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.

Archived old Fall-Spring tutorial abstracts: 00-01 01-02 02-03 03-04 04-05 05-06 06-07 07-08 08-09 09-10 10-11 11-12 12-13 13-14 14-15 15-16 16-17 17-18 18-19
Archived old Summer tutorial abstracts: 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2014 2015 2016 2017 2018 2019

Fall Tutorial 2019

Etale Cohomology

Étale cohomology is a way of assigning topological invariants to schemes. It originated from A. Grothendieck's program to prove the Weil conjectures and played a crucial role in the development of modern algebraic geometry. The étale cohomology of a field identifies with, as well as re-conceptualizes, its Galois cohomology. For general schemes, one can establish properties akin to topological cohomology theories after a substantial amount work. Notably, this allows us to apply the Lefschetz trace formula to the Frobenius endomorphism of a scheme over finite field to count its rational points. Besides being a useful invariant of schemes, étale cohomology gave birth to the theory of l-adic sheaves, whose formal structure served as a model for many subsequent discoveries of sheaf theories for schemes. The goal of this course is to build the theory of étale cohomology ``from scratch."


Altebraic geometry.

Contact: By Yifei Zhao,

Morse Theory

This is a course on Morse theory and some of its applications. We will start by developing the basic theory for finite-dimensional manifolds. After this we will pursue further topics depending on time and interest. Some possibilities are:
  • The Picard-Lefschetz formula.
  • Singularities of hypersurfaces.
  • Bott periodicity.
  • The h-cobordism theorem.
The standard reference is J. Milnor, Morse Theory. Princeton University Press, 1963.


The notion of a smooth manifold.

Contact: By Kevin Lin,

Spring Tutorials 2020

Classical Mechanics and Symplectic Geometry

Symplectic geometry is a modern and rapidly-developing field of mathematics that began with the study of the geometric ideas that underlie classical mechanics. This tutorial will begin similarly by introducing the Lagrangian and Hamiltonian formulations of classical mechanics and their resulting dynamical properties, before re-expressing them in the language of differential geometry, that is, in terms of symplectic manifolds. This new setting reveals that classical mechanics is really about symmetries in geometry; this will be illustrated by the detailed study of symmetries of mechanical systems, with many concrete examples drawn from physics such as orbital motion and rigid bodies; including Noether's theorem, canonical transformations and generating functions, and action-angle coordinates. The theory of manifolds and differential forms will be integrated throughout the tutorial and motivated from a physical perspective, from which it becomes most transparent and intuitive.


Multivariable calculus, and familiarity with open/closed sets and convergence as in 112 or 131 will be assumed, but no prior knowledge of physics or geometry will be required. We will finish with an informal introduction to the problems and techniques considered in modern symplectic topology, such as non-squeezing theorems and the use of J-holomorphic curves. There will be ample scope for students to follow their own interests, exploring connections to topics such as quantum mechanics, algebraic geometry, and dynamical systems.

Contact: Maxim Jeffs (

Mathematical Economics

Mathematics is a cornerstone of economic theory. From a mathematician's perspective, we will discuss both classical and contemporary works on economic theory, focusing on the validity of the assumptions, methods of proof, and the models' implications for society. Topics will be chosen with input from students, but tentatively include game theory, the study of strategic interactions between rational agents; decision theory, the study of how people make choices; and mechanism design, the study of creating systems of incentives to achieve desired societal outcomes (e.g., markets, voting systems, tax systems, climate-change policies, cryptoeconomic systems).


There are no specific prerequisites, and any prior experience with proofs would suffice. Students who are curious about how mathematical reasoning can be applied to better understand and improve the world would particularly find this class valuable.

Contact: Peter Park (
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