## Selected Presentations & Appearances

Bryant’s Laplacian flow is a geometric flow of closed positive 3-forms on a 7-manifold that aims to give a parabolic PDE approach to constructing Riemannian manifolds with holonomy $G_2$. My talk will discuss some recent developments and open questions in this area. Along the way I will try to highlight ways in which Laplacian flow has both some similar and some distinctive features compared to more well-known and better-understood geometric flows, like Ricci flow, mean curvature flow and Lagrangian mean curvature flow.

I will give a brief overview of some of the important developments in the construction of G2 holonomy metrics achieved during the lifetime of our Collaboration and mention some outstanding open problems.

https://scgp.stonybrook.edu/wp-content/uploads/2022/03/Computational-Differential-Geometry-1-scaled.jpg

Scope of meeting:

The workshop “Computational Differential Geometry and its Applications in Physics” grows out of recent work using machine learning techniques to solve geometric PDEs such as those determining Ricci-flat Kähler metrics in four and higher dimensions.

The mathematical focus will be on computational methods for Riemannian geometry: methods to represent and compare metrics, to find structures such as geodesics or minimal cycles, and to obtain explicit Einstein metrics, metrics of G2 and special holonomy and complex structures. The physics focus will be on using these explicit expressions for metrics, gauge connections, moduli potentials and so on to solve for physically relevant quantities in supergravity and string theory compactifications, such as Yukawa couplings and matter Kähler potentials in realistic superstring vacua. We also hope to stimulate discussion on the foundations of such work and the use of verified numerical results in rigorous proof.

The geometry of 3-forms in dimension 7 is very special, because of the existence of stable 3-forms, i.e. 3-forms whose orbit under GL(7) is an open subset of the space of all 3-forms. Such stable 3-forms turn out to be intimately related to the exceptional Lie group G_2, to the octonions and to so-called exceptional holonomy metrics, that is, Riemannian metrics whose holonomy group is contained in the group G_2. I will describe a geometric flow on closed 3-forms, due to Robert Bryant, called Laplacian flow that aims to produce Riemannian manifolds with holonomy group G_2. My talk will concentrate on certain special solutions to Laplacian flow called solitons: in particular I will describe recent constructions of non-compact shrinking, steady and expanding solitons in Laplacian flow. In other better-understood geometric flows, e.g. Ricci flow and mean curvature flow, solitons have played a key role in understanding singularity formation and hence in understanding the long-time behaviour of these flows. Time permitting I will make some comparisons with known solitons in Ricci flow and Lagrangian mean curvature flow. This is joint work with Johannes Nordström and also in part with Rowan Juneman (both at Bath).

Bryant’s Laplacian flow is a geometric flow of closed positive 3-forms on a 7-manifold that aims to give a parabolic PDE approach to constructing Riemannian manifolds with holonomy $G_2$. My talk will discuss some recent developments and open questions in this area. Along the way I will try to highlight ways in which Laplacian flow has both some similar and some distinctive features compared to more well-known and better-understood geometric flows, like Ricci flow, mean curvature flow and Lagrangian mean curvature flow.

I will give a brief overview of some of the important developments in the construction of G2 holonomy metrics achieved during the lifetime of our Collaboration and mention some outstanding open problems.

https://scgp.stonybrook.edu/wp-content/uploads/2022/03/Computational-Differential-Geometry-1-scaled.jpg

Scope of meeting:

The workshop “Computational Differential Geometry and its Applications in Physics” grows out of recent work using machine learning techniques to solve geometric PDEs such as those determining Ricci-flat Kähler metrics in four and higher dimensions.

The mathematical focus will be on computational methods for Riemannian geometry: methods to represent and compare metrics, to find structures such as geodesics or minimal cycles, and to obtain explicit Einstein metrics, metrics of G2 and special holonomy and complex structures. The physics focus will be on using these explicit expressions for metrics, gauge connections, moduli potentials and so on to solve for physically relevant quantities in supergravity and string theory compactifications, such as Yukawa couplings and matter Kähler potentials in realistic superstring vacua. We also hope to stimulate discussion on the foundations of such work and the use of verified numerical results in rigorous proof.

The geometry of 3-forms in dimension 7 is very special, because of the existence of stable 3-forms, i.e. 3-forms whose orbit under GL(7) is an open subset of the space of all 3-forms. Such stable 3-forms turn out to be intimately related to the exceptional Lie group G_2, to the octonions and to so-called exceptional holonomy metrics, that is, Riemannian metrics whose holonomy group is contained in the group G_2. I will describe a geometric flow on closed 3-forms, due to Robert Bryant, called Laplacian flow that aims to produce Riemannian manifolds with holonomy group G_2. My talk will concentrate on certain special solutions to Laplacian flow called solitons: in particular I will describe recent constructions of non-compact shrinking, steady and expanding solitons in Laplacian flow. In other better-understood geometric flows, e.g. Ricci flow and mean curvature flow, solitons have played a key role in understanding singularity formation and hence in understanding the long-time behaviour of these flows. Time permitting I will make some comparisons with known solitons in Ricci flow and Lagrangian mean curvature flow. This is joint work with Johannes Nordström and also in part with Rowan Juneman (both at Bath).

I will describe a geometric flow on 3-forms, due to Robert Bryant, called Laplacian flow that aims to produce Riemannian manifolds with holonomy group G2.

My talk will concentrate on certain special solutions to Laplacian flow called solitons: in particular I will describe a recent construction of non-compact shrinking, steady and expanding solitons in Laplacian flow all with asymptotically conical geometry. I will also describe an explicit complete steady soliton with one end, exponential volume growth and asymptotically constant negative scalar curvature.

We will report briefly on some of our recent results, numerical investigations (and resulting conjectures) and work still-in-progress on complete noncompact cohomogeneity one solitons in Bryant’s closed G2-Laplacian flow (G2-solitons for short). Our results include the construction of new complete noncompact (asymptotically conical) shrinking and expanding (gradient) G2-solitons on the total space of the bundle of anti-self dual 2-forms on S^4 and CP^2; we conjecture the existence of a 1-parameter family of asymptotically conical steady (gradient) G2-solitons on the total space of the bundle of anti-self dual 2-forms of CP^2 and describe some analytic and numerical evidence supporting this conjecture. In other more widely studied geometric flows (e.g. Ricci flow, codimension 1 mean curvature flow and Lagrangian mean curvature flow) solitons are key to understanding singularity formation and for attempting to continue the flow after the first singular time. We discuss very briefly how our results reveal both some similarities and differences with the behaviour of solitons currently known in Ricci flow and Kahler-Ricci flow.

I will give a (biased) overview of recent progress on the construction of complete noncompact metrics of exceptional holonomy. Along the way I will describe some of the most important historical developments since the field began (in the late 1970s). Throughout its history, the field has seen a fruitful back-and-forth between physicists and mathematicians, some of which I will describe. I will try to explain some of the similarities and differences between the more familiar special holonomy metrics — hyperkaehler and Calabi—Yau metrics— and the exceptional cases and holonomy, and why the latter are much more difficult to construct.

In the early 2000s M theorists predicted the existence of various new complete noncompact Riemannian metrics with holonomy group the compact exceptional Lie group G2. Recently mathematicians have constructed many, but by no means all, of these physically predicted metrics and also other metrics not necessarily anticipated by physics. It will turn out the construction of these complete noncompact metrics of exceptional holonomy relies on some of the most recent developments on constructing complete noncompact hyperkaehler and Calabi—Yau metrics with controlled asymptotic geometry

In this talk, Haskins will give an overview of some of the main recent developments in the construction of metrics with special and exceptional holonomy, concentrating on the progress made by members of the collaboration on resolving some long-standing open questions. He will indicate some of the symbiotic relationships that have developed between the two better understood cases of hyperKaehler 4-manifolds and Calabi-Yau 3-folds, and the more challenging case of 7-manifolds with holonomy G2.

One important unifying theme has been the more systematic study of special or exceptional holonomy metrics in either highly collapsed regimes or close to a suitable adiabatic limit: codimension one (circle) collapse of G2 holonomy metrics underpins the physical limit where M theory is modeled by Type IIA string theory; collapsed Calabi-Yau metrics with fibers close to flat tori appear in connection with degenerations of complex structure; K3 fibrations of Calabi-Yau 3-folds or coassociative K3 fibrations of G2 manifolds with small fiber size also both appear naturally. The latter adiabatic limit will be discussed in more detail in Simon Donaldson’s talk.

I will attempt to give a (biased) overview of recent progress on the construction of complete noncompact metrics of exceptional holonomy. Along the way I will describe some of the most important historical developments since the field began (in the late 1970s). Throughout its history, the field has seen a fruitful back-and-forth between physicists and mathematicians, some of which I will describe. I will try to explain some of the similarities and differences between the more familiar special holonomy metrics — hyperkaehler and Calabi—Yau metrics— and the exceptional cases G_2 and Spin_7 holonomy, and why the latter are much more difficult to construct.

In the early 2000s M theorists predicted the existence of various new complete noncompact Riemannian metrics with holonomy group the compact exceptional Lie group G_2. Very recently mathematicians have constructed many, but by no means all, of these physically predicted G_2 metrics and also other G_2 metrics not necessarily anticipated by physics. It will turn out the construction of these complete noncompact metrics of exceptional holonomy relies on some of the most recent developments on constructing complete noncompact hyperkaehler and Calabi—Yau metrics with controlled asymptotic geometry. If (by some miracle) time permits, I will describe some of the future prospects for the field.

In recent joint work with Lorenzo Foscolo and Johannes Nordstr\”om we gave an analytic construction of large families of complete circle-invariant $G_2$ holonomy metrics on the total space of circle bundles over a complete noncompact Calabi—Yau 3-fold with asymptotically conical geometry. The asymptotic models for the geometry of these $G_2$ metrics are circle bundles with fibres of constant length $l$, so-called asymptotically local conical (ALC) geometry. These ALC $G_2$ metrics can Gromov—Hausdorff collapse with bounded curvature to the given asymptotically conical Calabi—Yau 3-fold as the fibre length $l$ goes to $0$. A natural question is: what happens to these families of $G_2$ metrics as we try to make $l$ large? In general the answer to this question is not known, but in cases with sufficient symmetry we have recently been able to give a complete picture.

We give an overview of all these results and discuss some analogies with the class of asymptotically locally flat (ALF) hyperkaehler 4-manifolds. In particular we suggest that a particular $G_2$ metric we construct should be regarded as a $G_2$ analogue of the Euclidean Taub—NUT metric on the complex plane.

I will attempt to give a (biased) overview of recent progress on the construction of complete noncompact metrics of exceptional holonomy. Along the way I will describe some of the most important historical developments since the field began (in the late 1970s). Throughout its history, the field has seen a fruitful back-and-forth between physicists and mathematicians, some of which I will describe. I will try to explain some of the similarities and differences between the more familiar special holonomy metrics — hyperkaehler and Calabi—Yau metrics— and the exceptional cases G_2 and Spin_7 holonomy, and why the latter are much more difficult to construct.

In the early 2000s M theorists predicted the existence of various new complete noncompact Riemannian metrics with holonomy group the compact exceptional Lie group G_2. Very recently mathematicians have constructed many, but by no means all, of these physically predicted G_2 metrics and also other G_2 metrics not necessarily anticipated by physics. It will turn out the construction of these complete noncompact metrics of exceptional holonomy relies on some of the most recent developments on constructing complete noncompact hyperkaehler and Calabi—Yau metrics with controlled asymptotic geometry. If (by some miracle) time permits, I will describe some of the future prospects for the field.

In recent joint work with Lorenzo Foscolo and Johannes Nordstr\”om we gave an analytic construction of large families of complete circle-invariant G_2 holonomy metrics on the total space of circle bundles over a complete noncompact Calabi—Yau 3-fold with asymptotically conical geometry. The asymptotic models for the geometry of these G_2 metrics are circle bundles with fibres of constant length l, so-called asymptotically local conical (ALC) geometry. These ALC G_2 metrics can Gromov—Hausdorff collapse with bounded curvature to the given asymptotically conical Calabi—Yau 3-fold as the fibre length l goes to 0. A natural question is: what happens to these families of G_2 metrics as we try to make l large? In general the answer to this question is not known, but in cases with sufficient symmetry we have recently been able to give a complete picture.

We give an overview of all these results and discuss some analogies with the class of asymptotically locally flat (ALF) hyperkaehler 4-manifolds. In particular we suggest that a particular G_2 metric we construct should be regarded as a G_2 analogue of the Euclidean Taub—NUT metric on the complex plane.

During the week of October 23-27, 2017, the Simons Center for Geometry and Physics will host a workshop concerning canonical geometric structures on differentiable manifolds. Key researchers in differential geometry, gauge theory, complex geometry, symplectic geometry, and related disciplines will present overviews of their areas of expertise, and report on the current state of research. One lecture each day will be designated as a colloquium-style talk, intended to explain central themes to a broader audience of graduate students and non-experts.

We note with pleasure that Sir Simon Donaldson will turn 60 in 2017. Since Cambridge University will host a conference officially celebrating this event in August, we wish to avoid interfering with those festivities, and so will not describe our workshop as a “birthday conference.” However, it is an honor and a privilege to announce that our conference banquet will nonetheless be dedicated to the celebration of Sir Simon’s birthday, in the broader context of honoring his magnificent contributions to mathematics.

https://en.wikipedia.org/wiki/Geometry_Festival

This colloquium will tell the tale of the exceptional simple Lie group G_2: from its unexpected discovery in 1887 to the present, where it has come to play an important role both in Differential Geometry and in Theoretical Physics (M-theory). This is a tale suitable for the whole family (of mathematicians); several reversals of fortune and periods of exile will befall our hero en route to present day influence.

## Service to the Profession

This program sits at the intersection between differential geometry and analysis but also connects to several other adjacent mathematical fields and to theoretical physics. Differential geometry aims to answer questions about very regular geometric objects (smooth Riemannian manifolds) using the tools of differential calculus. A fundamental object is the curvature tensor of a Riemannian metric: an algebraically complicated object that involves 2nd partial derivatives of the metric. Many questions in differential geometry can therefore be translated into questions about the existence or properties of the solutions of systems of (often) nonlinear partial differential equations (PDEs). The PDE systems that arise in geometry have historically stimulated the development of powerful new analytic methods. In most cases the nonlinearity of these systems makes ‘closed form’ expressions for a solution impossible: instead more abstract methods must be employed.

The aim of this program is to study geometric and analytic aspects of special holonomy metrics, instanton bundles over such spaces and the calibrated submanifolds within them. Because these objects are Einstein metrics, Yang–Mills connections and minimal submanifolds respectively they fit into a wider geometric framework. But because each of them satisfies a first-order PDE system they enjoy various special properties. This program takes the position that the natural viewpoint, both geometrically and analytically, is to consider these three classes of closely-related geometric objects together. A common feature is that weak solutions to these higher-dimensional equations may have interesting non-isolated singular sets whose detailed geometric and analytic structure is still poorly understood. Improving our understanding of such (partial) regularity of solutions is a key component of the program.

In person/hybrid meeting of Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics. Videos for majority of talks available at https://sites.duke.edu/scshgap/geometry-topology-and-singular-special-holonomy-spaces-6-10-june-2022-freiburg-university/

The workshop will have two main themes:

(A) The Donaldson-Segal programme for defining enumerative invariants of compact G2-manifolds by counting G2 instantons, with correction terms from associative 3-folds. Analysis of G2-instanton moduli spaces, singularities of G2-instantons. Analysis of Seiberg-Witten type equations on 3-manifolds used to define correction terms in Donaldson-Segal programme. Related gauge theory problems, including singularities of Hermitian-Yang Mills connections.

(B) The Pantev-Toën-Vaquié-Vezzosi (PTVV) theory of shifted symplectic derived algebraic geometry, giving geometric structures on Calabi-Yau moduli spaces, and its applications to generalizations of Donaldson-Thomas theory of Calabi-Yau 3- and 4-folds. One particular aim is to present the theory in a way accessible to String Theorists, to encourage communication between mathematicians and physicists on this subject, and to facilitate interpretation of the implications of the theory in String Theory terms. Algebro-geometric enumerative invariants (and Floer theories, etc), particularly of Calabi-Yau manifolds, related to the PTVV theory: Donaldson-Thomas, Gromov-Witten, and Vafa-Witten invariants, Donaldson-Thomas type invariants of Calabi-Yau 4-folds, the Gopakumar-Vafa conjecture.

This program sits at the intersection between differential geometry and analysis but also connects to several other adjacent mathematical fields and to theoretical physics. Differential geometry aims to answer questions about very regular geometric objects (smooth Riemannian manifolds) using the tools of differential calculus. A fundamental object is the curvature tensor of a Riemannian metric: an algebraically complicated object that involves 2nd partial derivatives of the metric. Many questions in differential geometry can therefore be translated into questions about the existence or properties of the solutions of systems of (often) nonlinear partial differential equations (PDEs). The PDE systems that arise in geometry have historically stimulated the development of powerful new analytic methods. In most cases the nonlinearity of these systems makes ‘closed form’ expressions for a solution impossible: instead more abstract methods must be employed.

The aim of this program is to study geometric and analytic aspects of special holonomy metrics, instanton bundles over such spaces and the calibrated submanifolds within them. Because these objects are Einstein metrics, Yang–Mills connections and minimal submanifolds respectively they fit into a wider geometric framework. But because each of them satisfies a first-order PDE system they enjoy various special properties. This program takes the position that the natural viewpoint, both geometrically and analytically, is to consider these three classes of closely-related geometric objects together. A common feature is that weak solutions to these higher-dimensional equations may have interesting non-isolated singular sets whose detailed geometric and analytic structure is still poorly understood. Improving our understanding of such (partial) regularity of solutions is a key component of the program.

In person/hybrid meeting of Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics. Videos for majority of talks available at https://sites.duke.edu/scshgap/geometry-topology-and-singular-special-holonomy-spaces-6-10-june-2022-freiburg-university/

The workshop will have two main themes:

(A) The Donaldson-Segal programme for defining enumerative invariants of compact G2-manifolds by counting G2 instantons, with correction terms from associative 3-folds. Analysis of G2-instanton moduli spaces, singularities of G2-instantons. Analysis of Seiberg-Witten type equations on 3-manifolds used to define correction terms in Donaldson-Segal programme. Related gauge theory problems, including singularities of Hermitian-Yang Mills connections.

(B) The Pantev-Toën-Vaquié-Vezzosi (PTVV) theory of shifted symplectic derived algebraic geometry, giving geometric structures on Calabi-Yau moduli spaces, and its applications to generalizations of Donaldson-Thomas theory of Calabi-Yau 3- and 4-folds. One particular aim is to present the theory in a way accessible to String Theorists, to encourage communication between mathematicians and physicists on this subject, and to facilitate interpretation of the implications of the theory in String Theory terms. Algebro-geometric enumerative invariants (and Floer theories, etc), particularly of Calabi-Yau manifolds, related to the PTVV theory: Donaldson-Thomas, Gromov-Witten, and Vafa-Witten invariants, Donaldson-Thomas type invariants of Calabi-Yau 4-folds, the Gopakumar-Vafa conjecture.

I have been Managing Editor of Journal of the London Mathematical Society since August 2018. Earlier this year, Professor James Maynard at Oxford University, joined me as co Managing Editor.

All currently known construction methods of smooth compact G2-manifolds have been tied to certain singular G2-spaces, which in Joyce’s original construction are G2-orbifolds and in Kovalev’s twisted connected sum construction are complete G2-manifolds with cylindrical ends. By a slight abuse of terminology we also refer to the latter as singular G2-spaces, and in fact both construction methods may be viewed as desingularization procedures. In turn, singular G2-spaces comprise a (conjecturally large) part of the boundary of the moduli space of smooth compact G2-manifolds, and so their deformation theory is of considerable interest. Furthermore, singular G2-spaces are also important in theoretical physics. Namely, in order to have realistic low-energy physics in M-theory, one needs compact singular G2-spaces with both codimension 4 and 7 singularities according to Acharya and Witten. However, the existence of such singular G2-spaces is unknown at present. The aim of this workshop was to bring reserachers from special holonomy geometry, geometric analysis and theoretical physics together to exchange ideas on these questions.

This program seeks to connect recent developments and open questions in the theory of compact manifolds with special or exceptional holonomy (especially G_2 manifolds) with other areas of mathematics and theoretical physics: differential topology, algebraic geometry, (non compact) Calabi-Yau 3-folds, K3 surfaces, submanifold theory, J-holomorphic curves, gauge theory and M-theory in particular. One of the aims of the program is to bring to wider attention some of the problems in other parts of mathematics that have arisen recently in the theory of compact G_2 manifolds and to bring together experts from these disparate areas.

This is a research school aimed at beginning PhD student in geometry and topology. The aim of the research school will be to give a thorough introduction to G2 geometry, starting from fundamental material and progressing through to recent breakthroughs and current research in which the UK plays a leading role. The school will also introduce participants to topics of broader interest in algebra (e.g. representation theory), analysis (e.g. elliptic regularity), geometry (e.g. holonomy) and topology (e.g. characteristic classes). The course will also indicate some connections beyond mathematics to contemporary theoretical physics (M-theory)

This event was a combined 1-week summer school and a 1-week research workshop Ricci curvature: limit spaces and Kaehler geometry at ICMS Edinburgh. Speakers included: Robert Berman, Jeff Cheeger, Sir Simon Donaldson, Vincent Guedj, John Lott, Aaron Naber, Tashiki Mabuchi, Gang Tian and Burkhard Wilking.

I was the Editorial Advisor who handled submissions to the 4 London Mathematical Society journals (Bulletin, Journal, Proceedings and Transactions) in Differential Geometry and Geometric Analysis.

## Academic & Administrative Activities

Director of Graduate Studies, Duke Mathematics, 1 July 2023-30 June 2024.

Area lead for tenure-track hiring (coordinating applicants within Geometry and Topology), Fall 2022.

Member of London Mathematics Society Publications Nominating Group, 2020--2023.

Member of Duke Mathematics Graduate Admissions Committee, 2020.

Deputy Collaboration Director, Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics, July 2016-2023.

Member of Steering Committee, Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics, July 2016-2023.

Managing Editor, Journal of the London Mathematical Society, August 2018-.

Editorial Advisory Board, London Mathematical Society Journals, 2011-2016.

EPSRC Doctoral Training Allocations Panel, 2009.