New Assistant Professor – Stephan Wojtowytsch
I started my (academic) life as a pure mathematician. I received BSc and MSc degrees in mathematics from the University of Heidelberg in Germany, when I thought I was a differential geometer or a PDE analyst. During my undergraduate studies, I spent a year at the University of Bergen in Norway as part of an Erasmus exchange program. Here, my first publication grew out of an undergraduate research project (on sub-Lorentzian geometry, which is a generalization of the mathematical formalism underlying general relativity). The formative lecture of my undergraduate was functional analysis, and I followed the lecturer to Durham, UK to pursue a PhD program (and back to Freiburg, Germany, after two years). The focus of my thesis is ‘phase-field models for Willmore’s energy,’ i.e., computational methods for a curvature energy for the bending two-dimensional membranes in three-dimensional space.
After receiving a PhD from Durham University, I came to Pittsburgh for the first time as a postdoc at Carnegie Mellon University, followed by a postdoc in Princeton. During the second postdoctoral period, my interests shifted from applied analysis towards machine learning, particularly deep learning. I spent two years at Texas A&M University before joining the University of Pittsburgh.
I am interested in both applied analysis and the mathematical foundations of machine learning. In both fields, I enjoy exploring the role of geometry. Much of my recent work centers more around the question “how and why do neural networks work” rather than “how do I apply them to a specific problem,” although I am interested in that, too. Central themes in my work are approximation theory and parameter optimization/training. Often, the techniques in my work involve (ordinary, partial, stochastic) differential equations, functional analysis, statistical learning theory and some coding.
Faculty Spotlight: Bard Ermentrout
Bard Ermentrout, Distinguished University Professor of Computational Biology and Professor of Mathematics has been awarded a Collaborative Research in Computational Neuroscience Award with Prof. Josh Jacobs at Columbia University to study the role that traveling brain waves might play in human cognition.
Ermentrout, who has been a professor at Pitt for 41 years, uses differential equations to model biological patterns and behavior of many different physiological systems.
In this new project, the Jacobs lab is able get brain recordings from human subjects who have electrodes in their brains (to locate the source of epileptic seizures) while these subjects do various cognitive tasks that require short term memory (such as word lists or the locations of objects in a virtual environment). They find that people that more reliably recall a word from the list had waves that travel in a particular direction at the time that they see the word on the computer screen. Ermentrout's group is using mathematical modeling to explain the patterns and directions of the waves and how modulation such as attention or visual stimuli can alter these waves. He and his group are also offering hypotheses for how these waves could facilitate the transfer of information from one part of the brain (say the visual area) to another (such as the prefrontal cortex, where short term memory and integration across other brain areas occurs).
This award will partially support Ermentrout over the summer and will fund his PhD student who will do modeling and analysis of the equations that come out of the models. Better understanding of the electrical patterns in the brain could be useful in explaining why we need this activity for normal cognitive processes and how disruptions in the activity (such as happen in schizophrenia) can lead to deficits.
New Research
Structure-Preserving Finite Element Methods for Incompressible Flow on Smooth Domains and Surfaces, Michael Neilan
This project will develop numerical methods for solving equations modeling incompressible flow with applications such as predicting weather patterns, designing aircraft, and simulating blood flow. The primary objective is to design and analyze finite element methods (FEMs) that maintain key physical properties at the discrete level, specifically the conservation of mass and incompressibility of the fluid. Such FEMs possess several advantages over existing methods, including superior accuracy, robustness with respect to model parameters, and exact enforcement of multiple conservation laws. However, this class of FEMs is limited in their ability to handle various equation types and geometric domains. This research will overcome these limitations by developing new robust FEMs for incompressible fluid models that can be applied to a wider range of problems. It will focus on two main areas: improving existing FEMs for fluid flow on smooth domains and developing new FEMs for fluid flow on surfaces. The project will provide training opportunities for both undergraduate and graduate students.
Decision Dynamics in Cortico-Basal Ganglia-Thalamic Networks, Jonathan Rubin
Both extrinsic (e.g., sensory signals) and intrinsic (e.g., learned priors) factors determine how information flows through the brain as it accumulates towards eventual decisions. Theoretical and empirical findings point to the brain’s cortico-basal ganglia-thalamic (CBGT) circuits as playing a critical role in this noisy evidence accumulation process. The canonical CBGT framework features multiple action channels, each comprised of two structurally and functionally dissociable control streams, called the direct (facilitation) and indirect (suppression) pathways, that compete for control of thalamocortical dynamics. The effects of this adversarial pathway architecture can be represented algorithmically as regulating the parameters of a diffusing decision process as it moves through a two-dimensional information space, captured by frameworks like the drift diffusion model. This link suggests how CBGT circuits can determine the static state of decision policies, which can shift from exploratory to exploitative depending on environmental contingencies. This project will analyze the fine-grained temporal dynamics of the evaluation, action selection, and action control processes as they progress through the CBGT network, during both discrete and continuous decisions. To this end, we will shift to a higher-dimensional perspective on decision processes in CBGT circuits, modeling them as trajectories through an energy landscape that determines the flow of evidence processing within decisions as well as effects of learning on how the process evolves.
Evolutionary Disease Modeling to Study Community Reorganizations, Sabrina Streipert
Some of the most severe declines in amphibians and other taxa are caused by pathogens. However, trends in collected data suggest that some host populations rebound following disease-induced declines. For example, some populations of harlequin frogs (Atelopus varius) and common rocket frogs (Colostethus panamansis)—that are highly susceptible to infection and mortality from the fungus Batrachochytrium dendrobatidis (Bd)—are recovering after initial declines. The potential mechanisms of such disease recoveries and their effects on community structures are only beginning to be explored for epizoological systems. Nevertheless, the recognition that host diversity and species composition impact disease dynamics, and vice versa, is increasing. However, little attention has been paid to the role of rapid evolutionary processes, such as host defense strategies, on the restructuring of host communities. Such unexplored mechanisms may however explain reorganization events observed in some amphibian systems. Existing studies are also mainly concerned with long-term dynamics and ignore the significance of transient dynamics. We aim to close these gaps by studying the effects of rapid host evolution, habitat overlap, and competition on recovery trajectories and on community reorganization, following disease-induced declines. Our objective is to quantify the effects of community-level mechanisms on the shape of host recovery trajectories, capturing the impacts on transient dynamics. We further our study by identifying mechanisms that are responsible for a reorganization of communities after disease-induced declines. To learn more and/or get involved, contact the PI Sabrina Streipert (sas887@pitt.edu).