Speaker: Riccardo Pedrotti
Title: Towards a count of holomorphic sections of Lefschetz fibrations over the disc
Abstract: Given a positive factorisation of the identity in the mapping class group of a surface S, we can associate to it a Lefschetz fibration over S^2 with S as a regular fiber. Its total space X is a symplectic 4-manifold, so it is natural to ask what kind of invariants of X can be read off from this construction. I will report on an ongoing joint work with Tim Perutz, aimed at obtaining an explicit formula for counting holomorphic sections of a Lefschetz fibration over the disk, while keeping track of their relative homology classes. This is the first step in our program to give explicit formulas for the Donaldson-Smith invariants of a Lefschetz fibration which, thanks to a combination of results by Usher and Taubes, are equivalent to the SW invariants of X.
Institution: U Mass
Website: https://riccardopedrotti.github.io
Speaker: Franz Pedit
Abstract: I will explain how to construct examples of high genus minimal Lagrangian surfaces in $CP^2$ using a combination of methods from integrable systems, loop groups, and Higgs bundles.
Institution: U Mass
Website: https://www.umass.edu/mathematics-statistics/about/directory/franz-pedit
Speaker: Jie Min
Title: Contact cut graph and Weinstein L-invariant
Abstract: Curves on surfaces have been very effective in understanding topology of 3- and 4-manifolds, for example Heegaard diagrams, Lefschetz fibrations and trisection/multisection diagrams. I will talk about an analogous way to understand contact and symplectic topology in dimension 3 and 4, called the contact cut graph. We show that each path in the contact cut graph corresponds to a Weinstein domain, from which we define a new invariant for Weinstein domains called the Weinstein L-invariant. We also give some examples of L=0 and L arbitrarily large Weinstein domains. This is joint work with N. Castro, G. Islambouli, S. Sakalli, L. Starkston and A. Wu. (arXiv:2408.05340)
Institution: U Mass
Website: https://sites.google.com/site/jieminmath/home
Speaker: Joseph Palmer
Title: From Hamiltonian torus actions to integrable systems
Abstract: A complexity-one space on a symplectic 2n-manifold is the Hamiltonian action of a torus of dimension n-1. The momentum map for such an action can be thought of as n-1 real valued functions. On the other hand, an integrable system on such a manifold is the data of n functions. This motivates several natural questions: given a complexity-one space, when can an additional function be found to produce an integrable system? When can the resulting system be chosen to be toric? When can it be chosen to have no degenerate singularities? The case of when a circle action on a 4-manifold can be lifted to a toric integrable system was already completely understood by Karshon in 1999, but most other cases have remained open until relatively recently. In this talk, I will discuss answers to various versions of these questions, both in dimension four and higher, with a focus on the following question which I solved with Sonja Hohloch: what is the "nicest" class of integrable systems to which all complexity one spaces can be lifted in dimension four? Parts of this work are joint with Sonja Hohloch, Susan Tolman, and Jason Liu.
Institution: Amherst College
Website: https://sites.google.com/view/jpalmer
No seminar
Balillieul Lecture
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Speaker: Lorenzo Riva
Title: Topological field theories in algebraic topology and mathematical physics
Abstract: We learned a long time ago, for example from Thom's work, that there is a connection between bordism theory and homotopy theory that allows us to better understand smooth manifolds. One natural extension of the work of Thom is obtained by studying a higher category of bordisms and its associated representations, which for historical reasons are called (functorial) topological field theories (TFTs). This historical connection to mathematical physics has not ceased to be relevant: there are many classical and quantum field theories of mathematical interest which are conjectured to yield functorial TFTs that yield computable(ish) invariants of manifolds. In this talk we will explore one such example, given by the Rozansky-Witten models [Witten 1997] as categorified in [Kapustin, Rozansky, Saulina], and see how it works out in a fully coherent (i.e. infinity-categorical) setting.
Institution: Notre Dame
Website: https://sites.google.com/nd.edu/lorenzo-riva/home
Speaker: Agniva Roy
Title: Spinal open books and symplectic fillings with exotic fibers
Abstract: The question of understanding symplectic fillings of contact 3-manifolds has a rich history and deep connections to various aspects of low-dimensional topology, such as exotic 4-manifolds, mapping class groups of surfaces, complex geometry, algebraic singularities, among others. The technique of spinal open books, recently introduced by Lisi - Van Horn-Morris - Wendl, describes strong symplectic fillings of planar spinal manifolds in terms of foliations by pseudoholomorphic curves. These foliation descriptions, in the broadest generality, contain singular curves, which are Lefschetz-type singularities, and also a new phenomenon called exotic curves. In the Lefschetz-amenable setting, exotic curves disappear, and fillings can be classified as Lefschetz fibrations, depending on the number of singular curves. In joint work with Hyunki Min and Luya Wang, we give a topological description of the exotic curves in terms of identifying them with a local model, give a count of exotic fibers in any filling, and use these to classify symplectic fillings of certain planar spinal open books that are not Lefschetz-amenable.
Institution: Boston College
Website: https://sites.google.com/bc.edu/agniva-roy/home?authuser=0
Speaker: Ina Petkova
Title: GRID invariants and Lagrangian cobordisms
Abstract: Knot Floer homology is a powerful invariant of knots and links, developed by Ozsvath and Szabo in the early 2000s. Among other properties, it detects the genus, detects fiberedness, and gives a lower bound to the 4-ball genus. The original definition involves counting homomorphic curves in a high-dimensional manifold, and as a result the invariant can be hard to compute. In 2007, Manolescu, Ozsvath, and Sarkar came up with a purely combinatorial description of knot Floer homology for knots in the 3-sphere, called grid homology. Soon after, Ozsvath, Szabo, and Thurston defined invariants of Legendrian knots using grid homology. We show that the filtered version of these GRID invariants, and consequently their associated invariants in a certain spectral sequence for grid homology, obstruct decomposable Lagrangian cobordisms in the symplectization of the standard contact structure, strengthening a result of Baldwin, Lidman, and Wong. This is joint work with Jubeir, Schwartz, Winkeler, and Wong.
Institution: Dartmouth College
Website: https://math.dartmouth.edu/~ina/
Speaker: Jacob Garcia
Title: Characterizations of Stability via Morse Limit Sets
Abstract: An important example of Kleinian groups are the convex cocompact groups: every infinite order element of these groups is a loxodromic, and these groups are exactly the ones which admit Kleinian manifolds. A well known fact of convex cocompact groups is that they can be characterized exactly as the groups whose limit sets, on the visual boundary, are completely conical, or equivalently, completely horospherical. Convex cocompactness has been studied in the context of many non-hyperbolic spaces, such as mapping class groups, and has recently been generalized to the notion of subgroup stability. By using an analog of the visual boundary called the Morse boundary, a quasi-isometry invariant which "sees" hyperbolic directions for non-hyperbolic spaces, we show that subgroup stability is exactly classified by limit set conditions on the Morse Boundary which are analogous to the limit set conditions from the convex cocompact setting.
Institution: Smith College
Website: https://sites.google.com/view/jacobgarcia/home
Speaker: Robert Kusner
Title: Minimal Surfaces in Round Spheres and Balls
Abstract: A deep connection between extremal eigenvalue problems and minimal surfaces in round spheres or balls has emerged over the past decades, stemming from work of Nadirashvili and Fraser-Schoen. We use this to determine geometric properties of these surfaces, like a uniqueness theorem for embedded free boundary minimal annuli with antipodal symmetry, by showing the first eigenspace for many such surfaces coincides with the span of the ambient coordinate functions (generalizing the Yau and Fraser-Li conjectures). We also develop an equivariant version of eigenvalue optimization with sharp a priori eigenspace dimension bounds, letting us construct free boundary minimal surfaces of every topological type embedded in the round 3-ball, and many more closed minimal surfaces embedded in the round 3-sphere.
[Based on joint projects with Misha Karpukhin, Peter McGrath and Daniel Stern]
Institution: UMass Amherst & SLMath/MSRI
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Speaker: Hannah Turner
Title: The (fractional) Dehn twist coefficient and infinite-type surfaces
Abstract: The fractional Dehn twist coefficient (FDTC) is an invariant of a self-map of a surface which is some measure of how the map twists near a boundary component of the surface. It has mostly been studied for compact surfaces; in this setting the invariant is always a fraction. I will discuss work to give a new definition of the invariant which has a natural extension to infinite-type surfaces and show that it has surprising properties in this setting. In particular, the invariant no longer needs to be a fraction - any real number amount of twisting can be achieved! I will also discuss a new set of examples of (tame) big mapping classes called wagon wheel maps which exhibit irrational twisting behavior. This is joint work in progress with Diana Hubbard and Peter Feller.
Institution: Stockton University