Prof. Frank Sottile, Texas A&M, College Station, USA

Title: "The trace test in numerical algebraic geometry"

Numerical algebraic geometry uses tools from numerical analysis to study algebraic varieties on a computer. Its origins were in homotopy methods, which used Newton iterations and homotopy continuation to solve systems of equations. Early homotopy algorithms exploited combinatorial structures, such as multihomogeneity, for efficiency. In numerical algebraic geometry, a variety X is represented by a witness set, which is a linear section of X in a projective or affine space.

A fundamental step is to decompose a witness set for a variety X into subsets corresponding to the irreducible components of X. An algorithm for this numerical irreducible decomposition uses monodromy to compute a possible decomposition which is verified using the trace test.

In this talk I will introduce numerical algebraic geometry and witness sets, and describe numerical irreducible decomposition, including a new and elementary proof of the trace test. I will then explain versions of witness sets, the trace test, and numerical irreducible decomposition for multihomogeneous varieties X that take advantage of this structure.

This is joint work with Anton Leykin and Jose Rodriguez.

Sottile is a Professor of Mathematics at Texas A&M University. He holds a Masters degree from Cambridge University (1986) and a MSc. (1989) and Ph.D. from the University of Chicago (1994). His research interests are in algebraic combinatorics and the applications of algebraic geometry. He was the founding chair of the SIAM Activity Group on Algebraic Geometry and is a corresponding editor of the SIAM Journal on Applied Algebra and Geometry. He has published over 100 peer-reviewed articles and one book. He was a Churchill Scholar, held a NSF Career award, and is a Fellow of the American Mathematical Society.

Dr. Jürgen Gerhard, Senior Director of Research, Maplesoft

Title: What's New in Maple 2016

We will present some of the new features in Maple 2016, including data frames, Math apps, series and limit computations, symbolic integration and summation, symbolic PDE solving, Statistics, and more. See the Maple 2016 Research Datasheet.

Maplesoft (http://www.maplesoft.com) is the leading provider of high-performance software tools for engineering, science, and mathematics. Maplesoft's flagship product, Maple, combines the world's most powerful mathematics engine with an interface that makes it extremely easy to analyze, explore, visualize, and solve mathematical problems.

Jürgen Gerhard holds a PhD from University of Paderborn, Germany. He has been with Maplesoft since 2003, and is currently Senior Director of Research. His areas of interest are symbolic computation and its applications in engineering, and he is coordinating research and consulting projects in these areas at Maplesoft. Together with Joachim von zur Gathen, he is the co-author of the reference textbook "Modern Computer Algebra", which as of 2013 is in its 3rd edition.