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Dissecting Graphs

Presented by Ann Clifton (Louisiana Tech University)

Graphs are used to model a wide range of phenomena from social networks to the structure of chemical compounds. In a social network, we may want to divide the group into teams that share some special characteristics. In a chemical compound, we may be interested in certain types of substructures. How far can these divisions be taken while preserving the desired characteristics?We model this question by asking which graphs have an equitable dissection. An equitable dissection of a graph on n vertices is an iterative partitioning of the vertex set into two disjoint balanced subsets so that the induced subgraphs are connected. We say a graph is equitably dissectable if the iteration results in n isolated vertices. We will present some recent results and questions for future work.

A Taste of Equivariant Topology

Presented by Dr. Eric Hogle (ÐÓ°ÉÔ­´´)

Equivariant topology studies moving shapes. Some very simple questions about how shapes can move haven't been answered until quite recently. You might think it would be hard to understand research that hasn't even been published yet, but come give it a try! If you can imagine a donut, you can learn something only a handful of people on earth know.

Inventing the Future of Medical Imaging with Mathematics

Presented by Dr. Melody Alsaker (ÐÓ°ÉÔ­´´)

When most people think of medical imaging modalities such as CT, MRI, or ultrasound scans, they probably don't think about mathematicians being involved in the development of these important life-saving technologies. Medical imaging is part of a large and dynamic field of applied mathematics known as "inverse problems." This talk will provide an introduction to the idea of an inverse problem, and we will touch on the mathematics behind a few modern medical imaging technologies, including my own area of research, Electrical Impedance Tomography (EIT).

The Future of Governing Equations

Presented by J. Nathan Kutz (University of Washington)

Machine learning and AI algorithms are transforming a diverse number of fields in science and engineering. This is largely due their success in model discovery which turns data into reduced order models and neural network representations that are not just predictive, but provide insight into the nature of the underlying dynamical system that generated the data. We introduce a number of data-driven strategies, including targeted uses of deep learning, for discovering nonlinear multiscale dynamical systems, compact representations, and their embeddings from data. Importantly, data-driven architectures must jointly discover coordinates and parsimonious models in order to produce maximally generalizable and interpretable models of physics-based systems and processes.

Mathematical modeling of water flow, plankton, pollutants, and more: Are more complex models necessarily better?

by Lisa Lucas (Research General Engineer for the United States Geological Survey)

Abstract: A common refrain of environmental scientists is: “Environmental science isn’t rocket science. It’s harder than rocket science.” Understanding, predicting, and managing the workings of environmental systems is a grand challenge, due in no small part to the intricate interactions between physical, biological, and geochemical processes that are, individually, complex enough for whole careers to be spent deciphering them. Fortunately, technological advancements in field and laboratory instrumentation, remote sensing, and computing permit us to measure and mathematically model environmental systems with ever-increasing extent and resolution. Drawing on my and colleagues’ work in San Francisco and Chesapeake Bays, I will discuss both complex and super-simple modeling approaches, with an emphasis on the latter. I will show how simple, computationally trivial models---such as those that can be evaluated with a pencil and paper or a spreadsheet---can be surprisingly valuable for (1) diagnosing and learning about how complex environmental systems work, (2) integrating multiple physical, biological, and/or geochemical processes, (3) producing accurate results, and (4) providing valuable guidance for ecosystem management. I will draw on examples from environmental fluid mechanics, algal blooms, contaminant transport, and hypoxia in coastal and river systems.

Biography: Dr. Lucas is a Research Engineer and Eco-hydrodynamicist with the U.S. Geological Survey. Inhabiting the interface between environmental physics and biology in aquatic ecosystems, she studies how interactions between hydrodynamics and other physical and biogeochemical processes influence water quality and aquatic ecosystem function. She works in estuarine and river systems, implementing a broad range of mathematical modeling approaches spanning multi-dimensional numerical models to simple algebraic equations solvable with a hand calculator. Lisa’s work aims at improving understanding and prediction of how aquatic ecosystems work and, thereby, supporting informed ecosystem management. She loves collaborative, interdisciplinary science, as is required in her recent projects focusing on the modeling of harmful algal blooms, nutrients, salinity intrusion, and methyl mercury. Lisa studied Civil Engineering as an undergrad at the University of Notre Dame and did her Master’s and Ph.D. in Civil & Environmental Engineering at Stanford University.

Rithmomachia: A Proposal of Rules and Progressions for Victory

Presented by Rie Durnil (ÐÓ°ÉÔ­´´ Student)

Rithmomachia is an ancient mathematical strategy game that gradually disappeared after the Renaissance leaving inconsistent interpretations of the rules and the number of ways to achieve victories. As a basis for further research, Dr. Tomás Guardia and I propose a set of rules in hopes of creating an academic discussion to establish a final set that is both historically accurate and player friendly. Our current research on mathematic progressions within the game will also be discussed.

National Security Research and Careers in the US National Laboratories

by Aaron Luttman (Pacific Northwest National Laboratory)

The US Department of Energy maintains 17 national laboratories, located around the country, that employ over 40,000 scientific and technical staff across all STEM disciplines. Many of the laboratories, along with additional production plants and experimentation sites, focus on research and development tailored to solving the nation’s greatest national security challenges. Supporting national security missions is among the most fulfilling professional trajectories that a scientist or engineer can follow, and, in this presentation, we will discuss how to pursue technical careers in the national laboratories. In addition to being fulfilling, the actual science and technology being discovered, explored, and developed in the national laboratories is world-leading. For this discussion, we’ll highlight some of the mathematics, physics, chemistry, and engineering in nuclear fusion research, artificial intelligence and machine learning, and materials science to demonstrate how undergraduate and graduate students, as well as career staff scientists, in the national laboratories are driving scientific discovery to solve national security problems.

Dr. Aaron Luttman is a Senior Technical Advisor at Pacific Northwest National Laboratory (PNNL), where his work focuses on understanding emerging challenges in nuclear security and developing scientific approaches to addressing them, both in support of the US nuclear weapons stockpile and in support of nuclear nonproliferation research. He has degrees in mathematics from Purdue University, the University of Minnesota, and the University of Montana, and his research has focused on computer vision and image processing. Over the last 15+ years, Aaron was a university professor, a research scientist, a Senior Technical Advisor to the US National Nuclear Security Administration (NNSA) in Washington, D.C., a personnel manager of teams of scientists and engineers, and a program manager. He is active in the mathematics community, as an officer of the Mathematical Association of America’s (MAA) Special Interest Group on the mathematics of Business, Industry, and Government, as a member of the Society of Industrial and Applied Mathematics’ (SIAM) Committee on Applied Mathematics Education, and as a member of the Embry-Riddle Aeronautical University’s Industrial Advisory Board for Computational Mathematics. He is also an active industrial partner in undergraduate research, providing research projects to the MAA’s PIC Math program, to numerous REU programs, and to Montana State University’s Data Science program. Aaron was the 2011 Outstanding New Teacher at Clarkson University in Potsdam, NY, and he has received numerous awards for his scientific contributions to the NNSA. He was an MAA Distinguished Lecturer, a SIAM Visiting Lecturer, and was a featured mathematician in 101 Careers in Mathematics (3rd Ed.).

Spatial localization of eigenvectors (and waves)

Presented by Jeffrey Ovall (Director of Computation And Data Enabled Science Program at Portland State University)

Functions describing the behavior of (acoustic or electromagnetic) waves in space and time can often be expressed in terms of infinite sums in which each term is a product of a time-dependent "amplitude" and a space-dependent "standing wave". These standing waves are eigenvectors of an associated differential operator. It is known that properties of the differential operator, and geometric properties of the domain in which the problem is posed, can cause some eigenvectors to be highly concentrated in relatively small parts of the domain (i.e. nearly zero outside these regions). This phenomenon is referred to as localization of eigenvectors, and the connection between eigenvectors and waves implies that waves can also localize within certain frequency ranges. A better theoretical understanding of the mechanisms governing eigenvector localization, and practical algorithms for identifying such eigenvectors, are important in the design of materials that have desired acoustic or electromagnetic properties.

After making the connection between acoustic waves (the wave equation) and eigenvalue/eigenvector problems, we will describe and illustrate eigenvector localization through several examples, and highlight a few applications in which localization of waves is of interest.
Although there have been significant developments in the theoretical understanding of localization over the past decade or so, there is still much room for improvement, and even more so on the computational side. We will pose a few basic (but not necessarily simple) questions, before shifting the focus of the rest of the talk to the development of a new type of algorithm that is specifically designed to target localized eigenvectors.
Careers in Science presented by Provost Sacha Kopp (ÐÓ°ÉÔ­´´)

Co-sponsored by the Physics Department, Provost Kopp will talk about careers in science, how one gets started, what’s invigorating about doing science, and what are some options for careers post graduation. A Q&A session will follow the talk. Pizza and light beverages will be offered from 3:30pm - 4:00pm, followed by the talk from 4:00pm - 5:00pm. Bring your friends and classmates!