two such methods, the multiplicative and restricted additive Schwarz algorithms for overlapping domain decomposition, and discuss their implementation. Finally, we present a possible extension of this work to an agent-based modeling prototype currently being developed by the Air Force Research Laboratory's Autonomy Capability Team (ACT3).