Archived Summer Tutorials:  2007  2006  2005  2004  2003  2002  2001 
Welcome Message
The summer tutorial program offers some interesting mathematics to
those of you who will be in the Boston area during July and August.
Each tutorial will run for six weeks, meeting twice per week in the
evenings (so as not to interfere with day time jobs). The tutorials
will start early in July and run to mid August. The precise starting
dates and meeting times will be arranged for the convenience of the
participants once the tutorial rosters are set.
The format will be much like that of the termtime tutorials, with the
tutorial leader lecturing in the first few meetings and students
lecturing later on. Unlike the termtime tutorials, the summer
tutorials have no official Harvard status: you will not receive either
Harvard or concentration credit for them. Moreover, enrollment in the
tutorial does not qualify you for any Harvardrelated perks (such as a
place to live). However, the Math Department will pay each Harvard
College student participant a stipend of , and you can hand in
your final paper from the tutorial for you junior 5page paper
requirement for the Math Concentration.
The topics and leaders of the four tutorials this summer are:
A description of each topic is appended below. You can sign up for a
tutorial by emailing me at kronheim@math.harvard.edu. When you sign
up, please list at least one other choice in case your preferred
tutorial is either oversubscribed or undersubscribed. Places are
filled on a firstcome, firstserved basis. In the past, some
tutorials have filled up quickly.
If you have further questions about any given topic, contact the
tutorial leader via email. Please contact me if you have questions about the
administration of the tutorials.
Yours,
Peter Kronheimer
Quadratic Forms (Ana Caraiani, caraiani@math.harvard.edu)
Quadratic forms are some of the most beautiful objects in mathematics. In
this tutorial, we will study them mostly from the point of view of
number theory and use the opportunity to introduce some central ideas
and principles in the field.
We will introduce the padic rationals and prove that quadratic forms =
with rational coefficients satisfy the localglobal principle. Then we
will look at more instances in number theory where this principle arises
(primes in arithmetic progressions, twin primes). For integral quadratic
forms, we will prove the remarkable Fifteen Theorem which says that if
a quadratic form with integral matrix represents all positive integers
up to 15, then it represents all positive integers. We will also see how
binary quadratic forms are related to ideal theory in quadratic fields via
Gauss composition; this will prepare us to talk about recently discovered
higher composition laws. We might even have some fun with Rubik's cubes!
In student projects, we could explore the mysterious connection of higher
composition laws with the Dynkin diagrams of exceptional simple Lie groups
or learn about the classification of simplyconnected 4manifolds. We
may also investigate theta functions, which are constructed using
quadratic forms.
Prerequisites: The only prerequisites are basic algebra at the level taught in Math 122 and very elementary number theory as covered in the first part of Math 124 (factorization, prime numbers, congruences). Having some familiarity with rings, fields and field extensions would be helpful, but not required.
Pointless Topology (Sam Isaacson, sbisaacs@math.harvard.edu)
Traditionally, we think of spaces as a collection of points with some
added structure: namely, a collection of open sets satisfying a short
list of axioms. But what if we throw out the points? The open sets
of a topological space with the order imposed by inclusion forms a
locale, or generalized space; and the closedopen sets of a space form
a lattice under inclusion. Both the locale of open sets and the lattice
of closedopen sets have attached to them algebraic operations given by
union and intersection. In the 1930's, Marshall Stone inaugurated the
study of generalized spaces by proving a famous representation theorems:
compact Hausdorff totally disconnected spaces are precisely those spaces
determined by their lattice of closedopen sets.
We'll explore two consequences of Stone's work. First, we'll embark on
a systematic study of pointless topology. More general representation
theorems let us regard locales as a substitute for spaces. The chief
advantage of this perspective is that many results that required the axiom
of choice in classical pointset topology have constructive analogues.
Second, we'll study some of the consequences of Stone's work outside of
the localic world. Stone's insight that algebraic objects (lattices,
Boolean algebras, etc.) could act as substitutes for spaces is at the
heart of harmonic analysis and algebraic geometry with the GelfandNaimark
theorem and the notion of the spectrum of a commutative ring.
Prerequisites: The only prerequisite for this tutorial is a basic understanding of pointset topology  Math 131 or an equivalent will be sufficient.
The Model Theory of Fields Katy Körner (korner@math.harvard.edu) and Aaron Silberstein (asilbers@math.harvard.edu)
The properties of fields can be studied in a number of ways, some more
daunting than others; we seek to show students the value and friendliness
of a model theoretic approach.
Model theory is the study of mathematical structures. In order to study
model theory at all, students will be taken on a quick jog through the
basics of logic and set theory. The content will vary depending on the
background of the students  those who have taken 141 or 143 can, if
they choose, move quickly on to applications of model theory to their own
areas of mathematical interest, while those with no such background will
study notions of completeness and decidability, as well as exploring the
ordinals and cardinals through proving the equivalence of wellordering
and the axiom of choice. This is valuable for its own sake, serving to
give students some insight into the foundations on which the rest of
mathematics is built.
Galois theory is the study of the symmetry of both fields and equations.
First developed to determine the problem of how to solve a polynomial,
it has since developed into a deep and fruitful way of understanding
symmetries whether they arise in number theory, algebraic geometry,
topology, or differential equations. Emil Artin, in the first half of the
last century, made the discovery that the most fundamental questions in
Galois theory should be expressed in terms of logic; since then, Galois
theory has been a roiling synthesis of geometry, topology, arithmetic,
and logic, blurring the lines between these fields and allowing each to
provide insight into the other. After introducing Galois theory and some
of its many applications (including solving polynomial equations!) we
will delve into the model theory of fields and see how the model theory
of a field is very closely linked to its Galois theory. Projects will
then focus on aspects of logic and field theory, tailored to students'
interests  be they logic, number theory, topology, algebraic geometry,
or group theory.
Prerequisites: Algebra at the level of Math 122.
Symplectic Geometry (Alexandar Subotic, asubotic@math.harvard.edu)
Symplectic manifolds are a special class of smooth manifolds that
arises naturally in many problems in geometry. They are necessarily
evendimensional and in fact they are often important players in the big
research area of four dimensional geometry. In some ways they are similar
to complex manifolds (the manifolds of algebraic geometry) so they are
also thought of as being inbetween real and algebraic geometry. Finally
they are an essential part of geometry related to modern and classical
physics: mirror symmetry, string theory, and Hamiltonian mechanics.
This tutorial will provide a first introduction into symplectic
geometry. We will start with introducing the symplectic form and the
linear algebra associated to it. We will define symplectic manifolds,
and will give many examples along with the fundamental Darboux's theorem
on their local structure. We will introduce almost complex structures and
explain the relationship between symplectic and complex geometry. We will
explain hamiltonian mechanics and its relation to symplectic geometry,
and time permitting we will introduce the notion of a moment map.
Prerequisites: Basic concepts of manifolds, differential forms and vector fields.
Archive: Old Summer Tutorials, since 2001
Summer Tutorials:  2007  2006  2005  2004  2003  2002  2001 
