Math 595 (GMT) Fall 2015: Geometric Measure Theory Instructor: Jeremy Tyson Lecture Times and Location: MWF 9:00–9:50, 141 Altgeld Hall

Geometric measure theory considers the structure of Borel sets and Borel measures in metric spaces. In addition to its intrinsic interest, it has been a valuable tool for problems arising from real and complex analysis, harmonic analysis, PDE, and other fields. For instance, rectifiability criteria and metric curvature conditions played a key role in Tolsa’s resolution of the longstanding Painlev´eproblem on removable sets for bounded analytic functions. The classical setting for geometric measure theory is finite-dimensional Euclidean spaces. Contempo- rary trends in analysis and geometry in metric measure spaces, including analysis on fractals, motivate extensions of the subject to non-Riemannian and nonsmooth spaces. Sub-Riemannian spaces, particularly, the sub-Riemannian Heisenberg group, are an important testing ground and model for the general theory. In the first part of the course we will give an extended review of Euclidean geometric measure theory. Major topics to be covered include Hausdorff measure and dimension, density theorems, energy and ca- pacity methods, almost sure dimension distortion theorems, and tangent measures. Rectifiable sets and measures provide a rich measure-theoretic generalization of smooth differential submanifolds and their volume measures. We will give a short introduction to this important topic. In the second part of the course we will discuss ongoing efforts to extend this theory into non- Riemannian and general metric spaces. Motivation for such efforts will be indicated. We will emphasize notions of rectifiability and almost sure dimension distortion theorems in sub-Riemannian spaces.

Texts: There is no required textbook. There are many textbook treatments of Euclidean geomet- ric measure theory, e.g., [1] P. Mattila, Geometry of sets and measures in Euclidean space, Cambridge Univ. Press, 1995. [2] K. Falconer, The geometry of fractal sets, Cambridge University Press, 1985. [3] F. Morgan, Geometric measure theory: a beginner’s guide, 3rd ed., Academic Press, 2000. [4] K. Falconer, Fractal geometry: mathematical foundations and applications, 2nd ed., Wiley & Sons, 2003. [5] K. Falconer, Techniques in fractal geometry, Wiley & Sons, 1997. Our primary reference for the first part of the course will be [1]. In our discussion of rectifiability and density theorems we will follow the elegant treatment in [6] C. De Lellis, Rectifiable sets, densities and tangent measures, Z¨urich Lectures in Advanced Mathe- matics, European Math. Soc., 2008. Geometric measure theory in sub-Riemannian and metric spaces is an active area of research and no textbook treatment is yet available. Material will be drawn from various papers, including unpublished work. Basic aspects of analysis in metric spaces can be found in [7] L. Ambrosio and P. Tilli, Topics in analysis in metric spaces, Oxford Lecture series in mathematics and its applications, vol. 25, Oxford Univ. Press, 2004.

Prerequisites: Real and complex analysis (Math 540 and Math 542) are required. Some fluency with differential geometry (at the level of Math 518) would be helpful for understanding the context and motivation for the study of rectifiability, however, it is not necessary. Other closely related courses include harmonic analysis (Math 545) and functional analysis (Math 541). Grading policy: There will be no exams. Occasionally I will suggest homework exercises; these are not required, but working out the details may help to provide further motivation and context for the lectures. I will be happy to read and comment on solutions to these exercises. I hope to produce, by the end of the semester, a working set of lecture notes for the course. I will ask each student to type up a careful set of notes for 2-3 lectures. These notes should summarize the material discussed in the lectures on those days, but should also go deeper into the topic. I will provide references for each lecture, so that you have some resources to look at when preparing these written notes. An ability to write an articulate summary of an existing mathematical theory in your own words is an important skill for any working mathematician. At the end of the semester I will combine all of your notes into a unified document which will (ideally) be a detailed overview of the subject. Each of you will get a copy of this final document.

Tentative Syllabus

Section Topic Details 1 Review of measure theory abstract measure theory 3 lectures Lebesgue measure, Haar measure Hausdorff and packing measures 2 Covering and differentiation theorems basic covering theorem 4 lectures doubling metric spaces, covering thms differentiation of measures, densities 3 Contraction maps and IFS Banach Fixed Point Thm 3 lectures IFS and invariant sets open set condition and Hutchinson thm 4 Lipschitz maps Rademacher thm, Eilenberg inequality 4 lectures H¨oldermaps, nowhere diff’ble functions snowflaking of metric spaces, Assouad’s embedding thm 5 Energy and Capacity definitions 3 lectures Frostman’s Lemma, dimensions of product sets 6 Projection and slicing theorems O(n) and Grassmannians 4 lectures projection thms, self-similar sets with overlaps slicing theorems 7 Sobolev maps and dimension definitions of Sobolev functions and mappings 4 lectures Morrey–Sobolev inequality, Calder´ondiff’n thm distortion of Hausdorff dimension frequency of Hausdorff dimension distortion 8 Tangent measures and densities Tangent measures, Gromov–Hausdorff convergence 4 lectures uniform and uniformly distributed measures Marstrand’s density theorem Kowalski–Preiss theorem 9 Rectifiability rectifiable vs. purely unrectifiable sets 4 lectures structural decomposition of Borel sets characterizations of rectifiability, Preiss density thm 10 Sub-Riemannian geometry Heisenberg groups and Carnot groups 4 lectures metric and differential geometric structure Dimension Comparison Theorem 11 Sub-Riem geometric measure theory selected topics (TBD) 4 lectures Total 41 lectures