Fridays from 2:30-3:30pm in LGRT 1681 at UMass Amherst
Organizers: İnanç Baykur, Patricia Cahn, Miriam Kuzbary, Riccardo Pedrotti and Valentina Zapata Castro
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Speaker: ORGANISATION MEETING
Speaker: Riccardo Pedrotti
Title: An obstruction to the existence of sections for a Lefschetz fibration
Abstract: We characterise the presence of a section for a Lefschetz fibration over the sphere in terms of vanishing of an obstruction class living in a quotient of the fundamental group of the regular fiber. In the case of transitive LF we prove that such obstruction class always vanishes, therefore detecting a smooth section. Time permitting we will also show a computation of this obstruction class to the MCK fibration.
Institution: UMass Amherst
Website: https://riccardopedrotti.github.io
Speaker: Georgios Dimitroglou Rizell
Title: Non-regular concordances between stabilised Legendrian knots.
Abstract: We discuss a flexibility result that makes it possible to approximate totally real concordances in the symplectisation by Lagrangian concordances, after adding sufficiently many stabilisations of both signs to the Legendrians knots. The main application is the construction of Lagrangian concordances that are not regular in the sense of Eliashberg, and not even ribbon. This is joint work with R. Golovko.
Institution: Uppsala University
Website: https://www.uu.se/en/contact-and-organisation/staff?query=N7-1534
Baillieul Distinguished Lecture (https://www.umass.edu/mathematics-statistics/seminars/BDLS)
Speaker: Thomas Kindred
Title: How natural of a geometric operation is Murasugi sum?
Abstract: After presenting the basic definitions and a couple examples, the first result I'll share is that, although any Murasugi sum of $\pi_1$-essential spanning surfaces is $\pi_1$-essential, the same is not true for geometrically essential surfaces. The second topic I'll address is the question of when a Murasugi sum of an inessential spanning surface with another surface can be essential, where I have a few interesting examples in both directions. Lastly, I'll discuss how the first result adapts to dimension 4: any Murasugi sum of $\pi_1$-essential spanning solids is $\pi_1$-essential, and this result extends to generalized Murasugi sums, where the gluing is along any handlebody bounded by any Heegaard surface in a splitting standard 3-sphere in the ambient 4-manifold (rather than just along a 3-ball). To my mind, the most interesting unresolved aspect of the title question is whether Murasugi sum in 4D also respects $\pi_2$-essentiality. Actually, I don't even really know where to start thinking about the notion of $\pi_2$-essential spanning solids, so it will be exciting to have an opportunity to discuss this with the group.
Institution: Smith College
Website: https://thomaskindred.com/
Speaker: Yasin Karacan
Title: Generating Families, Augmentations, and Morse Complex n-Families
Abstract: For Legendrian knots, Fuchs and Rutherford proved a generating family gives an augmentation of the Chekanov–Eliashberg DGA whose linearized homology is generating family homology. For Legendrian surfaces, Rutherford and Sullivan showed that a generating family also yields an augmentation using Morse complex 2-families. I will introduce Morse complex n-families, which are defined for Legendrians with mild fronts in all dimensions. They produce augmentations of the Simplicial DGA. The guiding principle is flow lines persist until they break at composable flows, and the compactified spaces of flow lines encode the algebra. I will also discuss work in progress relating generating family homology to linearized Simplicial DGA homology. Some familiarity with Morse theory will be helpful.
Institution: University of Massachusetts
Website: https://yasinkaracan.github.io/
Speaker: Joshua Lehman
Title: Topologically flexible algebraic curves
Abstract: A surface S in a 4-manifold M is called flexible if any homeomorphism of S extends to a homeomorphism of M. As a special case of our results, we show that plane curves (algebraic curves in the complex projective plane) are flexible surfaces. Previously, flexibility of plane curves was only known in degrees at most 5. Our additional input is to study the equivariant intersection form on the exterior of S in M. In the right context, there are deep results which allow one to promote isometries of this form to homeomorphisms of the pair (M,S). We conclude by showing how the equivariant intersection form can also be used to obstruct flexibility. This is joint work with Tudur Lewis.
Institution: University of Notre Dame
Website: https://jlehman-math.github.io/
Speaker: Austin Konkel
Title: Inside Euclidean 3-space, which immersed 2-spheres are the boundary of some compact immersed 3-manifold?
Abstract: Suppose that you have any connected and orientable 3-manifold. Remove any open 3-ball. Then you can immerse what remains into Euclidean 3-space by a corollary of the theses of Eduard Stiefel and Tony Phillips. You wonder what this immersed boundary 2-sphere looks like because its complement inside 3-space can be filled in by the rest of the immersed 3-manifold. In this talk, we analyze this situation by defining a sheaf of finite groups over the immersed 2-sphere, whose first cohomology set contains an answer to the titled question and more.
Institution: Stony Brook University
Website: https://www.math.stonybrook.edu/include/cards/konkelaustin.html
Speaker: Erin Griffin
Title: Extending our Understanding of Ambient Obstruction Solitons
Abstract: Ambient obstruction solitons are fixed points of the ambient obstruction flow, a higher order geometric flow. In a recent paper, we define and investigate a generalization of solitons that we call extended ambient obstruction solitons. We show that a solution to the ambient obstruction flow that is fixed up to conformal diffeomorphism is an extended ambient obstruction soliton. On compact manifolds, this implies that ambient obstruction flat metrics are the only fixed points of the ambient obstruction flow up to conformal diffeomorphism. This construction allows us to ultimately show that all closed ambient obstruction solitons are ambient obstruction flat. We conclude by looking at how this notion applies to more explicit examples of non-compact Bach solitons and extended Bach solitons in dimension n=4. This work is joint with Rahul Poddar, Ramesh Sharma, and Will Wylie.
Institution: Mount Holyoke College
Website: https://sites.google.com/view/erin-griffin-math
Speaker: Friedrich Bauermeister
Title: Refocusing spacetimes need not be strongly refocusing
Abstract: Robert Low introduced the notion of a refocusing spacetime as part of his program to reconstruct spacetimes from their null geodesics, or “lightrays”, and their skies: the sets of all lightrays through individual points. For strongly causal spacetimes, the space of lightrays is naturally a contact manifold, and each sky is a Legendrian submanifold. The map sending a point to its sky is called the sky map. Low endowed the space of skies with a topology, which he called the “reconstructive topology”, and defined refocusing as those for which the sky map fails to be a homeomorphism. A key result in this program is the Legendrian Low conjecture, proved by Chernov and Nemirovski. It states that, for suitable globally hyperbolic spacetimes, two points are causally related exactly when their skies are Legendrian linked in the space of lightrays. One obstruction to being “suitable” is the possibility that two distinct points share the same sky. Such spacetimes are called strongly refocusing; equivalently, strong refocusing means that the sky map is not injective. In 2010, Chernov, Kinlaw, and Sadykov asked whether refocusing spacetimes must always be strongly refocusing, or equivalently whether injective sky maps must always be homeomorphisms. We answer these questions in the negative. In fact, every globally hyperbolic strongly refocusing spacetime of dimension at least three admits globally hyperbolic metrics that are refocusing but not strongly refocusing. We also define a new topology on the space of skies and a corresponding notion of refocusing, called Legendrian refocusing, which lies between refocusing and strong refocusing. I will explain why this new topology on the space of skies is in some ways more natural than Low’s reconstructive topology, and show that every globally hyperbolic Legendrian refocusing spacetime admits a globally hyperbolic strongly refocusing metric.
Institution: Dartmouth College
Website: https://sites.google.com/view/friedrich-bauermeister/home
Speaker: Alex Zupan
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Institution: University of Nebraska-Lincoln
Website: https://sites.google.com/view/alexzupan
Thanksgiving
Speaker: Mira Wattal
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Institution: Boston College
Website: https://www.bc.edu/bc-web/schools/morrissey/departments/math/people/grad-students/Mira-Wattal.html
Speaker: Daniel Alvarez-Gavela
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Institution: Brandeis University
Website: https://danielalvarezgavela.com/
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