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Extended Abstracts Fall 2013

Geometrical Analysis; Type Theory, Homotopy Theory and Univalent Foundations

herausgegeben von: Maria del Mar González, Paul C. Yang, Nicola Gambino, Joachim Kock

Verlag: Springer International Publishing

Buchreihe : Trends in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The two parts of the present volume contain extended conference abstracts corresponding to selected talks given by participants at the "Conference on Geometric Analysis" (thirteen abstracts) and at the "Conference on Type Theory, Homotopy Theory and Univalent Foundations" (seven abstracts), both held at the Centre de Recerca Matemàtica (CRM) in Barcelona from July 1st to 5th, 2013, and from September 23th to 27th, 2013, respectively. Most of them are brief articles, containing preliminary presentations of new results not yet published in regular research journals. The articles are the result of a direct collaboration between active researchers in the area after working in a dynamic and productive atmosphere.

The first part is about Geometric Analysis and Conformal Geometry; this modern field lies at the intersection of many branches of mathematics (Riemannian, Conformal, Complex or Algebraic Geometry, Calculus of Variations, PDE's, etc) and relates directly to the physical world, since many natural phenomena posses an intrinsic geometric content. The second part is about Type Theory, Homotopy Theory and Univalent Foundations.

The book is intended for established researchers, as well as for PhD and postdoctoral students who want to learn more about the latest advances in these highly active areas of research.

Inhaltsverzeichnis

Frontmatter

Erratum to: Univalent Categories and the Rezk Completion

Benedikt Ahrensm, Krzysztof Kapulkin, Michael Shulman

Geometrical Analysis

Frontmatter

A Positive Mass Theorem in Three Dimensional Cauchy–Riemann Geometry

Abstract

In this note we summarize the results from [6] on the positive mass problem in 3-dimensional CR (Cauchy–Riemann) geometry.

Jih-Hsin Cheng, Andrea Malchiodi, Paul Yang

On the Rigidity of Gradient Ricci Solitons

Abstract

A complete Riemannian manifold (M, g) is said to be a gradient Ricci soliton if there exists a smooth function \(f: M \rightarrow \mathbb{R}\) such that \(\displaystyle{ Rc + H_{f} =\lambda g, }\) where Rc denotes the Ricci tensor, H _f is the Hessian of the function f, and \(\lambda\) is a real number.

Manuel Fernández-López, Eduardo García-Río

Geometric Structures Modeled on Affine Hypersurfaces and Generalizations of the Einstein–Weyl and Affine Sphere Equations

Abstract

Affine hypersurface structures (AH structures) simultaneously generalize Weyl structures and abstract geometric structures induced on a nondegenerate co-oriented hypersurface in flat affine space. The aim of this note is to define equations for AH structures, called Einstein which, for Weyl structures, specialize to the usual Einstein Weyl equations, and, in the case of the AH structure induced on a hypersurface in flat affine space, recover the equations for affine spheres. Additionally, we indicate the simplest constructions of Einstein AH structures that do not arise in either of these manners.

Daniel J. F. Fox

Submanifold Conformal Invariants and a Boundary Yamabe Problem

Abstract

While much is known about the invariants of conformal manifolds, the same cannot be said for the invariants of submanifolds in conformal geometries. Codimension one embedded submanifolds (or hypersurfaces) are important for applications to geometric analysis and physics.

A. Rod Gover, Andrew Waldron

Variation of the Total Q-Prime Curvature in CR Geometry

Abstract

>Tom Branson introduced the concept of Q-curvature in conformal geometry, in connection with the study of conformal anomaly of determinants of conformally invariant differential operators. The definition can be generalized to CR manifolds via Fefferman’s conformal structure on a circle bundle over CR manifolds, see [2]. Using this correspondence, one can translate the properties of conformal Q-curvature to the CR analogue. However, there has been an important missing piece in this correspondence.

Kengo Hirachi

Conformal Invariants from Nullspaces of Conformally Invariant Operators

Abstract

This is an extended abstract of the talk Conformal invariants from nodal sets. The talk was based on joint work with Yaiza Canzani, Rod Gover and Raphaël Ponge, and the results appeared in the papers [2, 3]. The current abstract is an abbreviated version of [2].

Dmitry Jakobson

Rigidity of Bach-Flat Manifolds

Abstract

Bach-flat metrics were introduced in the study of a conformally invariant gravitational theory and has played important roles in general relativity and geometry. This metric is the most natural generalization of an Einstein metric.

Seongtag Kim

Uniformizing Surfaces with Conical Singularities

Abstract

We consider a class of singular equations, motivated by the problem of prescribing the Gaussian curvature, as well as from some models in theoretical physics such as self-dual Chern–Simons theory or Electroweak theory: our goal is to prove existence results by attacking the problem variationally, using suitable improvements of the Moser–Trudinger inequality.

Andrea Malchiodi

Recent Results and Open Problems on Conformal Metrics on ℝ n with Constant Q-Curvature

Abstract

We consider solutions to the equation \(\displaystyle{ (-\Delta )^{m}u = (2m - 1)!e^{2mu}\quad \text{in }\mathbb{R}^{2m}, }\) satisfying \(\displaystyle{ V:=\int _{\mathbb{R}^{2m}}e^{2mu(x)}dx <+\infty. }\) Geometrically, if u solves (1)–(2), then the conformal metric g _u: = e ^2u | dx | ² has Q-curvature \(Q_{g_{u}} \equiv (2m - 1)!\) and volume V (by | dx | ² we denote the Euclidean metric).

Luca Martinazzi

Isoperimetric Inequalities for Complete Proper Minimal Submanifolds in Hyperbolic Space

Abstract

The classical isoperimetric inequality for a domain \(\Sigma \subset \mathbb{R}^{k}\) with smooth boundary \(\partial \Sigma\) is \(\displaystyle{ k^{k}\omega _{ k}\mathrm{Vol}(\Sigma )^{k-1} \leq \mathrm{ Vol}(\partial \Sigma )^{k}, }\) where equality holds if and only if \(\Sigma\) is a ball in \(\mathbb{R}^{k}\).

Sung-Hong Min, Keomkyo Seo

Total Curvature of Complete Surfaces in Hyperbolic Space

Abstract

We present a Gauss–Bonnet type formula for complete surfaces in n-dimensional hyperbolic space \(\mathbb{H}^{n}\) under some assumptions on their asymptotic behaviour. As in recent results for Euclidean submanifolds (see Dillen–Kühnel [4] and Dutertre [5]), the formula involves an ideal defect, i.e., a term involving the geometry of the set of points at infinity.

Jun O’Hara, Gil Solanes

Constant Scalar Curvature Metrics on Hirzebruch Surfaces

Abstract

For each natural number m ≥ 0, a complex surface \(\Sigma _{m}\) called Hirzebruch surface is defined in [8].

Nobuhiko Otoba

Isoperimetric Inequalities for Extremal Sobolev Functions

Abstract

Let \(\Omega \subset \mathbf{R}^{n}\) be a bounded domain with boundary of class \(\mathcal{C}^{1}\). One can measure various geometric and physical quantities attached to \(\Omega\), such as volume, perimeter, diameter, in-radius, torsional rigidity, and principal frequency. The first chapter of [16] contains a long list of such interesting quantities, as well as their values for standard shapes such as disks, rectangles, strips, and triangles.

Jesse Ratzkin, Tom Carroll

Type Theory, Homotopy Theory, and Univalent Foundations

Frontmatter

Univalent Categories and the Rezk Completion

Abstract

When formalizing category theory in traditional set-theoretic foundations, a significant discrepancy between the foundational notion of “sameness” – equality – and its practical use arises: most category-theoretic concepts are invariant under weaker notions of sameness than equality, namely isomorphism in a category or equivalence of categories. We show that this discrepancy can be avoided when formalizing category theory in Univalent Foundations.

Benedikt Ahrens, Krzysztof Kapulkin, Michael Shulman

Covering Spaces in Homotopy Type Theory

Abstract

Covering spaces play an important role in classical homotopy theory, whose algebraic characteristics have deep connections with fundamental groups of underlying spaces. It is natural to ask whether these connections can be stated in homotopy type theory (HoTT), an exciting new framework coming with an interpretation in homotopy theory. This note summarizes the author’s attempt to recover the classical results (e.g., the classification theorem) so as to explore the expressiveness of the new foundation. Some interesting techniques employed in the current proofs seem applicable to other constructions as well.

Kuen-Bang Hou

Towards a Topological Model of Homotopy Type Theory

Abstract

The model of homotopy type theory in simplicial sets [7] has proven to be a grounding and motivating influence in the development of homotopy type theory. The classical theory of topological spaces has also proven to be motivational to the subject. Though the Quillen equivalence between simplicial sets and topological spaces provides, in some weak sense, a model in topological spaces, we explore the extent to which the category of topological spaces may be a more direct and strict model of homotopy type theory. We define a notion of model of homotopy type theory, and show that the category of topological spaces fully embeds into such a model.

Paige North

Made-to-Order Weak Factorization Systems

Abstract

For a cocomplete category M which satisfies certain “smallness” condition (such as being locally presentable), the algebraic small object argument defines the functorial factorization necessary for a “made-to-order” weak factorization system with right class

Emily Riehl

A Descent Property for the Univalent Foundations

Abstract

We present a version of the descent property [4, 5] which is formulated using families rather than morphisms. By the univalence axiom [3], there is an equivalence \((\sum _{Y:\text{Type}}Y \rightarrow X) \simeq (X \rightarrow \text{Type})\) for every type X [1]. A similar equivalence will hold for the kind of families over graphs we will study here: the equifibered families. This equivalence can be used to translate our simple version of the descent property back into the usual formulation of it.

Egbert Rijke

Classical Field Theory via Cohesive Homotopy Types

Abstract

In the year 1900, at the International Congress of Mathematics in Paris, David Hilbert stated his famous list of 23 central open questions of mathematics [7]. Among them, the sixth problem (see [3] for a review) is arguably the one that Hilbert himself regarded as the most valuable: “From all the problems in the list, the sixth is the only one that continually engaged [Hilbert’s] efforts over a very long period, at least between 1894 and 1932”, see [4].

Urs Schreiber

How Intensional Is Homotopy Type Theory?

Abstract

Martin-Löf’s Extensional Type Theory (ETT) has a straighforward semantics in the category Set of sets and functions and actually in any locally cartesian closed category with a natural numbers object (nno), e.g., in any elementary topos with a nno. Dependent products are interpreted by right adjoints to pullback functors, and extensional identity types are interpreted as diagonals in slice categories as explained, e.g., in [4].

Thomas Streicher

Titel: Extended Abstracts Fall 2013
herausgegeben von: Maria del Mar González
Paul C. Yang
Nicola Gambino
Joachim Kock
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-21284-5
Print ISBN: 978-3-319-21283-8
DOI: https://doi.org/10.1007/978-3-319-21284-5

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Erratum to: Univalent Categories and the Rezk Completion

Geometrical Analysis

Frontmatter

A Positive Mass Theorem in Three Dimensional Cauchy–Riemann Geometry

On the Rigidity of Gradient Ricci Solitons

Geometric Structures Modeled on Affine Hypersurfaces and Generalizations of the Einstein–Weyl and Affine Sphere Equations

Submanifold Conformal Invariants and a Boundary Yamabe Problem

Variation of the Total Q-Prime Curvature in CR Geometry

Conformal Invariants from Nullspaces of Conformally Invariant Operators

Rigidity of Bach-Flat Manifolds

Uniformizing Surfaces with Conical Singularities

Recent Results and Open Problems on Conformal Metrics on ℝ n with Constant Q-Curvature

Isoperimetric Inequalities for Complete Proper Minimal Submanifolds in Hyperbolic Space

Total Curvature of Complete Surfaces in Hyperbolic Space

Constant Scalar Curvature Metrics on Hirzebruch Surfaces

Isoperimetric Inequalities for Extremal Sobolev Functions

Type Theory, Homotopy Theory, and Univalent Foundations

Frontmatter

Univalent Categories and the Rezk Completion

Covering Spaces in Homotopy Type Theory

Towards a Topological Model of Homotopy Type Theory

Made-to-Order Weak Factorization Systems

A Descent Property for the Univalent Foundations

Classical Field Theory via Cohesive Homotopy Types

How Intensional Is Homotopy Type Theory?

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