Categorical metric approximability and applications to Lagrangian topology,
Geometry seminar,
St. Andrews University, 28th November 2025.
In this talk I will introduce a new notion of approximability for metric spaces that can be seen as a categorification of a concept introduced by Turing for metric groups in 1938 and as a generalization of total-boundedness. Categorical metric approximability relies on the theory of triangulated persistence categories introduced by Biran-Cornea-Zhang and allows for the definition of refinements of classical measurements of complexity of objects of triangulated categories such as categorical entropy. I will discuss approximablity of spaces of Lagrangian submanifolds and present some examples. If time permits, I will discuss applications to rigidity and complexity of Lagrangians, as well as potential relations to open problems in Lagrangian topology. This talk is based on joint work with Paul Biran and Octav Cornea.
In this talk I will introduce a new notion of approximability for metric spaces that can be seen as a categorification of a concept introduced by Turing for metric groups and as a generalization of total-boundedness. I will explain how recent technological advances in symplectic topology and persistence category theory allow us to talk about approximablity of spaces of Lagrangian submanifolds and discuss applications to rigidity and complexity of Lagrangians, as well as potential relations to open problems in Lagrangian topology. This talk is based on joint work with Paul Biran and Octav Cornea.
In this talk I will introduce a new notion of approximability for metric spaces that can be seen as a categorification of a concept introduced by Turing for metric groups and as a generalization of total-boundedness. I will explain how recent technological advances in symplectic topology and persistence category theory allow us to talk about approximablity of spaces of Lagrangian submanifolds and discuss applications to rigidity and complexity of Lagrangians, as well as potential relations to open problems in Lagrangian topology. This talk is based on joint work with Paul Biran and Octav Cornea.
Approximability for Lagrangian submanifolds, Symplectic geometry seminar, Stanford University, 20th October 2025.
In this talk I will introduce a new notion of approximability for metric spaces that can be seen as a categorification of a concept introduced by Turing for metric groups and as a generalization of total-boundedness. I will explain how recent technological advances in symplectic topology and persistence category theory allow us to talk about approximablity of spaces of Lagrangian submanifolds and discuss applications to rigidity and complexity of Lagrangians, as well as potential relations to open problems in Lagrangian topology. This talk is based on joint work with Paul Biran and Octav Cornea.
It is known that Fukaya categories are not filtered $A_\infty$ categories for arbitrary choices of the parameters needed for their construction, but only weakly-filtered. In this talk we will present a trick to construct classes of such parameters so that the associated Fukaya categories are filtered. Then, we will discuss how different choices of parameters will affect the persistence structure of Fukaya categories at the derived level. If time permits we will show some applications of the filtered structures on Fukaya categories.
"Junior" seminar talks at ETH
Introduction to Fukaya TPCs: algebra and geometry, Junior symplectic geometry seminar, ETH Zürich, 25th March 2025.
Computing Floer homology can be hard, computing Fukaya categories harder. We will motivate why it is useful to find a (split-)generating subset of objects in Fukaya categories mainly based on examples. In particular, we discuss generation of the wrapped Fukaya category of a cotangent bundle and split-generation of the Fukaya category of the torus. Moreover, we present a split-generation criterion due to Abouzaid by first going back to a linear version of it due to Biran-Cornea.
We will introduce the concept of Hamiltonian spaces and using the toric Darboux theorem we will show that moment maps are a rich source of Morse-Bott functions. As an application we will present how these Morse-Bott functions can be used in the study of recursive properties of symplectic toric manifolds.
We will discuss the topological origin of A_infinity-algebras by looking at loop spaces: The failure of the concatenation map to be associative can be measured in a rigorous way by an infinite sequence of higher homotopies. We will see how this allows us to recognise loop spaces and how it leads to the notion of A_infinity-algebras. In the second part we will introduce pearly Floer cohomology for certain exact Lagrangian immersions and provide some examples of computations. At the end, we will discuss the product structure on pearly cohomology and mention how to obtain an A_infinity-algebra associated to a Lagrangian immersion.
