Moderator:
Sajjad Lakzian (slakzian@ipm.ir)

First (the first session) we will have a brief review over characteristic classes and the index
theorem with emphasis on topological features of the problem. Then after (the second session) we
will transfer to the theory of quantum gauge fields and introduce the gauge anomalies. The physical
significance of gauge anomalies and their intimate correlation to the index theorem will be discussed
thoroughly.

We will briefly review essential background materials on Geometric and Analytic properties
of smooth spaces as they pertain to various curvature bounds. Then, we will explain how these properties
were used to define suitable notions of curvature bounds such as Alexandrov curvature bounds for geodesic
metric spaces and LottSturmVillani Ricci lower curvature bounds or metric measure spaces. We will
touch upon the fine properties of these spaces and eventually will talk about some of the recent and new
geometric analytic results in such spaces.

I will discuss the basics of 3manifold topology, and explain some of the tools that have been
used to address algorithmic questions about 3manifolds, such as normal surfaces.

We introduce the theory of quasilinear elliptic equations which arise in calculus of variations,
and the theory of fully nonlinear elliptic equations. We inspect the fundamental questions of existence,
uniqueness, and regularity of solutions. We also review the important notions of weak and viscosity
solutions.

In this lecture I will discuss the regularity theory of classical minimizing and stationary
harmonic maps. My ultimate goals is to cover the following material: Regularity of harmonic functions,
the notion of a harmonic map, Euler Lagrange and stationary equations for harmonic maps, monotonicity
formula for harmonic maps, classical stratification and quantitative stratification, main regularity results
for classical stationary harmonic maps.
Familiarity with measure theory and basic background in PDE will be assumed.

Cartan’s theory of equivalence deals with the determination of when two mathematical objects
are the same under a certain action of transformation groups .This theory is actually an intelligent
development of Sophus Lie’s theory of symmetries. Indeed, symmetries of a given geometric object can
be interpreted as the group of selfequivalences of it. Although this theory finds itself in the heart of
(differential) geometry, but it exhibits its applications in a wide variety of mathematical subjects such
as analysis, (computer) algebra, differential equations, variation problems, differential forms and many
others. As a clever observation of Cartan, he discovered that almost all of the equivalence problems can
be reformulated in terms of the geometric equivalence problem of coframes associated with them.
In this minicourse, after introducing some historical facts and required preliminaries (such as symmetries, invariants, transformation groups and etc), I will explain certain techniques of Cartan’s method for
solving equivalence problems. At first, I will concentrate on the equivalence problem of coframes, as the
fundamental part of the theory, in question. Next, we will see how it is extendable to general equivalence
problems.
After presenting major techniques and results concerning the theory of equivalence problems, I will
also explain some applications of it with particular attention to problems arising in (CauchyRiemann)
differential geometry and differential equations; although this will by no means restrict our discussion.

