Higher Algebra in Geometry

In these lectures, we will shed light on modern tools of higher algebra, where the traditional structures of algebra yield themselves only after controlled deformations. We will introduce infinity-categories, spectra, operads, and other standard tools of the last decade. The main applications will be to encode various higher-algebraic structures that inevitably arise in, and shed light on, geometry and topology. If time permits, we will illustrate how spectra naturally arise in geometric invariants.

The audience is imagined to consist of mathematicians interested in applications of infinity-categorical tools -- so a broad range of geometers (including topologists) and algebraists. From Lecture Two onward, I will assume basic knowledge of algebraic topology (e.g., the material of Hatcher) and homological algebra.

These lectures will be held between July 31 and August 10, each from 10:30 to 12:00, for a total of 8 lectures.

1st Week: Jul 31(mon), Aug 1(tue) - 3(thu)
- Introduction to ideas of higher algebra in geometry, for a general audience.
- Introduction to infinity-categories and to spectra.

2nd Week: Aug 7(mon) - 10(thu)
- Examples in geometry and topology, including invariants of Legendrian links and generating functions.
- Future Directions.

Profile:
Hiro Lee Tanaka is an assistant professor in the Department of Mathematics. After receiving his Ph.D. from Northwestern University and completing postdoctoral work at Harvard University, he conducted research at the Mathematical Sciences Research Institute in Berkeley, California, and at the Isaac Newton Institute in Cambridge, England. His research aims to fuse the higher structures in modern algebra with geometries emerging from both classical mechanics and supersymmetric field theories. Beyond research, Tanaka engages in efforts to create more equitable and supportive environments throughout the mathematics community.