ACMS Applied Math Seminar: Daniel Brake


Location: 127 Hayes-Healy

Daniel Brake
Postdoctoral Research Associate
Department of Applied and Computational Mathematics and Statistics

3:30 PM
127 Hayes-Healy

Parametrized Polynomial Systems, and Real Numerical Algebraic Geometry

Numerical Algebraic Geometry (NAG) is a set of theory and tools for studying systems of polynomials, and complements the symbolic methods.  It is distinctly geometric in flavor, and is well-suited to a number of applications in engineering and science. 

Systems of parametrized polynomial systems can be challenging to solve, both for symbolic methods and NAG, due to large number of symbolic objects, i.e. variables and parameters.  The ubiquitous Monte Carlo method, or plain mesh-like discretization, can be a useful tool to study parametrized systems.   Paramotopy, a program which solves systems over arbitrary samplings of a parameter space, has been used to study kinematic manipulators, chemical reaction networks, and biological multistability.  I will talk about its theoretical framework, and applications of the software.

A complement to parametrized systems, under-determined systems have positive-dimensional components to their varieties.  Over the complex numbers, components have generic behaviour, and are relatively easy to characterize, even if they have singularities.  Over the real numbers, however, positive-dimensional varieties are difficult to study, most namely because the reals are a set of measure zero in the complexes, and the theoretical basis for NAG doesn't hold.  However, recent theoretical algorithmic advances have provided us with tools to study 1- and 2-dimensional real sets.  In this vein, I will talk about Bertini Real, software which numerically `decomposes' real curves and surfaces in any number of variables.


















List of Speakers:

Apr. 23 Jason MacLean - Department of Neurobiology, University of Chicago
Apr. 16 Daniel Molzahn - Dow Sustainability Fellow, University of Michigan
Apr. 9 Zoltan Toroczkai - Physics
Apr. 2 Martina Bukač - Applied and Computational Mathematics and Statistics
Mar. 19 Daniel Brake - Applied and Computational Mathematics and Statistics
Feb. 19 Dhagash Mehta - Applied and Computational Mathematics and Statistics
Feb. 12 Holly Goodson - Chemistry and Biochemistry
Feb. 5 Joseph Powers - Aerospace and Mechanical Engineering
Jan. 27 Yunhua Xue - Nankai University and University of Dartmouth, Massachusetts
Jan. 22, 2015 Ava Mauro - Applied and Computational Mathematics and Statistics
Dec. 10 Nicolas Brunel - Department of Statistics and Neurobiology, University of Chicago
Dec. 3 Amy Buchmann - Applied and Computational Mathematics and Statistics
Nov. 19 Pinar Zorlutuna - Aerospace and Mechanical Engineering
Nov. 12 Ling Xu - Applied and Computational Mathematics and Statistics
Oct. 29 Joel Boerckel - Aerospace and Mechanical Engineering
Oct. 15 Zhangli Peng -  Aerospace and Mechanical Engineering
Oct. 8 Tim Weninger - Computer Science and Engineering
Oct. 1, 2014 Dervis Can Vural - Physics