Department of Mathematics Calendar
Martin Helmer, Effective Methods in Algebraic Geometry and Applications
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At its most basic, algebraic geometry studies algebraic varieties; that is, the solution sets of systems of polynomial equations. In this talk our focus is on developing a concrete understanding of the geometry and topology of varieties and using this understanding to obtain practical and effective computational methods. Such methods may then in turn be used to solve problems arising in science and engineering.
In particular, we will look at a method to compute Segre classes of pairs of varieties and explain how this gives rise to a new technique to test if one variety is contained in another.
As an application we will consider an engineering problem arising in mechanism design, known as Alt’s problem, which asks how many four-bar linkages there are whose coupler curve interpolates nine general points in the plane.
Finally, we will explore results which help to understand the geometric and topological influence of singular points (e.g., cusps, nodes, etc.) on a variety by giving relations between the relative global geometry of the variety and its local topology near the singularity. This will in turn provide novel approaches to computing certain invariants of sets of singular points inside a variety.