Instructor: Vyron Vellis (vyron.vellis "AT" uconn "DOT" edu), MONT 306
Lecture: MW 8:40 - 9:55 AM in OAK 441
Topics: In recent years, analysis and geometry have seen great advances where first-order differential calculus and geometric measure theory have been extended from the classical Euclidean or Riemannian setting to the realm of spaces without a priori smooth structure. In particular, there have been important advances in understanding the infinitesimal versus global behavior of Lipschitz functions and quasiconformal mappings in general settings. In this course we will cover a big part of Juha Heinonen's "Lectures on Analysis on Metric Spaces". Specifically, we will cover: Sobolev spaces, Poincare inequalities, Lipschitz functions and weak gradients, Quasiconformal and quasisymmetric mappings. These notions have naturally appeared in many fields of analysis and geometry such as geometric analysis, geometric function theory and sub-Riemannian geometry. The material will give the attendees a firm groundwork in analysis on metric spaces and will prepare them to study more substantial, related articles.
Prerequisite: Modern Analysis (or Real Analysis) and familiarity with measure theory.
Suggested texts:
J. Heinonen, Lectures on Analysis on Metric Spaces , Springer, 2001.
J. Tyson, J. Heinonen, N. Shanmugalingam, and P. Koskela, Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients , New Mathematical Monographs, 2015.
L.C. Evans and R.E. Gariepy, Measure Theory and Fine Properties of Functions , Studies in Advanced Mathematics, 1992.
J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings , Springer, 1971.
J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry , Acta. Math., 181, 1998.
January 28
Introduction to course, Sobolev spaces. Sobolev Embedding Theorem.
January 30
Equivalence of Sobolev Embedding Theorem for 1 ≤ p < n and manifolds satisfying an isoperimetric inequality.
February 1
Sobolev Embedding Theorem for p = ∞. Sobolev-Poincare inequality.
February 4
Maximal Function Theorem. Properties of Bessel's first potential.
February 4 (make-up)
Proof of Sobolev-Poincare inequality. Proof of Sobolev Embedding Theorem for p > n. Chain condition.
February 6
Hajłasz-Koskela Theorem for doubling metric measure spaces with the chain condition. Hajłasz-Sobolev spaces.
February 11
Equivalence of Hajłasz-Sobolev and Sobolev spaces for a class of euclidean domains. A PI inequality for Hajłasz-Sobolev spaces.
February 13
Lipschitz functions. Approximation of functions in M1,p by Lipschitz functions. A. e. differentiability of functions in W1,p when p > n.
February 18
Modulus of curve families and conformal modulus.
February 18 (make-up)
Two examples of modulus and an upper bound for modulus of metric rings which are upper Ahlfors regular.
February 20
Proof that 𝔹n and ℝn are not QC equivalent. Upper gradients, Newtonian spaces, Cheeger spaces. p-Capacity.
March 1 (make-up)
Equality of p-Capacity and p-modulus. Loewner spaces. Hausdorff dimension.
March 6
Examples of Loewner spaces. LLC condition and properties of Q-regular Q-Loewner spaces.
March 11
Quasiconvexity of Loewner spaces and an asymptotic bound on the Loewver function. (1,p)-Poincare inequality.
March 13
Quasiconvex, lower Q-regular spaces that satisfy (1,Q)-PI are Q-Loewner. Riemannian manifolds with nonnegative Ricci curvature.
March 25
Heisenberg group. Five analytic conditions and their equivalence in doubling geodesic metric measure spaces.
March 27
Proof that (V) implies (I) in the analytic equivalence theorem. Introduction to quasisymmetric maps.
April 1
Properties of quasisymmetric maps. Weakly quasisymmetric maps. Doubling spaces and Assouad dimension.
April 3
Equivalence of weakly quasisymmetric and quasisymmetric maps on doubling and connected spaces.
April 8
Doubling measures and their existence in complete doubling spaces.
April 10
Existence of doubling measures in complete doubling spaces.
April 15
Relation between quasisymmetric maps and doubling measures on the real line. Uniformly perfect spaces. Quasimetrics.
April 17
Characterization of spaces quasisymmetric equivalent to Ahlfors regular spaces. Uniformly disconnected spaces.
April 22
Metric doubling measures.
April 24
A metric doubling measure in ℝ3 that is no a QS pullback. Quasisymmetric thickness and existence of doubling measures.
April 29
The bi-Lipschitz embedding problem. Laakso graphs. Assouad's embedding theorem.
May 1
Proof of Assouad's embedding theorem.
May 8 (make-up)
Recent advances on the bi-Lipschitz embedding problem.