Tuesday, June 11
Tuesday, 8:30am-9:50am
Simon Thomas (Rutgers University),
Title: Measuring the relative complexity of mathematical problems
Abstract: In 1937, Reinhold Baer solved the classification problem for the additive subgroups of the rational numbers Q. Since then, despite the efforts of many mathematicians, no satisfactory classification has been found for the additive subgroups of Q ⊕ Q and it is natural to ask whether this problem is “genuinely more difficult”. In these talks, I will discuss a recently developed method for measuring the relative complexity of mathematical problems and will illustrate this approach with concrete problems from algebra and topology. In particular, I will explain why it would be a very bad idea to assign the classification problem for subgroups of Q⊕Q as the PhD thesis problem of a graduate student.
CurrentTrends-Thomas-2024.pdfCoffee break
Tuesday, 10:20am-11:40am
Henryk Iwaniec (Rutgers University),
Title: The distribution of prime numbers
Abstract: TBA
Lunch break
Abstract: The first non-trivial problem in Schubert calculus asks how many lines meet 4 fixed lines in general position in a 3-dimensional space. More generally, one can ask for the number of subspaces of a complex vector space that intersect other subspaces in specified dimensions. Grassmannian manifolds and their cohomology rings provide a modern framework for answering questions of this type. More generally, quantum cohomology can be used to determine the number of curves in a Grassmannian that pass through certain subsets. I will speak about this and how to use related combinatorial tools to do explicit computations.