The 2023-2024 Chern Lecture

Bernd UlrichSpeaker: Bernd Ulrich (Purdue University)

November 7, 8, 9

Title of the seriesEquisingularity, linkage, and the implicitization problem

This is a series of independent talks on three topics where algebra, singularity theory, and algebraic geometry come together.

Lecture 1: Tuesday, November 7, 4:10-5 pm, Moffett Library room 102

TitleEquisingularity and multiplicities.

A goal in equisingularity theory is to devise criteria for analytic sets to be "alike," especially when they occur in a family. Ideally, such criteria only depend on numerical invariants of the individual members rather than the total space of the family. It was Teissier's seminal insight to relate the known equisingularity conditions, such as the geometric conditions of a Whitney stratification, to the algebraic concept of multiplicity. Thus he was able to characterize the equisingularity of families of isolated hypersurface singularities. Any generalization beyond the case of isolated hypersurface singularities however, requires new notions of multiplicities. The talk will survey past and present work on this subject and explain the connection with multiplicity theory. 

Lecture 2. Wednesday, Nov 8, 4:10-5 pm, Tan Hall room 180
Title: Linkage of algebraic varieties and ideals.Linkage, or liaison, is a tool for classifying and studying varieties, ideals, and algebras. The theory has its origins in the nineteenth century, and it was recast in modern algebraic language and studied extensively since the 1970s. In this talk we will introduce the subject and lead up to more recent work on a generalization of linkage that arises naturally in intersection theory.

Lecture 3. Thursday, Nov 9, 4:10-5 pm, Evans Hall room 60
Title: The implicitization problem for algebraic varieties. 
The talk is concerned with a classical problem in elimination theory that has also been of interest in applied mathematics -- the problem of determining the equations of a variety that is given parametrically. A wide range of tools has been developed for this question, and we will describe some of them. We will also explain how knowing the types of singularities of a variety can inform the search for its implicit equations.

Bernd Ulrich received his doctoral degree in 1980 from the University of the Saarland in Germany. He became a Professor of Mathematics at Michigan State University in 1986 and moved to Purdue University in 2001. He spent three semesters at MSRI, where he held a Research Professorship in 2012/13, he was a Simons Foundation Fellow in 2013, and he will spend the Spring of 2024 at SLMath as a Clay Senior Scholar. He is a Fellow of the AMS, and he has graduated 25 doctoral students. His work is in commutative algebra and algebraic geometry, with an emphasis on topics in algebra that are motivated by geometry and singularity theory, such as liaison, intersection theory, residual intersection, equisingularity theory, multiplicity theory, and Rees algebras.