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 |
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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
Description: 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." Prerequisites: Altebraic geometry. Contact: By Yifei Zhao, yifei@math.harvard.edu |
Morse Theory
Classical Mechanics and Symplectic Geometry
Description: 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. Prerequisites: 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 (jeffs@math.harvard.edu) |
Mathematical EconomicsMathematics 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).Prerequisites: 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 (pspark@math.harvard.edu) |