# Scientific

## String Topology 3

A lecture titled "String Topology" by Katherine Poirier, New York City College of Technology. This is the 3rd in a series of 3.

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.

- Read more about String Topology 3
- 332 reads

## Floer Homology Fundamentals 6

A lecture titled "Floer Homology Fundamentals" by Nate Bottman, Max Planck. This is the 6th 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.

## String Topology 2

A lecture titled "String Topology" by Katherine Poirier, New York City College of Technology. This is the 2nd in a series of 3.

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.

- Read more about String Topology 2
- 447 reads

## Floer Homology Fundamentals 4

A lecture titled "Floer Homology Fundamentals" by Nate Bottman, Max Planck. This is the 4th in a series of 9.

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.

## Floer Homology Fundamentals 2

A lecture titled "Floer Homology Fundamentals" by Nate Bottman, Max Planck. This is the 2nd in a series of 9.

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.

## Floer Homology Fundamentals 1

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

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.

## 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.