Geometry
Workshop on
Geometry, PDE, and Applications
University of Isfahan
December 30-31, 2019 & January 4, 2020
Moderator: Sajjad Lakzian (slakzian@ipm.ir)
Lectures 1 & 2
Title: Geometry, Topology and Physics: Index Theorem
Instructor: Amir Abbass Varshovi (University of Isfahan, Iran)
Date: Monday, December 30, 2019
Time: 09:00 - 12:00
Venue: Department of Mathematical Sciences, Seminar Room 1
Abstract:

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.

Lectures 3 & 4
Title: Geometric Analysis in Singular Spaces
Instructor: Sajjad Lakzian (Isfahan University of Technology and IPM-Isfahan, Iran)
Date: Monday, December 30, 2019
Time: 14:00 - 17:00
Venue: Department of Mathematical Sciences, Seminar Room 1
Abstract:

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 Lott-Sturm-Villani 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.

Lectures 5 & 6
Title: 3-Manifold Topology and Algorithms
Instructor: Mehdi Yazdi (University of Oxford, UK)
Date: Tuesday, December 31, 2019
Time: 09:00 - 12:00
Venue: Department of Mathematical Sciences, Seminar Room 2
Abstract:

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

Lectures 7 & 8
Title: Quasilinear and Fully Nonlinear Elliptic Equations
Instructor: Mohammad Safdari (Sharif University of Technology, Iran)
Date: Tuesday, December 31, 2019
Time: 14:00 - 17:00
Venue: Department of Mathematical Sciences, Seminar Room 2
Abstract:

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.

Lectures 9 & 10
Title: Basic Theory of Harmonic Maps
Instructor: Zahra Sinaei (University of Massachusetts Amherst, USA)
Date: Saturday, January 4, 2020
Time: 09:00 - 12:00
Venue: TBA
Abstract:

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.

Lectures 11 & 12
Title: Cartan’s Theory of Equivalence and Its Application
Instructor: Masoud Sabzevari (Shahrekord University and IPM-Isfahan, Iran)
Date: Saturday, January 4, 2020
Time: 14:00 - 17:00
Venue: Department of Mathematical Sciences, Seminar Room 1
Abstract:

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 self-equivalences 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 mini-course, 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 (Cauchy-Riemann) differential geometry and differential equations; although this will by no means restrict our discussion.