Канал: Young Measures

Hyperbolic Filling, L5: quasisymmetry

Hyperbolic Filling, L5: quasisymmetry

Hyperbolic Filling, L4: uniformization

Hyperbolic Filling, L4: uniformization

Hyperbolic Filling, L3: Gromov Hyperbolicity

Hyperbolic Filling, L3: Gromov Hyperbolicity

Hyperbolic Filling, L2: Examples

Hyperbolic Filling, L2: Examples

Hyperbolic Filling, L1: Construction of the Graph

Hyperbolic Filling, L1: Construction of the Graph

Hyperbolic Filling of Metric Spaces

Hyperbolic Filling of Metric Spaces

The Epsilon-Delta in Absolute Continuity of Measures

The Epsilon-Delta in Absolute Continuity of Measures

Metric Trees and a Fun Fact About Triangles

Metric Trees and a Fun Fact About Triangles

Space with infinite length under all homeomorphisms, simpler example

Space with infinite length under all homeomorphisms, simpler example

Can Path Metric on a Compact Set be Non-Compact?

Can Path Metric on a Compact Set be Non-Compact?

Horizontality of Lipschitz curves in Heisenberg group

Horizontality of Lipschitz curves in Heisenberg group

Pansu Derivative

Pansu Derivative

Linear Maps on Heisenberg Group Preserve Orientation

Linear Maps on Heisenberg Group Preserve Orientation

Homogeneous Homomorphisms on Heisenberg Group 2/2

Homogeneous Homomorphisms on Heisenberg Group 2/2

Homogeneous Homomorphisms of Heisenberg Group 1/2

Homogeneous Homomorphisms of Heisenberg Group 1/2

Hausdorff Dimension of Heisenberg Group is 4.

Hausdorff Dimension of Heisenberg Group is 4.

Hausdorff Measure and Dimension of Heisenberg Group

Hausdorff Measure and Dimension of Heisenberg Group

13. Stoke’s Theorem

13. Stoke’s Theorem

2. Triple Integral in Spherical Ccordinates

2. Triple Integral in Spherical Ccordinates

14. Stoke’s Theorem Example

14. Stoke’s Theorem Example

9. Normal Vector and Tangent Plane to a Parametric Surface

9. Normal Vector and Tangent Plane to a Parametric Surface

8. What is a Surface? Three Answers.

8. What is a Surface? Three Answers.

3. Path Integrals

3. Path Integrals

11. Area of Parameterized Surfaces

11. Area of Parameterized Surfaces

7. Green’s Theorem to Compute Area

7. Green’s Theorem to Compute Area

6. Green’s Theorem

6. Green’s Theorem

12. Integrating Vectors on Surfaces — Flux

12. Integrating Vectors on Surfaces — Flux

5. Gradient Vector Fields, Path Independence

5. Gradient Vector Fields, Path Independence

4. Path Integrals and Vector Fields

4. Path Integrals and Vector Fields

10. Integrating Scalars Functions on Surfaces

10. Integrating Scalars Functions on Surfaces

1. Triple Integral Practice Exercise

1. Triple Integral Practice Exercise

15. Divergence Theorem

15. Divergence Theorem

Dilations Scale the Carnot Caratheodory Distance: proof

Dilations Scale the Carnot Caratheodory Distance: proof

Does the group dilation define a rectifiable path?

Does the group dilation define a rectifiable path?

Dilations (=scaling) in Heisenberg Groups as Homomorphisms

Dilations (=scaling) in Heisenberg Groups as Homomorphisms

Koranyi Metric On Heisenberg Group, vs Carnot Caratheodory Distance

Koranyi Metric On Heisenberg Group, vs Carnot Caratheodory Distance

Carnot Caratheodory Distance is Left Invariant

Carnot Caratheodory Distance is Left Invariant

Left Invariant Vectors in Heisenberg group—proof

Left Invariant Vectors in Heisenberg group—proof

Grushin Space — a non-Group Sub-Riemannian Manifold

Grushin Space — a non-Group Sub-Riemannian Manifold

What is a sub-Riemannian manifold?

What is a sub-Riemannian manifold?

Rectifiable Curves and Geodesics in Heisenberg Groups

Rectifiable Curves and Geodesics in Heisenberg Groups

Carnot-Caratheodory Distance Explicit Formula

Carnot-Caratheodory Distance Explicit Formula

Carnot-Caratheodory Distance Defines a Metric on Heisenberg Group

Carnot-Caratheodory Distance Defines a Metric on Heisenberg Group

Three Years of Advanced Content @youngmeasures

Three Years of Advanced Content @youngmeasures

Live Session @YoungMeasures

Live Session @YoungMeasures

Carnot-Caratheodory Distance on t-Axis, Horizontal lift of circles

Carnot-Caratheodory Distance on t-Axis, Horizontal lift of circles

Horizontal Lifts of Curves in Heisenberg Group

Horizontal Lifts of Curves in Heisenberg Group

Carnot-Caratheodory Distance and Projection of Horizontal Curves

Carnot-Caratheodory Distance and Projection of Horizontal Curves

The sub-Riemannian Aspect of Heisenberg Groups

The sub-Riemannian Aspect of Heisenberg Groups

Preface to Heisenberg Groups

Preface to Heisenberg Groups

Poincare Inequality-last lecture on Newtonian-Sobolev theory-Lecture 19

Poincare Inequality-last lecture on Newtonian-Sobolev theory-Lecture 19

Absolute Continuity of Newtonian Sobolev Functions on Rectifiable Paths-Lecture 18

Absolute Continuity of Newtonian Sobolev Functions on Rectifiable Paths-Lecture 18

Modulus of Paths and Applications

Modulus of Paths and Applications

Conformal Invariance of Modulus — Complete Proof

Conformal Invariance of Modulus — Complete Proof

Conformal Modulus in Annulus — part 2/2

Conformal Modulus in Annulus — part 2/2

Conformal Modulus in Annulus — part 1/2

Conformal Modulus in Annulus — part 1/2

Modulus of Paths Thru Origin is Zero. Proof.

Modulus of Paths Thru Origin is Zero. Proof.

Modulus of Path Families

Modulus of Path Families

Doubling Measures & Poincare Inequality-Lecture 17

Doubling Measures & Poincare Inequality-Lecture 17

Dirichlet Functions, Limit at infinity — Lecture 16

Dirichlet Functions, Limit at infinity — Lecture 16

Newtonian space on unit interval — Lecture 15

Newtonian space on unit interval — Lecture 15

Are Lipschitz Functions Sobolev? — Lecture 14

Are Lipschitz Functions Sobolev? — Lecture 14

No Rectifiable Curves? No Sobolev Functions — Lecture 13

No Rectifiable Curves? No Sobolev Functions — Lecture 13

Newtonian-Sobolev versus Sobolev Functions — Lecture 12

Newtonian-Sobolev versus Sobolev Functions — Lecture 12

Integration by Parts, Absolutely Continuous — Bonus Video 2

Integration by Parts, Absolutely Continuous — Bonus Video 2

Capacity and Modulus, the way ahead — Lecture 11

Capacity and Modulus, the way ahead — Lecture 11

$Non-zero N^{1,p} function with zero norm -- Lecture 10$

Non-zero N^{1,p} function with zero norm -- Lecture 10

Follow-up to Lecture 9; case p=1 — Lecture 9.5

Follow-up to Lecture 9; case p=1 — Lecture 9.5

Newtonian Sobolev Functions — Lecture 9

Newtonian Sobolev Functions — Lecture 9

Upper Gradient Along One Curve —Lecture 7

Upper Gradient Along One Curve —Lecture 7

Integrable Upper Gradient implies Absolute Continuity — Lecture 8

Integrable Upper Gradient implies Absolute Continuity — Lecture 8

What to Watch This Holiday

What to Watch This Holiday

Path Integrals on Metric Spaces — Lecture 6

Path Integrals on Metric Spaces — Lecture 6

Absolute Continuity on Curves — Lecture 5

Absolute Continuity on Curves — Lecture 5

Curves in Metric Spaces - Lecture 4

Curves in Metric Spaces - Lecture 4

Absolute Continuity of Measures and BV -- Bonus Video 1

Absolute Continuity of Measures and BV -- Bonus Video 1

Absolute Continuity and Integration—Lecture 3

Absolute Continuity and Integration—Lecture 3

Absolute Continuity--Lecture 2

Absolute Continuity--Lecture 2

Newtonian-Sobolev Functions - Lecture 1

Newtonian-Sobolev Functions - Lecture 1

Local Invertibility implies Global Invertibility -- Exam Question

Local Invertibility implies Global Invertibility -- Exam Question

Sequence of homeomorphisms blow up a point

Sequence of homeomorphisms blow up a point

Infinite Area under ALL homeomorphisms?

Infinite Area under ALL homeomorphisms?

What is a Young Measure?

What is a Young Measure?

Hello from University of Cincinnati

Hello from University of Cincinnati

Muckenhoupt Ap Weighted Sobolev Functions

Muckenhoupt Ap Weighted Sobolev Functions

Sobolev Inequality (Embedding) via Riesz Potential

Sobolev Inequality (Embedding) via Riesz Potential

Poincare Inequality via Riesz Potential

Poincare Inequality via Riesz Potential

Recover Multivariable Function from Gradient

Recover Multivariable Function from Gradient

Hajlasz Gradient via Riesz Potential Estimates

Hajlasz Gradient via Riesz Potential Estimates

Riesz Potential

Riesz Potential

Lebesgue Differentiation Theorem — Doubling Spaces

Lebesgue Differentiation Theorem — Doubling Spaces

Hardy Littlewood Maximal Function

Hardy Littlewood Maximal Function

Sobolev Extension Domains and Hajlasz-Sobolev space

Sobolev Extension Domains and Hajlasz-Sobolev space

Hajlasz-Sobolev Functions — analysis on metric spaces

Hajlasz-Sobolev Functions — analysis on metric spaces

Euclidean Sobolev Functions—Key Properties

Euclidean Sobolev Functions—Key Properties

Literature on Sobolev Spaces on Metric Spaces

Literature on Sobolev Spaces on Metric Spaces

Sobolev Spaces on Metric Measure Spaces, “History”

Sobolev Spaces on Metric Measure Spaces, “History”

Squares Glued along a Cantor Set — Poincare Inequality

Squares Glued along a Cantor Set — Poincare Inequality

Cantor Diamonds - Poincare Inequality

Cantor Diamonds - Poincare Inequality

Hello from Okinawa Japan! Typhoon Mawar!! 😳😳

Hello from Okinawa Japan! Typhoon Mawar!! 😳😳

Poincare Inequality on Two Triangles Sharing a Vertex

Poincare Inequality on Two Triangles Sharing a Vertex

Examples 1 — Metric Measure Spaces

Examples 1 — Metric Measure Spaces

Poincare Inequality on Metric Measure Spaces

Poincare Inequality on Metric Measure Spaces

Metric properties: rectifiable paths, Geodesic, Quasiconvex…

Metric properties: rectifiable paths, Geodesic, Quasiconvex…

Metric-Measure Spaces, Borel, doubling

Metric-Measure Spaces, Borel, doubling

What is Analysis… on Metric Spaces?

What is Analysis… on Metric Spaces?

Quasiworld Workshop, Helsinki 2023

Quasiworld Workshop, Helsinki 2023

Fribourg MAPS conference, 2023

Fribourg MAPS conference, 2023

American Mathematical Society Notices

American Mathematical Society Notices

I wrote my own Complex Analysis syllabus!

I wrote my own Complex Analysis syllabus!

Rectifiability—Wrap-up and Reference

Rectifiability—Wrap-up and Reference

Federer-Besicovitch Theorem | Rectifiability

Federer-Besicovitch Theorem | Rectifiability

Projection Theorem 2

Projection Theorem 2

Where to start Research

Where to start Research

Looman Menchoff Theorem

Looman Menchoff Theorem

Marstrand Projection Theorem 1

Marstrand Projection Theorem 1

Energies, Capacities, Hausdorff Dimension

Energies, Capacities, Hausdorff Dimension

Lectures by John Milnor, 1962

Lectures by John Milnor, 1962

Paper: Smooth Approximaton of Low Rank Maps

Paper: Smooth Approximaton of Low Rank Maps

Frostman Lemma Revisited | GMT

Frostman Lemma Revisited | GMT

Cantor Function Construction

Cantor Function Construction

dim AxB =1 but dim A = dim B = 0!

dim AxB =1 but dim A = dim B = 0!

Dimension of Very Critical Values | Arxiv

Dimension of Very Critical Values | Arxiv

Frostman’s Lemma, Application | GMT

Frostman’s Lemma, Application | GMT

Packing Dimension, Product Sets | GMT

Packing Dimension, Product Sets | GMT

Frostman Measure, Product Sets | GMT

Frostman Measure, Product Sets | GMT

Local Isodiametric Inequality 2| GMT

Local Isodiametric Inequality 2| GMT

Local Isodiametric Inequality | GMT

Local Isodiametric Inequality | GMT

Falconer: Regular Sets, Density | GMT

Falconer: Regular Sets, Density | GMT

Are Cantor Sets Countable?! | Wait a second!

Are Cantor Sets Countable?! | Wait a second!

Proof of Density Bounds | GMT

Proof of Density Bounds | GMT

Tools for Young Mathematicians

Tools for Young Mathematicians

Marstrand's Theorem on Existence of Densities

Marstrand's Theorem on Existence of Densities

Densities of Hausdorff Measures | GMT

Densities of Hausdorff Measures | GMT

Lebesgue Density Theorem | GMT

Lebesgue Density Theorem | GMT

Densities, Projections, Tangents | GMT, Rectifiability

Densities, Projections, Tangents | GMT, Rectifiability

Self-Similar Sets 2/2

Self-Similar Sets 2/2

Self-Similar Sets 1/2

Self-Similar Sets 1/2

Not Self Similar Cantor Sets | GMT Examples

Not Self Similar Cantor Sets | GMT Examples

The 4-Corner Cantor Set | GMT Examples

The 4-Corner Cantor Set | GMT Examples

The Two Branches of Geometric Measure Theory

The Two Branches of Geometric Measure Theory

Vitali Covering, but you have to pay for balls!

Vitali Covering, but you have to pay for balls!

Vitali Covering Lemma -- 5r Lemma

Vitali Covering Lemma -- 5r Lemma

300 Subscribers Special! Plan forward.

300 Subscribers Special! Plan forward.

Mappings between Cantor Sets — Coarea Formula

Mappings between Cantor Sets — Coarea Formula

Fat Cantor Sets — Smith-Voltera-Cantor

Fat Cantor Sets — Smith-Voltera-Cantor

Cantor sets with Hausdorff dimension 1 or 0

Cantor sets with Hausdorff dimension 1 or 0

Cantor Sets -- Basic Construction in R^1

Cantor Sets -- Basic Construction in R^1

Regular Trees as Metric Measure Spaces

Regular Trees as Metric Measure Spaces

Explicit Bijection [0,1] to [0,1)

Explicit Bijection [0,1] to [0,1)

On Intersection of Uncountably Many Measurable Sets

On Intersection of Uncountably Many Measurable Sets

On Intersection of an Uncountable Chain of Sets

On Intersection of an Uncountable Chain of Sets

Binary Tree vs All Finite Subsets of N

Binary Tree vs All Finite Subsets of N

Constructing Heisenberg Group from its Lie Algebra

Constructing Heisenberg Group from its Lie Algebra

Heisenberg Group 2

Heisenberg Group 2

The Heisenberg Group, as a metric space

The Heisenberg Group, as a metric space

Coarea Inequality__Paper 2022__an Example

Coarea InequalityPaper 2022an Example

Metric Trees, Factorization of Lipschitz Maps

Metric Trees, Factorization of Lipschitz Maps

Set with infinite length in all metrics!

Set with infinite length in all metrics!

Fields Medal 2022. I was there!!

Fields Medal 2022. I was there!!

Line Integral Along Rectifiable Curves in Metric Spaces

Line Integral Along Rectifiable Curves in Metric Spaces

Hausdorff-1 Measure of a Curve and Area Formula

Hausdorff-1 Measure of a Curve and Area Formula

Intrinsic Characterization of Curves

Intrinsic Characterization of Curves

Differentiability of Curves in Banach Spaces

Differentiability of Curves in Banach Spaces

Hopf-Rinow Theorem in Riemannian and Metric Spaces

Hopf-Rinow Theorem in Riemannian and Metric Spaces

Geodesic curves: Non-Riemannian examples

Geodesic curves: Non-Riemannian examples

Geodesics in Riemannian manifolds

Geodesics in Riemannian manifolds

Curves in Metric Spaces, Length Formula

Curves in Metric Spaces, Length Formula

Length of a curve formula, absolutely continuous curves

Length of a curve formula, absolutely continuous curves

Rectifiable Curves, arclength parameterization

Rectifiable Curves, arclength parameterization

When does the intermediate value property imply continuity?

When does the intermediate value property imply continuity?

Books on Geometric Measure Theory

Books on Geometric Measure Theory

Set of points at a constant distance to a set

Set of points at a constant distance to a set

Coarea Formula notes by Nicolaescu

Coarea Formula notes by Nicolaescu

Coarea Formula-Pop Quiz!

Coarea Formula-Pop Quiz!

Rectifiable Level Sets-Lipschitz Functions

Rectifiable Level Sets-Lipschitz Functions

Proof of Coarea Formula

Proof of Coarea Formula

Coarea Formula-Sard’s Theorem

Coarea Formula-Sard’s Theorem

Coarea Formula-Eilenberg’s Inequality

Coarea Formula-Eilenberg’s Inequality

Coarea Formula- Jacobian Determinant

Coarea Formula- Jacobian Determinant

Implicit Function Theorem

Implicit Function Theorem

Proof of Coarea Formula

Proof of Coarea Formula

Coarea Formula-Examples, Fubini type results

Coarea Formula-Examples, Fubini type results

Coarea Formula- Introduction

Coarea Formula- Introduction

Channel Overview, Coarea Formula on its way!

Channel Overview, Coarea Formula on its way!

Rectifiable Sets Are Almost C^1 Manifolds

Rectifiable Sets Are Almost C^1 Manifolds

Area Formula for C1 maps

Area Formula for C1 maps

Surface Measure on Riemannian Manifolds

Surface Measure on Riemannian Manifolds

C^1 Approximation of Lipschitz Maps

C^1 Approximation of Lipschitz Maps

Area Formula Corollaries

Area Formula Corollaries

Sard’s Theorem, Direct Proof

Sard’s Theorem, Direct Proof

Proof of Area Formula (geometric measure theory)

Proof of Area Formula (geometric measure theory)

Rademacher’s Theorem

Rademacher’s Theorem

Geometry of linear maps, Gram determinant, parallelepiped volume

Geometry of linear maps, Gram determinant, parallelepiped volume

Area Formula - Geometric Measure Theory

Area Formula - Geometric Measure Theory

Hausdorff and Lebesgue measure are equal (Proof)

Hausdorff and Lebesgue measure are equal (Proof)

Can self-maps of compact spaces increase distances? 2/2

Can self-maps of compact spaces increase distances? 2/2

Can self-maps of compact spaces increase distances? 1/2

Can self-maps of compact spaces increase distances? 1/2

The Hausdorff content

The Hausdorff content

Hausdorff measure and Lipschitz or Hölder maps

Hausdorff measure and Lipschitz or Hölder maps