Special Topics in Metric Geometry
GC Math 86500. Corequisite: Graduate Real Analysis.
Prerequisite: Undergraduate Real Analysis
Thursdays 6:30-8:30 pm at CUNYGG 365 Fifth Avenue
in C415A in the basement.
Professor Sormani: sormanic@gmail.com
GC Office Hours: Thursdays 5:30-6:00 in the classroom
We begin with a survey of a wide variety of metric spaces and a review of classical metric geometry. We then consider various notions of convergence for metric spaces including Gromov Lipschitz, Gromov Hausdorff, and Intrinsic Flat convergence. We will examine the conditions that imply convergence and subconvergence and properties that persist under convergence. Advanced students will be encouraged to conduct work related to metric spaces that arise in their own field of research, while first and second year students will be given easier metric spaces to study. Work that is completed may be of high enough quality to publish as a short paper. References include the textbook by Burago-Burago-Ivanov and a variety of articles published in the past decade.
Grades of A B C INC or F will depend upon the completion of projects (HW see below each lesson) that will be slowly assigned all semester long and collected as completed (due at the end of the semester). Everyone will be working on different projects although some may work in teams on some parts if they wish. Ideally publishable work will result although publishable work is not required for the A.
A precise grading scheme has been requested so it is as follows:
20% of grade is based on correct completion of Lesson 1 HW
20% of grade is based on correct completion of Lesson 2 HW
20% of grade is based on correct completion of Lesson 3 HW
20% of grade is based on correct completion of Lesson 5 HW
20% of grade is based on correct completion of one additional lesson HW or a project
Feedback will be given on work completed on time so that when it is resubmitted on finals week hopefully it will be perfect. Some advanced students skipped Lesson I because it was too easy. You may submit a more advanced hw lesson instead of lesson 1 HW if you wish. The goal is to choose to do HW on topics of most interest to you.
Materials, Resources and Accommodations:
Short article reviewing this area in the AMS Notices that I wrote recently.
Books and Articles: (will be linked to under each lesson)
Burago-Burago-Ivanov's text is highly recommended and well worth purchasing
Advice for Doctoral Students and some Prior Doctoral Students and Postdocs that worked with me
Accommodating Disabilities: CUNY is committed to providing access to all programs and curricula to all students.
Students with disabilities who may need classroom accommodations should let me know if there is something I can do.
Accommodating Holidays: If you have a holiday during a lesson or extra lesson, let me know, and something will be arranged.
Names/Gender: We will use last surnames in this class. You may call me Sormani or Professor.
Respect: All students will treat each other with respect and dignity. Let me know if you have concerns.
Schedule: (topics and links to related articles will appear as the semester progresses).
on the course webpage:
https://sites.google.com/site/professorsormani/metric-geometry-fall-2019
Aug 29: Lesson I: Metric Geometry, Balls, Open sets, Limits of points, Continuity, Lipschitz maps, bi-Lipschitz maps, Lipschitz convergence, Product spaces, Taxi products, Diameters, Projection maps,
HW: (1) Send me your proof that if f is lambda Lipschitz and x_j to x_infty then f(x_j) to f(x_infty) including the relationship between N for the x_j and N for the f(x_j). (2) Send me your proof of the relationship between diam(X) and diam(Y) when X and Y are lambda biLipschitz using the definition of diameter. (3) Also send me an email stating what courses and exams you've taken and are taking this year, and what kind of research they are interested in. In this way I can design your individual projects. Be sure to let me know if you have a background in Riemannian Geometry or Probability.
Sept 12: Lesson 2: Compactness, Uniform Compactness, Metric Completions, and Gromov’s Compactness Theorem
HW (1) Suppose a sequence of metric spaces X_j have biLipschitz maps $f_j:X_0 \to X_j$ with Lipschitz constants lambda_j
(1a) show that if lambda_j le K then N_j(r) can be bounded uniformly, relate explicitly to N_0(R) and give explicit construction of the nets.
(1b) go through the proof of Gromov’s theorem and describe the construction of the Limit space
(1c) show the limit space is (X_infty, d_infty) is biLip to (X_0,d_0)
(1d) what if lambda_j converge to 1?
(2) Consider X_j to be tori where one circle has fixed radius 1 and the other has radius R_j to 0
(2a) Find uniform N_j(r) for the sequence and explicit r-nets
(2b) go through the proof of Gromov’s theorem and describe the construction of the Limit space
(2c) show the limit space is the circle
(3) Consider X_j a sequence of metric spaces of your choosing and examine whether the sequence is uniformly compact of not.
(4) Take any metric space
(4a) Prove that N(r) can be taken to be the max number of disjoint balls of radius R_r.
(4b) What is the formula for R_r?
(4c) What if the maximum number of disjoint balls in a seqience of metric spaces diverges to infinity?
Sept 19: Lesson 3: Many Equivalent Definitions of the Gromov-Hausdorff Distance
(1) Write a proof that if the GH distance is < r then there is a 2r almost isometry
(this was done in class in a rambling way, so write it neatly)
(2) Write a proof that if there is an r almost isometry then the GH distance is <2r
(this was not quite finished in class since we only constructed Z and did not check the Hausdorff distance in Z)
(3) If the Gromov Lipschitz distance between two spaces is <r then estimate the GH distance
(4) Prove increasingly thin tori S^1xS^1_(1/j) GH converge to S^1 using almost isometries
(5) Choose a pair of compact metric spaces that you have studied in the past and estimate the GH distance between them.
Sept 26: Lesson 4: Pointed GH convergence, tangent cones at infinity, tangent cones at a point, and metric cones
Start thinking about limits of metric spaces you are interested in studying. With today’s lesson they do not need to be compact. Catch up on the first three projects. Be sure to email me ideas of metric spaces or ask me to recommend some to you while reminding me of your background. It is important that nobody quit the course yet, so resubmit work if you made many mistakes. Submit complete clear proofs only (not all the ideas you have on the way). Email me if you have not received the photos of the blackboards for every lesson.
Oct 3: Lesson 5: (no meeting) but study and do HW on Length Spaces, Geodesics, Midpoints, and the GH Arzela Ascoli Theorem
HW (1) (a) Prove that if X and Y contain r nets {x_1,...,x_N(r)} and {y_1,...,y_N(r)} and if there is a bijection between the r nets which is delta almost distance preserving |d_X(x_i,x_j)-d_Y(y_i,y_j)| <delta for all i,j le N(r), then d_GH(X,Y)< a function of r and delta. Give the formula for that function explicitly. Note in your proof you may try to construct Z and use the defn of GH convergence or you may try to proof there is an epsilon almost isometry between X and Y. (b) Recall that when we proved Gromov’s compactness theorem we proved there exist r_j nets with r_j to 0 in X_j and in X and there are bijections between these r_j nets which are delta_j almost distance preserving with delta_j to 0. Use Part (a) to prove this implies X_j GH to X. (c) Use this to prove a sequence of metric spaces you choose converges to a specific limit.
(2) Suppose X_j GH to X so there are r_j almost isometries from X to X_j with r_j to 0, (a) Prove that if X has an R net with N points then
X_j have R_j nets with N points too finding a formula for R_j depending on r_j and R. (b) Use this property to find a sequence of metric spaces with no GH limit and prove this.
(3) Suppose X_j GH to X so there are r_j almost isometries from X to X_j with r_j to 0, (a) prove that for any p,q in X we can find p_i and q_i in X_i such that d_i(p_i,q_i) to d(p,q). (b) prove that if this p_i and q_i have a midpoint x_i then p and q have a midpoint too. (c) conclude that the GH limits of compact length spaces are compact length spaces (see textbook for more about length spaces. (d) apply this to a sequence of metric spaces that are length spaces.
(5) Suppose X_j GH to X and Y_j GH to Y all compact and suppose F_j:X_j to Y_j are Lipschitz with Lipschitz constant K. Use the almost isometries phi_j:X to X_j and psi_j:Y_j to Y to define H_j=psi_j F_j phi_j: X to Y. (a) is H_j Lipschitz? Find a counter example or prove it. If it isn’t Lipschitz what is know about H_j? (b) let S= {x_1,x_2,...,x_N,...} be a countable dense set in X such that x_1,...,x_N(r) is an r net. Prove that one can find a subsequence of H_j converging to some function H:S to Y and that H is Lipschitz K (c) prove that you can extend H to X uniquely as a Lipschitz K function. This is called the GH Arzela Ascoli Theorem.
Oct 10: Lesson 6: Lengths, geodesics, Lipschitz Maps, Hausdorff Measures, Rectifiable Metric Spaces, Hausdorff Dimension,
Read about Hausdorff measures of sets in Euclidean space in Morgan’s Geometric Measure Theory
Read this proof of Rademacher’s Theorem that Lipschitz functions are differentiable almost everywhere
(1) Show the cube is a 2rectifiable metric space with 6 charts
(2) Show the tori are 2 rectifiable metric spaces and the circle is a 1 rectifiable metric space
(3) Show that graphs with finite collections of edges are 1 rectifiable and their Hausdorff 1 measure is the sum of the lengths of the edges.
(4) Show that the taxi square is 2 rectifiable with 1 chart (very easy).
(5) Show that if two metric spaces are bi Lipschitz and one is rectifiable then the other is too, and estimate the Hausdorff measures of the two spaces in terms of the biLip constant. Does Gromov Lip convergence imply Hausdorff measures converge?
Oct 17: Lesson 7: Metric Spaces with boundary: Integral Current Spaces
Today’s lesson’s primary reference is Sormani-Wenger’s original JDG paper
For a proof that Lipschitz functions on Euclidean space are differentiable almost everywhere see this
For more about differential forms see Calculus on Manifolds by Spivak
The following HW should be doable based only on our lesson and vector calc
(1) (a) draw a few finite directed graphs with weighted edges and find their boundaries.
(b) explain why any weighted directed graph with finitely many edges of length 1 is an integral current space.
(c) consider the metric product of a finite weighted graph with an interval, explain why it is an integral current space and find its boundary.
(d) this time do the same but take the isometric product of a finite graph with a circle.
(2) If you have studied Riemannian geometry show that any oriented Riemannian manifold with boundary is an integral current space.
(3) Prove that if F:X to Y is bi Lipschitz and (X,d,T) is an integral current space, then (Y, d_Y, F_# T) is an integral current space. If the biLip const is K (finds charts and check pts of positive density). What is the relationship between the weighted volumes?
(4) If (X,d,T) is and integral currents space, show (X,d,kT)is an integral current space if k is an integer. What is the relationship between the weighted volumes?
Oct 24: Lesson 8: Guest Lecture An introduction of Alexandrov Geometry by Prof Nan Li (City Tech)
Abstract: This is an introductory talk about Alexandrov Geometry. We will begin with the motivations and then focus on the following topics: (1) definitions and examples, (2) the globalization of local Alexandrov spaces, (3) dimensions, (4) tangent cones and singular/regular sets, (5) the stratification structures.
Chapter 4 of the Burago-Burago-Ivanov textbook is a good resource for this topic.
HW: Prove that if X_j are Alexandrov spaces with curvature bounded below by 0 and if X_j GHto X_infty then X_infty is an Alexandrov space with curvature bounded below by 0.
Do four problems of your choice from Chapter 4 of Burago-Burago-Ivanov.
Oct 31: Class meets Online this week pdf of slides and ask questions by email or skype
Lesson 9: Smocked Metric Spaces
HW Read https://arxiv.org/pdf/1906.03403.pdf which introduces this notion rigorously and requires no background.
You do not need to read the proofs in complete detail. Try to get a sense of the whole paper. At the end of the paper are some open problems. Let me know if you are interested in doing any of them. Note that a “pulled thread” is a “pulled string”.
(1) Is a pulled string space a length space? Is it a geodesic space?
(2) Is a pulled string space an Alexandrov Space with curvature bounded above or below or neither?
(3) is a smocked metric space a length space? Is it a geodesic space?
(4) Is a smocked space an Alexandrov space with curvature bounded above or below or neither?
Nov 7: Lesson 10: Introduction to Smooth and Intrinsic Flat (SWIF) Convergence
Go through the slides of this talk:``Converging Sequences of Metric Spaces”(AMS Sampler) (slides)
Today’s lesson’s primary reference is Sormani-Wenger’s original JDG paper
Papers about intrinsic flat convergence can be found here
The lesson is being taught introducing every concept needed and only exactly what is necessary. Email me for photos of the boards.
(1) Show that the intrinsic flat limit of a sequence of spheres of radii decreasing to 0 is the 0 space.
(2) Show that the sequence of collapsing tori also converges in the intrinsic flat sense to the 0 space
(3)(a) Use the infimum defn of the intrinsic flat distance and intrinsic flat convergence to prove M_j converges in the intrinsic flat sense to M iff there exists metric spaces Z_j and distance preserving maps phi_j:M_j to Z_j and psi_j:M to Z_j and integral currents A_j and B_j in Z_j whose masses converge to 0 such that the boundary of B_j is A_j plus the difference between the images of the M_j and M viewed as currents.
(3)(b) Show that if M_j converges in the intrinsic flat sense to M then the boundaries converge to the boundary.
(4) Are the metric spaces you are interested in studying naturally integral current spaces?
Nov 14: Lesson 11: Metric Measure Convergence and the Wasserstein Distance
Sturm has a paper introducing his approach to metric measure convergence
This is worth reading if you have an interest in probability or metric measure spaces
Examples of Converging Sequences of Riemannian Manifolds
Warped products with Brian Allen see preprint
Sewing with Jorge Basilio and Jozef Dodziuk see paper
Nov 21: Lesson 12: Gromov- Hausdorff, Intrinsic Flat Convergence, and embeddings into a common Z
Gromov’s Groups of Polynomial Growth paper Section 6
Sormani’s Intrinsic Flat Arzela-Ascoli paper or preprint
No meeting on Thanksgiving
Dec 5: Lesson 13: Guest Lecture: Geometric Group Theory by Prof Jason Behrstock (GC and Lehman)
Gromov’s Groups of Polynomial Growth paper
(1) (easy) prove two generating sets give quasi isomorphic word metrics
(2) (easy) prove that if two metric spaces X and Y are quasi isomorphic then rescaling them by a fixed constant R they are still quasi isomorphic.
(3) (challenge) try to give a proof that any group quasi isomorphic to Z^2 has a finite index subgroup isomorphic to Z^2 without using Gromiv’s Theorem. If you do this one send it to both Prof Behrstock and Prof Sormani
Dec 12: Lesson 14: Students meet to complete course projects
Hand in all completed and corrected homework for me to determine your final grade. You should have completed the homework assignments from Lessons 1-5 plus one additional lesson or a project. You may also choose to submit more challenging hw in place of the easier lessons as you wish. Last chance for feedback before final submission was the weekend of Dec 6-8. Final deadline to submit completed work is Thursday Dec 19 in the classroom at 6:30 pm or by email.
Dec 19: Lesson 15: Rigidity and Almost Rigidity
See DoCarmo’s Riemannian Geometry textbook and Burago-Burago-Ivanov textbook
See Shen-Sormani on the Topology of Manifolds with Nonnegative Ricci Curvature
See Wei Manifolds with a Lower Ricci Curvature Bound for a survey of Ricci Rigidity and Almost Rigidity
See Sormani Scalar Curvature and Intrinsic Flat Convergence
Submission of Final Projects by email or in person.
We might continue with a reading seminar in the Spring where students present more details. I will meet with my research team in January if you wish to join in.