Information for Professors admitting students from Iran

Given that state-imposed internet blackouts are happening more frequently in Iran, this could be very challenging in application season (please see the netblocks website). In the case communication is cut off, please let me know by email if you need to contact a prospective student, especially from Sharif University.

Geometric measure theory

Here are some resources I used over the years to learn geometric measure theory. These include books, lecture notes, papers and surveys. What works best for you is a matter of personal taste, and what I wrote below reflects my own opinion.
  • Lecture Notes by A.Pigati: I was the TA for the second half of a summer school held by SLmath on June 17-26 2024. The second week was an introduction to geometric measure theory taught by Alessandro Pigati and we used the following Lecture notes (written by Alessandro):
  • Sets of finite perimeter: Considering surfaces of codimension one as boundaries of sets of finite perimeter, was pioneered by Renato Caccioppoli in 1927. They are also named Caccioppoli sets.
    • Sets of Finite Perimeter and Geometric Variational Problems, Francesco Maggi. link: This book builds from the basics, with lots of examples and proofs. It is my favorite one to start learning from. The pre-requisites are multidimensional calculus and a bit of functional analysis.
  • Currents
    • Cartesian Currents in the Calculus of Variations I: Mariano Giaquinta, Giuseppe Modica, Jiri Soucek. link: This book is intended for the (specialized) topic of Cartesian currents, however it is an excellent introduction to the theory of currents. It also contains a very nice crash course on measure theory and real analysis.
    • Geometric integration theory, Steven Krantz, Harold Parks. link: This book starts from the basics of measure theory and it is a great reference for (a detailed proof of) the epsilon regularity theorems for integral currents. It is written in the modern language of GMT and it is very accessible.
    • De-Giorgi's epsilon regularity theorem: There are multiple references where one can learn, one already mentioned above. Here I will put some more:
      • Center manifold: A case study, Camillo De Lellis, Emanuele Nunzio Spadaro: [Appendix A], arxiv link.
  • Varifolds
    • On the First Variation of a Varifold, William K. Allard. link: Perhaps the most cited paper in this field. In this paper Allard introduces the concept of a varifold and proves the compactness and regularity theorems (among other results).
    • Introduction to Geometric Measure Theory, Leon Simon. link: This book introduces Geometric measure theory in a condensed text and it is a great place to begin for slightly more advanced graduate students. It also presents a detailed version of the regularity theorem of Allard, with a more modern notation compared to the original paper of Allard.
    • Allard's interior regularity theorem: An invitation to stationary Varifolds, Camillo De Lellis. link. This is a gentle introduction to the theory of integral varifolds, especially the Allard regularity theorem. It contains a soft presentation of the proof of the tilt-excess decay lemma.

Ginzburg-Landau theory

Here you can find a series of resources to learn Ginzburg-Landau and Yang-Mills-Higgs theory.
  • Non-magnetic GL:
    Under construction.
    • Ginzburg-Landau Vortices, Fabrice Bethuel, Haïm Brezis, Frédéric Hélein. link: In this book the authors pioneer the study of Ginzburg-Landau vortices in two dimensions. Among the many results is the Gamma convergence, and the derivation of the renormalized energy. It is a great place to start learning the theory.
    • Eta-ellipticity lemma and the GL model in higher dimensions:
  • Yang-Mills-Higgs/abelian Higgs:
    Under construction.