Seminar Notes
Minimality of Planes in Normed Spaces
- Based on Burago, D., & Ivanov, S. (2012). Minimality of planes in normed spaces. Geometric and Functional Analysis, 22(3), 627-638.
- The main purpose of this paper is to prove that planar discs minimize the area among orientable surfaces with the same boundary in every finite-dimensional normed space \(V\), it is proved by the method of calibration forms.
- Seminar Notes-Minimality of Planes in Normed Spaces
Differentiability of Lipschitz functions
Talk one
- Based on Alberti, G., Csörnyei, M., & Preiss, D. (2010). Differentiability of Lipschitz functions, structure of null sets, and other problems. In Proceedings of the International Congress of Mathematicians 2010 (ICM 2010) (In 4 Volumes) Vol. I: Plenary Lectures and Ceremonies Vols. II–IV: Invited Lectures (pp. 1379-1394).
- This talk is based on the motivation of strengthening the Rademacher’s theorem. We review the characteristics of the non-differentiability sets of Lipschitz functions on Euclidean space and introduce the definition of the tangent fields and several results of it.
- Seminar Notes-Differentiability of Lipschitz functions
Talk two
- Based on Davies, R. O. (1952, April). On accessibility of plane sets and differentiation of functions of two real variables. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 48, No. 2, pp. 215-232). Cambridge University Press.
- We first prove a beautiful result that given any measurable plane set \(E\) with finite measure, we can construct a set \(L\) of lines that covers \(E\) without increasing the measure, and then give some remarks on the theories introduced in talk one.
- Seminar Notes-On accessibility of Plane Sets
Notes of Optimal Transportation
Finishing this notes is one of the things I want to do most in Spring 2022 (:_」∠_)
This notes based on the course: Optimal Transportation in Beihang University instructed by Prof. Xiangyu Liang.
The textbook for the course is Villani, C. (2021). Topics in optimal transportation (Vol. 58). American Mathematical Soc..