**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 |