James D. Brunner

Graduate Student
University of Wisconsin
Department of Mathematics

Office: Van Vleck 520
Email: jdbrunner@math.wisc.edu

My work at Los Alamos National Laboratory was recently featured in SIAM News.

Research Interests

I am interested in dynamical systems and applied mathematics. My research is on chemical reaction network theory, the use of mathematical models to analyze large biochemical reaction networks. My advisor is Gheorghe Craciun. For an introduction to chemical reaction network theory, I recommend these lecture notes from Prof. Jeremy Gunawardena. Also, take a look at Martin Feinberg's lecture notes.

Chemical reaction network theory deals with dynamical models for sets of chemical and biochemical reactions. Large networks involving hundreds of reactions are the rule rather than the exception in physiology and medicine. To analyze and understand such networks, chemical reaction network theory seeks to draw conclusions from network properties that can be determined, such as structure and connectivity. This is done by the analysis of a broad class of deterministic and stochastic differential equations, solutions of which simulate time course data from the corresponding chemical network. These differential equations are determined by data and a hypothesized chemical mechanism, and therefore form a bridge between data and understanding.

I study dynamical systems in order to develop tools that can be readily applied to individual models. An understanding of the general class of models provides an understanding of individual models as well as the similarities between models. My current focus is on the global behavior chemical reaction networks, specifically the characteristic of permanence. Informally, a permanent chemical network has the property that it never "uses up" any chemical that is involved. Permanence is an interesting property not only for its physical interpretation, but also for what it tells us about the mathematical system. Permanence can help to characterize global behavior of a system, including the existence and global stability of equilibrium states and periodic solutions. The purpose of my current work is to create a characterization of permanent networks. Such a characterization could then be used to determine whether any given model of the right type is permanent.

The rigorous mathematical study of dynamical systems provides the tools which make understanding mathematical models and simulation of biological and biochemical phenomenon possible. Understanding a class of models helps us analyze mathematical models used in individual instances. For a more details on my research interests, please read my research statement.

About Me

I am a graduate student at UW Madison studying applied mathematics. I am interested in the use of applied mathematics to solve real biological problems, and about the mathematics that can be inspired by these problems. I have worked in the past with the University of Michigan math department, the Virginia Bioinformatics Institute at Virginia Tech, and the department of cancer biology at the Wake Forrest School of Medicine. I received a B.S. in mathematics from the University of Michigan in 2012, with an academic minor in Biology.

My current work is in chemical reaction network theory with Prof. Gheorghe Craciun. I am organizing the graduate applied math seminar, which meets Fridays at 1:00 pm. I am also organizing this year's Mega Math Meet, a competition for local 5th and 6th grade students.

I grew up in Novi, Michigan, a suburb of Detroit. I am a die hard fan of the Tigers, Red Wings, Pistons, and Lions, as well as the University of Michigan Wolverines. I enjoy swimming, running, playing my guitar, and camping. I highly recommend vacationing in northern Michigan, especially at Sleeping Bear National Lake Shore and Mackinac Island. I am a brother of Triangle Fraternity, the fraternity of scientists, architects and engineers.

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