1 code implementation • 22 Dec 2020 • Anton Petrunin, Sergio Zamora Barrera
These notes are designed for those who either plan to work in differential geometry, or at least want to have a good reason not to do it.
History and Overview Differential Geometry 53-01, 53A04, 53A05, 53C45
1 code implementation • 20 Sep 2020 • Nina Lebedeva, Anton Petrunin
We give another proof of Toyoda's theorem that describes 5-point subpaces in CAT(0) length spaces
Metric Geometry Differential Geometry 53C23, 30L15, 51F99
1 code implementation • 20 Jul 2020 • Anton Petrunin
We discuss domestic affairs of metric spaces, keeping away from any extra structure.
Metric Geometry 51F99, 97G10
1 code implementation • 22 Feb 2020 • Alexander Lytchak, Anton Petrunin
We construct short retractions of a CAT(1) space to its small convex subsets.
Differential Geometry Metric Geometry 53C20 (Primary) 53C23, 53C44 (Secondary)
1 code implementation • 20 Mar 2019 • Stephanie Alexander, Vitali Kapovitch, Anton Petrunin
This book includes material up to the definition of dimension.
Differential Geometry Metric Geometry 53C23, 53C45, 30L99
1 code implementation • 17 Dec 2018 • Anton Petrunin
This is a supplement for "Pearls in graph theory" -- a textbook written by Nora Hartsfield and Gerhard Ringel.
Combinatorics History and Overview
1 code implementation • 18 Jun 2018 • Andrei Petrovich Kiselyov, Nikodim Ershov, Anton Petrunin, Sergei Tabachnikov
The book is a unique phenomenon in Russian geometry education.
History and Overview 97-01, 51M04
1 code implementation • 10 Mar 2018 • Anton Petrunin
We discuss a recursive formula for number of spanning trees in a graph.
History and Overview Combinatorics 97K30
1 code implementation • 26 Nov 2017 • Nina Lebedeva, Anton Petrunin, Vladimir Zolotov
We define a new type of metric comparison similar to the comparison of Alexandrov.
Differential Geometry Metric Geometry 53C20, 30L99, 46B85, 53C45, 51K10
1 code implementation • 30 Jul 2017 • Anton Petrunin, Stephan Stadler
A surface which does not admit a length nonincreasing deformation is called metric minimizing.
Differential Geometry Metric Geometry 53C45, 53C43, 53C23, 30L05
1 code implementation • 24 Jul 2017 • Vitali Kapovitch, Anton Petrunin, Wilderich Tuschmann
We present a condition for towers of fiber bundles which implies that the fundamental group of the total space has a nilpotent subgroup of finite index whose torsion is contained in its center.
Algebraic Topology Differential Geometry 57R19, 55R10, 53C20
1 code implementation • 7 Jul 2017 • Anton Petrunin, Stephan Stadler
We prove an analog of the Schoen-Yau univalentness theorem for saddle maps between discs.
Differential Geometry Metric Geometry 53C45, 53C43, 53C23
1 code implementation • 12 May 2017 • Vitali Kapovitch, Alexander Lytchak, Anton Petrunin
We relate the existence of many infinite geodesics on Alexandrov spaces to a statement about the average growth of volumes of balls.
Differential Geometry Metric Geometry 53C20, 52A15, 53C23
1 code implementation • 16 Feb 2017 • Arseniy Akopyan, Anton Petrunin
We review the theory of intrinsic geometry of convex surfaces in the Euclidean space and prove the following theorem: if the surface of a convex body K contains arbitrary long closed simple geodesics, then K is an isosceles tetrahedron.
Differential Geometry History and Overview 53-01, 53C45, 52A15, 53A05
1 code implementation • 12 Jan 2017 • Stephanie Alexander, Vitali Kapovitch, Anton Petrunin
The idea is to demonstrate the beauty and power of Alexandrov geometry by reaching interesting applications with a minimum of preparation.
Differential Geometry Algebraic Topology Metric Geometry 53-01, 53C23, 53C45
1 code implementation • 3 Jan 2017 • Alexander Gil, Anton Petrunin
We discuss a couple of examples of Markov chains.
History and Overview Probability
1 code implementation • 19 Nov 2014 • Anton Petrunin
Here I give a description of Alexandrov 4-point comparison via quadratic forms and then propose a natural 5-point condition which might have some future.
Metric Geometry Differential Geometry 53C45, 51K10
1 code implementation • 24 Dec 2013 • Dima Panov, Anton Petrunin
In the latter case we prove that the ramification is CAT[0] if the metric is non-negatively curved.
Geometric Topology Metric Geometry 51K10 (Primary), 53C23, 57N65, 20F36 (Secondary)
1 code implementation • 1 Apr 2013 • Anton Petrunin
The following is a compilation of some techniques in Alexandrov's geometry which are directly connected to convexity.
Differential Geometry 53C21 (53C20 53C45 58E05)
1 code implementation • 7 Feb 2013 • Anton Petrunin
The book is designed for a semester-long course in Foundations of Geometry and meant to be rigorous, conservative, elementary and minimalist.
History and Overview Metric Geometry 51-01, 51M10, 51M05
1 code implementation • 27 Dec 2010 • Stephanie Alexander, Vitali Kapovitch, Anton Petrunin
We give a simplified proof of the generalized Kirszbraun theorem for Alexandrov spaces, which is due to Lang and Schroeder.
Differential Geometry 53C23 53C45
1 code implementation • 30 Mar 2010 • Anton Petrunin
Here I show compatibility of two definition of generalized curvature bounds --- the lower bound for sectional curvature in the sense of Alexandrov and lower bound for Ricci curvature in the sense of Lott--Villani--Sturm.
Metric Geometry Differential Geometry Optimization and Control 53C23 (49Q20)
1 code implementation • 29 Mar 2010 • Anton Petrunin
I consider compact metric spaces which admit intrinsic isometries to Euclidean d-space.
Metric Geometry Differential Geometry 30L05 (Primary) 53C (Secondary)
1 code implementation • 1 Jun 2009 • Anton Petrunin
Problems for the graduate students who want to improve problem solving skills in geometry.
History and Overview Differential Geometry Metric Geometry 53Cxx, 51Kxx, 53Axx
1 code implementation • 14 Jun 2005 • Vitali Kapovitch, Anton Petrunin, Wilderich Tuschmann
We show that almost nonnegatively curved m-dimensional manifolds are, up to finite cover, nilpotent spaces in the sense of homotopy theory and have C(m)-nilpotent fundamental groups.
Differential Geometry 53C20
