Lecture series: Hirzebruch-Riemann-Roch as a categorical trace.
Professor Dennis Gaitsgory (Harvard University)
Location: Bonn, Germany.
Dates: January 10-13, 2017.
- Tuesday January 10:
- 04:30 PM - 06:30 PM.
- Vivatsgasse 7, Hörsaal MPIM
- Wednesday January 11:
- 02:00 PM - 04:00 PM.
- Endenicher Allee 60, Room 1.016 (Lipschitz hall)
- Friday January 13:
- 12:30 PM - 02:30 PM.
- Vivatsgasse 7, Hörsaal MPIM
Abstract
Let \(X\) be a smooth proper scheme over a field of characteristic 0, and let \(E\) be a
vector bundle on \(X\). The classical Hirzebruch-Riemann-Roch says that the Euler
characteristic of the cohomology \(H^*(X,E)\) equals \(\int_X \text{ch}(E) \; \text{Td}(X)\).
Thus, HRR is an equality of numbers, i.e., elements of a set. In these talks,
we will explain a proof of HRR that uses the hierarchy
\[\{2-\text{categories}\} \rightarrow \{1-\text{categories}\} \rightarrow \{\text{Vector spaces}\} \rightarrow \{\text{Numbers}\}.\]
I.e., the origin of HRR will be 2-categorical. The procedure by which we
go down from 2-categories to numbers is that of *categorical trace*.
However, in order to carry out our program, we will need to venture into
the world of higher categories: the 2-category we will be working with
consists of DG-categories, the latter being higher categorical objects.
And the process of calculation of the categorical trace will involve derived
algebraic geometry: the key geometric player will be the self-intersection
of the diagonal of \(X\), a.k.a. the inertia (derived) scheme of \(X\).
So, this series of talks can be regarded as providing a motivation for studying
higher category theory and derived algebraic geometry: we will use them
in order to prove an equality of numbers. That said, we will try to make these
talks self-contained, and so some necessary background will be supplied.
Video recordings are available at https://www.mpim-bonn.mpg.de/node/7032
News
Corona: Maßnahmen im Mathematik-Zentrum
Mathemacher des Monats Dezember 2020: Dr. Antje Kiesel und Dr. Thoralf Räsch
Bonner Mathematik belegt bei Shanghai Ranking den 1. Platz in Deutschland und weltweit den 13. Platz
Prof. Georg Oberdieck erhält Heinz Maier-Leibnitz-Preise 2020
Prof. Daniel Huybrechts erhält gemeinsam mit Debarre, Macri und Voisin ERC Synergy Grant
Prof. Peter Scholze erhält Verdienstorden der Bundesrepublik Deutschland
Prof. Dr. Valentin Blomer wurde zum Mitglied der Academia Europaea gewählt
Prof. Peter Scholze erhält Fields-Medaille 2018