Tag - Differential geometry

Bertrand Toën: Geometric quantization for shifted symplectic structures

4th July, 2024

The purpose of this talk is to present an ongoing work on geometric quantization in the setting of shifted symplectic structures. I will start by recalling the various notions involved as well as the results previously obtained by James Wallbridge, who constructed the prequantized (higher) categories of a given integral shifted symplectic structure. I will then explain our main result so far: the construction of the shifted analogues of the Kostant–Souriau prequantum operators, which will be realized as a "Poisson module over a Poisson category" (a categorification of the notion of a Poisson module over a Poisson algebra). This will be obtained by means of deformation theory arguments for categories of sheaves in the setting of (derived) differential geometry. If time permits, I will discuss further aspects associated to the notion of polarizations of shifted symplectic structures.

Sahana Vasudevan: Triangulated Surfaces in Moduli Space

29th April, 2024

Triangulated surfaces are Riemann surfaces formed by gluing together equilateral triangles. They are also the Riemann surfaces defined over the algebraic numbers. Brooks, Makover, Mirzakhani and many others proved results about the geometric properties of random large genus triangulated surfaces, and similar results about the geometric properties of random large genus hyperbolic surfaces. These results motivated the question: how are triangulated surfaces distributed in the moduli space of Riemann surfaces, quantitatively? I will talk about results related to this question.

Rudolf Zeidler: Metric inequalities under lower scalar curvature bounds

25th March, 2024

We will explain geometric situations where a lower bound on the scalar curvature of a Riemannian manifold leads to quantitative distance estimates and rigidity results. The study of these has been prompted by several conjectures of Gromov from the recent years. Intuitively, these results can be seen as analogues for scalar curvature of comparison geometry statements such as the Bonnet-Myers theorem for Ricci curvature. However, unlike classical comparison geometry involving stronger curvature conditions, such results for scalar curvature typically rely on an additional topological assumption such as the non-existence of positive scalar curvature metrics on certain submanifolds. Along the way we will thus also provide a brief introduction to obstructions to the existence of positive scalar curvature metrics on closed manifolds.

Peng Lu: Conformal Bach flow

10th August, 2023

We introduce conformal Bach flow and establish its well-posedness on closed manifolds. We also obtain its backward uniqueness. To give an attempt to study the long-time behaviour of conformal Bach flow, assuming that the curvature and the pressure function are bounded, global and local Shi's type L2-estimate of derivatives of curvatures are derived. To make the talk more accessible, we will spend some time to survey on high-order parabolic curvature flow.

Michael Entov: Kähler-type and Tame Embeddings of Balls into Symplectic Manifolds

5th May, 2023

A symplectic embedding of a disjoint union of domains into a symplectic manifold M is said to be of Kähler type (respectively tame) if it is holomorphic with respect to some (not a priori fixed) integrable complex structure on M which is compatible with (respectively tamed by) the symplectic form. I'll discuss when Kähler-type embeddings of disjoint unions of balls into a closed symplectic manifold exist and when two such embeddings can be mapped into each other by a symplectomorphism. If time permits, I'll also discuss the existence of tame embeddings of balls, polydisks and parallelepipeds into tori and K3 surfaces.

Yi Wang: Yamabe flow of asymptotically flat metrics

10th October, 2022

In this talk, we will discuss the behaviour of the Yamabe flow on an asymptotically flat (AF) manifold. We will first show the long-time existence of the Yamabe flow starting from an AF manifold and discuss the uniform estimates on manifolds with positive Yamabe constant. This would allow us to prove global weighted convergence along the Yamabe flow on such manifolds. We also prove that the flow will diverge if the Yamabe constant is non-positive. It turns out that the blowup behaviour of the flow starting from manifolds with non-positive Yamabe constant is explicit. We will discuss some further work in this direction.

Tashi Walde: Higher Segal spaces via Higher Excision

20th May, 2021

Starting from the classical Segal spaces, Dyckerhoff and Kapranov introduced a hierarchy of what they call higher Segal structures. While the first new level (2-Segal spaces) has been well studied in recent years, not much is known about the higher levels and the hierarchy as a whole.

In this talk I explain how this hierarchy can be understood conceptually in close analogy to the manifold calculus of Goodwillie and Weiss. I describe a natural 'discrete manifold calculus' on the simplex category and on the cyclic category, for which the polynomial functors are precisely the higher Segal objects. Furthermore, this perspective yields intrinsic categorical characterizations of higher Segal objects in the spirit of higher excision.

Michael Ching: Tangent ∞-categories and Goodwillie calculus

8th April, 2021

In 1984 Rosický introduced tangent categories in order to capture axiomatically some properties of the tangent bundle functor on the category of smooth manifolds and smooth maps. Starting in 2014 Cockett and Cruttwell have developed this theory in more detail to emphasize connections with cartesian differential categories and other contexts arising from computer science and logic.

In this talk I will discuss joint work with Kristine Bauer and Matthew Burke which extends the notion of tangent category to ∞-categories. To make this generalization we use a characterization by Leung of tangent categories as modules over a symmetric monoidal category of Weil-algebras and algebra homomorphisms. Our main example of a tangent ∞-category is based on Lurie's model for the tangent bundle to an ∞-category itself. Thus we show that there is a tangent structure on the ∞-category of (differentiable) ∞-categories. This tangent structure encodes all the higher derivative information in Goodwillie’s calculus of functors, and sets the scene for further applications of ideas from differential geometry to higher category theory.

Asaf Yekutieli: Triple Product of Maass Forms

20th December, 2020

Maass forms are a particular class of smooth functions defined on a hyperbolic Riemann surface. In the special case of Riemann surfaces associated with a congruence subgroup, it is often the case that results concerning Maass forms bear witness to the existence of profound arithmetic relations. Our main goal is to describe the problem of estimating the triple product functional, explain its significance, and illustrate the representation theoretical techniques employed by Bernstein and Reznikov to make progress. If time permits, we shall discuss non-Archimedean instances of the above theory. I will not be assuming familiarity with any of the abovementioned notions.