In search of quantum geometry by Pranav Pandit

COLLOQUIUM

IN SEARCH OF QUANTUM GEOMETRY

SPEAKER: Pranav Pandit (ICTS - TIFR, Bengaluru)

DATE: Mon, 29 November 2021, 15:30 to 17:00

VENUE: Online and Ramanujan Lecture Hall

RESOURCES

ABSTRACT
Notions of geometry have evolved throughout the history of mathematics, often in parallel with developments in physics. I will discuss a recent chapter in this story. This chapter is deeply interwoven with ideas from string theory. In the type of geometry that I will describe, higher categories play the role of geometric spaces. The theory will be illustrated by examples drawn from diverse areas of mathematics.

Table of Contents (powered by https://videoken.com)
0:00:00 In search of quantum geometry
0:02:37 OUTLINE
0:04:32 EUCLID
0:06:26 ANALYTIC GEOMETRY
0:07:27 Riemannian Geometry & Symplectic Geometry
0:14:44 Symplectic manifold & Kahler manifold
0:17:04 Smooth projective varieties are Kahler.
0:18:09 DISCRETE VS CONTINUOUS GEOMETRY
0:26:29 HOMOLOGY
0:28:37 DE RHAM COHOMOLOGY
0:32:02 DE RHAM ALGEBRA (A* (M), d)
0:32:43 RATIONAL HOMOTOPY THEORY
0:37:49 HOMOTOPY ASSOCIATIVITY
0:39:25 Special cases: Ordinary associative algebras & Differential graded algebras
0:39:45 Geometric interpretation:
0:43:52 A-MODEL
0:46:39 B-MODEL
0:47:57 ALGEBRAIC EXAMPLE
0:48:43 GEOMETRY IN HIGHER CATEGORIES
0:48:48 HODGE THEORY
0:50:48 Mirror symmetry for Hodge diamonds
0:52:03 HOMOLOGICAL MIRROR SYMMETRY
0:53:45 C A infinity-category -Hochschild chain complex HH. (C) -S1
0:56:09 MODULI SPACES
0:57:49 Question: What is the noncommutative analogue of a Kahler metric?
0:59:06 SUMMARY
1:10:45 Q&A
1:11:04 HOMOLOGICAL MIRROR SYMMETRY