No seminar (dies academicus)
Discussion of possible topics
Transverse tori in Engel manifolds
We will give an introduction to Engel structures and discuss recent progress in the classification of Engel manifolds and their submanifolds.
Transversality for simple holomorphic curves and the slice theorem
I will review the necessary prerequisites for my talk next week.
Firstly, I'll show that simple holomorphic curves are generically
Fredholm regular; i.e. "transversality holds for simple curves". The
proof is standard, but will become much more interesting in the
equivariant case. Secondly, we will review the slice theorem for smooth
Lie group actions on Banach manifolds --- with an eye towards
infinite-dimensional applications. If time permits, we might see how
these are related.
Equivariant transversality for simple holomorphic curves
Consider a closed symplectic manifold with a compact Lie group G
acting symplectically. What can we say about the moduli space of
holomorphic curves on M, w.r.t. a generic G-equivariant almost
complex structure? We should not expect it to be a manifold (after all,
transversality and symmetry are famously incompatible). However, we can
hope for a clean intersection condition: the moduli space decomposes
into disjoint strata which are smooth manifolds of explicitly computable
dimension. I will focus on the case of simple curves (which is already surprisingly
interesting): I'll explain how to decompose the moduli space into
iso-symmetric strata and prove that each stratum is smooth.
Introduction to Khovanov homology and its variants
In this talk, we will discuss the construction
and properties of Khovanov homology, a link invariant introduced by
Khovanov in 1999. We will discuss some of its applications in low
dimensional topology, symplectic and contact geometry.
seminar cancelled due to that virus, you know the one
Khovanov homology of fibered-positive links and its cables
In this talk, we will discuss properties of fibered-positive
links and prove a vanishing theorem of Khovanov homology for such links and cables
obtained from them. Fibered-positive links are a proper subclass of links whose
associated open book decomposition induces the unique tight contact
structure on S3.
Untangling Khovanov homology and its geometrization
Khovanov homology was introduced by Mikhail Khovanov at the end of last century.
This link invariant, which categorifies the Jones polynomial, provides
geometric and topological information about the link. In particular, it is an unknot detector.
Some years later, Lipshitz and Sarkar introduced the Khovanov spectrum,
a refinement of Khovanov homology: given a link diagram D,
they provide a method to obtain a space whose stable homotopy type is a
link invariant, and whose cohomology is isomorphic to the Khovanov homology
of the link represented by D.
In this talk we will present some simplifications of the Khovanov spectrum for some specific
gradings. As a consequence, we will obtain properties of the Khovanov homology of several families of links.
Higher algebra of A-infinity algebras in Morse theory
Generic transversality and non-transversality in symmetric Morse theory
In standard Morse theory on a compact manifold M, for every
Morse function f on M a generic Riemannian metric g satisfies that (f,g)
is Morse-Smale, i.e. stable and unstable manifolds intersect
transversally for every pair of critical points. If we impose an
additional finite symmetry on M, i.e. if a finite group G acts on M by
diffeomorphisms such that f and g are G-invariant, then this statement
fails: there are Morse functions f such that (f,g) is never
Morse-Smale, for any g.
We will show that nontheless there exist for any manifold M and any
group G always G-invariant Morse functions f such that generic
G-invariant Riemannian metrics give Morse-Smale pairs (f,g), and we will
show that whenever the action by G is not free, then there also always
exist G-invariant Morse functions such that (f,g) is for no G-invariant
BMS Seminar Room (exceptional location)
Hamiltonian actions on Floer homology and Fukaya categories
I will talk about some algebraic structures arising on
Lagrangian Floer homology and Fukaya categories in the presence of a
Hamiltonian action of a compact Lie group.
On the one hand, this should be related to a conjecture of Teleman,
motivated by Homological Mirror Symmetry. On the other hand, our
motivation comes from gauge theory in low dimensions, and is part of a
program aimed at recasting Donaldson-Floer theory into an extended
More speculatively, we also expect these structures to arise in higher
dimensional gauge theory, following the Donaldson-Segal program, and
in relation with the Moore-Tachikawa field theories.
This is based on two joint works in progress, one with Paul Kirk, Mike
Miller-Eismeier and Wai-Kit Yeung, and another with Alex Hock and
Constrainahedra and the Fukaya category of Lagrangian torus fibrations
I will begin by describing two pieces of the context
for this talk: first, the symplectic (A-infinity,2)-category (Symp),
which is the natural setting for building functors between Fukaya
categories; and second, Lagrangian torus fibrations, which are the
central geometric objects in SYZ mirror symmetry. Next, I will explain
my construction with Daria Poliakova of a family of polytopes called
constrainahedra, which we introduced in order to define the notion of
a monoidal A-infinity category. Finally, I will describe work-in-progress
that aims to equip the Fukaya category of a Lagrangian torus fibration with
a monoidal A-infinity structure, which should be mirror to the tensor
product of sheaves. This is based on past and ongoing work with Daria
Poliakova and Mohammed Abouzaid, including arXiv:2208.14529 and arXiv:2210.11159.
On the Lagrangian capacity of
four-dimensional convex toric domains
The Lagrangian capacity is a symplectic capacity defined by
Cieliebak and Mohnke. In this talk, via a neck-stretching argument
of Cieliebak-Mohnke, we compare the Lagrangian and McDuff-Seigel
capacities of Liouville domains. As a consequence, we show that
the Lagrangian capacity of a 4-dimensional convex toric domain
is equal to its diagonal. This positively settles a conjecture
of Cieliebak and Mohnke for the Lagrangian capacity of the
Note for new students: If you think you might be interested in this seminar
but have neither attended before nor spoken with Profs. Mohnke, Wendl, or Walpuski about it,
it is a good idea to get in touch with one of us ahead of time!