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

## Spring Tutorials 2018

### Curves on Algebraic Surfaces

 Description: The main goal of the tutorial is to go through the material in Mumford's classic Lectures on curves on an algebraic surface. 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: Prerequisites: Basic algebraic geometry. Contact: Ziquan Yang, zqyang@math.harvard.edu

### Forcing

 Description: Cantor's Theorem states that the real numbers are uncountable. In other words there is no bijection between N and R. We say the cardinality of N is smaller than that of R, and write |N| < |R|. The structure of infinite cardinalities is the domain of set theory. One of the simplest questions one could ask about this structure is whether there are cardinals between |N| and |R|. Equivalently, is there an uncountable set of real numbers that not in bijection with R? This question is called the Continuum Problem. The statement that there is no such set is called the Continuum Hypothesis. In the 1940s, Gödel showed that it is impossible to disprove the Continuum Hypothesis using the generally accepted axioms of mathematics, ZFC. Then in the early 1960s, Cohen introduced the technique of forcing, and used it to show that it is also impossible to prove the Continuum Hypothesis using ZFC. Together, these theorems show that the Continuum Problem cannot be solved on the basis of the accepted axioms of mathematics. Forcing is now the most important non-elementary method in set theory. The technique is remarkably general and has been used to establish the unsolvability of hundreds of problems not only in set theory but also in other branches of mathematics including algebra, analysis, and topology. The plan for this course is to start with a self-contained but quick review axiomatic set theory. Then we will introduce forcing, prove the general Forcing Theorems, and prove the unsolvability of the Continuum Problem. 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, goldberg@math.harvard.edu

## Fall Tutorial 2017

### Arithmetic of Elliptic Curves

 Description: 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, zyao@math.harvard.edu)