Robert Bryant

The Bonnet problem in Euclidean surface theory is well-known: Given a metric g on an oriented surface M and a function H, when (and in how many ways) can (M,g) be isometrically immersed in R3 with mean curvature H? For generic data (g,H), such an isometric immersion is impossible and, in the case that it does exist, the immersion is unique. Bonnet showed that, aside from the famous case of surfaces of constant mean curvature, there is a finite dimensional moduli space of (g,H) for which the space of such Bonnet immersions has positive dimension.

The corresponding problem in affine theory (a favorite topic of Eugenio Calabi) is still not completely solved. After reviewing the Euclidean results on this problem by O. Bonnet, J. Radon, É. Cartan, and A. Bobenko, I will give a report on affine analogs of these results. In particular, I will consider the classification of the data (g,H) for which the space of affine Bonnet immersions has positive dimension, showing a surprising connection with integrable systems in the case of data with the highest possible dimension of solutions.

The Bonnet problem in Euclidean surface theory is well-known: Given a metric g on an oriented surface M^2 and a function H, when (and in how many ways) can (M,g) be isometrically immersed in R^3 with mean curvature H? For generic data (g,H), such an isometric immersion is impossible and, in the case that it does exist, the immersion is unique. Bonnet showed that, aside from the famous case of surfaces of constant mean curvature, there is a finite dimensional moduli space of (g,H) for which the space of such Bonnet immersions has positive dimension.

The corresponding problem in affine theory (a favorite topic of Eugenio Calabi) is still not completely solved. After reviewing the Euclidean results on this problem by O.Bonnet, J.Radon, E.Cartan, and A.Bobenko, I will give a report on affine analogs of these results. In particular, I will consider the classification of the data (g,H) for which the space of \emph{affine Bonnet immersions} has positive dimension, showing a surprising connection with integrable systems in the case of data with the highest possible dimension of solutions.

This introductory talk will survey a history of the origins and theory of holonomy and its applications since its inception in the works of J. A. Schouten and E ́. Cartan, the early classification works of Berger and Simons, and the modern era where local and global techniques from PDE and geometry have made enormous advances, bringing us up to the century mark.

Abstract: Familiar mechanical phenomenon, such as driving and parking a car, rolling a ball, and even the ability of falling cats to land on their feet (usually) are examples of an underlying mathematical concept that, in the 19th century, became to be known as `holonomy'. As it became better understood, mathematicians and physicists began to realize that holonomy underlay many different phenomena, from the everyday situations mentioned above to understanding the curvature of space in Einstein's theory of general relativity. Currently, holonomy lies at the heart of both deep mathematical objects and high-energy physical theories, such as string theory and the still-mysterious M-theory, on which many theoretical physicists would like to base a 'theory of everything'.

In this talk, after discussing some holonomic phenomena in everyday life, I'll explore their underlying commonality and their appearance in more advanced situations and try to provide some insight into why this idea has turned out to be so fundamental.

This was a pair of lectures of Bonnet surfaces in Euclidean and affine geometry, delivered on July 24 (Part I) and July 25 (Part II). The description was

The first talk on the Euclidean case is more classical and, in particular, will give the history. The second talk will be about the affine Bonnet problem. Since both talks are about surfaces in 3-space, they will be fairly visual and won’t involve a lot of abstract machinery.

Abstract: The concept of 'holonomy', which was founded in mechanics, entered differential geometry almost 100 years ago, with the work of J. Schouten and E. Cartan. Its importance in applications to complex geometry was realized almost immediately, for metrics on complex varieties for which the complex structure is parallel, the Kahler metrics, are central to applications in algebraic geometry. This naturally raised the question of which holonomy groups were possible, and Berger introduced his classification in the 1950s. This led to milestone works by S.-s. Chern, E. Calabi, and S.-T. Yau in the following decades. There remained two mysterious special possibilities, in dimensions 7 and 8. In the 1980s, when these possibilities were proven to exist, they opened the way to the possibility of a vast generalization of string theory in theoretical physics.

In this talk, I will report on what has been happening in this area in the following 40 years up to the present day, where the ties with theoretical physics and geometric analysis have become ever deeper, raising fascinating questions with tantalizing partial answers.

Abstract: Familiar mechanical phenomenon, such as driving and parking a car, rolling a ball, and even the ability of falling cats to land on their feet (usually) are examples of an underlying mathematical concept that, in the 19th century, became to be known as `holonomy'. As it became better understood, mathematicians and physicists began to realize that holonomy underlay many different phenomena, from the everyday situations mentioned above to understanding the curvature of space in Einstein's theory of general relativity. Currently, holonomy lies at the heart of both deep mathematical objects and high-energy physical theories, such as string theory and the still-mysterious M-theory, on which many theoretical physicists would like to base a 'theory of everything'.

In this talk, after discussing some holonomic phenomena in everyday life, I'll explore their underlying commonality and their appearance in more advanced situations and try to provide some insight into why this idea has turned out to be so fundamental.

Abstract: The 6-sphere has a well-known G2-invariant almost complex structure and its pseudoholomorphic curves have interesting properties: They are minimal (though not homologically minimizing) and the cone on such a curve is an associative 3-fold whose local singular structure is well-understood. I will briefly survey some of the basic known results on the geometry of such pseudoholomorphic curves, in particular exploring their similarities and differences with holomorphic curves in CP3. Using a connection with the G2-invariant holomorphic 2-plane field on the complex 5-quadric discovered by Cartan, I will explain how one can construct pseudoholomorphic curves with some prescribed singularities. If time permits, I will discuss some of what I know about the Gromov compactification of certain moduli spaces of pseudoholomorphic curves in the 6-sphere.

Abstract: Following Sophus Lie's pioneering work on continuous transformation groups and their applications to differential equations, the structure theory of the finite (dimensional) case was well-developed, but the story in the infinite (dimensional) case was much more obscure. In the first decade of the 20th century, Élie Cartan's research on the infinite case was profound but beset with significant difficulties. By mid-century, geometers were ready to begin addressing these difficulties. In 1965, Singer and Sternberg revisited Cartan's classification in an influential paper published in Journal d'Analyse Mathématique, completing and clarifying Cartan's work and laying the foundation for our modern understanding of the so-called `infinite groups'. In this lecture, I will explain what the interesting issues were and still are and explain how they relate to our current understanding of diffeomorphism groups.

The mathematical concept of 'holonomy' was developed in the late 19th and early 20th centuries in a number of different contexts, and it was found to lie at the base of many everyday phenomena. Anyone who has ever tried to control the orientation of a 3-dimensional object on a computer screen using a trackball has dealt with the problem of trying to control three parameters (yaw, pitch, and roll) with an interface that can only accept two inputs (the direction and speed of rotation of the ball). That one can do this (and many other similar feats, such as parallel parking a car or a trailer or, for a cat, the ability to turn itself in the air to land on its feet) is the phenomenon of 'non-holonomy' of mechanical systems.

Making this somewhat vague concept precise, so that it can be put to use, has occupied engineers and mathematicians for more than a century. It turns out to be deeply geometrical in nature. Even as simple a system as a ball rolling over a surface without slipping or twisting turns out to have surprising connections with other parts of mathematics, including the so-called 'exceptional' groups. Holonomy is used to detect the curvature of spacetime, and constraints on it are used to describe systems that are important in string theory and particle physics.

I will start by recalling the basic geometry of G2-structures on 7-manifolds and their relation with special holonomy in dimension 7. Then, I’ll discuss a natural evolution defined for closed G2-structures, known as the closed G2 Laplacian flow, that is a natural analog of Ricci flow for Riemannian metrics. After discussing the short-time existence and uniqueness question, I will discuss the local ‘generality’ of solitons for this flow in E ́. Cartan’s sense, introducing a natural exterior differential system whose integral manifolds correspond to these solitons. Finally, I will discuss the special case of gradient solitons for this flow, and the remaining question of their generality.

An analytic system of PDE can be shown to have solutions provided that it is involutive, a property that generalizes and includes both the Cauchy-Kowalevskaya case and the Froebenius case. This was the main tool used by Élie Cartan in his study of transformation groups and latter works in differential geometry. For systems that are not involutive, Cartan devised a process of prolongation whose goal was to either uncover obstructions to existence of solutions of a given system or to replace it with an; 'equivalent; system that was involutive. Cartan tried to prove that prolongation did indeed achieve its goal, but was not able to do so. In one of his first major works, M. Kuranishi gave a sufficient condition for the prolongation process to succeed.

In this talk, I will review the historical background of Kuranishi's fundamental result and discuss some interesting geometric problems (some solved, some still open) where prolongation has turned out to be useful, even essential, in solving (or at least making progress in understanding) the problems described.

ABSTRACT: The 6-sphere has a well-known G_2-invariant almost complex structure and its pseudoholomorphic curves have interesting properties: They are minimal (though not homologically minimizing) and the cone on such a curve is an associative 3-fold whose local singular structure is well-understood. I will briefly survey some of the basic known results on the geometry of such pseudoholomorphic curves, in particular exploring their similarities and differences with holomorphic curves in \mathbb{CP}^3. Using a connection with the G_2(\mathbb{C})-invariant holomorphic 2-plane field on the complex 5-quadric discovered by Cartan, I will explain how one can construct pseudoholomorphic curves with some prescribed singularities. I will discuss some of what I know about the Gromov compactification of certain moduli spaces of pseudoholomorphic curves in the 6-sphere.

For G2-structures on 7-manifolds, there is a natural analog of the Ricci-flow studied in Riemannian geometry, namely, one considers a 1-parameter family σ = σ(t) of G2-structures on a given 7-manifold that satisfies the equation
dσ/dt = ∆σ with a specified initial G2-structure σ(0) = σ0. When the 1-parameter family σ moves by diffeomorphism and scaling, we say that σ is a soliton for the G2-Laplacian flow. The most interesting case is when the initial G2-structure is closed. In this talk, I will describe some of what is known about the existence and local generality of solitons for this flow, concluding with a discussion of the still-unsolved problem of the generality of the gradient solitons, which are of great interest in the theory of G2-structures.

As the Director of the Simons Collaboration on Special Holonomy, I made this presentation at the Simons Foundation. A video of the presentation is available at the link above.

This talk served as an introduction to the meeting, including background on the area of special holonomy and an overview of the fundamental existence results, progress made by our collaboration (and others) and what we see as the major goals and challenges in current research in special holonomy.

This was a plenary lecture at a week-long NSF-supported conference at the University of Michigan whose purpose was to highlight the work of and facilitate networking among LG&BTQ mathematicians.

Abstract: For closed G_2-structures on 7–manifolds, there is a natural analog of the Ricci-flow for Riemannian metrics. In this talk, after giving a brief introduction to the geometry of G_2–structures, I will discuss what we know about this flow, including short time existence, convergence, and the existence of solitons.

This was a lecture during the Abel Prize Week that introduced a lay audience to the work of Karen Uhlenbeck, the 2019 Abel Prize Recipient.

Abstract: Ever since the Greeks, the challenges of understanding limits and infinities have fascinated us, ultimately leading to the development of calculus and much of modern mathematics. When does a limit exist and in what sense? How do we capture these notions in geometric and intuitive ways? Professor Uhlenbeck's work provides fundamental ideas for how to interpret situations in which one would like to take a limit of a set of geometric objects and interpret the result in useful ways. I will try to give a sense of what the challenges are and how Uhlenbeck's ideas provide answers to questions that mathematicians and physicists have been asking for many years. At the end, I will give a sense of how influential her work has been and continues to be.

Three lectures were given, one for high school students, one for undergraduate students, and one for graduate students and faculty

"Mathematical Mysteries of the Ellipse"
Thursday, March 28, 2019, 4:00pm-5:00pm, 130 Nicholson Hall
High-School-Level Lecture

"Geometry Old and New: From Euclid to String Theory"
Friday, March 29, 2019, 2:30pm-3:30pm, 130 Howe-Russell Geoscience Complex
Undergraduate-Level Lecture

"The Best Possible Shapes of Surfaces"
Friday, March 29, 2019, 4:10pm-5:10pm, 130 Howe-Russell Geoscience Complex
Graduate-Level Lecture

Much of classical mathematics involves finding a configuration or shape that provides an optimum solution to a problem. For example, it has long been known (though a rigorous proof took quite a while to find) that the surface of least area enclosing a given volume is a round sphere. There are many other ways to measure surfaces, though, and finding 'the' surface that optimizes a given 'measurement' (subject to some given constraints) remains a challenging problem that has motivated some of the deepest recent work in the mathematics of geometric shapes.
In this talk, I will explain some of the classic ways to measure shapes of surfaces and relate this to classical problems involving surface area (soap films and bubbles) and total curvature as well to as recent progress by myself and others on these important optimization problems.

The notion of groups of motions and symmetries of objects is a fundamental one in mathematics, and it is nowhere more important than dealing with continuous motions and and problems in physics and geometry. I will start by discussing familiar objects, such as the group of rotations in space and higher dimensions, and move on to other groups, mainly groups of linear transformations, so that we can stay as concrete as possible. After developing the theory of continuous groups and seeing how these objects can be studied using multivariable calculus and linear algebra, we will then apply these concepts to study some mechanical systems and the surprising connection between symmetries and conservation laws that is so fundamental in modern mathematics and physics.

In this talk, I will describe a couple of cases in which Finsler geometry leads to connections with special holonomy. In particular, I will describe how a Finsler metric with constant positive flag curvature induces a canonical Kahler metric on its space of geodesics and how certain such Finsler metrics determine (and are determined by) spaces carrying a torsion-free connection whose holonomy is 'exotic' in the sense that it is not on the lists of the currently accepted groups that can be holonomy of torsion-free connections.

Abstract: An algebraic curve in the projective plane (or, more generally in a higher dimensional projective space) is said to be 'self-dual' if it is projectively equivalent to its dual curve (after, possibly, an automorphism of the curve). Familiar examples are the nonsingular conics (or, more generally, rational normal curves in higher dimensions) and the 'binomial curves' y^a = x^b, but there are many more such curves, even in the plane.

I'll survey some of the literature on these curves, particularly in the plane and 3-space, and some of what is known about their classification and moduli, including their connection with contact curves in certain contact 3-folds, some of which are singular. I'll also provide what appear to be some new examples of these curves.

Familiar mechanical phenomenon, such as driving and parking a car, rolling a ball, and even the ability of falling cats to land on their feet (usually) are examples of an underlying mathematical concept that, in the 19th century, became to be known as `holonomy'. As it became better understood, mathematicians and physicists began to realize that holonomy underlay many disparate phenomena, from the everyday situations mentioned above to understanding the curvature of space in Einstein's theory of general relativity. Nowadays, holonomy lies at the heart of both deep mathematical objects and high-energy physical theories, such as string theory and the still-mysterious M-theory, on which many theoretical physicists would like to base a 'theory of everything.
In this talk, after discussing some holonomic phenomena in everyday life, I'll explore their underlying commonality and their appearance in more advanced situations and try to provide some insight into why this idea has turned out to be so fundamental.

There are various methods known now for constructing more-or-less explicit metrics with special holonomy; most of these rely on assumptions of symmetry and/or reduction. Another promising method for constructing special solutions to look for metrics that satisfy algebraic curvature conditions. This often leads to a study of structure equations
that satisfy an overdetermined system of PDE, sometimes involutive sometimes not, and the theory of exterior differential systems is particularly well-suited for analyzing these problems.

In this talk, I will describe the ideas and the underlying techniques needed from the theory of exterior differential systems, illustrate the application in the most basic cases, and describe the results so far.
A similar program has been implemented for finding explicit calibrated submanifolds of the associated geometries and, time permitting, I will describe some of this work and the current results.

This talk surveys progress in the past year on classifying the algebraically special associative submanifolds in R^7, in particular, the ones for which the second fundamental form has nontrivial symmetries, and metrics with special holonomy whose curvature tensors are algebraically special.

This was an invited lecture at an NSF-CBMS conference in Bozeman, MT on Topological and Geometric Methods in QFT. I was recruited to lecture on G-structures and their applications in understanding geometric structures arising in quantum field theory.

When Marcel Berger classified the possible holonomies of Riemannian metrics in 1954, all but a handful of the possibilities were already known to exist and those covered important cases, such as Kahler geometry (which corresponds to metrics admitting a parallel complex structure). In the following years, great advances were made, such as Calabi's construction of what are now called hyper-kohler metrics and Yau's solution of the Calabi conjecture, proving the existence of Ricci-flat Kahler metrics (which also admit a parallel holomorphic volume form) in important special cases.

This left two so-called "exceptional cases" in dimensions 7 and 8, which were not even known to exist locally until 1984. In the intervening years, these metrics have made an appearance in theoretical high-energy physics and mathematicians, particularly Dominic Joyce, have developed new techniques for constructing compact examples. In recent years, mathematicians exploring these constructions and their connections with gauge theory have made significant advances in our understanding of these mysterious but beautiful geometries.

In this talk, I will give an introduction to the subject of special holonomy followed by a survey of recent important results and the prospects for further progress and their applications.

A Riemannian manifold $(M,g)$ is said to be {\it curvature-homogeneous} if it is homogeneous to second order, i.e., if, for any two points in $M$ the Riemannian curvature tensors are equivalent under some isometry of the two tangent spaces. Of course, a locally homogeneous metric is curvature-homogeneous, but the converse is not true in dimensions greater than $2$. (For a surface (i.e., in dimension $2$) curvature-homogeneity is equivalent to having constant Gaussian curvature, and such metrics are, of course, all classified locally and they are locally homogeneous.) Already in dimension $3$, there are many questions about the existence and generality of curvature homogeneous metrics, even locally. In this case, curvature-homogeneity is equivalent to having the eigenvalues of the Ricci curvature be constant, which is a system of partial differential equations on the metric.

In this talk, I review what is known about such metrics in dimension $3$, particularly the work of O. Kowalski and his collaborators during the 1990s. I will show that, for certain values of the eigenvalues of the Ricci tensor, these partial differential equations are integrable by the Darboux’ method, which yields some surprising relations with classical subjects, such as the theory of holomorphic curves in the complex projective plane and the geometry of contact curves in the $3$-sphere.

This was the second of two lectures that I gave in the Swarthmore College Distinguished Lecture Series, the Dresden Lectures. This was an introduction to Finsler geometry and, particularly Finsler surfaces for a lay audience, covering their physical motivations and explaining a number of interesting phenomena.

This was the first of two public lectures in the Swarthmore Distinguished Lecture Series, the Dresden Lectures. This lecture was to introduce the concept of holonomy and how it appears in modern mathematics

I presented a survey on my work on the properties of rigid curves in Engel structures, and I constructed examples of compact Engel manifolds that supported complex and/or symplectic structures.

This was an invited plenary address. I introduced the geometry of Finsler manifolds, gave examples, and explained some of the recent results of myself and others in the field.

There are various methods known now for constructing more-or-less explicit metrics with special holonomy; most of these rely on assumptions of symmetry and/or reduction. Another promising method for constructing special solutions is provided by the strategy of looking for metrics that satisfy algebraic curvature conditions. This method often leads to a study of structure equations that satisfy an overdetermined system of partial differential equations, sometimes involutive sometimes not, and the theory of exterior differential systems is particularly well-suited for analyzing these problems. In this talk, I will describe the ideas and the underlying techniques needed from the theory of exterior differential systems, illustrate the application in the most basic cases, and describe the landscape for the research needed to carry out this program. A similar program is envisioned for finding special calibrated submanifolds of the associated geometries and, if time permits, I will describe some of this work and the initial results.

These two lectures will describe some of the techniques that have been found useful for explicitly finding the integrals of exterior differential systems arising in geometry. These include Darboux’ method and its generalizations, compatible reductions, integrable extensions and Bäcklund transformations, conservation laws, and methods connected with analysis of the characteristic variety. Emphasis will be placed on illustrative examples and computations using the structure equations of Cartan. The first lecture will begin with a brief summary of the basics of exterior differential systems (EDSs) including involutivity, Cartan-Kähler theory, and the characteristic variety. This will be followed by an introduction to EDS formulations of some geometric problems via Cartan’s structure equations, illustrated by examples. The second lecture will focus mainly on examples, including minimal submanifolds, pseudo-holomorphic curves, Willmore geometry, and prescribed curvature problems in Riemannian and Finsler geometry.

Opening Lecture at the Conference: A Riemannian manifold $(M,g)$ is said to be {\it curvature-homogeneous} if it is homogeneous to second order, i.e., if, for any two points in $M$, the Riemannian curvature tensors are equivalent under some isometry of the two tangent spaces. Of course, a locally homogeneous metric is curvature-homogeneous, but the converse is not true in dimensions greater than $2$. Already in dimension $3$, there are many unanswered questions about the existence and generality of curvature homogeneous metrics, even locally. In this case, curvature-homogeneity is equivalent to having the eigenvalues of the Ricci curvature be constant, which is a system of partial differential equations on the metric. In this talk, I will review what is known about such metrics in dimension $3$, particularly the work of O. Kowalski and his collaborators during the 1990s and, more recently, Schmidt and Wolfson. I will show that, for certain values of the eigenvalues of the Ricci tensor, these partial differential equations are integrable by Darboux’ Method, which yields some surprising relations with classical subjects, such as the theory of holomorphic curves in the complex projective plane and contact curves in the boundary of the complex 2-ball.

A Riemannian manifold $(M,g)$ is said to be {\it curvature-homogeneous} if it is homogeneous to second order, i.e., if, for any two points in $M$, the Riemannian curvature tensors are equivalent under some isometry of the two tangent spaces. Of course, a locally homogeneous metric is curvature-homogeneous, but the converse is not true in dimensions greater than $2$. (For a surface (i.e., in dimension $2$) curvature-homogeneity is equivalent to having constant Gaussian curvature, and such metrics are, of course, all classified locally and they are locally homogeneous.) Already in dimension $3$, there are many unanswered questions about the existence and generality of curvature homogeneous metrics, even locally. In this case, curvature-homogeneity is equivalent to having the eigenvalues of the Ricci curvature be constant, which is a system of partial differential equations on the metric.

In this talk, I will review what is known about such metrics in dimension $3$, particularly the work of O. Kowalski and his collaborators during the 1990s and, more recently, Schmidt and Wolfson. I will show that, for certain values of the eigenvalues of the Ricci tensor, these partial differential equations are integrable by Darboux’ Method, which yields some surprising relations with classical subjects, such as the theory of holomorphic curves in the complex projective plane and contact curves in the boundary of the complex 2-ball.

When H(x) is a real-valued function of a real variable, the 2-step recursion relation x_{i+1} = H(x_i) - x_{i-1} is said to be n-periodic if it is periodic with period n for all initial conditions (x_0,x_1). These n-periodic recursion relations and their generalizations for certain values of n turn out to have close connections with interesting problems in both dynamical systems and in the theory of cluster algebras. In this talk, I will explain what is known about the classification (up to a natural notion of equivalence) of such recursion relations and their surprising relationship with differential geometry, cluster algebras, and the theory of overdetermined differential equations. I won’t assume that the audience knows any differential geometry, just basic multi-variable calculus, and the emphasis will be on describing the interesting results rather than on technical details.

A sequence of functions $f = (f_i)$ ($-\infty < i < \infty$) on a surface $S$ is said to be \emph{equi-areal} (or sometimes, \emph{equi-Poisson}) if it satisfies the relations $$ df_{i-1}\wedge df_i = df_i\wedge df_{i+1}\ (\not=0) $$ for all $i$. In other words, the successive pairs $(f_i,f_{i+1})$ are local coordinates on $S$ that induce the same area form on $S$, independent of $i$. One says that $f$ is \emph{$n$-periodic} if $f_i = f_{i+n}$ for all $i$. The $n$-periodic equi-areal sequences for certain values of $n$ turn out to have close connections with interesting problems in both dynamical systems and in the theory of cluster algebras. In this talk, I will explain what is known about the classification (up to a natural notion of equivalence) of such sequences and their surprising relationships with differential geometry, cluster algebras, and the theory of overdetermined differential equations. I won’t assume that the audience knows much differential geometry, just basic multi-variable calculus, and the emphasis will be on describing the interesting results rather than on technical details.

The problem of determining the possible holonomy groups of $G$-structures with prescribed conditions has had a long and distinguished history, and the works of A. Lichnerowicz have played a fundamental role in this story. Many of these questions, at least at the local level, can be understood as problems of determining whether a given set of \emph{structure equations} (in the sense of \'Elie Cartan) has a solution. More explicitly, given a vector space~$V$ of dimension~$n$ and a submanifold~$A\subset V\otimes\Lambda^2(V^\ast)$, one wishes to know when there exists an $n$-manifold~$M$, a $V$-valued coframing~$\omega:TM\to V$, and a mapping~$a:M\to A$ satisfying $$ \mathrm{d}\omega = a (\omega\wedge\omega). $$ More generally, one wishes to know how to classify or describe such data~$(M,\omega,a)$ up to local equivalence. (The case when $A$ is a single point is, of course, resolved by the third fundamental theorem of Lie.) This problem and its natural generalizations (to be discussed in the talk) encompasses a vast array of problems in differential geometry. I will discuss analysis of this problem via the tools of Cartan-K\"ahler theory and exterior differential systems and will describe some of its applications to problems in holonomy as well as problems involving geometrically natural PDE that arise in various contexts, including mathematical relativity.

This set of 3 lectures developed techniques using exterior differential systems for understanding $G$-structures whose torsion and curvature satisfy geometrically defined conditions. It included a lecture on the basics of exterior differential systems and then the remaining two lectures were applications of those ideas to problems in understanding the generality of $G$-structures satisfying specified geometric conditions.

As is well-known, the holonomic system of a ball rolling without slipping or twisting on a plane is described by a 2-plane field with growth vector (2,3,5), but this is not the famous plane field discovered by Cartan and Engel, whose local automorphism group is of dimension 14. However, the case of a ball of radius 1 rolling without slipping or twisting over a ball of radius 3 does turn out to be locally isomorphic to the Cartan/Engel 2-plane field. Recently, Nurowski and An found examples of surfaces whose holonomic system when rolling over a plane *is* the Cartan/Engel 2-plane field, and I have shown that, for one surface rolling over another (where the two surfaces have distinct Gauss curvatures), the associated 2-plane field cannot be of Cartan/Engel type unless at least one of the surfaces has constant Gauss curvature. I will report on this result and some other related geometric constructions of 2- and 3-plane fields whose equivalence with the flat model is of interest.

The mechanical system of one rigid surface rolling over another without twisting or slipping is a staple of non-holonomic mechanics and has been studied from a number of different points of view. The differential equations that describe this motion turn out to be a special case of a system of PDE studied by E ́lie Cartan in 1910. Remarkably, Cartan showed that such systems can have a symmetry group with dimension as large as 14 (and that, in this case, the symmetry group is isomorphic to the exceptional group G2). For example, it turns out that a sphere of radius 1 rolling over a sphere of radius 3 belongs to this highly symmetric case. In recent years, there have been some surprising developments; Nurowski and An have discovered a remarkable convex surface in 3-space whose differential constraints that describe its rolling over the flat plane have G2-symmetry. In this talk, I will describe the history of this problem, the geometry that goes into its study, and the recent develop- ments, including some results of my own that provide progress in classifying the pairs of surfaces whose rolling constraints have exceptional symmetry.

Given a closed, convex curve C in the plane, a billiard path on C is a polygon P inscribed in C such that, at each vertex v of P, the two edges of P incident with v make equal angles with the tangent line to C at v. (Intuitively, this is the path a billiard ball would follow on a frictionless pool table bounded by C.) For 'most' convex curves C, there are only a finite number of triangular billiard paths on C, a finite number of quadrilateral billiard paths, and so on. Obviously, when C is a circle, there are infinitely many closed billiard n-gons inscribed in C, but, surprisingly, the same is true when C is an ellipse. (This is a famous theorem due to Chasles.) The interesting question is whether there are other curves, besides ellipses, for which this is true. In this talk, I'll discuss these phenomena and show how they are related to the geometry of nonholonomic plane fields (which will be defined and described).

The notion of 'holonomy' in mechanical systems has been around for more than a century and gives insight into daily operations as mundane as steering and parallel parking and in understanding the behavior of balls (or more general objects) rolling on a surface with friction. A sample question is this: What is the best way to roll a ball over a flat surface, without twisting or slipping, so that it arrives at at given point with a given orientation? In geometry and physics, holonomy has turned up in many surprising ways and continues to be explored as a fundamental invariant of geometric structures. In this talk, I will illustrate the fundamental ideas in the theory of holonomy using familiar physical objects and explain how it is also related to group theory and symmetries of basic geometric objects.

While the local generality of metrics with special holonomy is now well-understood and the existence of complete or even compact examples is well-established, the construction of explicit examples remains a challenge. An approach that has been followed with great success is to look for examples with large symmetry groups, for this often reduces the problem to studying systems of ordinary differential equations. A different approach is to look for examples whose curvature tensors satisfy some algebraically natural conditions, for this imposes higher order (partial) differential equations on the solutions, and so the methods of exterior differential systems can be brought to bear to analyze the resulting overdetermined systems of equations. This can sometimes yield solutions with relatively high cohomogeneity and may be expected to provide interesting characterizations of some of the known solutions as well. In this talk, I will report on work in progress along these lines, particularly in the low dimensions, in which the holonomy can be either $SU(2)$, $SU(3)$, or $G_2$. I will also show how these methods can be used to study similar problems in certain geometries with torsion.

http://www.claymath.org/events/invitation-geometry-and-topology-g2 <p> This was a series of 5 lectures introducing the audience to the fundamentals of special holonomy metrics. The lectures were entitled: 1.Special Holonomy in Riemannian Geometry 2. Differential forms and spinors 3. Local existence and generality 4. Algebra and geometry of G_2 and Spin(7) 5. Closed G_2 structures

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The committee also serves as an oversight committee for the associate editors of Mathematical Reviews and, in conjunction with the administrative staff, reviews the editorial functions of the editors.

The Conference Board of the Mathematical Sciences (CBMS) is an umbrella organization consisting of nineteen professional societies all of which have as one of their primary objectives the increase or diffusion of knowledge in one or more of the mathematical sciences. Its purpose is to promote understanding and cooperation among these national organizations so that they work together and support each other in their efforts to promote research, improve education, and expand the uses of mathematics.

The AAAS-AMS Invited Address Selection Committee will select the lecturer for the AAAS-AMS Invited Address to be given each year at the Joint Mathematics Meetings.

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.

Yuri Tschinkel is the Executive Director of the ICM 2026 Organizing Committee, and I will be serving as a member of this committee, in charge of arranging the non-scientific support for the ICM 2026.

I chaired the committee to review the programs of Mathematics and Theoretical Computer Science at the Austrian Institute of Science and Technology. This is a review of their research and training program that is conducted every 10 years.

The study of the geometry of higher dimensional spaces, important in many applications, both theoretical and applied, has led to an understanding of their properties in terms of *holonomy*, a way of describing global effects of curvature. Those geometric spaces with ‘special’ (i.e., reduced) holonomy have come to play a fundamental role in partial differential equations, algebraic geometry, calculus of variations, topology, and theoretical physics, often revealing connections between these subjects that are yielding new insights in both mathematics and physics.
The 2022 annual meeting of the Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics will highlight and explain the progress in our knowledge of spaces with special holonomy that has been made in recent years by the Collaboration, focusing on the construction of new examples and the definition of geometric invariants that distinguish such structures when they are defined on the same underlying manifold; the increasing role of solitons and limits of Ricci-flat spaces; moduli spaces and BPS invariants; and connections with physics, particularly the new areas of higher form symmetries. We will also describe goals of our continuing research program as well as the challenges that lie ahead.

I serve on the Local Organizing Committee that is planning the 2026 International Congress of Mathematicians, to be held in Philadelphia, PA, July 23–30, 2026. (This is the quadrennial meeting of the International Mathematical Union. This will be the first time in 40 years that the ICM has been held in the US.)

The Ricci flow has played a decisive role in geometry and topology during the forty years since it was first introduced by Richard Hamilton. Other geometric flow equations have also led to impressive results, and all of these developments have co-evolved as interdependent parts of a thriving intellectual ecosystem. The workshop will bring established international experts in this vibrant area into contact with promising young researchers who are developing new approaches to the subject. The workshop will thus simultaneously celebrate the stunning past achievements of geometric flows, and provide a forum for the development of new ideas by a new generation of geometric analysts.

The link explains the committee's charge

This is the NASEM umbrella organization for the U.S. National Committees in the various different disciplines. I am the alternate representative of the US National Committee on Mathematics. The 'main' member for USNCM is Eric Friedlander.

I was a member of the U.S. Committee to bid for the 2026 International Congress of Mathematicians. Our committee was appointed by the U.S. National Committee for Mathematics to prepare a bid for this quadrennial international congress, which would be presented at the 2022 International Congress of Mathematicians (ICM). We proposed to host the 2026 ICM in Philadelphia, PA in July 2026 after spending several months scouting venues. We presented our proposal at the 2022 ICM in August, and our proposal was accepted by the Executive Committee of the International Mathematical Union. (I am now a member of the planning Committee for the 2026 ICM.)

This was the search committee for the new NSF Assistant Director for Mathematics and the Physical Sciences.

The USNC/M promotes the advancement of the mathematical sciences in the United States and throughout the world and effects appropriate participation in all activities of the International Mathematical Union (IMU) through the National Academy of Sciences, which adheres to IMU on behalf of mathematical scientists of the United States. The IMU serves to build global interatction among research communities and promote advances in the mathematical sciences at the international level. This committee serves to advise the National Acadmies in all matters pertaining to IMU, a member union of the International Science Council (ISC).

As Section Chair for Section 11 (Mathematics), it is my responsibility to conduct the annual election voting for Section 11 and to keep our Section informed of items of interest to professional mathematicians.

This was the committee that recruited and nominated candidates for the new Director of the Division of Mathematical Sciences

About the Foundation

The first phase of EDGE was launched in 1998 by Sylvia Bozeman and Rhonda Hughes to strengthen the ability of female students to successfully complete graduate programs in the mathematical sciences. As the number of EDGE participants grew so did its mentoring efforts. Now EDGE is much more than a summer bridge program. EDGE programs provide ongoing support for women pursuing careers in the mathematical sciences at several critical stages of their careers. In 2013, in response to an overwhelming push from EDGE participants, we established the Sylvia Bozeman and Rhonda Hughes EDGE Foundation to support these established programs.

Project Euclid's Advisory Board provides strategic, programmatic, and fiscal guidance for the project, as well as advocacy to the stakeholders and communities served by Project Euclid.

The membership of the board represents the diverse communities that Project Euclid serves: mathematicians and statisticians (as both authors and readers of scientific literature), partner publishers, and librarians.

Math Community Services ; In November 2013, I was elected to be the American Mathematical Society President for the years 2015 & 2016. This means that I will serve on the Council of the AMS for 4 years, the first year (2014) as President-elect, the next two years (2015 & 2016) as President, and then the fourth year (2017) as Past President.

The Advisory Committee for Mathematical and Physical Sciences (MPSAC) provides advice and recommendations to the National Science Foundation's programs within the Directorate for Mathematical and Physical Sciences (MPS).

Math Community Services

This is the committee that, each year, helps the AAAS manage the nomination and election of new members to its Class I, Section 1 (Mathematics)

FIMU is a US-based non-profit organization that receives donations to support its programs and services to benefit the international mathematical community.