Summer Graduate School

We will discuss various geometric inequalities motivated by famous existence  theorems of various minimal objects in differential geometry proven by  topological methods. Let M be a closed Riemannian manifold. Quantitative  versions of such theorems as the existence of a periodic geodesic on M due to  A. Fet and L. Lusternik, the existence of infinitely many geodesics between an  arbitrary pair of points on M (J. P. Serve) and the existence of three simple  closed geodesics ona Riemannian 2-sphere (L. Lusternik and L. Schnirelmann)  will be presented.

We will begin with a discussion of surfaces, next explore how the results for  surfaces can be generalized to curvature-free estimates on higher dimensional  manifolds. We will next discuss geometric inequalities that involve curvature  bounds. If time permits, we will also talk about the case of non-compact  complete manifolds with some geometric constraints, like finite volume.

In terms of the prerequisites, in addition to Do Carmo's Riemannian Geometry, I  would expect students to know some fundamentals of Algebraic Topology, such as  Homology and Homotopy Groups, which can be found in Hatcher's textbook.

The minicourse will start with a quick introduction, essentially from scratch, to currents  in metric spaces in the sense of Ambrosio-Kirchheim. This will be followed by a proof  of the isoperimetric filling inequality of Euclidean type for cycles in CAT(0) spaces. This important inequality is due to Federer-Fleming for Euclidean space and to Gromov and Wenger in the general case. Some applications will be discussed. If time permits,  an improvement of the isoperimetric inequality for cycles of dimension greater than or  equal to the asymptotic rank of the underlying CAT(0) space, also due to Wenger, will be sketched. This pertains to notions of higher-rank hyperbolicity studied recently in work of Kleiner, the lecturer, and others.

We shall look at Gromov’s classification of group actions on hyperbolic spaces and I’ll give a brief tour of how it is being used to study outer automorphism groups of free groups using relative free factor complexes.

We will discuss various geometric inequalities motivated by famous existence  theorems of various minimal objects in differential geometry proven by  topological methods. Let M be a closed Riemannian manifold. Quantitative  versions of such theorems as the existence of a periodic geodesic on M due to  A. Fet and L. Lusternik, the existence of infinitely many geodesics between an  arbitrary pair of points on M (J. P. Serve) and the existence of three simple  closed geodesics ona Riemannian 2-sphere (L. Lusternik and L. Schnirelmann)  will be presented.

We will begin with a discussion of surfaces, next explore how the results for  surfaces can be generalized to curvature-free estimates on higher dimensional  manifolds. We will next discuss geometric inequalities that involve curvature  bounds. If time permits, we will also talk about the case of non-compact  complete manifolds with some geometric constraints, like finite volume.

In terms of the prerequisites, in addition to Do Carmo's Riemannian Geometry, I  would expect students to know some fundamentals of Algebraic Topology, such as  Homology and Homotopy Groups, which can be found in Hatcher's textbook.

The minicourse will start with a quick introduction, essentially from scratch, to currents  in metric spaces in the sense of Ambrosio-Kirchheim. This will be followed by a proof  of the isoperimetric filling inequality of Euclidean type for cycles in CAT(0) spaces. This important inequality is due to Federer-Fleming for Euclidean space and to Gromov and Wenger in the general case. Some applications will be discussed. If time permits,  an improvement of the isoperimetric inequality for cycles of dimension greater than or  equal to the asymptotic rank of the underlying CAT(0) space, also due to Wenger, will be sketched. This pertains to notions of higher-rank hyperbolicity studied recently in work of Kleiner, the lecturer, and others.

Let MM be a non-simply connected Riemannian manifold. The least length of a closed curve on MM that is not contractible to a point is called the systole of MM. Yu. Burago and V. Zalgaller and, independently, J. Hebda proved that the systole of any closed Riemannian surface MM does not exceed 2Area(M)−−−−−−−−√2Area(M).  In 1983 M. Gromov discovered that for a special class of essential manifolds the systole does not exceed c(n)volume(M)1nc(n)volume(M)1n, where nn denotes the dimension. His paper established connections between the systolic inequality and higher codimension isoperimetric inequalities in Banach spaces, introduced new natural metric invariants of Riemannian manifolds and became a starting point for development of the area of systolic geometry. We are going to sketch essential elements of Gromov's proof and then review some newer developments in the study of systoles including a much simpler proof of Gromov's result recently found by P. Papazoglou. Other topics include isoperimetric inequalities for Hausdorff contents and upper bounds for the systole in terms of Hausdorff contents discovered by Y. Liokumovich, B. Lishak, R. Rotman and the speaker.

The minicourse will cover some aspects of metric and analytical structure on boundaries of Gromov hyperbolic spaces, applications to rigidity, and open problem.

Recommended preparatory reading:
(1) Quasi-isometries and the Milnor-Svarc lemma.  Bridson-Haefliger I.8; Drutu-Kapovich
8.1-8.3.
(2) Gromov hyperbolic spaces: definitions, examples, Morse lemma on stability of
quasigeodesics, definition of the boundary.  Bridson-Haefliger.  III.H.1, III.H.3; Drutu-
Kapovich 11.1, 11.10, 11.11, 11.13.
(3) The theorems of Rademacher and Stepanov, Section 3 in Lectures on Lipschitz analysis,
Heinonen, available here: 
http://www.math.jyu.fi/research/reports/rep100.pdf#page=18

The main goal will be to present a proof that mapping class groups have finite asymptotic dimension. This will give me a good excuse to talk about projection complexes, asymptotic dimension, curve complexes and subsurface projections. Most of this will be self-contained, with few "black boxes".

Reading list:
Hyperbolic groups and spaces, from the standard books like
Bridson-Haefliger or Kapovich-Drutu
Some familiarity with mapping class groups, e.g. the first 3 sections
of Farb-Margalit

Hierarchical hyperbolicity provides a common framework to work with various classes of spaces and groups such as mapping class groups, Teichmueller space, and cubical groups, as well as many fundamental groups of 3-manifolds, Artin groups, etc. I will explain what a hierarchically hyperbolic structure is and try to convey a picture of what a hierarchically hyperbolic space looks like.

Knot theory is divided into several subfields. One of these is hyperbolic knot theory, which is focused on the hyperbolic structure that exists on many knot complements. Another branch of knot theory is concerned with invariants that have connections to 4-manifolds, for example the knot signature and Heegaard Floer homology. In my talk, I will describe a new relationship between these two fields that was discovered with the aid of machine learning. Specifically, we show that the knot signature can be estimated surprisingly accurately in terms of hyperbolic invariants. We introduce a new real-valued invariant called the natural slope of a hyperbolic knot in the 3-sphere, which is defined in terms of its cusp geometry. Our main result is that twice the knot signature and the natural slope differ by at most a constant times the hyperbolic volume divided by the cube of the injectivity radius. This theorem has applications to Dehn surgery and to 4-ball genus. We will also present a refined version of the inequality where the upper bound is a linear function of the volume, and the slope is corrected by terms corresponding to short geodesics that have odd linking number with the knot. My talk will outline the proofs of these results, as well as describing the role that machine learning played in their discovery.

The minicourse will cover some aspects of metric and analytical structure on boundaries of Gromov hyperbolic spaces, applications to rigidity, and open problem.

Recommended preparatory reading:
(1) Quasi-isometries and the Milnor-Svarc lemma.  Bridson-Haefliger I.8; Drutu-Kapovich
8.1-8.3.
(2) Gromov hyperbolic spaces: definitions, examples, Morse lemma on stability of
quasigeodesics, definition of the boundary.  Bridson-Haefliger.  III.H.1, III.H.3; Drutu-
Kapovich 11.1, 11.10, 11.11, 11.13.
(3) The theorems of Rademacher and Stepanov, Section 3 in Lectures on Lipschitz analysis,
Heinonen, available here: 
http://www.math.jyu.fi/research/reports/rep100.pdf#page=18

The main goal will be to present a proof that mapping class groups have finite asymptotic dimension. This will give me a good excuse to talk about projection complexes, asymptotic dimension, curve complexes and subsurface projections. Most of this will be self-contained, with few "black boxes".

Reading list:
Hyperbolic groups and spaces, from the standard books like
Bridson-Haefliger or Kapovich-Drutu
Some familiarity with mapping class groups, e.g. the first 3 sections
of Farb-Margalit

The combination of Kazhdan’s property (T) and negative curvature typically limits the amount of outer automorphisms. Indeed, it is a result of Paulin that every property (T) hyperbolic group has a finite outer automorphism group. Belegradek and Szczepan ́ski extends Paulin’s result to property (T) relatively hyperbolic groups. We prove that for every countable group Q there is an acylindrically hyperbolic group G such that Out(G) = Q. Therefore the combination of property (T) and acylindrical hyperbolicity is much more flexible in terms of outer automorphisms.

The minicourse will cover some aspects of metric and analytical structure on boundaries of Gromov hyperbolic spaces, applications to rigidity, and open problem.

Recommended preparatory reading:
(1) Quasi-isometries and the Milnor-Svarc lemma.  Bridson-Haefliger I.8; Drutu-Kapovich
8.1-8.3.
(2) Gromov hyperbolic spaces: definitions, examples, Morse lemma on stability of
quasigeodesics, definition of the boundary.  Bridson-Haefliger.  III.H.1, III.H.3; Drutu-
Kapovich 11.1, 11.10, 11.11, 11.13.
(3) The theorems of Rademacher and Stepanov, Section 3 in Lectures on Lipschitz analysis,
Heinonen, available here: 
http://www.math.jyu.fi/research/reports/rep100.pdf#page=18

The main goal will be to present a proof that mapping class groups have finite asymptotic dimension. This will give me a good excuse to talk about projection complexes, asymptotic dimension, curve complexes and subsurface projections. Most of this will be self-contained, with few "black boxes".

Reading list:
Hyperbolic groups and spaces, from the standard books like
Bridson-Haefliger or Kapovich-Drutu
Some familiarity with mapping class groups, e.g. the first 3 sections
of Farb-Margalit

In many models of random groups, the groups are typically hyperbolic.  I'll survey some of what's known about these groups, with a particular focus on the properties of their boundaries at infinity.