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
Previous talks
2025
March 27: Dylan Cant (Université Paris-Saclay)
Title: Spectral diameter and Lagrangian capacity
Abstract: I will present some aspects of two papers (arXiv:2408.07214 and arXiv:2409.14142) concerning spectral invariants for compactly supported Hamiltonian systems in Cn. First, I will explain our discovery in arXiv:2408.07214 that the spectral diameter of a ball B(a) (of capacity a) is half of that of a cylinder Z(a). Thus the spectral diameter is not a normalized capacity. Then I will explain how this fact is used to prove certain cases of the conjecture that: any closed Lagrangian L in a ball B(1) in Cn, n>1, bounds holomorphic disks with area at most 1/2 (note that the Lagrangian lift of RPn-1 inside the boundary of B(1) bounds disks with area 1/2, so the conjecture cannot be improved). Our proof is based on a delicate version of the Lagrangian control property which incorporates spectral invariants, boundary depth, and the minimal area of holomorphic disks.
March 20: Adrien Currier (University of Nantes)
Title: Exact Lagrangians in cotangent bundles with locally conformally
symplectic structure
Abstract: First considered by Lee in the 40s, locally conformally
symplectic (LCS) geometry appears as a generalization of symplectic
geometry which allows for the study of Hamiltonian dynamics on a wider
range of manifolds while preserving the local properties of symplectic
geometry. After a long period of hibernation (especially as far as the
topological aspect is concerned), this topic has recently seen renewed
interest. However, to this day, the field of LCS topology remains vastly
unexplored.
In this talk, we will introduce the various objects of LCS geometry and
their behavior through both definitions and examples. We will then
explore some questions around an LCS version of the nearby Lagrangian
conjecture and some of the connections between LCS and contact
geometry.
March 13: Zuyi Zhang (Peking University)
Title: Immersed Lagrangian Floer theory, Lagrangian compositions, and bounding cochains on surfaces
Abstract: In this talk, I will first give the background of immersed Lagrangian Floer theory based on character varieties. Then I will focus on the relation between immersed quilted Lagrangian Floer theory and the usual immersed Lagrangian Floer theory when they are related by a Lagrangian correspondence, in the topological aspect. I will use many examples to explore this relation. If time permitted, I will give some examples about bounding cochains.
February 27: Tobias Ekholm (UU)
Title: The skein valued topological vertex
Abstract: We count holomorphic curves in complex 3-space with boundaries on three special Lagrangian solid tori. The count is valued in the HOMFLYPT skein module of the union of the tori. Using 1-parameter families of curves at infinity, we derive three skein valued operator equations that determines the count uniquely. The solution count agrees with the topological vertex from topological string theory. The talk reports on joint work with P. Longhi and V. Shende.
February 20: Leon Menger (University of Notre Dame)
Title: The Fukaya–Morse Category: Enhancements and Field-Theoretic Approach
Abstract: The Fukaya–Morse category is an A-infinity-(pre-)category proposed by Fukaya in 93. It makes use of multiple Morse functions and their flow lines to represent higher Massey products and is intimately tied to the rational homotopy type of a manifold. In this talk we will go over its construction, highlight some of its issues and talk about a recent proposal by Chekeres, Losev, Mnev and Youmans to tackle them. We will then discuss further enhancements of this category to Morse–Bott functions and graphs with one loops. Time permitting I will sketch how the category arises as a particular effective field theory.
January 23: Russell Avdek (Sorbonne Université)
Title: Transverse stabilization in dim>3 contact topology
Abstract: We'll define stabilization for codim=2 contact manifolds of dim>3 contact manifolds so that a contact manifold is overtwisted iff its "standard contact unknot" is a stabilization. The talk will outline a proof that for n>1 there are two (smoothly unknotted) embedding of the standard contact 2n-1 sphere into the standard contact 2n+1 sphere which are formally contact isotopic, but which are not contact isotopic.
2024
December 13: Alice Hedenlund (Norwegian University of Science and Technology)
Title: Conditional Convergence Revisited
Abstract: Spectral sequences have proven to be immensely powerful tools in modern mathematics. One of the main questions when working with spectral sequences is whether the spectral sequence is computing what we want it to compute. This is the question of convergence. At the inception of the subject, dealing with this question usually involved imposing quite severe finiteness conditions, but as the subject developed it became apparent that better considerations were needed. One groundbreaking point of view is Boardman's notion of conditional convergence. The concept of conditional convergence is very useful in that it allows one to deduce strong convergence from conditions which are entirely internal to the spectral sequence in question, and which in many cases are easy to check.
From a structural point of view, convergence is often treated a bit abusely. We speak of a spectral sequence being convergent, even though technically speaking convergence is extra data, and not just a property of the spectral sequence. Moreover, the notion of conditional convergence is not even internal to spectral sequences. In other words, given a spectral sequence, with no information on how it arose, the question “Does the spectral sequence converge conditionally?” does not even
make sense. In this talk, I instead propose the two notions derived weak and derived strong convergence which recaptures Boardman's work, but are more structurally well-behaved. This is joint work in progress with Achim Krause and Thomas Nikolaus.
December 12: Guillaume Laplante-Anfossi (Southern Denmark University)
Title: A universal characterization of uncurving
Abstract: Given a curved A-infinity algebra and a bounding cochain (a.k.a Maurer-Cartan element), one can twist the former by the latter to get a new A-infinity algebra where the curvature is zero. This uncurving procedure plays a central rôle in Floer theory, where Fukaya categories are naturally curved objects. The goal of this talk will be to give a universal characterization of the procedure, using the theory of operads: we will show that the operad cMC, whose algebras are curved A-infinity algebras endowed with a Maurer-Cartan element, together with its natural morphism to the A-infinity operad, is terminal in a certain comma category. This category does not contain the Maurer-Cartan equation, thus the theorem provides a way to « rediscover » it as the answer to a universal algebraic problem. This is joint work with Adrian Petr and Vivek Shende.
December 6: Jennifer Brown (University of Edinburgh)
Title: Quantum Decorated Character Varieties
Abstract: Character varieties parametrize G-local systems on manifolds, for G a reductive group. It's sometimes natural to consider manifolds equipped with a patchwork of local systems for different groups, leading to a need for defects in character varieties. When the patchwork consists of G and its maximal torus T, the resulting moduli space is called a decorated character variety. These appear in the definition of cluster coordinates, in the study of irregular connections, and in the construction of a knot invariant known as the A-polynomial. In all of these cases there is physical motivation to understand the quantized version. We will introduce character varieties and explain a skein theoretic approach to quantizing them with or without defects. The quantization of the A-polynomial will serve as our main motivation and a running example. This is based on joint work in progress with David Jordan.
November 28: Maciej Borodzik (University of Warsaw)
Title: Concordance implies regular homotopy
Abstract: I will sketch a proof of the statement in the title, valid in any dimension. The key tool is immersed Morse theory and the path lifting lemma, which we introduce. This is a joint work with Mark Powell and Peter Teichner.
November 14: Alexander Berglund (Stockholm University)
Title: Poincaré duality fibrations and Kontsevich's Lie graph complex
Abstract: I will talk about certain higher algebraic structure, governed by Kontsevich's Lie graph complex, that can be associated to an oriented fibration with Poincaré duality fiber. The main construction is a generalized fiber integration map associated to each Lie graph homology class and the main result is that this gives a faithful representation of Lie graph homology. I will discuss how this leads to new possible interpretations of Lie graph homology classes as obstructions to, on one hand, smoothness of Poincaré duality fibrations, and, on the other hand, the existence of Poincaré duality algebra resolutions of the cochains of the total space as a dg module over the cochains of the base space.
November 4: Ezra Getzler (Northwestern University)
Title: The Fedosov product on differential forms and flat connections on differential graded manifolds
Abstract: The Hochschild-Kostant-Rosenberg theorem relates negative cyclic homology of the functions on a differential graded manifold M to the de Rham complex of M. The natural A-infinity structure on negative cyclic homology corresponds to an A-infinity structure on the de Rham complex called the Fedosov product. (This product agrees with the usual one if either argument is closed, but it is not graded symmetric in general.) In this talk, we study the analogue of flat connections for this product: these are in bijection with flat connections for the usual product on the classical locus but appear to be different in general. The non-commutative analogue of the Gauss-Manin connection is our main example.
October 17: Maxime Fairon (University of Burgundy)
Title: Compatible Poisson structures on multiplicative quiver varieties
Abstract: Any multiplicative quiver variety is endowed with a Poisson structure constructed by M. Van den Bergh through reduction from a Hamiltonian quasi-Poisson structure. The smooth locus of this variety carries a corresponding symplectic form defined by D. Yamakama through quasi-Hamiltonian reduction. In this talk, I want to explain how to include this Poisson structure as part of a larger pencil of compatible Poisson structures on the multiplicative quiver variety. The pencil is defined by reduction from a pencil of (non-degenerate) Hamiltonian quasi-Poisson structures, whose construction can be adapted to various frameworks, e.g. in relation to character varieties. I will start by explaining the simpler analogous situation that leads to a pencil of Poisson structures on (additive) quiver varieties. This is based on arXiv:2310.18751.
October 10: Daniil Mamaev (London School of Geometry and Number Theory)
Title: Relative Wrapped Fukaya Categories of Surfaces
Abstract: Relative wrapped Fukaya categories of surfaces arise naturally in homological mirror symmetry in two ways. As mirrors to some relative curves and, conjectured by Lekili-Segal, as mirrors to the total spaces of algebraic torus fibrations over bordered surfaces. After giving motivating examples I will sketch an explicit construction of a triangulated relative wrapped Fukaya category of a surface directly from geometric data, without the use of triangulated envelopes or A-infinity quotients. Then I will explain why the curves with isotopic Legendrian lifts to the contactisation give quasi-isomorphic objects in the category and state a generation criterion that in particular implies that in the non-relative case the newly constructed category is equivalent to the topological Fukaya category of Haiden-Katzarkov-Kontsevich.
September 26: Yin Li (UU)
Title: Audin's conjecture beyond vector spaces
Abstract: The original conjecture of Audin, proved by Cieliebak-Mohnke, asserts that any Lagrangian torus in the symplectic vector space bounds a holomorphic disc of Maslov index 2. In this talk, we will show that the same conclusion holds for Lagrangian tori in low degree affine hypersurfaces in C^n. More precisely, we prove that any closed, oriented, aspherical Lagrangian submanifold in a Liouville manifold with finite first Gutt-Hutchings capacity bounds a holomorphic disc of Maslov index 2. The proof relies on three ingredients: Fukaya-Irie's de Rham model of algebraic string topology, Cohen-Ganatra's idea in the construction of the Cieliebak-Latschev map, and Irie's technique of finite energy approximation of virtual fundamental chains.
September 5: Lukas Nakamura (UU)
Title: The worldsheet skein D-module and basic curves on Lagrangian fillings of the Hopf link conormal
Abstract: We propose a skein-valued lift of the D-module associated to a link in S^3, which is the quotient of the skein of the cylindrical lift of the unit conormal of the link by an ideal which in the classical limit reduces to the defining ideal of the augmentation variety of the link. The Gromov-Witten partition function of a Lagrangian filling can then be viewed as a morphism from the D-module to the skein of the filling. This imposes constraints on the partition function which for some links can be used to determine the partition function uniquely.
I will then exemplify this general framework in the particular case of the Hopf link. For every point of the augmentation variety, we identify a corresponding Lagrangian filling whose partition function is uniquely determined by the constraints. We obtain explicit expressions for the partition functions in terms of contributions of basic holomorphic curves, which can be viewed as skein-valued versions of the Gopakumar-Vafa formula and the knots-quivers correspondence. This is joint work with T. Ekholm and P. Longhi.
August 29: Tianyu Yuan (Peking University)
Title: Bulk deformed Lagrangian Floer theory on symmetric products
Abstract: We discuss some progress on orbifold Lagrangian Floer theory. In the case of symmetric products, it is modeled by the higher-dimensional Heegaard Floer homology (HDHF) and is related to symplectic Khovanov homology. In this talk, we discuss HDHF of cotangent fibers and its relation to various Hecke algebras. This is joint work with Ko Honda, Roman Krutowski and Yin Tian.
August 15: Alejandro Vicente (Hebrew University of Jerusalem)
Title: On symplectic embedding problems via Lagrangian fibrations
Abstract: In this talk I will talk about how to study certain symplectic embedding problems in sufficiently symmetric manifolds with the use of Lagrangian fibrations. In particular I will give ideas on how to compute the Gromov width of disk cotangent bundles of ellipsoids of revolution and talk about some interesting open problems that could be worked with similar ideas. This is based on joint work with Brayan Ferreira and Vinicius Ramos, as well as undergoing work with Santiago Achig-Andrango and Renato Vianna.
May 30: Louis Hainaut (Stockholm University)
Title: A classifying space for the handlebody mapping class group
Abstract: In recent work join with Dan Petersen, we described a model for the classifying space of the mapping class group of 3-dimensional handlebodies. This model allows us to detect the existence of many classes in the cohomology of this group, in a similar way as in the work of Chan-Galatius-Payne about the surface mapping class group. I will first explain some background about the handlebody mapping class group and Teichmuller space, and then explain the construction of our model. In the remaining time I will explain the connection between our model and the tropical moduli space of curves.
May 23: Trygve Poppe Oldervoll (NTNU)
Title: An A-infinity structure on generating function homology
Abstract: Generating functions are a tool for producing Lagrangians in cotangent bundles. To a pair of generating functions, one can associate a Morse theoretic invariant known as generating function homology. This is analogous to the Floer homology of the associated Lagrangians. In Floer theory, there are higher operations defined by counting rigid holomorphic polygons. In Morse theory, one can construct higher operations by counting rigid flow trees. This was made precise for standard Morse theory on a compact manifold by Fukaya and Oh. Generalizing the work of Myer, I will explain how to construct such operations for generating function homology.
April 25: Lukas Woike (University of Burgundy)
Title: Applications of factorization homology in quantum algebra
Abstract: Factorization homology allows us to integrate algebras over the little n-disks operad over n-dimensional manifolds. After giving a brief introduction to factorization homology, I will explain how factorization homology, in the case n=2, can be used to classify systems of representations of mapping class groups of surfaces called modular functors. The methods can be understood as a far-reaching generalization of skein theory. This is joint work with Adrien Brochier.
April 18: Tanushree Shah (Chennai Mathematics Institute)
Title: Giroux torsion in contact structures on 3-manifolds
Abstract: I will start by introducing contact structures. They come in two flavors: tight and overtwisted. Classification of overtwisted contact structures is well understood as opposed to tight contact structures. There infinitely many Tight structures iff the manifold is toroidal. Tight contact structures have been completely classified on very few toroidal 3 manifolds. We will look at what more classification results can we hope to get using the current techniques and what is far-fetched.
April 18: Marco Golla (Nantes Université)
Title: Line arrangements with only triple points
Abstract: Line arrangements are unions of lines in a (projective or affine) space over some field and they are studied from a combinatorial or algebro-geometrical standpoint. (Non-trivial) complex line arrangements always have at least a double or a triple point, a point where exactly two or three lines meet. Does there exist 13 lines in the complex projective plane that only meet at triple points? I will discuss some topological ideas around this question and higher-degree variants.
April 11: Du Pei (University of Southern Denmark)
Title: 4d Symplectic Duality
Abstract: Symplectic duality can be understood as a non-trivial relation between the Coulomb and Higgs branches of 3d N=4 quantum field theories. In this talk, I will present some emerging evidence for an analogous relation between the Coulomb and Higgs branches for 4d theories.
April 4 : Santiago Achig (UU)
Title: Singular Lagrangian Torus Fibrations on the smoothing of algebraic cones.
Abstract: In this talk, we explore a new approach to special Lagrangian fibrations on the smoothing of Gorenstein singularities, initially introduced by Gross and further analyzed in the context of the SYZ conjecture of mirror symmetry. Utilizing global coordinates connected to Altmann's characterization of the smoothing, our methodology not only provides alternative proofs for existing theorems about these fibrations but also expands on Symington’s work by creating a convex base diagram in higher dimensions. Additionally, we apply techniques from Pascaleff and Tonkonog to recover Lau's calculation of the potential for certain monotone fibers and investigate the non-displaceability of these fibers. This presentation includes a series of illustrative examples and concludes with an overview of future research directions.
March 21: Stefan Behrens (Bielefeld University)
Title: Generalized Froyshov invariants
Abstract: Froyshov invariants are rational valued invariants of 3-manifolds with the same rational homology as the 3-sphere. They can be extracted from monopole Floer theory and contain information about the topology of 4-manifolds that bound a given such 3-manifold. I will discuss a framework to obtain generalized Froyshov invariants using Seiberg-Witten-Floer homotopy types and various tools from equivariant stable homotopy theory. This is work in progress with Tyrone Cutler.
March 20: Shah Faisal (Humboldt University of Berlin)
Title: Extremal Lagrangian tori in convex toric domains
Abstract: Cieliebak and Mohnke define the symplectic area of a Lagrangian submanifold of a symplectic manifold as the minimal positive symplectic area of a 2-disk in the symplectic manifold with a boundary on the Lagrangian. I will explain that every Lagrangian torus that maximizes the symplectic area among the Lagrangian tori in the standard symplectic unit ball must lie entirely on the boundary of the ball. This answers a question attributed to L. Lazzarini and settles a conjecture of Cieliebak and Mohnke in the affirmative. I will also explain to which extent this statement is true for general toric domains.
February 22: Milica Ðukic (UU)
Title: A deformation of Chekanov-Eliashberg dg-algebra for Legendrian knots
Abstract: Inspired by symplectic field theory and string topology, we
introduce a chain complex for any Legendrian knot, whose homology is an
invariant of the knot up to Legendrian isotopy. The chain complex is
obtained by deforming Chekanov-Eliashberg differential using
pseudoholomorphic annuli in the symplectization. We show how to compute
the invariant combinatorially for any Legendrian knot in R^3.
February 21: Rémi Leclercq (Université Paris-Saclay)
Title: Essential loops of Hamiltonian homeomorphisms
Abstract: In 1987, Gromov and Eliashberg showed that if a sequence of diffeomorphisms preserving a symplectic form C⁰ converges to a diffeomorphism, the limit also preserves the symplectic form -- even though this is a C¹ condition. This result gave rise to the notion of symplectic homeomorphisms, i.e. elements of the C⁰-closure of the group of symplectomorphisms in that of homeomorphisms, and started the study of "continuous symplectic geometry".
In this talk, I will present recent progress in understanding the fundamental group of the C⁰-closure of the group of Hamiltonian diffeomorphisms in that of homeomorphisms. More precisely, I will explain a sufficient condition which ensures that certain essential loops of Hamiltonian diffeomorphisms remain essential when seen as "Hamiltonian homeomorphisms". I will illustrate this method (and its limits) on toric manifolds, namely complex projective spaces, rational products of 2-spheres, and rational 1-point blow-ups of CP².
Our condition is based on (explicit) computation of the spectral norm of loops of Hamiltonian diffeomorphisms which is of independent interest. For example, in the case of 1-point blow-ups of CP², I will show that the spectral norm exhibits a surprising behavior which heavily depends on the choice of the symplectic form. This is joint work with Vincent Humilière and Alexandre Jannaud.
February 8th: Alberto Cavallo (IMPAN)
Title: Transverse links in Stein fillable contact 3-manifolds
Abstract: We study the behavior of different versions of the Ozsváth-Szabó tau-invariant for holomorphically fillable links in Stein domains. More specifically, we relate the Hedden's version of the invariant, which needs the assumption that our links live in a contact 3-manifold with non-vanishing contact invariant, with the one introduced by Grigsby, Ruberman and Strle, which on the other hand only depends on the pair link-Spin^c 3-manifold and is then a purely topological invariant. This is joint work with Antonio Alfieri.
The main goal of the talk is to describe how our work allows us to recover results about properly embedded holomorphic curves, such as the slice-Bennequin inequality and the relative Thom conjecture, and to find new restrictions on the topology of Stein fillings of certain 3-manifolds. In particular, building on a result of Mark and Tosun, we show that a Brieskorn 3-sphere, with its canonical orientation, never bounds a rational homology 4-ball Stein filling; confirming a conjecture of Gompf.
February 1: Matt Magill (UU)
Title: Functorial QFT and Thom spectra
Abstract: Thanks to work of Atiyah and Segal in the late '80s, it has been understood that quantum field theories furnish us with representations of bordism categories. Some particularly interesting QFTs are "anomalous" theories. In this talk, I will spend some time reviewing these ideas, then show how (in certain settings) the language of Thom spectra can be used to classify anomalous theories.
January 25: Lisa Lokteva (UU)
Title: Graph Manifolds with Rational Homology Ball Fillings
Abstract: The subfield of low-dimensional topology colloquially called "3.5-dimensional topology" studies closed 3-manifolds through the eyes of the 4-manifolds that they bound. This talk focusses on Casson's question of which rational homology 3-spheres bound rational homology 4-balls. Since rational homology 3-spheres bounding rational homology 4-balls is a rare phenomenon, we will discuss how to construct examples.