Department of Mathematics FAS Harvard University One Oxford Street Cambridge MA 02138 USA Tel: (617) 495-2171 Fax: (617) 495-5132

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
Archived old Summer tutorial abstracts: 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2014 2015 2016 2017

Fall Tutorial 2017

Arithmetic of Elliptic Curves

This tutorial is an introduction to the arithmetic of elliptic curves. After introducing several equivalent definitions of elliptic curves, we will prove the Mordell-Weil theorem, which states that for an elliptic curve defined over a number field, the set of rational points forms a finitely generated Abelian group. Then we will go into the theory of complex multiplication, and hopefully will have time to discuss more advanced topics including Selmer groups, Tate curves, etc. The first half of the material overlaps heavily with the tutorial in the previous year.
Prerequisites: Students should be familiar with basic algebraic number theory. Having taking the undergraduate algebraic geometry class would be helpful, but not required. Though it might be a good idea to take this and algebraic geometry as the same time.
Contact: Zijian Yao,

Springs Tutorial 2017

Infinitary Methods in Mathematics

The field of set theory was born out of Cantor's discovery of infinite ordinal and cardinal numbers in the late 19th century. These number systems are now central to almost all modern research in mathematical logic. The subject of this tutorial, however, is the applications of the theory of ordinal and cardinal numbers outside of mathematical logic. The course will therefore touch on several areas of mathematics, including algebra, topology, analysis, game theory, and combinatorics, with the common theme that the proofs will make essential use of infinitary techniques. Prerequisites: The course assumes no prior knowledge of set theory or logic, but a basic knowledge of abstract algebra and point-set topology are recommended. Math 122 and 131 more than suffice. Contact: Gabriel Goldberg,

Curves on algebraic surfaces

The main goal of the tutorial is to go through the material in Mumford's classic "Lectures on curves on an algebraic surfaces". This tutorial could be a natural continuation of a first course in algebraic geometry. The book is a great gateway to learn about many important ideas in algebraic geometry and see them used in practice. Let me just name a few: Riemann-Roch, Hilbert schemes, Picard schemes, nilpotent elements, moduli problems, deformation theory, etc. On the other hand, Mumford strived for the book to be self-contained the book in fact starts from an introduction to schemes and cohomology, so prerequisites are minimal. Of course, we could skip this part depending on the preference of audience. Unlike most courses on algebraic surfaces, this course is not about birational classification. It is mainly about linear series on algebraic surfaces. Prerequisites: Basic algebraic geometry. Contact: Ziquan Yang,