Department of Mathematics FAS Harvard University One Oxford Street Cambridge MA 02138 USA Tel: (617) 495-2171 Fax: (617) 495-5132
For Summer tutorials, see this page.

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Spring Tutorials 2019

Large Deviations Principle and Its Applications

Large deviations principle (LDP) is a theory investigating probability of rare events, especially, the asymptotic behavior of the exponential decay of the small probabilities. The first LDP result was proved by Herald Cram ́er for modeling the insurance business in early 1900’s. Later, more results were proved in many different type of problems among mathematics, statistics and statistical mechanics. A unified formalization of LDP was developed by Varadhan in 1966, and have become an important tool in probability theory since then. In this tutorial, we will mainly go through LDP in finite dimensional spaces, and introduce the applications of finite dimensional LDP in Markov systems, statistics and statistical mechanics. Later, we will go into the application of LDP into numerical simulation, namely Importance Sampling (IS). Topics may include, but are not limited to:
  1. LDP for finite dimensional space
  2. LDP for finite state Markov chains
  3. Gibbs conditioning principle
  4. Importance Sampling

Students' backgrounds and interests will be taken into consideration for later topics. Prerequisites: Basic probability theory and comfort with dealing with math proofs are required. Math 55A,B or 112 is required. Math 114 is recommended.
Contact: Yixiang Mao (

Category Theory

Category theory is central to the study of modern mathematics. Most of the subjects which have been profoundly influenced are, however, rather technical and typically lie outside of the standard undergraduate curriculum. On the other hand, insights gained from a categorical point of view can be very useful when learning mathematics at any level. We will introduce the subject by building on many familiar examples from basic algebra and topology. We will explain the meaning of categories, functors, natural transformations, limits, colimits, Yoneda’s lemma, representability, and adjunctions. In order to make our exposition more meaningful and enjoyable, each definition will be followed by an explicit example in a category that the students are acquainted with. Once the basis of the subject has been thoroughly developed, we will venture into more advanced topics. Potential topics may include, but are not limited to:
  1. Abelian categories and homological algebra,
  2. Model categories,
  3. Higher categories.

Students' backgrounds and interests will be the main parameters taken into consideration when choosing topics. Prerequisites: Math 122 and Math 131.

Contact: Danny Shi (

Fall Tutorial 2018

Infinite Combinatorics

Infinite combinatorics is the study of infinite trees, graphs, partitions, and more generally the infinite generalizations of the objects that arise in finite combinatorics. Compared with their finite counterparts, these structures present problems of a very different nature, which are often impossible to solve without resorting to hypotheses beyond the usual axioms of mathematics and therefore involve deep interactions with the foundations of mathematics. This course will survey several problems in infinite combinatorics and the set theoretic hypotheses that arise in their solutions.

Possible topics: Erdös's partition calculus, the non-stationary ideal, ultrafilters, Suslin's problem, the Singular Cardinals Problem, large cardinal axioms, the Continuum Hypothesis, Martin's Axiom, the Axiom of Determinacy.


Contact: Gabriel Goldberg,