Past OxPDE Short Courses

The aim of this course is to give an introduction to the regularity theory of non-smooth spaces with lower bounds on the Ricci Curvature. This is a quickly developing field with motivations coming from classical questions in Riemannian and differential geometry and with connections to Probability, Geometric Measure Theory and Partial Differential Equations.

In the lectures we will focus on the non collapsed case, where much sharper results are available, mainly adopting the synthetic approach of the RCD theory, rather than following the original proofs.

The techniques used in this setting have been applied successfully in the study of Minimal surfaces, Elliptic PDEs, Mean curvature flow and Ricci flow and the course might be of interest also for people working in these subjects.

Aimed: An introduction to the nonlinear theory of hyperbolic PDEs, as well as its close connections with the other areas of mathematics and wide range of applications in the sciences.

Courserequirements: Basicmathematicalanalysis.

Examination and grading: The exam will consist in the presentation of some previously as- signed article or book chapter (of course the student must show a good knowledge of those issues taught during the course which are connected with the presentation.).

SSD: MAT/05 Mathematical Analysis
Aim: to make students aware of smooth and non-smooth controllability results and of some

applications in various fields of Mathematics and of technology as well.

Course contents:

Vector fields are basic ingredients in many classical issues of Mathematical Analysis and its applications, including Dynamical Systems, Control Theory, and PDE’s. Loosely speaking, controllability is the study of the points that can be reached from a given initial point through concatenations of trajectories of vector fields belonging to a given family. Classical results will be stated and proved, using coordinates but also underlying possible chart-independent interpretation. We will also discuss the non smooth case, including some issues which involve Lie brackets of nonsmooth vector vector fields, a subject of relatively recent interest.

Bibliography: Lecture notes written by the teacher.

Aimed: An introduction to the nonlinear theory of hyperbolic PDEs, as well as its close connections with the other areas of mathematics and wide range of applications in the sciences.

Aimed: An introduction to the nonlinear theory of hyperbolic PDEs, as well as its close connections with the other areas of mathematics and wide range of applications in the sciences.

Aimed: An introduction to the nonlinear theory of hyperbolic PDEs, as well as its close connections with the other areas of mathematics and wide range of applications in the sciences.

Many PDE models encode fundamental physical, geometric and topological structures. These structures may be lost in discretisations, and preserving them on the discrete level is crucial for the stability and efficiency of numerical methods. The finite element exterior calculus (FEEC) is a framework for constructing and analysing structure-preserving numerical methods for PDEs with ideas from topology, homological algebra and the Hodge theory. 

In this seminar, we present the theory and applications of FEEC. This includes analytic results (Hodge decomposition, regular potentials, compactness etc.), Hodge-Laplacian problems and their structure-preserving finite element discretisation, and applications in electromagnetism, fluid and solid mechanics. Knowledge on geometry and topology is not required as prerequisites.

References:

1. Arnold, D.N.: Finite Element Exterior Calculus. SIAM (2018) 

2. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1 (2006) 

3. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bulletin of the American Mathematical Society 47(2), 281–354 (2010) 

4. Arnold, D.N., Hu, K.: Complexes from complexes. Foundations of Computational Mathematics (2021)

Many PDE models encode fundamental physical, geometric and topological structures. These structures may be lost in discretisations, and preserving them on the discrete level is crucial for the stability and efficiency of numerical methods. The finite element exterior calculus (FEEC) is a framework for constructing and analysing structure-preserving numerical methods for PDEs with ideas from topology, homological algebra and the Hodge theory. 

In this seminar, we present the theory and applications of FEEC. This includes analytic results (Hodge decomposition, regular potentials, compactness etc.), Hodge-Laplacian problems and their structure-preserving finite element discretisation, and applications in electromagnetism, fluid and solid mechanics. Knowledge on geometry and topology is not required as prerequisites.

References:

1. Arnold, D.N.: Finite Element Exterior Calculus. SIAM (2018) 

2. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1 (2006) 

3. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bulletin of the American Mathematical Society 47(2), 281–354 (2010) 

4. Arnold, D.N., Hu, K.: Complexes from complexes. Foundations of Computational Mathematics (2021)

Many PDE models encode fundamental physical, geometric and topological structures. These structures may be lost in discretisations, and preserving them on the discrete level is crucial for the stability and efficiency of numerical methods. The finite element exterior calculus (FEEC) is a framework for constructing and analysing structure-preserving numerical methods for PDEs with ideas from topology, homological algebra and the Hodge theory. 

In this seminar, we present the theory and applications of FEEC. This includes analytic results (Hodge decomposition, regular potentials, compactness etc.), Hodge-Laplacian problems and their structure-preserving finite element discretisation, and applications in electromagnetism, fluid and solid mechanics. Knowledge on geometry and topology is not required as prerequisites.

References:

1. Arnold, D.N.: Finite Element Exterior Calculus. SIAM (2018) 

2. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1 (2006) 

3. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bulletin of the American Mathematical Society 47(2), 281–354 (2010) 

4. Arnold, D.N., Hu, K.: Complexes from complexes. Foundations of Computational Mathematics (2021)

Many PDE models encode fundamental physical, geometric and topological structures. These structures may be lost in discretisations, and preserving them on the discrete level is crucial for the stability and efficiency of numerical methods. The finite element exterior calculus (FEEC) is a framework for constructing and analysing structure-preserving numerical methods for PDEs with ideas from topology, homological algebra and the Hodge theory. 

In this seminar, we present the theory and applications of FEEC. This includes analytic results (Hodge decomposition, regular potentials, compactness etc.), Hodge-Laplacian problems and their structure-preserving finite element discretisation, and applications in electromagnetism, fluid and solid mechanics. Knowledge on geometry and topology is not required as prerequisites.

References:

1. Arnold, D.N.: Finite Element Exterior Calculus. SIAM (2018) 

2. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1 (2006) 

3. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bulletin of the American Mathematical Society 47(2), 281–354 (2010) 

4. Arnold, D.N., Hu, K.: Complexes from complexes. Foundations of Computational Mathematics (2021)

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 innity 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. Feerman, 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 dierentiable functions dened 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 dierentiability 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 dierentiable functions dened 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: @email

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 innity 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. Feerman, 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 dierentiable functions dened 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 dierentiability 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 dierentiable functions dened 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: @email

Lecture notes and the manuscript for Lecture 2.

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 innity 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. Feerman, 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 dierentiable functions dened 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 dierentiability 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 dierentiable functions dened 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: @email

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 innity 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. Feerman, 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 dierentiable functions dened 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 dierentiability 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 dierentiable functions dened 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: @email

Abstract, lecture 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.

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.

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.

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.

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.

Lecture 1:  Introduction and Preliminaries

• Regularity of solutions
• Ergodicity
• Pathwise approach to SPDE

Literature: [Hai09, DKM⁺09, DPZ96, Hai14, GIP15]

References

[DKM⁺09] Robert Dalang, Davar Khoshnevisan, Carl Mueller, David Nualart, and Yimin Xiao. A minicourse on stochastic partial differential equations, vol- ume 1962 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2009.

[DPZ96] G. Da Prato and J. Zabczyk. Ergodicity for Infinite Dimensional Systems. London Mathematical Society Lecture Note Series. Cambridge University Press, 1996.

[DPZ14] Giuseppe Da Prato and Jerzy Zabczyk. Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2 edition, 2014.

[Eva10] Lawrence Craig Evans. Partial Differential Equations. American Mathe- matical Society, 2010.

[GIP15] Massimiliano Gubinelli, Peter Imkeller, and Nicolas Perkowski. Paracon- trolled distributions and singular PDEs. Forum Math. Pi, 3:75, 2015.

[Hai09]  Martin Hairer.  An Introduction to Stochastic PDEs.  Technical  report, The University of Warwick / Courant Institute, 2009. Available at: http://hairer.org/notes/SPDEs.pdf

[Hai14] M. Hairer. A theory of regularity structures. Inventiones mathematicae, 198(2):269–504, 2014.

[Par07] Etienne  Pardoux. Stochastic  partial  differential  equations.  https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.405.4805&rep=rep1&type=pdf  2007.

[PR07] Claudia Pr´evˆot and Michael R¨ockner. A concise course on stochastic partial differential equations. Springer, 2007.

The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.

Prof. Franco Rampazzo ‘Mathematical Control Theory’ (Department of Mathematics of the University of Padova, as part of Oxford Padova connection) TT 2021
Aimed at: Any DPhil students with interest in learning about Mathematical Control Theory
Course Length:     24 hours total (to be in English) 
Dates and Times:  starts 2 March 2021