## Search

## Topological Quantum Field Theory

Invariants of $4$-manifolds from Khovanov-Rozansky link homology.

*Paul Wedrich*, Max Planck Institute and University of Bonn.

## Abstract

Ribbon categories are $3$-dimensional algebraic structures that control quantum link polynomials and that give rise to $3$-manifold invariants known as skein modules. I will describe how to use Khovanov-Rozansky link homology, a categorification of the $\operatorname{\mathfrak{gl}}(N)$ quantum link polynomial, to obtain a $4$-dimensional algebraic structure that gives rise to vector space-valued invariants of smooth $4$-manifolds. The technical heart of this construction is the newly established functoriality of Khovanov-Rozansky homology in the $3$-sphere. Based on joint work with Scott Morrison and Kevin Walker https://arxiv.org/abs/1907.12194.

## Colloquium

Imaginary time flows to reality.

*João Pimentel Nunes*, Instituto Superior Técnico, CAMGSD, Universidade de Lisboa.

## Abstract

The mathematical expressions of the idea of *quantization* are a source of rich interdisciplinary relations between different areas in geometry and other subjects such as analysis and representation theory. In this colloquium, after a gentle description of some structures of modern geometry and of the quantization problem, we will describe “flows in imaginary time” and will give an idea of their role in quantization (in particular, in so-called **real** polarizations) and Kahler geometry. Finally, we will give a light description of some recent results.

## Mathematical Relativity

Theoretical cosmology.

*Alan Coley*, Dalhousie University.

## Abstract

This will be an informal talk on open problems in theoretical cosmology, suitable for a (relatively) general audience.

## String Theory

Integrability in and beyond AdS/CFT.

*João Caetano*, Simons Center for Geometry and Physics.

## Abstract

In this talk, I am going to review some aspects of the current state of the art of Integrability in the AdS/CFT correspondence and beyond. I will first review a general nonperturbative approach to compute multipoint correlation functions of local operators in the $N=4$ SYM theory which allows us to explore the theory even beyond the planar level. In the second part, I will describe my recent work about exploring deformations of $N=4$ SYM by irrelevant operators, which revives an old attempt of generalizing the AdS/CFT correspondence. Here integrability seems to also play an important role and opens the door for its application for non-conformal field theories.

## Partial Differential Equations

Compensated compactness and under constant rank constraints.

*André Guerra*, University of Oxford.

## Abstract

In this talk we will discuss the fundamental aspects of the theory of compensated compactness in the general A-free framework developed by Murat and Tartar, under the assumption that A has constant rank. We prove sharp weak continuity results for the compensated compactness quantities and we show that they are precisely the nonlinear quantities with Hardy space integrability, thus proposing an answer to a question raised by Coifman-Lions-Meyer-Semmes. This gives a link between compensated compactness and compensated regularity.

## Analysis, Geometry, and Dynamical Systems

From the porous medium model to the porous medium equation.

*Renato de Paula*, Instituto Superior Técnico.

## Abstract

The aim of this seminar is to present an overview of the porous medium model and its hydrodynamic equation, the porous medium equation. We will focus on exploring the main characteristics of this equation and how we can see it from the particle system's point of view.

## Functional Analysis, Linear Structures and Applications

$C^\ast$-algebras of Bergman type operators with piecewise slowly oscillating coefficients.

*Yuri Karlovich*, Universidad Autónoma del Estado de Morelos.

## Abstract

Given a simply connected domain $U$ in the complex plane with a piecewise Dini-smooth boundary which admits a finite set of Dini-smooth corners, we study the $C^\ast$-algebra $B_U$ generated by the Bergman and anti-Bergman projections acting on the Lebesgue space $L^2(U)$ and by the operators of multiplication by piecewise continuous functions that slowly oscillate at points of the domain boundary. Applying the Allan-Douglas local principle, the limit operators techniques and the Kehe Zhu results on $\operatorname{VMO}_{\partial}$ functions, we construct a Fredholm symbol calculus for the $C^\ast$-algebra $B_U$ and establish a Fredholm criterion for the operators $A\in B_U$.

The talk is based on joint work with E. Espinoza-Loyola.

## Mathematical Relativity

Mode stability for the Teukolsky equation on extremal Kerr black hole spacetimes.

*Rita Teixeira da Costa*, University of Cambridge.

## Abstract

We prove that there are no exponentially growing modes nor modes on the real axis for the Teukolsky equation on extremal Kerr black hole spacetimes. While the result was previously known for subextremal spacetimes, we show that the proof for the latter cannot be extended to the extremal case as the nature of the event horizon changes radically in the extremal limit.

Finally, we explain how mode stability could serve as a preliminary step towards understanding boundedness, scattering and decay properties of general solutions to the Teukolsky equation on extremal Kerr black holes.

## Partial Differential Equations

A two-phase free boundary problem ruled by the infinity laplacian.

*Damião Júnio Araújo*, Universidade Federal da Paraíba.

## Abstract

In this talk we discuss some regularity aspects for viscosity solutions of a non-variational two-phase free boundary problem ruled by the infinity Laplacian. Joint work with E. Teixeira (University of Central Florida - USA) and J. M. Urbano (Universidade de Coimbra - PT).

## Mathematical Relativity

Conformally flat slices of asymptotically flat spacetimes.

*Miguel Duarte*, Instituto Superior Técnico.

## Abstract

For mathematical convenience initial data sets in numerical relativity are often taken to be conformally flat. Employing the dual-foliation formalism, we investigate the physical consequences of this assumption. Working within a large class of asymptotically flat spacetimes we show that the ADM linear momentum is governed by the leading Lorentz part of a boost even in the presence of supertranslation-like terms. Following up, we find that in spacetimes that are asymptotically flat, and admit spatial slices with vanishing linear momentum that are sufficiently close to conformal flatness, any boosted slice can not be conformally flat. Consequently there are no conformally flat boosted slices of the Schwarzschild spacetime. This confirms the previously anticipated explanation for the presence of junk-radiation in Brandt-Bruegmann puncture data.

This seminar is joint with CENTRA, and will take place on the Physics Department (seminar room, 2nd floor).

## Mathematical Relativity

To be announced.

*Jorge V. Rocha*, ICCUB.

## String Theory

KdV Charges in $2D$ CFT.

*Simon Ross*, Durham University.

## Abstract

Two-dimensional CFTs have an infinite set of commuting conserved charges, known as the quantum KdV charges, built out of the stress tensor. We compute the thermal correlation functions of the these KdV charges on a circle. We show that these correlation functions are given by quasi-modular differential operators acting on the torus partition function.