Past OxPDE Short Courses

28 May 2021
10:00
to
12:00
Professor Radu Ignat

Further Information: 

3 x 2 hour Lectures via Zoom (see email of 10th May 2021 for details)

Aimed at: The course is addressed to postgraduate students, postdocs and other members of the Mathematical Institute. It is an introduction to concentration phenomenon around vortices in Ginzburg-Landau type problems. The aim is to present topological methods (based on Jacobian, winding number...) that allow for detection of vortices and computation of the interaction energy between them. The purpose of this course is to analyse vortex singularities appearing in Ginzburg-Landau type problems.

The lecture will be via Zoom and the link was also sent out in an email on 10th May 

 

 

 

26 May 2021
11:00
to
12:30
Dr. Krzysztof Ciosmak

Further Information: 

4 x 1.5 hour Lectures 

Aimed at: any DPhil students with interest in learning about extensions of functions. 

 

Suggested Pre-requisites: Suitable for OxPDE students, but also of interests to functional analysts, geometers, probabilists, numerical analysts and anyone who has a suitable level of prerequisite knowledge.

 

Abstract

Abstract. The aim of the course is to present several results on extensions of functions. Among the most important are Kirszbraun's and Whitney's theorems.
They provide powerful technical tools in many problems of analysis. One way to view these theorems is that they show that there exists an interpolation
of data with certain properties. In this context they are useful in computer science, e.g. in clustering of data (see e.g. [26, 23]) and in dimension reduction (see e.g. [15]).

1. Syllabus
Lecture 1. McShane's theorem [25], Kirszbraun's theorem [18, 31, 35], Kneser- Poulsen conjecture [19, 29, 16].
Lecture 2. Whitney's covering and associated partition of unity, Whitney's ex-tension theorem [37, 12, 33].
Lecture 3. Whitney's theorem { minimal Lipschitz extensions [22].
Lecture 4. Ball's extension theorem, Markov type and cotype [6].

2. Required mathematical background
Markov chains, Hilbert spaces, Banach spaces, metric spaces, Zorn lemma

3. Reading list
The reading list consists of all the papers cited above, lecture notes [27], and parts of books [36, 8].

4. Assesment
Students will be encouraged to give a short talk on a topic related to the content of the course. Suggested topics include:
(1) Brehm's theorem [10],
(2) continuity of Kirszbraun's extension theorem [20],
(3) Kirszbraun's theorem for Alexandrov spaces [21, 1],
(4) two-dimensional Kneser-Poulsen conjecture [9],
(5) origami [11],
(6) absolutely minimising Lipschitz extensions and in nity Laplacian [17, 32,
34, 2, 3, 5, 4],
(7) Fenchel duality and Fitzpatrich functions [30, 7],
(8) sharp form of Whitney's extension theorem [13],
(9) Whitney's extension theorem for Cm [14],
(10) Markov type and cotype calculation [27, 6, 28], 

(11) extending Lipschitz functions via random metric partitions [24, 27].

References
1. S. Alexander, V. Kapovitch, and A. Petrunin, Alexandrov meets Kirszbraun, 2017.
2. G. Aronsson, Minimization problems for the functional supx F(x; f(x); f0(x)), Ark. Mat. 6 (1965), no. 1, 33{53.
3. , Minimization problems for the functional supx F(x; f(x); f0(x))(ii), Ark. Mat. 6 (1966), no. 4-5, 409{431.
4. , Extension of functions satisfying lipschitz conditions, Ark. Mat. 6 (1967), no. 6, 551{561.
5. , Minimization problems for the functional supx F(x; f(x); f0(x))(iii), Ark. Mat. 7 (1969), no. 6, 509{512.
6. K. Ball, Markov chains, Riesz transforms and Lipschitz maps, Geometric & Functional Analysis GAFA 2 (1992), no. 2, 137{172.
7. H. Bauschke, Fenchel duality, Fitzpatrick functions and the extension of rmly nonexpansive mappings, Proceedings of the American Mathematical Society 135 (2007), no. 1, 135{139. MR 2280182
8. Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, Colloquium publications (American Mathematical Society) ; v. 48, American Mathematical Society, Providence, R.I., 2000 (eng).
9. K. Bezdek and R. Connelly, Pushing disks apart { the Kneser-Poulsen conjecture in the plane, Journal fur die reine und angewandte Mathematik (2002), no. 553, 221 { 236.
10. U. Brehm, Extensions of distance reducing mappings to piecewise congruent mappings on Rm, J. Geom. 16 (1981), no. 2, 187{193. MR 642266
11. B. Dacorogna, P. Marcellini, and E. Paolini, Lipschitz-continuous local isometric immersions: rigid maps and origami, Journal de Mathematiques Pures et Appliques 90 (2008), no. 1, 66 { 81.
12. L. C. Evans and R. F. Gariepy, Measure theory and ne properties of functions; Rev. ed., Textbooks in mathematics, ch. 6, CRC Press, Oakville, 2015.
13. C. L. Fe erman, A sharp form of Whitney's extension theorem, Annals of Mathematics 161 (2005), no. 1, 509{577. MR 2150391
14. , Whitney's extension problem for Cm, Annals of Mathematics 164 (2006), no. 1, 313{359. MR 2233850
15. L.-A. Gottlieb and R. Krauthgamer, A nonlinear approach to dimension reduction, Weizmann Institute of Science.
16. M. Gromov, Monotonicity of the volume of intersection of balls, Geometrical Aspects of Functional Analysis (Berlin, Heidelberg) (J. Lindenstrauss and V. D. Milman, eds.), Springer Berlin Heidelberg, 1987, pp. 1{4.
17. R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Archive for Rational Mechanics and Analysis 123 (1993), no. 1, 51{74.
18. M. Kirszbraun,  Uber die zusammenziehende und Lipschitzsche Transformationen, Fundamenta Mathematicae 22 (1934), no. 1, 77{108 (ger).
19. M. Kneser, Einige Bemerkungen uber das Minkowskische Flachenma, Archiv der Mathematik 6 (1955), no. 5, 382{390.
20. E. Kopecka, Bootstrapping Kirszbraun's extension theorem, Fund. Math. 217 (2012), no. 1, 13{19. MR 2914919
21. U. Lang and V. Schroeder, Kirszbraun's theorem and metric spaces of bounded curvature, Geometric & Functional Analysis GAFA 7 (1997), no. 3, 535{560. MR 1466337
22. E. Le Gruyer, Minimal Lipschitz extensions to di erentiable functions de ned on a Hilbert space, Geometric and Functional Analysis 19 (2009), no. 4, 1101{1118. MR 2570317
23. J. Lee, Jl lemma and Kirszbraun's extension theorem, 2020, Sublinear Algorithms for Big Data Lectues Notes, Brown University.
24. J. R. Lee and A. Naor, Extending Lipschitz functions via random metric partitions, Inventiones mathematicae 160 (2005), no. 1, 59{95.
25. E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12, 837{842. MR 1562984
26. A. Naor, Probabilistic clustering of high dimensional norms, pp. 690{709. 

27. , Metric embeddings and Lipschitz extensions, Princeton University, Lecture Notes, 2015.
28. A. Naor, Y. Peres, O. Schramm, and S. Sheeld, Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces, Duke Math. J. 134 (2006), no. 1, 165{197.
29. E. T. Poulsen, Problem 10, Mathematica Scandinavica 2 (1954), 346.
30. S. Reich and S. Simons, Fenchel duality, Fitzpatrick functions and the Kirszbraun{Valentine extension theorem, Proceedings of the American Mathematical Society 133 (2005), no. 9, 2657{2660. MR 2146211
31. I. J. Schoenberg, On a Theorem of Kirzbraun and Valentine, The American Mathematical Monthly 60 (1953), no. 9, 620{622. MR 0058232
32. S. Sheeld and C. K. Smart, Vector-valued optimal Lipschitz extensions, Communications on Pure and Applied Mathematics 65 (2012), no. 1, 128{154. MR 2846639
33. E. Stein, Singular integrals and di erentiability properties of functions, ch. 6, Princeton University Press, 1970.
34. P. V. Than, Extensions lipschitziennes minimales, Ph.D. thesis, INSA de Rennes, 2015.
35. F. A. Valentine, A Lipschitz condition preserving extension for a vector function, Amer. J. Math. 67 (1945), 83{93. MR 0011702
36. J. H. Wells and L. R. Williams, Embeddings and extensions in analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete ; Bd. 84, Springer-Verlag, Berlin, 1975 (eng).
37. H. Whitney, Analytic extensions of di erentiable functions de ned in closed sets, Transactions of the American Mathematical Society 36 (1934), no. 1, 63{89. MR 1501735 

University of Oxford, Mathematical Institute and St John's College, Oxford, United Kingdom
E-mail address: Krzysztof.Ciosmak@maths.ox.ac.uk

The join button will be published on the right (Above the view all button) 30 minutes before the seminar starts (login required).

25 May 2021
11:00
to
12:30
Dr. Krzysztof Ciosmak

Further Information: 

4 x 1.5 hour Lectures 

Aimed at: any DPhil students with interest in learning about extensions of functions. 

Suggested Pre-requisites: Suitable for OxPDE students, but also of interests to functional analysts, geometers, probabilists, numerical analysts and anyone who has a suitable level of prerequisite knowledge.

 

Abstract

Abstract. The aim of the course is to present several results on extensions of functions. Among the most important are Kirszbraun's and Whitney's theorems.
They provide powerful technical tools in many problems of analysis. One way to view these theorems is that they show that there exists an interpolation
of data with certain properties. In this context they are useful in computer science, e.g. in clustering of data (see e.g. [26, 23]) and in dimension reduction (see e.g. [15]).

1. Syllabus
Lecture 1. McShane's theorem [25], Kirszbraun's theorem [18, 31, 35], Kneser- Poulsen conjecture [19, 29, 16].
Lecture 2. Whitney's covering and associated partition of unity, Whitney's ex-tension theorem [37, 12, 33].
Lecture 3. Whitney's theorem { minimal Lipschitz extensions [22].
Lecture 4. Ball's extension theorem, Markov type and cotype [6].

2. Required mathematical background
Markov chains, Hilbert spaces, Banach spaces, metric spaces, Zorn lemma

3. Reading list
The reading list consists of all the papers cited above, lecture notes [27], and parts of books [36, 8].

4. Assesment
Students will be encouraged to give a short talk on a topic related to the content of the course. Suggested topics include:
(1) Brehm's theorem [10],
(2) continuity of Kirszbraun's extension theorem [20],
(3) Kirszbraun's theorem for Alexandrov spaces [21, 1],
(4) two-dimensional Kneser-Poulsen conjecture [9],
(5) origami [11],
(6) absolutely minimising Lipschitz extensions and in nity Laplacian [17, 32,
34, 2, 3, 5, 4],
(7) Fenchel duality and Fitzpatrich functions [30, 7],
(8) sharp form of Whitney's extension theorem [13],
(9) Whitney's extension theorem for Cm [14],
(10) Markov type and cotype calculation [27, 6, 28], 

(11) extending Lipschitz functions via random metric partitions [24, 27].

References
1. S. Alexander, V. Kapovitch, and A. Petrunin, Alexandrov meets Kirszbraun, 2017.
2. G. Aronsson, Minimization problems for the functional supx F(x; f(x); f0(x)), Ark. Mat. 6 (1965), no. 1, 33{53.
3. , Minimization problems for the functional supx F(x; f(x); f0(x))(ii), Ark. Mat. 6 (1966), no. 4-5, 409{431.
4. , Extension of functions satisfying lipschitz conditions, Ark. Mat. 6 (1967), no. 6, 551{561.
5. , Minimization problems for the functional supx F(x; f(x); f0(x))(iii), Ark. Mat. 7 (1969), no. 6, 509{512.
6. K. Ball, Markov chains, Riesz transforms and Lipschitz maps, Geometric & Functional Analysis GAFA 2 (1992), no. 2, 137{172.
7. H. Bauschke, Fenchel duality, Fitzpatrick functions and the extension of rmly nonexpansive mappings, Proceedings of the American Mathematical Society 135 (2007), no. 1, 135{139. MR 2280182
8. Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, Colloquium publications (American Mathematical Society) ; v. 48, American Mathematical Society, Providence, R.I., 2000 (eng).
9. K. Bezdek and R. Connelly, Pushing disks apart { the Kneser-Poulsen conjecture in the plane, Journal fur die reine und angewandte Mathematik (2002), no. 553, 221 { 236.
10. U. Brehm, Extensions of distance reducing mappings to piecewise congruent mappings on Rm, J. Geom. 16 (1981), no. 2, 187{193. MR 642266
11. B. Dacorogna, P. Marcellini, and E. Paolini, Lipschitz-continuous local isometric immersions: rigid maps and origami, Journal de Mathematiques Pures et Appliques 90 (2008), no. 1, 66 { 81.
12. L. C. Evans and R. F. Gariepy, Measure theory and ne properties of functions; Rev. ed., Textbooks in mathematics, ch. 6, CRC Press, Oakville, 2015.
13. C. L. Fe erman, A sharp form of Whitney's extension theorem, Annals of Mathematics 161 (2005), no. 1, 509{577. MR 2150391
14. , Whitney's extension problem for Cm, Annals of Mathematics 164 (2006), no. 1, 313{359. MR 2233850
15. L.-A. Gottlieb and R. Krauthgamer, A nonlinear approach to dimension reduction, Weizmann Institute of Science.
16. M. Gromov, Monotonicity of the volume of intersection of balls, Geometrical Aspects of Functional Analysis (Berlin, Heidelberg) (J. Lindenstrauss and V. D. Milman, eds.), Springer Berlin Heidelberg, 1987, pp. 1{4.
17. R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Archive for Rational Mechanics and Analysis 123 (1993), no. 1, 51{74.
18. M. Kirszbraun,  Uber die zusammenziehende und Lipschitzsche Transformationen, Fundamenta Mathematicae 22 (1934), no. 1, 77{108 (ger).
19. M. Kneser, Einige Bemerkungen uber das Minkowskische Flachenma, Archiv der Mathematik 6 (1955), no. 5, 382{390.
20. E. Kopecka, Bootstrapping Kirszbraun's extension theorem, Fund. Math. 217 (2012), no. 1, 13{19. MR 2914919
21. U. Lang and V. Schroeder, Kirszbraun's theorem and metric spaces of bounded curvature, Geometric & Functional Analysis GAFA 7 (1997), no. 3, 535{560. MR 1466337
22. E. Le Gruyer, Minimal Lipschitz extensions to di erentiable functions de ned on a Hilbert space, Geometric and Functional Analysis 19 (2009), no. 4, 1101{1118. MR 2570317
23. J. Lee, Jl lemma and Kirszbraun's extension theorem, 2020, Sublinear Algorithms for Big Data Lectues Notes, Brown University.
24. J. R. Lee and A. Naor, Extending Lipschitz functions via random metric partitions, Inventiones mathematicae 160 (2005), no. 1, 59{95.
25. E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12, 837{842. MR 1562984
26. A. Naor, Probabilistic clustering of high dimensional norms, pp. 690{709. 

27. , Metric embeddings and Lipschitz extensions, Princeton University, Lecture Notes, 2015.
28. A. Naor, Y. Peres, O. Schramm, and S. Sheeld, Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces, Duke Math. J. 134 (2006), no. 1, 165{197.
29. E. T. Poulsen, Problem 10, Mathematica Scandinavica 2 (1954), 346.
30. S. Reich and S. Simons, Fenchel duality, Fitzpatrick functions and the Kirszbraun{Valentine extension theorem, Proceedings of the American Mathematical Society 133 (2005), no. 9, 2657{2660. MR 2146211
31. I. J. Schoenberg, On a Theorem of Kirzbraun and Valentine, The American Mathematical Monthly 60 (1953), no. 9, 620{622. MR 0058232
32. S. Sheeld and C. K. Smart, Vector-valued optimal Lipschitz extensions, Communications on Pure and Applied Mathematics 65 (2012), no. 1, 128{154. MR 2846639
33. E. Stein, Singular integrals and di erentiability properties of functions, ch. 6, Princeton University Press, 1970.
34. P. V. Than, Extensions lipschitziennes minimales, Ph.D. thesis, INSA de Rennes, 2015.
35. F. A. Valentine, A Lipschitz condition preserving extension for a vector function, Amer. J. Math. 67 (1945), 83{93. MR 0011702
36. J. H. Wells and L. R. Williams, Embeddings and extensions in analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete ; Bd. 84, Springer-Verlag, Berlin, 1975 (eng).
37. H. Whitney, Analytic extensions of di erentiable functions de ned in closed sets, Transactions of the American Mathematical Society 36 (1934), no. 1, 63{89. MR 1501735 

University of Oxford, Mathematical Institute and St John's College, Oxford, United Kingdom
E-mail address: Krzysztof.Ciosmak@maths.ox.ac.uk

The join button will be published on the right (Above the view all button) 30 minutes before the seminar starts (login required).

21 May 2021
10:00
to
12:00
Professor Radu Ignat

Further Information: 

3 x 2 hour Lectures via Zoom - see email from 17th May for details 

Aimed at: The course is addressed to postgraduate students, postdocs and other members of the Mathematical Institute. It is an introduction to concentration phenomenon around vortices in Ginzburg-Landau type problems. The aim is to present topological methods (based on Jacobian, winding number...) that allow for detection of vortices and computation of the interaction energy between them. The purpose of this course is to analyse vortex singularities appearing in Ginzburg-Landau type problems.

The lecture will be via Zoom and the link for Lecture 2 was sent out in an email on 17th May. 

 

 

 

 

Abstract

Lecture notes and the manuscript for Lecture 2.

19 May 2021
11:00
to
12:30
Dr. Krzysztof Ciosmak

Further Information: 

4 x 1.5 hour Lectures 

Aimed at: any DPhil students with interest in learning about extensions of functions. 

Suggested Pre-requisites: Suitable for OxPDE students, but also of interests to functional analysts, geometers, probabilists, numerical analysts and anyone who has a suitable level of prerequisite knowledge.

 

Abstract

Abstract. The aim of the course is to present several results on extensions of functions. Among the most important are Kirszbraun's and Whitney's theorems.
They provide powerful technical tools in many problems of analysis. One way to view these theorems is that they show that there exists an interpolation
of data with certain properties. In this context they are useful in computer science, e.g. in clustering of data (see e.g. [26, 23]) and in dimension reduction (see e.g. [15]).

1. Syllabus
Lecture 1. McShane's theorem [25], Kirszbraun's theorem [18, 31, 35], Kneser- Poulsen conjecture [19, 29, 16].
Lecture 2. Whitney's covering and associated partition of unity, Whitney's ex-tension theorem [37, 12, 33].
Lecture 3. Whitney's theorem { minimal Lipschitz extensions [22].
Lecture 4. Ball's extension theorem, Markov type and cotype [6].

2. Required mathematical background
Markov chains, Hilbert spaces, Banach spaces, metric spaces, Zorn lemma

3. Reading list
The reading list consists of all the papers cited above, lecture notes [27], and parts of books [36, 8].

4. Assesment
Students will be encouraged to give a short talk on a topic related to the content of the course. Suggested topics include:
(1) Brehm's theorem [10],
(2) continuity of Kirszbraun's extension theorem [20],
(3) Kirszbraun's theorem for Alexandrov spaces [21, 1],
(4) two-dimensional Kneser-Poulsen conjecture [9],
(5) origami [11],
(6) absolutely minimising Lipschitz extensions and in nity Laplacian [17, 32,
34, 2, 3, 5, 4],
(7) Fenchel duality and Fitzpatrich functions [30, 7],
(8) sharp form of Whitney's extension theorem [13],
(9) Whitney's extension theorem for Cm [14],
(10) Markov type and cotype calculation [27, 6, 28], 

(11) extending Lipschitz functions via random metric partitions [24, 27].

References
1. S. Alexander, V. Kapovitch, and A. Petrunin, Alexandrov meets Kirszbraun, 2017.
2. G. Aronsson, Minimization problems for the functional supx F(x; f(x); f0(x)), Ark. Mat. 6 (1965), no. 1, 33{53.
3. , Minimization problems for the functional supx F(x; f(x); f0(x))(ii), Ark. Mat. 6 (1966), no. 4-5, 409{431.
4. , Extension of functions satisfying lipschitz conditions, Ark. Mat. 6 (1967), no. 6, 551{561.
5. , Minimization problems for the functional supx F(x; f(x); f0(x))(iii), Ark. Mat. 7 (1969), no. 6, 509{512.
6. K. Ball, Markov chains, Riesz transforms and Lipschitz maps, Geometric & Functional Analysis GAFA 2 (1992), no. 2, 137{172.
7. H. Bauschke, Fenchel duality, Fitzpatrick functions and the extension of rmly nonexpansive mappings, Proceedings of the American Mathematical Society 135 (2007), no. 1, 135{139. MR 2280182
8. Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, Colloquium publications (American Mathematical Society) ; v. 48, American Mathematical Society, Providence, R.I., 2000 (eng).
9. K. Bezdek and R. Connelly, Pushing disks apart { the Kneser-Poulsen conjecture in the plane, Journal fur die reine und angewandte Mathematik (2002), no. 553, 221 { 236.
10. U. Brehm, Extensions of distance reducing mappings to piecewise congruent mappings on Rm, J. Geom. 16 (1981), no. 2, 187{193. MR 642266
11. B. Dacorogna, P. Marcellini, and E. Paolini, Lipschitz-continuous local isometric immersions: rigid maps and origami, Journal de Mathematiques Pures et Appliques 90 (2008), no. 1, 66 { 81.
12. L. C. Evans and R. F. Gariepy, Measure theory and ne properties of functions; Rev. ed., Textbooks in mathematics, ch. 6, CRC Press, Oakville, 2015.
13. C. L. Fe erman, A sharp form of Whitney's extension theorem, Annals of Mathematics 161 (2005), no. 1, 509{577. MR 2150391
14. , Whitney's extension problem for Cm, Annals of Mathematics 164 (2006), no. 1, 313{359. MR 2233850
15. L.-A. Gottlieb and R. Krauthgamer, A nonlinear approach to dimension reduction, Weizmann Institute of Science.
16. M. Gromov, Monotonicity of the volume of intersection of balls, Geometrical Aspects of Functional Analysis (Berlin, Heidelberg) (J. Lindenstrauss and V. D. Milman, eds.), Springer Berlin Heidelberg, 1987, pp. 1{4.
17. R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Archive for Rational Mechanics and Analysis 123 (1993), no. 1, 51{74.
18. M. Kirszbraun,  Uber die zusammenziehende und Lipschitzsche Transformationen, Fundamenta Mathematicae 22 (1934), no. 1, 77{108 (ger).
19. M. Kneser, Einige Bemerkungen uber das Minkowskische Flachenma, Archiv der Mathematik 6 (1955), no. 5, 382{390.
20. E. Kopecka, Bootstrapping Kirszbraun's extension theorem, Fund. Math. 217 (2012), no. 1, 13{19. MR 2914919
21. U. Lang and V. Schroeder, Kirszbraun's theorem and metric spaces of bounded curvature, Geometric & Functional Analysis GAFA 7 (1997), no. 3, 535{560. MR 1466337
22. E. Le Gruyer, Minimal Lipschitz extensions to di erentiable functions de ned on a Hilbert space, Geometric and Functional Analysis 19 (2009), no. 4, 1101{1118. MR 2570317
23. J. Lee, Jl lemma and Kirszbraun's extension theorem, 2020, Sublinear Algorithms for Big Data Lectues Notes, Brown University.
24. J. R. Lee and A. Naor, Extending Lipschitz functions via random metric partitions, Inventiones mathematicae 160 (2005), no. 1, 59{95.
25. E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12, 837{842. MR 1562984
26. A. Naor, Probabilistic clustering of high dimensional norms, pp. 690{709. 

27. , Metric embeddings and Lipschitz extensions, Princeton University, Lecture Notes, 2015.
28. A. Naor, Y. Peres, O. Schramm, and S. Sheeld, Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces, Duke Math. J. 134 (2006), no. 1, 165{197.
29. E. T. Poulsen, Problem 10, Mathematica Scandinavica 2 (1954), 346.
30. S. Reich and S. Simons, Fenchel duality, Fitzpatrick functions and the Kirszbraun{Valentine extension theorem, Proceedings of the American Mathematical Society 133 (2005), no. 9, 2657{2660. MR 2146211
31. I. J. Schoenberg, On a Theorem of Kirzbraun and Valentine, The American Mathematical Monthly 60 (1953), no. 9, 620{622. MR 0058232
32. S. Sheeld and C. K. Smart, Vector-valued optimal Lipschitz extensions, Communications on Pure and Applied Mathematics 65 (2012), no. 1, 128{154. MR 2846639
33. E. Stein, Singular integrals and di erentiability properties of functions, ch. 6, Princeton University Press, 1970.
34. P. V. Than, Extensions lipschitziennes minimales, Ph.D. thesis, INSA de Rennes, 2015.
35. F. A. Valentine, A Lipschitz condition preserving extension for a vector function, Amer. J. Math. 67 (1945), 83{93. MR 0011702
36. J. H. Wells and L. R. Williams, Embeddings and extensions in analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete ; Bd. 84, Springer-Verlag, Berlin, 1975 (eng).
37. H. Whitney, Analytic extensions of di erentiable functions de ned in closed sets, Transactions of the American Mathematical Society 36 (1934), no. 1, 63{89. MR 1501735 

University of Oxford, Mathematical Institute and St John's College, Oxford, United Kingdom
E-mail address: Krzysztof.Ciosmak@maths.ox.ac.uk

The join button will be published on the right (Above the view all button) 30 minutes before the seminar starts (login required).

18 May 2021
11:00
to
12:30
Dr. Krzysztof Ciosmak

Further Information: 

4 x 1.5 hour Lectures 

Aimed at: any DPhil students with interest in learning about extensions of functions. 

Suitable for OxPDE students, but also of interest to probabilists, functional analysts,  geometers and numerical analysts, and has a suitable level of prerequisites knowledge for people from those specialisms to join. 

Abstract

Abstract. The aim of the course is to present several results on extensions of functions. Among the most important are Kirszbraun's and Whitney's theorems.
They provide powerful technical tools in many problems of analysis. One way to view these theorems is that they show that there exists an interpolation
of data with certain properties. In this context they are useful in computer science, e.g. in clustering of data (see e.g. [26, 23]) and in dimension reduction (see e.g. [15]).

1. Syllabus
Lecture 1. McShane's theorem [25], Kirszbraun's theorem [18, 31, 35], Kneser- Poulsen conjecture [19, 29, 16].
Lecture 2. Whitney's covering and associated partition of unity, Whitney's ex-tension theorem [37, 12, 33].
Lecture 3. Whitney's theorem { minimal Lipschitz extensions [22].
Lecture 4. Ball's extension theorem, Markov type and cotype [6].

2. Required mathematical background
Markov chains, Hilbert spaces, Banach spaces, metric spaces, Zorn lemma

3. Reading list
The reading list consists of all the papers cited above, lecture notes [27], and parts of books [36, 8].

4. Assesment
Students will be encouraged to give a short talk on a topic related to the content of the course. Suggested topics include:
(1) Brehm's theorem [10],
(2) continuity of Kirszbraun's extension theorem [20],
(3) Kirszbraun's theorem for Alexandrov spaces [21, 1],
(4) two-dimensional Kneser-Poulsen conjecture [9],
(5) origami [11],
(6) absolutely minimising Lipschitz extensions and in nity Laplacian [17, 32,
34, 2, 3, 5, 4],
(7) Fenchel duality and Fitzpatrich functions [30, 7],
(8) sharp form of Whitney's extension theorem [13],
(9) Whitney's extension theorem for Cm [14],
(10) Markov type and cotype calculation [27, 6, 28], 

(11) extending Lipschitz functions via random metric partitions [24, 27].

 

References
1. S. Alexander, V. Kapovitch, and A. Petrunin, Alexandrov meets Kirszbraun, 2017.
2. G. Aronsson, Minimization problems for the functional supx F(x; f(x); f0(x)), Ark. Mat. 6 (1965), no. 1, 33{53.
3. , Minimization problems for the functional supx F(x; f(x); f0(x))(ii), Ark. Mat. 6 (1966), no. 4-5, 409{431.
4. , Extension of functions satisfying lipschitz conditions, Ark. Mat. 6 (1967), no. 6, 551{561.
5. , Minimization problems for the functional supx F(x; f(x); f0(x))(iii), Ark. Mat. 7 (1969), no. 6, 509{512.
6. K. Ball, Markov chains, Riesz transforms and Lipschitz maps, Geometric & Functional Analysis GAFA 2 (1992), no. 2, 137{172.
7. H. Bauschke, Fenchel duality, Fitzpatrick functions and the extension of rmly nonexpansive mappings, Proceedings of the American Mathematical Society 135 (2007), no. 1, 135{139. MR 2280182
8. Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, Colloquium publications (American Mathematical Society) ; v. 48, American Mathematical Society, Providence, R.I., 2000 (eng).
9. K. Bezdek and R. Connelly, Pushing disks apart { the Kneser-Poulsen conjecture in the plane, Journal fur die reine und angewandte Mathematik (2002), no. 553, 221 { 236.
10. U. Brehm, Extensions of distance reducing mappings to piecewise congruent mappings on Rm, J. Geom. 16 (1981), no. 2, 187{193. MR 642266
11. B. Dacorogna, P. Marcellini, and E. Paolini, Lipschitz-continuous local isometric immersions: rigid maps and origami, Journal de Mathematiques Pures et Appliques 90 (2008), no. 1, 66 { 81.
12. L. C. Evans and R. F. Gariepy, Measure theory and ne properties of functions; Rev. ed., Textbooks in mathematics, ch. 6, CRC Press, Oakville, 2015.
13. C. L. Fe erman, A sharp form of Whitney's extension theorem, Annals of Mathematics 161 (2005), no. 1, 509{577. MR 2150391
14. , Whitney's extension problem for Cm, Annals of Mathematics 164 (2006), no. 1, 313{359. MR 2233850
15. L.-A. Gottlieb and R. Krauthgamer, A nonlinear approach to dimension reduction, Weizmann Institute of Science.
16. M. Gromov, Monotonicity of the volume of intersection of balls, Geometrical Aspects of Functional Analysis (Berlin, Heidelberg) (J. Lindenstrauss and V. D. Milman, eds.), Springer Berlin Heidelberg, 1987, pp. 1{4.
17. R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Archive for Rational Mechanics and Analysis 123 (1993), no. 1, 51{74.
18. M. Kirszbraun,  Uber die zusammenziehende und Lipschitzsche Transformationen, Fundamenta Mathematicae 22 (1934), no. 1, 77{108 (ger).
19. M. Kneser, Einige Bemerkungen uber das Minkowskische Flachenma, Archiv der Mathematik 6 (1955), no. 5, 382{390.
20. E. Kopecka, Bootstrapping Kirszbraun's extension theorem, Fund. Math. 217 (2012), no. 1, 13{19. MR 2914919
21. U. Lang and V. Schroeder, Kirszbraun's theorem and metric spaces of bounded curvature, Geometric & Functional Analysis GAFA 7 (1997), no. 3, 535{560. MR 1466337
22. E. Le Gruyer, Minimal Lipschitz extensions to di erentiable functions de ned on a Hilbert space, Geometric and Functional Analysis 19 (2009), no. 4, 1101{1118. MR 2570317
23. J. Lee, Jl lemma and Kirszbraun's extension theorem, 2020, Sublinear Algorithms for Big Data Lectues Notes, Brown University.
24. J. R. Lee and A. Naor, Extending Lipschitz functions via random metric partitions, Inventiones mathematicae 160 (2005), no. 1, 59{95.
25. E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12, 837{842. MR 1562984
26. A. Naor, Probabilistic clustering of high dimensional norms, pp. 690{709. 

27. , Metric embeddings and Lipschitz extensions, Princeton University, Lecture Notes, 2015.
28. A. Naor, Y. Peres, O. Schramm, and S. Sheeld, Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces, Duke Math. J. 134 (2006), no. 1, 165{197.
29. E. T. Poulsen, Problem 10, Mathematica Scandinavica 2 (1954), 346.
30. S. Reich and S. Simons, Fenchel duality, Fitzpatrick functions and the Kirszbraun{Valentine extension theorem, Proceedings of the American Mathematical Society 133 (2005), no. 9, 2657{2660. MR 2146211
31. I. J. Schoenberg, On a Theorem of Kirzbraun and Valentine, The American Mathematical Monthly 60 (1953), no. 9, 620{622. MR 0058232
32. S. Sheeld and C. K. Smart, Vector-valued optimal Lipschitz extensions, Communications on Pure and Applied Mathematics 65 (2012), no. 1, 128{154. MR 2846639
33. E. Stein, Singular integrals and di erentiability properties of functions, ch. 6, Princeton University Press, 1970.
34. P. V. Than, Extensions lipschitziennes minimales, Ph.D. thesis, INSA de Rennes, 2015.
35. F. A. Valentine, A Lipschitz condition preserving extension for a vector function, Amer. J. Math. 67 (1945), 83{93. MR 0011702
36. J. H. Wells and L. R. Williams, Embeddings and extensions in analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete ; Bd. 84, Springer-Verlag, Berlin, 1975 (eng).
37. H. Whitney, Analytic extensions of di erentiable functions de ned in closed sets, Transactions of the American Mathematical Society 36 (1934), no. 1, 63{89. MR 1501735 

 

University of Oxford, Mathematical Institute and St John's College, Oxford, United Kingdom
E-mail address: Krzysztof.Ciosmak@maths.ox.ac.uk

 

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14 May 2021
10:00
to
12:00
Professor Radu Ignat

Further Information: 

3 x 2 hour Lectures via Zoom (see email on 10th May 2021 for details)

Aimed at: The course is addressed to postgraduate students, postdocs and other members of the Mathematical Institute. It is an introduction to concentration phenomenon around vortices in Ginzburg-Landau type problems. The aim is to present topological methods (based on Jacobian, winding number...) that allow for detection of vortices and computation of the interaction energy between them. The purpose of this course is to analyse vortex singularities appearing in Ginzburg-Landau type problems.

The lecture will be via Zoom and the link has also been emailed out separately on 10th May. 

 

 

 

Abstract

Abstractlecture notes and the manuscript for Lecture 1


References
[1] F. Bethuel, H. Brezis, F. Helein, Ginzburg-Landau vortices, Birkhauser, Boston, 1994.
[2] H. Brezis, L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries,
Selecta Math. (N.S.) 1 (1995), 197{263.
[3] R. Ignat, R.L. Jerrard, Renormalized energy between vortices in some Ginzburg-Landau models
on 2-dimensional Riemannian manifolds, Arch. Ration. Mech. Anal. 239 (2021), 1577{1666.
[4] R. Ignat, L. Nguyen, V. Slastikov, A. Zarnescu, On the uniqueness of minimisers of Ginzburg-
Landau functionals, Ann. Sci. Ec. Norm. Super. 53 (2020), 589{613.
[5] R.L. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals, SIAM J. Math. Anal.
30 (1999), 721-746.
[6] R.L. Jerrard, H.M. Soner, The Jacobian and the Ginzburg-Landau energy, Calc. Var. PDE 14
(2002), 151-191.
[7] E. Sandier, Lower bounds for the energy of unit vector elds and applications J. Funct. Anal.
152 (1998), 379-403.
[8] E. Sandier, S. Serfaty, Vortices in the magnetic Ginzburg-Landau model, Birkhauser, 2007.

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28 April 2021
10:00
to
11:30
Dr. Avi Mayorcas

Further Information: 

Structure: 4 x 1.5hr Lectures 

Lecture 4: Further Topics and Directions (time permitting)

  • Regularity of solutions
  • Ergodicity
  • Pathwise approach to SPDE

 

Abstract

The course will aim to provide an introduction to stochastic PDEs from the classical perspective, that being a mixture of stochastic analysis and PDE analysis. We will focus in particular on the variational approach to semi-linear parabolic problems, `a  la  Lions. There will also be comments on  other models and approaches.

  Suggested Pre-requisites: Suitable for OxPDE students, but also of interests to functional analysts, geometers, probabilists, numerical analysts and anyone who has a suitable level of prerequisite knowledge.

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27 April 2021
10:00
to
11:30
Dr. Avi Mayorcas

Further Information: 

Structure: 4 x 1.5hr Lectures 

Lecture 3: Variational Approach to Parabolic SPDE

  • Itˆo’s formula in Hilbert spaces
  • Variational approach to monotone, coercive SPDE
  • Concrete examples
Abstract

The course will aim to provide an introduction to stochastic PDEs from the classical perspective, that being a mixture of stochastic analysis and PDE analysis. We will focus in particular on the variational approach to semi-linear parabolic problems, `a  la  Lions. There will also be comments on  other models and approaches.

  Suggested Pre-requisites: The course is broadly aimed at graduate students with some knowledge of PDE theory and/or stochastic  analysis. Familiarity with measure theory and functional analysis will be useful.

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21 April 2021
10:00
to
11:30
Dr. Avi Mayorcas

Further Information: 

Structure: 4 x 1.5hr Lectures 

Lecture 2: Variational Approach to Deterministic PDE

  • Variational approach to linear parabolic equations
  • Variational approaches to non-linear parabolic equations
Abstract

The course will aim to provide an introduction to stochastic PDEs from the classical perspective, that being a mixture of stochastic analysis and PDE analysis. We will focus in particular on the variational approach to semi-linear parabolic problems, `a  la  Lions. There will also be comments on  other models and approaches.

  Suggested Pre-requisites: The course is broadly aimed at graduate students with some knowledge of PDE theory and/or stochastic  analysis. Familiarity with measure theory and functional analysis will be useful.

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