July 31 (Mon) at 10:30 - August 10 (Thu) at 12:00, 2023 (JST)
  • Hiro Lee Tanaka (Assistant Professor, Department of Mathematics, Texas State University, USA)
Taketo Sano

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.

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.


  1. Jacob Lurie, Higher Topos Theory (PDF 4.8MB), doi: 10.1515/9781400830558
  2. Jacob Lurie, Higher Algebra (PDF 6.9MB)
  3. Kerodon - an online resource for homotopy-coherent mathematics
  4. Jacob Lurie, Hiro Lee Tanaka, Associative algebras and broken lines, arXiv: 1805.09587
  5. Jacob Lurie, On the Classification of Topological Field Theories, arXiv: 0905.0465
  6. Oleg Lazarev, Zachary Sylvan, Hiro Lee Tanaka, The infinity-category of stabilized Liouville sectors, arXiv: 2110.11754
  7. Araminta Amabel, Artem Kalmykov, Lukas Müller, Hiro Lee Tanaka, Lectures on Factorization Homology, Infinity-Categories, and Topological Field Theories, arXiv: 1907.00066
  8. David Ayala, John Francis, Hiro Lee Tanaka, Factorization homology of stratified spaces, arXiv: 1409.0848
  9. David Nadler, Hiro Lee Tanaka, A stable infinity-category of Lagrangian cobordisms, arXiv: 1109.4835
  10. David Gepner, An Introduction to Higher Categorical Algebra, arXiv: 1907.02904

This is a closed event for scientists. Non-scientists are not allowed to attend. All scientists, including those outside RIKEN, are welcome to attend, please register using the registration form.

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