Schedule, Notes/Handouts & Videos

Abstract

Imagine you are visiting Honolulu and have a packed schedule of activities. Your map shows a dot for every site to visit, the best poke spot, and a secret bay to snorkel with sea turtles. Using the roads connecting these dots as edges of a graph, you wish to find all the minimal routes connecting these destinations that avoid forming a cycle. A matroid holds the key! In this mini-course we will get our hands dirty defining, computing, and exploring various perspectives of matroids. We will narrow in on realizable matroids, those arising from linear systems, and in particular, a well-behaved family of realizable matroids called Positroids. By exploring the many combinatorial objects associated with Positroids, we will touch briefly on their far-reaching implications in other areas of mathematics and science. This is geared towards undergraduate students and will assume some familiarity with linear algebra.

Link to presentation slides: https://www.academia.edu/40810771/Modernmath2019_minicourse

In the geosciences, it is oen necessary to simulate physical processes that are impractical or impossible to observe directly such as subsurface water ux. e classic approach to simulation and prediction for many years has been development, calibration, and validation of physics-based models; however these approaches have notable limitations including introduction of user and model bias, sparse observational datasets, not to mention computational burden. Over the past few decades applications of articial intelligence, most notably deep neural networks (DNNs), have grown in popularity due to advances in computer hardware in conjunction with the availability of remotely sensed data sets. A DNN is a layered network of densely interconnected information-processing nodes trained to recognize the relationship between model inputs and desired outputs to make predictions as a weighted linear combination of inputs.
Here, a DNN was used to map input parameters (soil/land-use characteristics, weather) from the Soil & Water Assessment Tool (SWAT) to remotely sensed soil moisture from NASA’s Soil Moisture Active Passive (SMAP) satellite. e objective of this research was to accurately predict regional soil moisture corresponding to temporally synchronous weather observations/forecasts using a DNN. Predicted soil moisture may then be used as a parameter for risk assessments (e. g. ooding and crop viability), providing near-real-time, high-resolution results.

Motivated by recent advances in technology for medical imaging and high-throughput genotyping, we consider an imaging genetics approach to discover relationships between the interplay of genetic variation and environmental factors and measurements from imaging phenotypes. We propose an image-on-scalar regression method, in which the spatial heterogeneity of gene-environment interactions on imaging responses is investigated via an ultra-high-dimensional spatially varying coefficient model (SVCM). Bivariate splines on triangulations are used to represent the coefficient functions over an irregular two-dimensional (2D) domain of interest. For the proposed SVCMs, we further develop a unified approach for simultaneous sparse learning (i.e., G×E interaction identification) and model structure identification (i.e., determination of spatially varying vs. constant coefficients). Our method can identify zero, nonzero constant and spatially varying components correctly and efficiently. The estimators of constant coefficients and varying coefficient functions are consistent and asymptotically normal. The performance of the method is evaluated by Monte Carlo simulation studies and a brain mapping study based on the Alzheimer's Disease Neuroimaging Initiative (ADNI) data.

Abstract

Imagine you are visiting Honolulu and have a packed schedule of activities. Your map shows a dot for every site to visit, the best poke spot, and a secret bay to snorkel with sea turtles. Using the roads connecting these dots as edges of a graph, you wish to find all the minimal routes connecting these destinations that avoid forming a cycle. A matroid holds the key! In this mini-course we will get our hands dirty defining, computing, and exploring various perspectives of matroids. We will narrow in on realizable matroids, those arising from linear systems, and in particular, a well-behaved family of realizable matroids called Positroids. By exploring the many combinatorial objects associated with Positroids, we will touch briefly on their far-reaching implications in other areas of mathematics and science. This is geared towards undergraduate students and will assume some familiarity with linear algebra.

Link to presentation slides: https://www.academia.edu/40810771/Modernmath2019_minicourse

In 1970, Cheeger established lower bounds on the first eigenvalue of the Laplacian on compact Riemannian manifolds in terms of a certain isoperimetric problem. The analogous problem on domains of Euclidean space has generated much interest in recent years, due in part to its connections to capillarity theory, image processing, and landslide modeling. In this talk, based on joint work with Leonardi and Saracco, we give an explicit characterization of minimizers in this isoperimetric problem for a very general class of planar domains.

We explore the dynamics of a charged particle inside of a bounded region under the influence of a magnetic field. We discuss how the shape of the region might make it easier (or harder) for us to trap the particle to its interior using a magnetic field. The system we will discuss provides a simplified mathematical model for the interior of fusion reactor devices and particle accelerators.