This seminar series is jointly organised by the Vrije Universiteit Amsterdam (Thomas Rot , Leiden (Federica Pasquotto) and Utrecht (Alvaro del Pino Gomez). The seminar aims to introduce a wide audience (starting at a master level) into various research areas in differential topology and geometry. Every meeting is a 2-hour minicourse starting with an introduction to the field and ending with a discussion of open problems. Previous knowledge about the topic is not assumed. We strongly encourage master students with an interest in topology and geometry to attend. Please send an email to spam@math.leidenuniv.nl to be added to the mailing list. However, change spam into f.pasquotto. Past Lectures will be posted on our Youtube channel

Abstract: The efforts of many mathematicians during the eighties, nineties, and early two-thousands culminated in a clear path of action for the classification problem in 3-dimensional contact topology, both in the global version (contact structures) and the relative version (Legendrians/transverse links). However, the classification problem for families of objects (contact structures, contactomorphisms, Legendrians, transverse links) has remained mysterious, both in the case of tight and overtwisted contact 3-manifolds. Very recently, Eliashberg and Mishachev proved the contractibility of the space of tight contact structures standard at infinity in the Euclidean 3-space, giving a new impulse to the field.

In this series of talks, we will provide a panoramic overview of the classification problems for families in 3-dimensional contact topology. In the first introductory talk, we will review the general strategy to tackle the problem, culminating with the microfibration trick for embedding spaces. In the second talk, we will explore several instances where the trick applies, i.e., where a complete classification can be addressed, and we will use it to study the new subclass of strongly overtwisted contact structures, which is governed by a parametric h-principle. Some parts of these talks will be based on joint works with Martínez-Aguinaga, Min, Muñoz-Echániz, and Presas.

Abstract: The first part of the talk surveys the main properties of complex symplectic structures, the analogue of symplectic structures in the context of complex manifolds. We will show how they are related to many other geometric structures, for instance hyperkähler, hypercomplex and hypersymplectic. Complex symplectic structures on compact manifolds suffer the same problem as their real analogues: there is no general theorem that constructs them! Therefore, in the second part of the talk, we will present some construction techniques of complex symplectic structures on compact quotients of nilpotent Lie groups. The talk is based on joint works with M. Freibert, A. Gil García, A. Latorre, B. Meinke, and N. Tardini.

Abstract: In the first talk, I will review the classical work of Atiyah and Singer describing the homotopy type of the family of Dirac operators on a spin Riemannian manifold, and its consequences regarding metrics of positive scalar curvature. In the second talk, I will discuss how one can exploit the Seiberg-Witten equations and Floer theory to obtain more detailed information about the structure of the family in the case of a three-manifold for which the spectral gap of the Hodge Laplacian on coexact 1-forms is large compared to the curvature. For concreteness, we will have a special focus on the case of the n-torus throughout the talks.

Ever since Alan Weinstein's foundational 1983 paper, the local structure of Poisson manifolds has been a central problem in Poisson Geometry. Among other results, the paper contains the Splitting Theorem for Poisson structures, the formulation of the linearization problem and several conjectures, some of which have been solved soon thereafter by Jack Conn. In this talk, first I will explain these classical linearization theorems and some open problems. Then I will focus on recent work with my former PhD student, Florian Zeiser, about singularities with isotropy sl_2(C).

Every knot leaves a trace in the 4-dimensional world. The trace of a knot is the smooth 4-manifold obtained by attaching a 2-handle to the 4-ball along a knot in the 3-sphere. In the first part of the talk, will carefully introduce these notions and discuss a strategy to disprove the smooth 4-dimensional Poincare conjecture by finding knot traces with certain exotic properties. The second part of the talk is devoted to certain methods and results that might be useful to find such exotic knot traces.

Homotopy theory has proven to be a robust tool for studying non-homotopical questions about manifolds; for example, surgery theory addresses manifold classification questions using homotopy theory. In joint work with Sarah Yeakel, we are developing a program to study manifold topology via isovariant homotopy theory. I'll explain what isovariant homotopy theory is and how it relates to the study of manifolds via their configuration spaces, and talk about an application to fixed point theory.

The spectral flow is an integer-valued homotopy invariant for paths of selfadjoint Fredholm operators that was invented by Atiyah, Patodi and Singer in the seventies, for investigating elliptic differential operators on manifolds. Later it was used by Floer in the construction of his famous homology groups, and became a well known invariant in global and symplectic analysis as well as mathematical physics in the decades since.

In the first part of this lecture, we introduce some basic definitions and concepts of functional analysis like, e.g., Fredholm operators, spectra and functional calculus. The second part deals with different topologies on the space of selfadjoint Fredholm operators, for which the spectral flow induces an isomorphism between their fundamental groups and the integers.

Taking Klein bottles as a guiding theme, I shall discuss some aspects of 3-dimensional geometric topology and contact dynamics. Starting from Samelson's beautiful proof that hypersurfaces in Euclidean space are orientable, we will consider the question which 3-manifolds admit embedded Klein bottles,
and how to visualise such embeddings.

This provides an opportunity to learn some basic terms of 3-manifold topology (prime vs. irreducible manifolds, (in-)compressible surfaces, Dehn fillings). I shall then view this in the context of contact dynamics, where Klein bottles arise, albeit rarely, as critical surfaces of integrable Reeb flows.

These lectures are based on joint work with Norman Thies, Jakob Hedicke, and Murat Sağlam.

The Ray--Singer analytic torsion is a spectral invariant associated with the deRham complex on closed Riemannian manifolds.
A celebrated result due to Cheeger, M\"uller and Bismut--Zhang asserts
that it essentially coincides with the Reidemeister torsion, a
topological invariant.

Recently, analytic torsion has been extended to two kinds of filtered
geometries: contact manifolds and (2,3,5) geometries aka generic rank
two distributions in dimension five.

For these two geometries one can define analytic torsion by replacing
the deRham complex with the Rumin complex, a complex of higher order
differential operators which is intrinsic to the filtered geometry.
There are ongoing efforts to compare these new invariants with the
Ray--Singer torsion.

The first part of this minicourse will offer an introduction to analytic
torsion and the extension to contact manifolds due to Rumin and
Seshadri.

In the second part I will introduce (2,3,5) geometries, present some of
their intriguing features, and discuss their analytic torsion.

Second part of the lecture

I will give an introduction to Steenrod squares and the classical Hopf invariant one problem. Then I will talk about a generalisation of the classical theorem that orientable manifolds have even Euler characteristic unless the dimension is a multiple of 4. This generalization to higher orientable manifolds leads to a new open problem that extends the Hopf invariant one problem.

First part of the minicourse

Second part of the minicourse

Since the 1980s, Bill Thurston has done fundamental work in apparently quite different areas of mathematics: geometry of 3-manifolds, geometry of surface automorphisms, and dynamics of branched covers of the 2-sphere. In all contexts,
Thurstons theorems have fundamental importance and are the cornerstones of ongoing research. These results are surprisingly closely connected both in statements and in proofs. In all three areas, the statements can be expressed as follows: either a topological object has a geometric structure (the manifold is geometric, the surface automorphism has Pseudo-Anosov structure, a branched cover of the sphere respects the complex structure), or there is a well defined topological-combinatorial obstruction. Moreover, all three theorems are proved by an iteration process in a finite dimensional Teichmu
ller space (this is a complex space that parametrizes Riemann surfaces of finite type).

The goal of this minicourse is to discuss Thurstons theorems (up to some level of detail) and the common machinery (Teichmuller theory) in a unified way, so as to highlight many of the analogies between the results. We may also have a glimpse at the modern research in holomorphic dynamics continuing the legacy of Bill Thurston.

Freedman's disk embedding theorem is a cornerstone of 4-manifold topology. In order to appreciate the result, one has to be aware that manifold topology is divided into low and high dimensions. The border runs somewhere in dimension 4. The high dimensional world has powerful tools to tackle classification problems for manifolds, known as surgery theory and the s-cobordism theorem. A key feature of high dimensional manifolds is the Whitney trick, which allows to make certain geometric and algebraic intersection counts match. Unfortunately, the standard Whitney trick is not available in dimension 4 and it is known that the high-dimensional machinery does not apply to smooth 4-manifolds. In contrast, Freedman's theorem makes the Whitney trick available for a class of topological 4-manifolds (including all simply connected ones), thereby opening the gates for surgery and s-cobordism techniques. The goal of the minicourse is to explain the statement of the disk embedding theorem, to indicate why it is important, and to give at least a rough idea of the proof.

See this Quanta magazine article about the book Stefan coauthored about this topic.

The theory of C*-algebras offers an elegant setting for many problems in mathematics and physics. In view of Gelfand duality, their study is often referred to as non-commutative topology: general noncommutative C*-algebras are interpreted as non-commutative spaces. Many classical geometric and topological concepts can be translated into operator algebraic terms, leading to the so-called noncommutative geometry (NCG) dictionary, which I will describe in this talk.

The Dirac operator on a compact Riemannian manifold is a suitable "square root" of the Laplacian. One hand it allows one to recover geometric information such as the Riemannian distance function and various curvature tensors. On the other hand it encodes the fundamental class in K-homology. These two observations motivate the paradigm of operator algebraic non-commutative geometry as formulated by Connes. In this picture, the notion of manifold is replaced by a C*-algebra represented on a Hilbert space, together with a suitable unbounded self-adjoint operator. This formalism emerges in various areas of mathematics and physics. In this seminar we will sketch the main philosophical ideas and discuss some applications.

In 1915 Albert Einstein introduced a new theory of gravity
that no longer models gravitation as a force on a fixed geometric
background but instead views it as a inherent geometric property of a
four-dimensional manifold that marries space and time (and matter).
Through this theory emerged a lot of new and initially confusing
phenomena, such as singularities and black holes, whose inevitable
existence was firmly established by the incompleteness theorems of
Penrose and Hawking in the 1960s mathematically - and in recent years
also through observations by gravitational waves and the event horizon
telescope. Many other beautiful ideas, deep mathematical results and
exciting conjectures lie at the heart of the general theory of
relativity until today.

In the first part of this minicourse I will give a brief overview of the
vast mathematics that is involved in and contributed to our
understanding of the general theory of relativity. In the second part I
will describe old and new ideas to model the motion of particles in
general relativity with the help of naked singularities and the
conservation of energy-momentum a la Noether. Some basic background in differential geometry, in particular Riemannian
Geometry, and partial differential equations is useful.

Each open stable differential relation R imposed on smooth maps of manifolds determines cohomology theories k and h; the cohomology theory k describes invariants of solutions of R, while h describes invariants of so-called stable formal solutions of R. We prove the bordism version of the h-principle: The cohomology theories k and h are equivalent for a fairly arbitrary open stable differential relation R. Furthermore, we determine the homotopy type of h. Thus, we show that for a fairly arbitrary open stable differential relation R, the machinery of stable homotopy theory can be applied to perform explicit computations and determine invariants of solutions.

The lecture .

Slides first talk.

Slides second talk.

It can be very helpful to understand what kind of vector
bundles over a given manifold exist. Wether of producing good
(counter)examples or to understand your manifold better in some sense.

The task of classifying vector bundles over a fixed manifold can be
arbitrary difficult, especially in higher dimensions. Therefore in this
talk we would like to concentrate ourselves to low dimensions, where
vector bundles can be uniquely described by known (and computable)
invariants.

The case of 5-dimensional manifolds is of particular interest since it
cannot be treated like the other low dimensional cases. Here we will
discover that the classification (under certain mild assumptions) is
given by homotopy classes of maps from the 5-manifold into the 4-sphere.
The entirety of such classes are usually called a cohomotopy group. And by
probably the most beautiful construction in differential topology, the
Pontryagin-Thom construction, it is possible to understand the above
mentioned cohomotopy group in a very geometrical way.

One of the main mysteries in symplectic geometry is the existence of both flexible and rigid phenomena. On the rigid side, J-holomorphic curve invariants can distinguish different symplectic structures on the same smooth manifold, while on the flexible side, h-principles imply that certain special structures that are diffeomorphic must actually be symplectomorphic (and that J-holomorphic curve invariants vanish). An important construction in classical topology associates to a space X and prime p, a new "localized" space Xp whose homotopy and homology groups are obtained from those of X by inverting p. I will discuss a symplectic analog of this construction and explain how it sheds light on the flexibility-rigidity dichotomy. This talk is based on joint work with Z. Sylvan and H. Tanaka.

Knot Theory studies the classification of knots and links. One of the greatest breakthroughs in the area was the Jones polynomial, introduced by V. Jones in 1984, as it was the first polynomial knot invariant capable of distinguishing a knot from its mirror image. At the end of the past century, M. Khovanov introduced Khovanov homology as a categorification of the Jones polynomial. More precisely, this link invariant is a bigraded homology whose Euler characteristic is the Jones polynomial of the link. In the first part of the talk we will provide an overview of these two invariants and present some recent developments on the geometric realization of Khovanov homology.

The second part of the talk will be devoted to the Alexander polynomial, introduced by J. W. Alexander (and later re-discovered by J. Conway in terms of skein relations), which was the first polynomial invariant, and to Knot-Floer homology, introduced by P. OzsvÃƒÂ¡th and Z. SzabÃƒÂ³ around 2002 as a categorification of the Alexander polynomial. We will explain the original definition, based on holomorphic curves, and the more recent combinatorial definition, based on grid-diagrams.

This talk will begin with a simple introduction to cellular sheaves as a generalized notion of a network of algebraic objects. With a little bit of geometry, one can often define a Laplacian for such sheaves. The resulting Hodge theory relates the geometry of the Laplacian to the algebraic topology of the sheaf. By using this sheaf Laplacian as a diffusion operator, we will be able to do dynamics on sheaves, which leads to decentralized methods for computing sheaf cohomology. This talk will be grounded in examples arising in applications, with a particular focus on social networks and opinion dynamics, with problems of consensus and polarization as being especially well-suited to sheaf-theoretic analysis.

Both talks

Notes for the first talk

Notes for the second talk

Here you can find a paper Gabriele wrote for the DDTG.

Here is a nice "what is" introduction to systolic geometry.

In their classical formulation, systolic inequalities aim to study
Riemannian metrics on a givencompact surface by looking at the length of
the shortest non-constant periodic geodesic, also known as the systole.
One of the fundamental questions in the field is to find an upper bound,
independent of the metric, for the systolic length when the metric has
unit area.

After briefly discussing this question in general, we will concentrate
on the case of the two-sphere. Here an interesting class of metrics pops
up: Those for which all geodesics are systoles. Abbondandolo, Bramham,
Hryniewicz and Salomao showed recently that these so-called Zoll metrics
locally maximize the systolic length in the space of Riemannian metrics
on the two-sphere.

Their proof uses symplectic techniques and will lead us, in the second
talk, to consider an analogue of systoles and of Zoll metrics for
contact hypersurfaces inside symplectic manifolds. In the contact world,
global upper bounds for the systole do not hold but Zoll hypersurfaces
still remain local maximizers for the systolic length.

These phenomena are related to the famous Viterbo conjecture in
symplectic geometry about the capacity of convex domains in euclidean
space, which will finally bring us to the frontier of current research.

First lecture

Second lecture

The slides

Topological complexity is an integer-valued homotopy invariant of topological spaces that is motived by the motion planning problem from robotics and that was introduced by Michael Farber in 2003. I will give an overview over its definition and its most important properties before presenting some explicit computations and connections to other parts of topology and geometry.

Afterwards I will present a variety of results on the topological complexity of configurdation spaces that naturally occur in robotics in the study of simultaneous motion planning for multiple robots moving in the same workspace that are supposed not to collide while performing their tasks.
In the end, I will present some recent results and outline some current directions of research.

First lecture

Second lecture

Lecture notes (Warning: Large file (99 mb))

In the first part of the seminar we will discuss what are sub-Riemannian structures,
where do they come from and what are some of the major properties and open
questions, using some famous examples to drive the discussion.
For this part of the talk I will only assume some basic knowledge of Riemannian
geometry and calculus of variation.

In the second part of the talk, we will discuss recent advances in sub-Riemannian
spectral geometry, focussing in particular on the research around the meaning(s)
of intrinsic sub-Laplacians, their self-adjointness and what we can learn from studying
their spectra. The discussion will be structured to provide the main ideas and touch
upon some of the major open questions. For this part of the talk I will assume
some knowledge of functional analysis and operator theory.

Given the limited amount of time and to try and keep the talk understandable to
the broadest audience, I will stay away from the most complex technicalities as
much as possible. In any case, I will provide plenty of references for the interested
participants and the more experienced audience.