Thursdays 11:45-1:45 pm

Room 5417

**Professor:** Christina Sormani

**Topics:**
Lipschitz Maps, Hausdorff Measure and Dimension,
Length Spaces, Hyperbolic Space, Alexandrov Spaces,
Fundamental Groups of these Spaces,
Gromov-Hausdorff Convergence, Gromov's Compactness Theorem,
Quasi Isometries, Gromov Hyperbolic Spaces, Gromov-Hausdorff Limits,
Tangent Cones at Infinity, and Metric Measure Spaces.
Applications to Riemannian Geometry, the work of Perelman and
Complex Analysis will be discussed.

**Text:** Burago-Burago-Ivanov's
"A Course in Metric Geometry" AMS Graduate Studies in Mathematics
Volume 33. See this link
for typos. It's easiest to order this book from the AMS.

**Prerequisites:** Graduate Topology and Real Analysis. Differential
Geometry is a corequisite.

**Meetings**:

**Sept 1: An Introduction***Covered 1.1-1.6 of BBI, defined quasi-isometries and epsilon almost isometries and epsilon nets. Be sure to read 1.3 on your own.***Sept 8: Compact Spaces, Hausdorff Measures and Dimensions**

BBI 1.6-1.7 and Morgan's Geometric Measure Chap 1

*Homework assigned from BBI Ch 1 due on Sept 15: 1.4.9, 1.5.2, 1.5.8, 1.6.10, 1.6.13,***Sept 15: Length Structures, Strictly Intrinsic Metrics**BBI Chapter 2.*Read 2.1-6, Do exercises 2.1.5, 2.1.8, 2.1.9, 2.1.2 and rewrite the proof of Lemma 2.6.1 with appropriate pictures.***Sept 22: Metric Measure Spaces and Ricci Curvature***to prepare students for Max von Renesse's Talks on Tuesday Sept 27.*

Handwritten lecture notes for today in pdf contains quite a few exercises.

A nicely written proof of the Volume Comparison Theorem by Zhu is here**Sept 29: Constructions of Spaces: Gluing and Covering Spaces***I will cover 3.1, 3.3, 3.4 in class, (3.2 will be next week)*

The definition of delta covers can be found in an article by Guofang Wei and I in AMS Transactions and as a preprint in the arxiv.

The Covering Spectrum can be found in our JDG paper also available as a preprint in the arxiv.

*Read 3.1, 3.3 and 3.4/ Do 3.1.13, 3.1.14, 3.1.19, 3.3.9, 3.4.6, 3.4.7, 3.4.8, 3.4.9,***Oct 6: Quasi Isometries and Graphs***Christine Abreu Suzuki will present her thesis on quasi isometries during the first hour. ([BBI] 8.3)*

Jozef Dodziuk will present Cayley graphs, word metrics and combinatorial Laplacians the second hour. ([BBI] 3.2)*Read 3.2, Do 3.2.12, 3.2.19, 3.2.21, 3.2.24 Read 8.3, Do 8.3.7, 8.3.8, 8.3.13, 8.3.16, 8.3.17, 8.3.22*- Oct 13: Yom Kippur
**Oct 20: Cones, Warped Products, and Spaces of Bounded Curvature,**

BBI 3.6 and 4.1*Read 3.6: Do 3.6.3 note the tori are taxicab here, 3.6.23, show the standard sphere is the circle warped by sin(t) over the interval [0,pi], show the hyperbolic plane is the circle warped by sinh(t) over the interval [0, infty), 3.6.24, show that any radially symmetric surface*

x^{2}+ y^{2}= r(z)^{2}with r(z)>0

is a warped product of a line with a circle with a warping function

f(t) such that f(t(z))=r(z) when t(z) is arclength of (z, r(z)) from 0 to z.

Do 3.6.24, 3.6.28, 3.6.29, Prove Prop 3.6.30, Do 3.6.36, 3.6.40,**Oct 27: Chap 4 Alexandrov Spaces***Two important recent construction articles by Alexander and Bishop are posted here and here for future reading. A nice survey of the Alexandrov Spaces by Otsu can be found here.*

Read 4.1-3, Do 4.1.7, 4.1.11. 4.1.14, 4.2.8, 4.2.10, 4.3.8-
**Nov 3: Class Cancelled**(extra class added Tue Nov 22 2pm) -
**Nov 10: Chapter 7 Gromov's Compactness Theorem**

We will review various equivalent definitions of Gromov-Hausdorff Convergence and then prove Gromov's Precompactness Theorem (7.4.15) mentioned on Sept 22 and prove that Gromov-Hausdorff limits of length spaces are length spaces (7.5.1). Together, these form Gromov's Compactness Theorem. See also Gromov's "Metric Structures..." Chapter 5.

Homework: Read Chapter 7 except the last subsection 7.5.2 on the topology of Gromov-Hausdorff limits which has many errors, do 7.3.2, 7.3.4, 7.3.5, 7.3.13, 7.3.14, 7.3.15, 7.3.19, 7.3.23, 7.3.24, 7.3.26, 7.3.31, -
**Nov 17: 7.5.1: Gromov-Hausdorff limits: when they are compact, when they are length spaces, and how every length space can be achieved as a limit of graphs**

Homework: 7.4.5, 7.4.6, 7.4.14, 7.4.17

Section 7.5.9 on the topology of limit spaces has serious errors, see instead:

"Hausdorff Convergence and Universal Covers" by Sormani and Wei in Transactions of the American Mathematical Society 353 (2001), no. 9, 3585--3602. One can prove BBI Exer 7.5.9 using these techniques, but BBI Exer 7.5.11 is false. There is an example of compact simply connected length spaces X_{i}which converge to a compact length space that is not simply connected in this Sormani-Wei paper and also elsewhere. The correction to Exer 7.5.11 given on Y. Burago's website is to assume the limit space $X$ is locally simply connected in which case students who read this paper should be able to complete the exercise. **Nov 22: (Tuesday 2pm Makeup lesson in Room 8405) A discussion of currents (see Frank Morgan's Geometric Measure Theory)**

**At 4pm Stefan Wenger (CIMS) will speak on Alexandrov Spaces for the Differential Geometry Seminar**. Tea will be served at 3:30pm in the math lounge.- Nov 24: Thanksgiving
**Dec 1: Tangent Cones at Infinity and Structure**

8.1-2, a discussion of work by Cheeger-Colding, Sormani-Wei, Menguy, Otsu

Homework: 7.3.6,**Dec 6 (Tuesday): Talk by Kenneth Clarkson (Bell Labs)**2:30 pm Differential Geometry Seminar on metric spaces applied to real world problems.-
**Dec 8: Convergence of Geodesics and the Covering Spectrum and a quick review of the Hyperbolic Space**

See 5.3, particularly pages 174-5, for a review of the Hyperbolic Plane, but beware some serious typos. You may wish to consult other sources. The most serios typo is on page 175 line 11 should say:-
d(a,p)+d(b,p) < d(a,b) +4 or d(a,p) + d(c,p) < d(a,c) +4

_{2}([ab] union [ac]).

See "The Covering Spectrum of a Compact Length Space" by Sormani and Wei, Journal of Differential Geometry 66 (2004) 647-689. -
**Dec 15: Gromov Hyperbolic Spaces***Cancelled Makeup in the Spring*

8.4 in [BBI] is vague and has many typos. On page 286 he uses the term delta almost degenerate without being careful to label that it is relative to p and then lines 9 and 10 need opposite inequalities. I am going through this carefully and will post more corrections as we proceed. A nice alternate reference is Kapovich's Lectures on Geometric Group Theory. Gromov Hyperbolic Spaces" by Vaisala has a nice exposition when one assumes the space is only intrinsic. **Final Assignment (due Thursday Dec 22 at 11:45 am):**The final assignment is a collection of 20 problems selected from the homework completed, submitted and resubmitted during the semester as listed below. Please gather it into a single packet (possibly including extra problems) and add a cover sheet with the list of which 10 correctly completed problems should be used for the grade:- Four out of the five problems assigned on Sept 8,
- Four out of the five problems assigned on Sept 15,
- Four of the eight problems assigned on Sept 29
- One problem from 3.2 and two from 8.3 assigned on Oct 6
- Two of the problems from Oct 20
- One problem from Oct 27
- Two of the problems from Nov 10 - 17.

Be careful to explain you've done this on the cover sheet.

The problems will be given grades of 0-5 based on correctness of the proofs with the usual grades of A - F given as follows:

A 90-100, B 80-90, C 70-80, D 60-70, F 0-60