This seminar series is jointly organised by the universities of Amsterdam (Thomas Rot and Rob Vandervorst), 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 three-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.

The seminars will be held online due to the pandemic. Zoom information can be found below. Lectures will be posted on our Youtube channel .

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. Ozsváth and Z. Szabó 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

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.