# Scientific

## Floer Homology Fundamentals 1

A lecture titled "Floer Homology Fundamentals"by Catherine Cannizzo, SCGP. This is the 1st in a series of 9.

General Description:

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

The goal of the summer school is to provide participants the tools in symplectic geometry and stable homotopy theory required to work on Floer homotopy theory. Students will come away with a basic understanding of some of the key techniques, questions, and challenges in both of these fields. The summer school may be particularly valuable for participants with a solid understanding of one of the two fields who want to learn more about the other and the connections between them.

## Multiple mixing for SL(2,R)

We present a special case of an argument of Mozes that mixing implies mixing of all orders for certain Lie groups.

## Recent results in interface motions in the framework of optimal transport: Lecture 3

In the lectures we will discuss recent results obtained on interface motions in the framework of optimal transport. We intend to (time allowing) discuss the following problems:

The Hele-Shaw type flows in the context of tumor growth. Here the flow describe the growth of tumor cells with contact inhibition. The tumor cells then form a congested zone, which evolves by the pressure generated by the constraint on maximal density. We start with a simple mechanical model, and discuss the effects of nutrients and surface tension in the context of minimizing movements. While the well-posedness would be established by minimizing movements, we will also explore qualitative properties of solutions such as regularity of the interface.

The Stefan problem, in the framework of optimal stopping time. Our focus will be on the well-posedness of the supercooled Stefan problem, which describes freezing of supercooled fluid. The interface between the fluid and ice, as it freezes, exhibits a high degree of irregularity. Our goal is to introduce a notion of solutions that are physically meaningful and stable. We will start with a quick introduction of the necessary background on the optimal stopping time between probability measures. We will establish the well-posedness, and discuss qualitative behavior of solutions.

## Recent results in interface motions in the framework of optimal transport: Lecture 2

In the lectures we will discuss recent results obtained on interface motions in the framework of optimal transport. We intend to (time allowing) discuss the following problems:

The Hele-Shaw type flows in the context of tumor growth. Here the flow describe the growth of tumor cells with contact inhibition. The tumor cells then form a congested zone, which evolves by the pressure generated by the constraint on maximal density. We start with a simple mechanical model, and discuss the effects of nutrients and surface tension in the context of minimizing movements. While the well-posedness would be established by minimizing movements, we will also explore qualitative properties of solutions such as regularity of the interface.

The Stefan problem, in the framework of optimal stopping time. Our focus will be on the well-posedness of the supercooled Stefan problem, which describes freezing of supercooled fluid. The interface between the fluid and ice, as it freezes, exhibits a high degree of irregularity. Our goal is to introduce a notion of solutions that are physically meaningful and stable. We will start with a quick introduction of the necessary background on the optimal stopping time between probability measures. We will establish the well-posedness, and discuss qualitative behavior of solutions.

## Recent results in interface motions in the framework of optimal transport: Lecture 1

In the lectures we will discuss recent results obtained on interface motions in the framework of optimal transport. We intend to (time allowing) discuss the following problems:

The Hele-Shaw type flows in the context of tumor growth. Here the flow describe the growth of tumor cells with contact inhibition. The tumor cells then form a congested zone, which evolves by the pressure generated by the constraint on maximal density. We start with a simple mechanical model, and discuss the effects of nutrients and surface tension in the context of minimizing movements. While the well-posedness would be established by minimizing movements, we will also explore qualitative properties of solutions such as regularity of the interface.

The Stefan problem, in the framework of optimal stopping time. Our focus will be on the well-posedness of the supercooled Stefan problem, which describes freezing of supercooled fluid. The interface between the fluid and ice, as it freezes, exhibits a high degree of irregularity. Our goal is to introduce a notion of solutions that are physically meaningful and stable. We will start with a quick introduction of the necessary background on the optimal stopping time between probability measures. We will establish the well-posedness, and discuss qualitative behavior of solutions.

## A variational approach to the regularity theory for optimal transportation: Lecture 3

In this mini-course, we shall explain the variational approach to regularity

theory for optimal transportation introduced in [8]. This approach does

completely bypass the celebrated regularity theory of Caffarelli [2], which is

based on the regularity theory for the Monge-Amp ere equation as a fully

nonlinear elliptic equation with a comparison principle. Nonetheless, one

recovers the same partial regularity theory [5, 4].

The advantage of the variational approach resides in its robustness regarding

the regularity of the measures, which can be arbitrary measures [7][Theorem

1.4], and in terms of the problem formulation, e.g. by its extension to almost

minimizers [10]. The former for instance is crucial in order to tackle the

widely popular matching problem [3, 1] e.g. the optimal transportation between

(random) point clouds, as carried out in [7, 6, 9]. The latter is convenient

when treating more general than square Euclidean cost functions.

The variational approach follows de Giorgi’s philosophy for minimal surfaces.

At its core is the approximation of the displacement by the gradient of a

harmonic function. This approximation is based on the Eulerian formulation of

optimal transportation, which reveals its strict convexity and the proximity to

the $H^{-1}$-norm. In this mini-course, we shall give a pretty self-contained

derivation of this harmonic approximation result, and establish applications to

the matching problem.

#### References

- [1] L. Ambrosio, F. Stra, D. Trevisan: A PDE approach to a 2-dimensional

matching problem. Probab. Theory Relat. Fields 173, 433–477 (2019). - [2] L.A. Caffarelli: The regularity of mappings with a convex potential.

Journal of the American Mathematical Society 5 (1992), no. 1, 99–104. - [3] S. Caracciolo, C. Lucibello, G. Parisi, G. Sicuro: Scaling hypothesis for

the Euclidean bipartite matching problem. Physical Review E, 90(1), 2014. - [4] G. De Philippis, A. Figalli: Partial regularity for optimal transport

maps. Publications Mathématiques. Institut de Hautes Études Scientifiques

121 (2015), 81–112. - [5] A. Figalli, Y.-H. Kim: Partial regularity of Brenier solutions of the

Monge-Amépre equation. Discrete and Continuous Dynamical Systems (Series A)

28 (2010), 559–565. - [6] M. Goldman, M. Huesmann: A fluctuation result for the displacement in the

optimal matching problem. arXiv e-prints, May 2021. arXiv:2105.02915. - [7] M. Goldman, M. Huesmann, F. Otto: Quantitative linearization results for

the Monge-Amp`ere equation. Communications on Pure and Applied Mathematics

(2021). - [8] M. Goldman, F. Otto: A variational proof of partial regularity for optimal

transportation maps. Annales Scientifiques de l’Ećole Normale Supérieure.

Quatriéme Série 53 (2020), no. 5, 1209–1233. - [9] M. Huesmann, F. Mattesini, F. Otto: There is no stationary cyclically

monotone Poisson matching in 2d. arXiv e-prints, September 2021.

arXiv:2109.13590. - [10] F. Otto, M. Prod’homme, T. Ried: Variational approach to regularity of

optimal transport maps: general cost functions. (English summary) Ann. PDE 7

(2021), no. 2, Paper No. 17, 74 pp.

## A variational approach to the regularity theory for optimal transportation: Lecture 2

In this mini-course, we shall explain the variational approach to regularity

theory for optimal transportation introduced in [8]. This approach does

completely bypass the celebrated regularity theory of Caffarelli [2], which is

based on the regularity theory for the Monge-Amp ere equation as a fully

nonlinear elliptic equation with a comparison principle. Nonetheless, one

recovers the same partial regularity theory [5, 4].

The advantage of the variational approach resides in its robustness regarding

the regularity of the measures, which can be arbitrary measures [7][Theorem

1.4], and in terms of the problem formulation, e.g. by its extension to almost

minimizers [10]. The former for instance is crucial in order to tackle the

widely popular matching problem [3, 1] e.g. the optimal transportation between

(random) point clouds, as carried out in [7, 6, 9]. The latter is convenient

when treating more general than square Euclidean cost functions.

The variational approach follows de Giorgi’s philosophy for minimal surfaces.

At its core is the approximation of the displacement by the gradient of a

harmonic function. This approximation is based on the Eulerian formulation of

optimal transportation, which reveals its strict convexity and the proximity to

the $H^{-1}$-norm. In this mini-course, we shall give a pretty self-contained

derivation of this harmonic approximation result, and establish applications to

the matching problem.

#### References

- [1] L. Ambrosio, F. Stra, D. Trevisan: A PDE approach to a 2-dimensional

matching problem. Probab. Theory Relat. Fields 173, 433–477 (2019). - [2] L.A. Caffarelli: The regularity of mappings with a convex potential.

Journal of the American Mathematical Society 5 (1992), no. 1, 99–104. - [3] S. Caracciolo, C. Lucibello, G. Parisi, G. Sicuro: Scaling hypothesis for

the Euclidean bipartite matching problem. Physical Review E, 90(1), 2014. - [4] G. De Philippis, A. Figalli: Partial regularity for optimal transport

maps. Publications Mathématiques. Institut de Hautes Études Scientifiques

121 (2015), 81–112. - [5] A. Figalli, Y.-H. Kim: Partial regularity of Brenier solutions of the

Monge-Amépre equation. Discrete and Continuous Dynamical Systems (Series A)

28 (2010), 559–565. - [6] M. Goldman, M. Huesmann: A fluctuation result for the displacement in the

optimal matching problem. arXiv e-prints, May 2021. arXiv:2105.02915. - [7] M. Goldman, M. Huesmann, F. Otto: Quantitative linearization results for

the Monge-Amp`ere equation. Communications on Pure and Applied Mathematics

(2021). - [8] M. Goldman, F. Otto: A variational proof of partial regularity for optimal

transportation maps. Annales Scientifiques de l’Ećole Normale Supérieure.

Quatriéme Série 53 (2020), no. 5, 1209–1233. - [9] M. Huesmann, F. Mattesini, F. Otto: There is no stationary cyclically

monotone Poisson matching in 2d. arXiv e-prints, September 2021.

arXiv:2109.13590. - [10] F. Otto, M. Prod’homme, T. Ried: Variational approach to regularity of

optimal transport maps: general cost functions. (English summary) Ann. PDE 7

(2021), no. 2, Paper No. 17, 74 pp.

## A variational approach to the regularity theory for optimal transportation: Lecture 1

In this mini-course, we shall explain the variational approach to regularity

theory for optimal transportation introduced in [8]. This approach does

completely bypass the celebrated regularity theory of Caffarelli [2], which is

based on the regularity theory for the Monge-Amp ere equation as a fully

nonlinear elliptic equation with a comparison principle. Nonetheless, one

recovers the same partial regularity theory [5, 4].

The advantage of the variational approach resides in its robustness regarding

the regularity of the measures, which can be arbitrary measures [7][Theorem

1.4], and in terms of the problem formulation, e.g. by its extension to almost

minimizers [10]. The former for instance is crucial in order to tackle the

widely popular matching problem [3, 1] e.g. the optimal transportation between

(random) point clouds, as carried out in [7, 6, 9]. The latter is convenient

when treating more general than square Euclidean cost functions.

The variational approach follows de Giorgi’s philosophy for minimal surfaces.

At its core is the approximation of the displacement by the gradient of a

harmonic function. This approximation is based on the Eulerian formulation of

optimal transportation, which reveals its strict convexity and the proximity to

the $H^{-1}$-norm. In this mini-course, we shall give a pretty self-contained

derivation of this harmonic approximation result, and establish applications to

the matching problem.

#### References

- [1] L. Ambrosio, F. Stra, D. Trevisan: A PDE approach to a 2-dimensional

matching problem. Probab. Theory Relat. Fields 173, 433–477 (2019). - [2] L.A. Caffarelli: The regularity of mappings with a convex potential.

Journal of the American Mathematical Society 5 (1992), no. 1, 99–104. - [3] S. Caracciolo, C. Lucibello, G. Parisi, G. Sicuro: Scaling hypothesis for

the Euclidean bipartite matching problem. Physical Review E, 90(1), 2014. - [4] G. De Philippis, A. Figalli: Partial regularity for optimal transport

maps. Publications Mathématiques. Institut de Hautes Études Scientifiques

121 (2015), 81–112. - [5] A. Figalli, Y.-H. Kim: Partial regularity of Brenier solutions of the

Monge-Amépre equation. Discrete and Continuous Dynamical Systems (Series A)

28 (2010), 559–565. - [6] M. Goldman, M. Huesmann: A fluctuation result for the displacement in the

optimal matching problem. arXiv e-prints, May 2021. arXiv:2105.02915. - [7] M. Goldman, M. Huesmann, F. Otto: Quantitative linearization results for

the Monge-Amp`ere equation. Communications on Pure and Applied Mathematics

(2021). - [8] M. Goldman, F. Otto: A variational proof of partial regularity for optimal

transportation maps. Annales Scientifiques de l’Ećole Normale Supérieure.

Quatriéme Série 53 (2020), no. 5, 1209–1233. - [9] M. Huesmann, F. Mattesini, F. Otto: There is no stationary cyclically

monotone Poisson matching in 2d. arXiv e-prints, September 2021.

arXiv:2109.13590. - [10] F. Otto, M. Prod’homme, T. Ried: Variational approach to regularity of

optimal transport maps: general cost functions. (English summary) Ann. PDE 7

(2021), no. 2, Paper No. 17, 74 pp.

## Optimal Transport for Machine Learning: Lecture 3

Optimal transport (OT) has recently gained lot of interest in machine learning. It is a natural tool to compare in a geometrically faithful way probability distributions. It finds applications in both supervised learning (using geometric loss functions) and unsupervised learning (to perform generative model fitting). OT is however plagued by the curse of dimensionality, since it might require a number of samples which grows exponentially with the dimension. In this course, I will explain how to leverage entropic regularization methods to define computationally efficient loss functions, approximating OT with a better sample complexity. More information and references can be found on the website of our book “Computational Optimal Transport”.

## Optimal Transport for Machine Learning: Lecture 2

Optimal transport (OT) has recently gained lot of interest in machine learning. It is a natural tool to compare in a geometrically faithful way probability distributions. It finds applications in both supervised learning (using geometric loss functions) and unsupervised learning (to perform generative model fitting). OT is however plagued by the curse of dimensionality, since it might require a number of samples which grows exponentially with the dimension. In this course, I will explain how to leverage entropic regularization methods to define computationally efficient loss functions, approximating OT with a better sample complexity. More information and references can be found on the website of our book “Computational Optimal Transport”.