Introduction to Research Seminar

AMS Intro to Research Seminar is now online! If you need the Zoom link, email amsmath@umn.edu.

3/10/21 - Speaker: Jeff Calder

Title: PDEs, Random Walks and Graph-Based Learning

Abstract: The talk will be an introduction to some recent research at the intersection of PDEs, random walks, and graph-based learning. We'll discuss various forms of regularization in graph-based semi-supervised learning (Laplacian, p-Laplacian, re-weighted Laplacians, and Poisson learning), and discuss how connections between graph-PDEs and continuous partial differential equations can be used for analysis and development of new algorithms.

3/17/21 - Speaker: Tyler Lawson

4/21/21 - Speaker: Gilad Lerman

12/9/20 - Speaker: Ru-Yu Lai

Title: Introduction to Inverse Problems

Abstract: Inverse problems are the opposites of direct problems. They are problems where the cause for a desired or an observed effect are to be determined from the outputs. Applications of inverse problems can be found in our daily lives and are closely connected to a wide range of scientific studies including geophysics, solar physics, medical imaging, and many others. For example, in geophysics, they are used to locate resistivity anomalies, to monitor mixtures of conductive fluids by surface electrodes, and to determine the substructure of the Earth by the traveling information of waves on the surface. In this talk, I will introduce mathematical background and several interesting questions for inverse problems.

12/2/20: Ask a Postdoc Panel!

11/11/20: Binary Day!

10/14/20 - Speaker: Jasmine Foo (**special time: 1:30pm**)

Title: Cancer evolution and personalized medicine

Abstract: In this talk I will introduce some examples of mathematical models of cancer evolution and how they can be leveraged to develop approaches for personalized cancer therapies.

10/7/20: No seminar, department fellowship meeting

9/30/20: Special session - What Would Bonny Do?

9/23/20 - Speaker: Dan Spirn

Title: Cross fields and thin filaments

Abstract: In this Intro to Research seminar, I will describe two directions of current research: generation of -cross fields and the study of thin structures in viscous fluids. In both cases, I will describe some interesting directions of study.

n-cross fields are locally defined orthonormal coordinate systems that are invariant with respect to the reordering and inversions, and they are used to generate of high quality numerical meshes, to model crystalline lattice structures in advanced materials design and to efficiently define computer graphics shapes. To generate optimal n-cross fields, we develop algorithms that combine tools from algebraic topology, geometric measure theory, the calculus of variations, and partial differential equations. These methods generate smooth n-cross fields, outside of (conjectured) co-dimension two rectifiable sets.

Thin filaments and membranes in viscous fluids thin filaments arise in many important situations, including the modeling of microscopic swimmers and microfluidic devices, where numerical methods for these problems typically break down as the filaments or membranes get very thin. The corresponding equations are interesting integro-differential equations that are only beginning to be studied from a rigorous point of view.

9/16/20 - Speaker: Paul Carter

Title: Vegetation patterns in dryland ecosystems

Abstract: In water-limited regions, competition for water resources results in the formation of vegetation patterns; on sloped terrain, one finds that the vegetation typically aligns in stripes or arcs. The dynamics of these patterns can be modeled by reaction-diffusion PDEs describing the interplay of vegetation and water resources, where sloped terrain is modeled through advection terms representing the downhill flow of water. We focus on one such model in the 'large-advection' limit, and we prove the existence of traveling planar stripe patterns using analytical and geometric techniques. We also discuss implications for the stability of the resulting patterns, as well as the appearance of curved stripe solutions.