ÐÓ°ÉÔ­´´

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 Uni