# 2016 Undergraduate Summer School

The UCLA Logic Center plans to hold a three-week summer school for undergraduates in 2016, from Sunday July 10 to Saturday July 30.

The goal of the summer school is to introduce future mathematicians to
central results and techniques from mathematical logic. Courses are very
intensive, and reach advanced material, at a graduate level.
They are designed to
*not* require specific background in logic, but they do require
high mathematical sophistication, for example from upper division or graduate
courses in analysis or algebra. The summer school courses serve as good
introduction to the kind of work that students of mathematics
can expect in graduate school.

Each course in the summer school will meet daily for two hours of lecture, and one hour of guided problem solving in small groups. Each student takes two of the three courses offered. In addition to the six daily hours of course work (not to mention endless hours of extra work on challenging problem sets over evenings and weekends) there will be lectures on topics of current research, social events, and planned outings in the area.

Thanks to an NSF grant (DMS-1044604) and Logic Center support we provide admitted students with a stipend of $3,000, travel allowance up to $500, and dormitory housing at UCLA for no charge (double occupancy rooms, breakfast and dinner included).

The application procedure is
described below. The application deadline is
Friday **March 4**, at 9pm Pacific Time.

The summer school does not lead to formal course credit, but we will be happy to provide letters of recommendation for students who do well in the summer school. The letters will be written by Prof. Itay Neeman in consultation with the instructors.

### Courses:

**Cardinal arithmetic**

Instructor: John Susice

Questions concerning the behavior of the cardinal exponential function date back to Cantor's original investigations in set theory in the 1870s, and attained prominence at the turn of the century when the Continuum Hypothesis was listed as the first of Hilbert's problems. Although the CH was proved independent of ZFC by Cohen in the early 1960s, and subsequent work of Easton extended this result to show that the behavior of the exponential function was essentially completely undetermined for the regular cardinals (in ZFC), Silver surprised the set-theoretic community in 1975 by showing that the behavior of the exponential function below a singular cardinal of uncountable cofinality can determine the value of the exponential function at the cardinal itself. Galvin-Hajnal proved a related result the same year, and Shelah extended Silver's results to cardinals of countable cofinality using his celebrated pcf theory in the 1980s. This course will begin with the classical results of Cantor, König, and Bukovsky on cardinal arithmetic and the definability of cardinal exponentiation in terms of the gimel function. We'll then move quickly through results of Ramsey and Erdos-Rado in the partition calculus, and prove the theorems of Silver and Galvin-Hajnal. Finally, the course will culminate in a brisk introduction to Shelah's pcf theory and a bound of Shelah's on the exponential of the first limit cardinal.

**Forcing and independence in set theory**

Instructor: Zach Norwood

Set-theoretic forcing is a technique originally introduced by Paul Cohen to prove that the axiom of choice and the continuum hypothesis are independent of the classical axioms of mathematics. Since then it has led to an explosive growth in research, and has been used by set theorists to prove literally hundreds of independence results. In this course we will introduce students to the forcing technique by first studying some of the combinatorial consequences of Martin's Axiom, an axiom which when added to ZFC makes it possible to prove consistency results by means very similar to those used in forcing. We will then introduce forcing itself and show how it can be applied to settle the independence of the continuum hypothesis.

**o-minimality, variations, and applications**

Instructor: James Freitag

An infinite totally ordered structure is called o-minimal if every definable set (in one dimension) is a finite union of points and intervals. There is a deep structure theory of definable sets in o-minimal structures, and there are mathematically rich o-minimal structures. In this course, we will develop the theory of o-minimality from the beginning, building up a structure theory of definable sets and providing numerous examples. We will also study variants of o-minimality and applications to differential equations and number theory if time permits.

### To apply:

Applications are collected through the American Mathematical
Society's MathPrograms.org. Complete the online
**cover
sheet and application form** for the summer school.
You will be asked to provide the following:

- Choice, and alternate choice, of two out of the three courses offered in the summer school.
- Contacts for two references.
- A description of your educational background, your current interests and goals, and your reasons for wanting to attend the summer school.
- A brief description of the three most advanced pure mathematics courses you have taken.
- A copy of your undergraduate transcripts.

We are looking for students who are ready for graduate level material, and for challenging problems at a graduate level. At a minimum we expect applicants to have done very well in upper division courses in abstract mathematics, for example analysis and abstract algebra. Additional advanced work is a plus.

You may upload unofficial transcripts, or scanned images of official transcripts. Your references should be written by people who can comment on your academic abilities, especially in pure mathematics.

You will be sent an email confirmation of your application, and
your letter-writers will be contacted once you apply. You can use
MathPrograms.org to track the submission of reference letters, edit
or revise your application until the application deadline,
and have reminders
sent to your letter-writers if necessary. We do **not** accept
any material by mail. All material must be uploaded
on MathPrograms.org.

Your application should be complete, and supporting letters uploaded,
by **Friday March 4** at
9pm Pacific Time. Admission decisions will be emailed in March.
If you have deadlines for other programs
let us know
and we will try
to make an early decision on your application if possible.

The summer school is for undergraduates, currently in their Junior year or earlier. Women and members of underrepresented minorities are especially encouraged to apply. Both US and foreign students may apply. Foreign students who are not at a US university should be at a stage equivalent to the US Junior year or earlier.

Inquiries on matters which are not addressed above may be directed to logicschool@math.ucla.edu.

### Links to previous years' summer schools:

The 2015 Summer School.

The 2014 Summer School.

The 2013 Summer School.

The 2012 Summer School.

The 2010 Summer School.

The 2009 Summer School.