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# The Kapustin-Witten equations

## Rafe Mazzeo Stanford University

One of the new four-dimensional gauge theories is a set of equations discovered by Kapustin and Witten in 2005. This theory involves a boundary condition which in turn involves a knot K in a 3-manifold. A later conjecture by Gaiotto and Witten states that a sequence of numerical counts of elements of the moduli space of solutions determines the Jones polynomial of the knot. While this remains open, quite a lot has been discovered. I will briefly survey these developments, including the basic structural and regularity theory of these equations, which was joint work with Witten, and a fairly complete resolution of this problem in the dimensionally reduced case, on the product of a Riemann surface and a half-line, which was joint work with S. He.

Acceso a la sala; access to the room

Sala virtual

# Short-time existence for the network flow

## Mariel Sáez Pontificia Universidad Católica de Chile

The network flow is a system of parabolic differential equations that describes the motion of a family of curves in which each of them evolves under curve-shortening flow. This problem arises naturally in physical phenomena and its solutions present a rich variety of behaviors. The goal of this talk is to describe some properties of this geometric flow and to discuss an alternative proof of short-time existence for non-regular initial conditions. The methods of our proof are based on techniques of geometric microlocal analysis that have been used to understand parabolic problems on spaces with conic singularities. This is joint work with Jorge Lira, Rafe Mazzeo, and Alessandra Pluda.

Password of the virtual room: 808148

Seminario online.

# Approximation theorems for the Schrödinger equation and the reconnection of quantum vortices in Bose-Einstein condensates

## Alberto Enciso Consejo Superior de Investigaciones Científicas

The Gross--Pitaevskii equation is a nonlinear Schrödinger equation that models the behavior of a Bose-Einstein condensate. The quantum vortices of the condensate are defined by the zero set of the wave function at time $t$. In this talk we will present recent work about how these quantum vortices can break and reconnect in arbitrarily complicated ways. As observed in the physics literature, the distance between the vortices near the breakdown time, say $t = 0$, scales like the square root of $t$: it is the so-called $t^{1/2}$ law. At the heart of the proof -- which ultimately entails understanding the evolution of curves in space -- lies a remarkable global approximation property for the linear Schrödinger equation. The talk is based on joint work with Daniel Peralta-Salas.

# Integrable systems methods for surfaces and new families of constant mean curvature surfaces in $\mathbb{R}^3$

## Eduardo Mota Sánchez University College Cork

THIS SEMINAR HAS BEEN POSTPONED BECAUSE OF THE UNIVERSITY POLICY CONCERNING THE CORONAVIRUS OUTBREAK. A NEW DATE WILL BE ANNOUNCED AS SOON AS POSSIBLE.

In this lecture I will outline the integrable systems technique for CMC surfaces, but with a view at some other cases. Then I will explain some recent developments in the construction of certain families of CMC surfaces using this setup. In particular, we start with a $2\times 2$ Cauchy problem to which we associate a scalar second order differential equation. The singularities in this ODE correspond to the ends in the resulting surface. Particularly, regular singularities produce asymptotically Delaunay ends while irregular singularities produce irregular ends. Our aim is to discuss global issues such as period problems and asymptotic behavior involved in the construction of CMC surfaces in $\mathbb{R}^3$ arising from the family of Heun's differential equations.

Seminario 1ª planta - IEMath-GR

# The Classification of Semigraphical Translators for Mean Curvature Flow

## Francisco Martín Serrano Universidad de Granada

We say that a surface is semigraphical if it is properly embedded, and, after removing a discrete collection of vertical lines, it is a graph. In this talk, we provide a nearly complete classification of semigraphical translators.

Seminario de la primera planta, IEMath

# A gravitational collapse singularity theorem that improves Penrose's

## Ettore Minguzzi Università degli Studi di Firenze

The global hyperbolicity assumption present in gravitational collapse singularity theorems is in tension with the quantum mechanical phenomenon of black hole evaporation. In this work I show that the causality conditions in Penrose's theorem can be almost completely removed. As a result, it is possible to infer the formation of spacetime singularities even in absence of predictability and hence compatibly with quantum field theory and black hole evaporation.

Seminario 1A Planta, IEMath

# Holomorphicity of real Kaehler submanifolds

## Sergio Chion Aguirre IMPA, Río de Janeiro

I will discuss the subject of real Kaehler submanifolds, that is, isometric immersions $f\colon M^{2n}\to\mathbb{R}^{2n+p}$ of a Kaehler manifold $(M^{2n},J)$ of complex dimension $n\geq 2$ into Euclidean space with codimension $p$. In particular, I will present a recent result that shows that for codimension $2p\leq 2n-1$ generic rank conditions on the second fundamental form of $f$ imply that the submanifold has to be minimal. In fact, for codimension $p\leq 11$ we have a stronger conclusion, namely, that $f$ must be holomorphic with respect to some complex structure in the ambient space.

This is joint work with A. de Carvalho and M. Dajczer.

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# Summer School "Geometric Analysis and Calibrated Geometries"

## Zürich (Switerzland)

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Invited speakers:

Single talks:

Maria Colombo, EPFL
Guido De Philippis, SISSA
Mark Haskins, Duke University
Alessandro Pigati, ETH Zürich
Antoine Song, Princeton University
Thomas Walpuski, Michigan State University
Lu Wang, University of Wisconsin-​Madison
Brian White, Stanford University
Xin Zhou, University of California, Santa Barbara

4-​hour minicourses:

Costante Bellettini, University College London
Bo Berndtsson, Chalmers University of Technology
Jake Solomon, The Hebrew University of Jerusalem
Luca Spolaor, MIT

# X International Meeting on Lorentzian Geometry

## Córdoba (Spain)

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The meeting is intended for all kind of researchers with interest on Lorentz Geometry and its applications to General Relativity.
For PhD students the meeting will represent an ideal way to have their first contact with current research topics on the area. Furthermore, an advanced course given by an expert on the area will be organized.
For senior researchers, these meetings represent an ideal place where to exhibit their latest results and to create new ways of collaboration.

# Geometric Analysis in Castro

## Castro (Italy)

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The conference will be on complex, Riemannian, metric geometry and geometric analysis. The recent results and developments of these research fields will be presented and discussed.

# Dynamical Aspects of Pseudo-Riemannian Geometry

## Braga (Portugal)

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This conference will focus on recent progress in Pseudo-Riemannian Geometry and, in particular, in Lorentzian Geometry.

One of the goals of the conference is to explore the natural intervention/interaction of dynamical systems in several questions of Pseudo-Riemannian geometry.