Geometry and Topology seminar series
The seminars are usually held on Thursdays at 13:15 in room Å64119 and on Zoom.
For more info contact the organiser: Alex Takeda
Upcoming seminars
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December 2023

–16:15
Ångströmlaboratoriet, Lägerhyddsvägen 1
GT seminar: "Homotopy rigidity of nearby Lagrangian cocores"
Johan Asplund will give a seminar talk titled "Homotopy rigidity of nearby Lagrangian cocores"

–14:15
Ångströmlaboratoriet, Lägerhyddsvägen 1
GT seminar: "A cohomological formula for Reidemeister torsion"
Amina Abdurrahman will give a seminar talk titled "A cohomological formula for Reidemeister torsion".
PREVIOUS TALKS
7 December: Amanda Hirschi (University of Cambridge)
Title: Global Kuranishi charts and a localisation formula in symplectic GW theory
Abstract: I will briefly describe the construction of a global Kuranishi chart for moduli spaces of stable pseudoholomorphic maps and show how this allows for a straightforward definition of sympletic GW invariants. Subsequently, I will describe how to extend this to the equivariant setting and sketch the proof of a localisation formula for the equivariant GW invariants of a Hamiltonian manifold. This is partially joint work with Mohan Swaminathan.
23 November: Oscar Kivinen (Aalto University)
Title: The HilbvsQuot conjecture
Abstract: The OblomkovRasmussenShende conjecture relates the (a, q, t)graded KhovanovRozansky homology of the link of a plane curve singularity to the topology of its punctual Hilbert schemes. While the t=1 limit is a theorem of Maulik, and much is understood in the q=1 limit, almost no examples incorporating all three gradings exist. In work with Trinh, we introduce a different kind of Quot scheme and show that an analogous conjecture for these Quot schemes does hold for torus knots and (n, nk)torus links. This motivates a surprising conjecture about how two kinds of Quot scheme on the same curve are related by a motivic substitution of variables, without reference to links or link homology.
9 November: Agustin Moreno (Heidelberg University)
Title: Symplectic geometry of Anosov flows in dimension three
Abstract: In this talk, I will illustrate how symplectic geometry can be leveraged in the study of uniformly hyperbolic dynamics. Namely, every Anosov flow on a threemanifold M has an associated nonWeinstein Liouville domain V=[1,1]xM, whose invariants (e.g. symplectic cohomology, RabinowitzFloer cohomology, wrapped Fukaya category) are homotopy invariants of the flow. Moreover, every orbit of the flow gives an exact Lagrangian cylinder, all of which are nontrivial and independent of each other in the Fukaya category, such that the A_infty category they generate does not satisfy Abouzaid’s generation criterion, in contrast to the Weinstein case. Time permitting, I will discuss the classical algebraic cases, their symplectic invariants, and further results concerning closed Lagrangians.
2 November: Paolo Ghiggini (Université GrenobleAlpes)
Title: A relative exact sequence for Lagrangian Floer homology
Abstract: Some years ago Chantraine, DimitroglouRizell, Golovko and I defined a
version of Lagrangian Floer homology for Lagrangian cobordisms also
known as Cthulhu homology. I will show that concatenation of cobordisms
induce a long exact sequence in Cthulhu homology. A similar exact
sequence was proved by Cieleibak and Oancea if the negative ends of the
cobordisms are filled.
26 October: David Kern (KTH Stockholm)
Title: Symmetric 2Segal conditions for shifted cotangent groupoids
Abstract: Algebraic (derived) stacks are generalised spaces that can be presented
as quotients of geometric groupoids. Similarly, shifted symplectic
stacks admit atlases given by symplectic groupoids. In this talk, based
on joint work in progress with Damien Calaque, we describe such an atlas
for shifted cotangent bundles. It is obtained through a careful study of the structure of groupoid
objects as certain Calabi–Yau algebras in spans, and functoriality of
the cotangent stack construction producing Lagrangian correspondences
from spans.
19 October: Georgios Dimitroglou Rizell (Uppsala University)
Title: Restrictions on the augmentation variety from the relative CalabiYau structure of the ChekanovEliashberg algebra (Part II: bulk deformations)
Abstract: We describe the relative CalabiYau property and resulting restrictions on the augmentation variety in the setting of bulk deformations. This is a continuation of the previous talk on the topic.
12 October: Axel Husin (Uppsala University)
Title: Floer theory with generalized local systems
Abstract: A path local system is a dgfunctor from the path category of a topological space to the category of chain complexes. We will define Lagrangian intersection Floer theory with coefficients in path local systems, and use it to construct a functor between categories of path local systems defined on transversally intersecting Lagrangians. In the case when the ambient symplectic manifold is the cotangent bundle of a closed manifold and one Lagrangian is closed and exact and the other is the base, a local computation will show that the functor we defined is an equivalence.
28 September: Georgios Dimitroglou Rizell (Uppsala University)
Title: Restrictions on the augmentation variety from the relative CalabiYau structure of the ChekanovEliashberg algebra
Abstract: The ChekanovEliashberg DGA of a Legendrian in the boundary of a subcritical Weinstein domain with loop space coefficients admits a relative CalabiYau structure (as defined by Brav and Dyckerhoff). This is a generalisation of Sabloff duality that holds for its finite dimensional representations. We describe notions of the fundamental class in the general setting of the relative CYstructure, and show how it can be used to give constraints on the dimension of the augmentation variety. The main application is a constraint on the dimension of the variety of augmentations of the Knot contact homology algebra of a smooth knot. This is joint work in progress with Legout.
14 September: Alex Takeda (Uppsala University)
Title: PreCY structures and explicit algebraic analogs of string topology
Abstract: I will explain how using the notion of preCY structures and a certain
graphical formalism, one can write down chainlevel representatives for some
operations on Hochschild homology that resemble those from string topology,
and that we conjecture to be a new algebraic way of calculating the Chas
Sullivan product and GoreskyHingston coproduct. It turns out that the natural
tool for assembling all this data is Efimov's definition of the "categorical
formal punctured neighborhood of infinity", which generalizes the category of
perfect complexes near the divisor at infinity in a compactification of an
algebraic variety and is related, on the other side of mirror symmetry, to
Rabinowitz Fukaya categories. This talk is about joint work with Manuel Rivera
and Zhengfang Wang.
17 August: Roman Golovko (Charles university)
Title: On a few special Legendrian submanifolds.
Abstract: We will discuss three types of examples of Legendrian submanifolds in high dimensions that
1) realize an arbitrary finitely generated abelian group in linearized Legendrian contact homology,
2) admit an infinite family of diffeomorphic, but not Hamiltonian isotopic exact Lagrangian fillings,
3) admit an augmentation, whose nongeometricity can not be obstructed by the isomorphism of SeidelEkholmDimitroglou Rizell
21 June: Adrian Petr (University of Southern Denmark)
Title: Legendrian contact homology in prequantization bundles
Abstract: The ChekanovEliashberg DGalgebra is an invariant associated to Legendrians in contact manifolds. It has first been introduced in $R^3$, and then into the general framework of Symplectic Field Theory. In this talk, I will try to explain the relation between Legendrian contact homology in the total space of a circle bundle and Floer theory in the base through a Fukayacategory perspective. Part of this is work in progress with Noémie Legout.
15 June: Michael Sullivan (UMass Amherst)
Title: Chain homotopy data for Legendrians and Lagrangians
Abstract: I'll introduce chain homotopy data, also known as Brown's twisted complexes. Then I'll show how they arise for Floertype theories associated to Legendrians and Lagrangians. Joint work with various people. Results are either old or speculative.
12 June: Álvaro del Pino Gomez (Utrecht university)
Title: Overtwistedness of maximally nonintegrable distributions of rank 2
Abstract: A distribution is maximally nonintegrable if its sheaf of tangent vector fields has "as many nontrivial Lie brackets as possible". This condition defines a partial differential relation on the space of distributions that turns out to be rather nontrivial to tackle.
In this talk I will focus on distributions of rank 2. In this case, the maximal nonintegrability condition is beyond the applicability of the classic hprinciple approaches due to Gromov. However, the 3dimensional situation (i.e. for 3dimensional contact structures) was addressed successfully by Eliashberg in 1989 using a socalled removal of singularities approach. He proved that there is a class of 3dimensional contact structures, called overtwisted, that can be completely classified in homotopical terms.
In 2018, Vogel and myself proved an analogous statement in dimension 4 (i.e. for socalled Engel structures). The proof we gave recovers Eliashberg's statement as well, suggesting that Engel and contact overtwistedness are concrete incarnations of a more general phenomenon.
In this talk I will discuss the following conjecture: maximally nonintegrable distributions of rank2 admit an overtwisted class that satisfies the complete hprinciple. My goal will be to outline a strategy of proof generalising the 4dimensional approach. The crucial technical step is a flexibility statement for submanifolds transverse to a bracketgenerating distribution.
This is ongoing work with F. ter Haar and F.J MartínezAguinaga.
26 May: Inanc Baykur (UMass Amherst)
Title: Exotic 4manifolds with signature zero
Abstract: We are going to talk about our recent construction of small symplectic 4manifolds with signature zero, which include the smallest closed simplyconnected exotic 4manifolds with signature zero known to date. Our novel examples are derived from explicit Lefschetz fibrations, with spin and nonspin monodromies. Joint work with N. Hamada.
11 May: Georgios Dimitroglou Rizell (Uppsala university)
Title: A relative CalabiYau structure for Legendrian contact homology and applications to the augmentation variety
Abstract: We show how Sabloff duality in linearized Legendrian contact homology can be generalised to a relative CalabiYau structure of the Legendrian contact homology DGA, as defined by Brav and Dyckerhoff. We also discuss the generalised notion of the fundamental class and give applications, including constraints on the dimension of the augmentation variety. The structure is established through the acyclicity of a version of Rabinowitz Floer Homology for Legendrian submanifolds with coefficients in the ChekanovEliashberg DGA. This is joint work in progress with Legout.
4 May: Tobias Ekholm (Uppsala university)
Title: Counting bare curves VI
Abstract: Followup of the talk held on April 13.
27 April: Lukas Nakamura (Uppsala university)
Title: A metric on the contactomorphism group of an orderable contact manifold.
Abstract: We define a pseudometric on the contactomorphism group of a cooriented contact manifold M which is bounded from above by the ShelukhinHofer metric and nondegenerate if, in addition, M is orderable. If M is closed, we show that its metric topology agrees with the interval topology of Chernov and Nemirovsky, thereby positively answering their question of whether the interval topology of an orderable contact manifold is Hausdorff. We discuss similar results on the universal cover of the contactomorphism group and on spaces of Legendrians. If time permits, we explain connections to Hedicke's recently defined Lorentzian distance function.
13 April: Tobias Ekholm (Uppsala university)
Title: Counting bare curves V
Abstract: (This is a followup of the talks which were given in the last few months). In a series of talks we will discuss how to count bare holomorphic curves with Lagrangian boundary conditions in a CalabiYau 3fold. Here bare means that the curves does not have any symplectic area zero components. The main application and motivation for this is the construction of invariant skein valued counts of open curves. In this first talk we will explain why the usual approaches to GromovWitten counts does not work to define skein valued counts and present one of the key technical results for bare counts: we finds a class of perturbations for the CauchyRiemann equations that has the property that if a ghost bubbles forms in the limit of sequence of bare curves then it leaves a trace (e.g. vanishing complex derivative) on the limiting positive area curve on which it is attached. In later talks we will construct a perturbation scheme using polyfold language with perturbations of this kind that gives rise to solutions of the perturbed CauchyRiemann equations that admit forgetful maps. The talks report on joint work with Vivek Shende.
30 March: Olof Bergvall (Mälardalens university)
Title: Point sets in projective spaces and moduli spaces
Abstract: Intuitively, a collection of points on a variety is in "general position" if there is no "unexpected" subvariety passing through them. In this talk, I will discuss configurations of points in general position in projective spaces and various constructions related to them. In particular, I will discuss their relation to moduli spaces and related structures.
23 March: Jack Smith (Cambridge)
Title: Hochschild cohomology of the Fukaya category via Floer cohomology with coefficients
Abstract: The Hochschild cohomology of the Fukaya category is an important symplectic invariant, and plays a central role in the generation criterion of Abouzaid and Sheridan. Unfortunately, it is usually very difficult to calculate directly. I will describe new results relating it to a more geometric and computable object, namely Floer cohomology of a Lagrangian with enriched coefficients, and discuss some applications. Perhaps surprisingly, matrix factorisations make an important appearance.
9 March: Nick Kuhn (University of Oslo)
Title: Degenerations of stable sheaves on fibered surfaces
Abstract: In the paper "Floer homology and algebraic geometry", Donaldson raised the question of constructing an algebraic theory for moduli spaces of vector bundles on algebraic surfaces which are singular along a nodal divisor. A succesful candidate for such a theory would enable one to compute sheaftheoretic invariants of a smoothing of the singular surface in terms of relative invariants of its normalization, similar to the degeneration formulas in GromovWitten or DonaldsonThomas theory.
I will present work in progress on a good theory of sheaves on degenerations that works when the surface degenerates together with a fibration to a curve. This opens the door for many interesting computations  for example, one can recover and strengthen results about the geometry of moduli spaces on elliptic surfaces.
23 February: Jian Qiu (Uppsala university)
Title: Quantization via branes and minimal resolution
Abstract: I want to explain in this talk the framework of quantization using branes, as developed by GukovWitten and later refined by GaiottoWitten. This approach embeds the symplectic manifold to be quantized into a bigger space (of double the dimension) as a brane, and the open strings ending on the brane play the role of the operators acting on the Hilbert space.
The open strings are expected to be quantized via the Amodel, while in practice the open string wave functions are identifiable as holomorphic functions on the larger space and therefore many algebraic approach can be applied for its quantization and bypass the difficult Amodel. I will apply Kontsevich’s deformation quantization to deform the algebra of operators. This approach would have been impractical without further tools such as the minimal bialgebra resolution and some recently developed techniques for deformation of quiver algebras.
I will run you through some examples, such as the quantization of Kleinian singularity, in the A_1 case, the quantization gives the Verma module of Usu(2) and integrality condition for the arameters, which Is unusual for deformation quantization. Further examples include T^*CP^2, which is useful in the context of generalized Kahler structure.
Future applications include the quantization of character varieties, Higgs moduli space etc.
16 February: Paolo Ghiggini (Institut Fourier Grenoble)
Title: Heegaard Floer homology as a Floer field theory and genus two mutations
Abstract: I will show that the hat version of Heegaard Floer homology can be cast into a modified version of Wehrheim and Wodward's Floer field theory: roughly speaking, a 2+1 TQFT with values in compact Fukaya categories. The main difference from previous unpublished work by Lekili and Perutz is that we use a "cylindrical reformulation" of the Fukaya category of a symmetric product allowing us to avoid quilted Floer homology. As an application, I will show that the total rank of Heegaard Floer homology is not changed by a surgery operation called "genus two mutation". This is a work in progress in collaboration with Ina Petkova.
9 February: Tasos Moulinos (Université Paris 13)
Title: A tour through the topological Ktheory of dg categories
Abstract: I will discuss some of my past work on the topological Ktheory of dg categories. This is an invariant which shares some formal properties with algebraic Ktheory but which lands in the infinity category of KUmodule spectra.
In particular it is a “localizing invariant of complexlinear differential graded categories” which outputs a KUmodule spectrum. I will begin the talk with motivations coming by way of Hodge theoretic mirror symmetry, and will proceed to describe the basic construction, originally due to Blanc. I will then describe a variant of this construction from previous work, relative to any base complex scheme, together with applications of such a construction towards computations in twisted Ktheory, and towards the theory of variations of Hodge structures. Time permitting, I will describe some open questions.
2 February: Tobias Ekholm (Uppsala university)
Title: Counting bare curves IV
Abstract: This is a follow up of the talk held on November 24th.
12 January: Noah Porcelli (Imperial College London)
Abstract: Lagrangian Floer theory is a useful tool for studying the structure of the homology of Lagrangian submanifolds. In some cases, it can be used to detect more we show it can detect the framed bordism class of certain Lagrangians and in particular recover a result of AbouzaidAlvarezGavelaCourteKragh about smooth structures on exact Lagrangians in cotangent bundles of spheres. The main technical tool we use is Large's recent construction of a stablehomotopical enrichment of Lagrangian Floer theory.
This is based on joint workinprogress with Ivan Smith.
Past seminars from previous years