MA 396: Theory of large deviations and related topics (Fall 2019)
Instructor: Anirban Basak 
Email: anirban.basak@icts.res.in 
Office location: L14, IISc Mathematics Department. 
Office hours: By appointment. 
Class time and location: TuTh 2.003.30 PM, LH3, IISc Mathematics department. First class is on August 01. 
Prerequisite: This is a graduate level topics course in probability theory. Graduate level measure theoretic probability will be useful, but not a requirement. The course will be accessible to advanced undergraduates who have had sufficient exposure to probability. 
Course outline: Large deviations provide quantitative estimates of the probabilities of rare events in (highdimensional) stochastic systems.
The course will begin with general foundations of the theory of large deviations and will cover classical large deviations techniques such as Sanov's Theorem, Cramér's Theorem, GärtnerEllis Theorem, Contraction principle, Varadhan's Integral Lemma, and Bryc's Inverse Varadhan Lemma. The latter half of the course will focus on some recent developments. Topics include large deviations in the context of random matrices, large deviations for subgraphs counts in random graphs (both dense and sparse regime), nonlinear large deviations, and low complexity of Gibbs measures. If time allows selected topics from Spin Glasses will be introduced. 
Suggested books: 
Tentative Weekly Schedule:

References on large deviations for sums of stretched exponential random variables:
A. V. Nagaev, Integral Limit Theorems Taking Large Deviations into Account when Cramér’s Condition Does Not Hold.
N. Gantert, K. Ramanan, and F. Rembart, Large deviations for weighted sums of stretched exponential random variables.
Reference on the necessity of \(0 \in {\mathcal D}_\Lambda^\circ\) in Cramér's Theorem on \(\mathbb{R}^d\) with \(d \ge 2\):
I. H. Dinwoodie, A Note on the Upper Bound for I.I.D. Large Deviations.
References on large deviations of subgraph counts in sparse ErdősRényi graphs: