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Morse Theory
Description:
Morse theory is the study of the topology of smooth manifolds
by looking at smooth functions. It turns out that a "generic" function
can reflect quite a lot of information of the background manifold.
In Morse theory, such "generic" functions are called "Morse functions".
By definition, a Morse function on a smooth manifold is a smooth
function whose Hessians are non-degenerate at critical points. One can
prove that every smooth function can be perturbed to a Morse function,
hence we think of Morse functions as being "generic".
Roughly speaking, there are two different ways to study the topology
of manifolds using a Morse function. The classical approach is to
construct a cellular decomposition of the manifold by the Morse
function. Each critical point of the Morse function corresponds to a
cell, with dimension equals the number of negative eigenvalues of the
Hessian matrix. Such an approach is very successful and yields lots of
interesting results. However, for some technical reasons, this method
cannot be generalized to infinite dimensions. Later on people developed
another method that can be generalized to infinite dimensions. This new
theory is now called "Floer theory".
In the tutorial, we will start from the very basics of differential
topology and introduce both the classical and Floer-theory approaches
of Morse theory. Then we will talk about some of the most important and
interesting applications in history of Morse theory. Possible topics
include but are not limited to: Smooth h-Cobordism Theorem, Generalized
Poincare Conjecture in higher dimensions, Lefschetz Hyperplane Theorem,
and the existence of closed geodesics on compact Riemannian manifolds,
and so on. If students are interested, we can also talk about other
closely related topics such as the Conley index theory, Morse-Bott theory,
equivariant homology, or Bott periodicity.
The background and interest of students will be the first priority when
choosing topics for the tutorial.
Prerequisites:
Although we will briefly review the basic
terminologies of manifolds, it is highly recommended that the students
have some familiarity with the concepts of smooth manifolds, tangent and
cotangent bundles, and tangent maps. Prior knowledge in homology theory
is not required but would be helpful.
Contact: Boyu Zhang (bzhang@math.harvard.edu)
Partitions, Young Diagrams and Beyond
Quotation:
The theory of partitions is one of the very few branches of
mathematics that can be appreciated by anyone who is endowed with little
more than a lively interest in the subject. Its applications are found
wherever discrete objects are to be counted or classified, whether in
the molecular and the atomic studies of matter, in the theory of numbers,
or in combinatorial problems from all sources. Gian-Carlo Rota
Description:
A partition of n is a finite weakly decreasing sequence of
positive integers with a sum equal to n. It can be visualized as a Young
diagram: a collection of cells arranged in left-justified rows with row
lengths given by elements of the sequence.
Despite such elementary description, Young diagrams occur in a variety of interplays between combinatorial and algebraic structures, related in particular to group representation theory, algebra of symmetric polynomials, and beautiful identities with series. They lead to surprising connections; in particular the knowledge about Young diagrams and representations of the symmetric group sheds light on questions such as:
How many times do you need to shuffle a deck of cards to make it close to random? Or: What can we say about the length of a longest increasing subsequence of a random permutation? Such interplays and their consequences will be the focus of this tutorial.
Prerequisites: Only basic linear algebra and interest in
combinatorics. All other notions will be introduced during the course.
Contact: Konstantin Matveev (kmatveev@math.harvard.edu)
Fall Tutorial 2015
Category Theory
Description: Category theory is central to the study of modern mathematics. Most of the
subjects which have been profoundly influenced are, however, rather technical and typically
lie outside of the standard undergraduate curriculum. On the other hand, insights gained
from a categorical point of view can be very useful when learning mathematics at any level.
We will introduce the subject by building on many familiar examples from basic algebra
and topology. We will explain the meaning of categories, functors, natural transformations,
limits, colimits, Yoneda's lemma, representability, and adjunctions. In order to make our
exposition more meaningful and enjoyable, each definition will be followed by an explicit
example in a category that the students are acquainted with.
Once the basis of the subject has been throughly developed, we will venture into more
advanced topics. Potential topics may include, but are not limited to:
(1) Abelian categories and homological algebra,
(2) Model categories,
(3) Higher categories.
Students' backgrounds and interests will be the main parameters taken into consideration when choosing topics.
Prerequisites: Math 122 and Math 131.
Contact: Danny Shi (dannyshi@math.harvard.edu)