Fridays from 2:30-3:30pm in LGRT 1681 at UMass Amherst
Organizers: İnanç Baykur, Patricia Cahn, Miriam Kuzbary and Riccardo Pedrotti
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Speaker: ORGANISATION MEETING
Speaker: Patricia Cahn
Title: Rep-Tiles
Abstract: An n-dimensional rep-tile is a compact submanifold of R^n that can be decomposed into isometric re-scaled copies of itself, with non-overlapping interiors. We give a complete isotopy classification of rep-tiles in all dimensions. This is joint work with Blair, Kjuchukova, and Schwartz.
Institution: Smith College
Website: http://www.science.smith.edu/~pcahn/
Speaker: Riccardo Pedrotti
Title: On sections of Lefschetz fibrations over the disk
Abstract: I'll report on an ongoing project, joint work with J. Hillman, aimed at finding criteria for the existence of sections on a given Lefschetz fibration over a surface. We will start by presenting a nice algebraic criterion for the existence of sections in a surface bundle and explain what goes wrong if we try to apply it to the more general Lefschetz fibration case. The question of when a nullhomotopic loop in the boundary of a Lefschetz fibration over the disk can be extended to a section over the whole disk is one such subtle issue. Our computations suggest that working with continuous or smooth sections leads to different answers.
Institution: University of Massachusetts Amherst
Website: https://riccardopedrotti.github.io
Speaker: Abhishek Mallick
Title: Instanton Floer homology, involutions, and exotica
Abstract: Instanton Floer homology together with the Donaldson-invariant has played a central role in understanding many exotic phenomena in 4-manifold topology. In this talk, I will discuss some recent applications of instanton Floer homology to studying exotica via the TQFT-like structures within the theory together with actions of symmetry. In particular, we will discuss exotic smooth structures and exotic disks surviving stabilization by projective planes. This is based on a joint work with Alfieri, Dai, and Taniguchi and an ongoing work with Dai and Taniguchi.
Institution: Dartmouth College
Website: https://abhishekmallickmath.github.io
Speaker: Thomas Kindred
Title: Murasugi sum and arborescence in 3D and 4D
Abstract: Many invariants of knots and links in $S^3$ can be understood in terms of spanning surfaces---compact surfaces embedded in $S^3$ with boundary equal to a given knot or link. Murasugi sum, also called generalized plumbing, is a particular way of gluing together two spanning surfaces along a disk. I will begin by describing how these gluings work and how they improve our understanding of at least some of the following: the Alexander polynomial, the genus of a knot, fiberedness, incompressibility and the fundamental group, Tait's flyping conjecture, Kauffman states, state surfaces, Khovanov homology, and Heegaard Floer homology. Next, I will explain how plumbing unknotted annuli and Mobius bands in the pattern of a tree yields several important classes of links: 2-bridge, pretzel, Montesinos, and arborescent. The 2-bridge case is particularly rich, as described in a beautiful paper by Hatcher and Thurston, and I may go into some detail about current work on this topic. Finally, I will describe other current work that adapts Murasugi sum to an operation on spanning solids in $S^4$. Expect lots of pictures!
Institution: Smith College
Website: https://thomaskindred.com
Speaker: Laura Wakelin
Title: Dehn surgery functions are never injective.
Abstract: For any fixed rational number p/q, Dehn surgery gives a map from the set of knots in the 3-sphere to the set of closed orientable 3-manifolds. In 1978, Gordon conjectured that these maps are never injective. I will briefly discuss some results which demonstrate non-injectivity for some special cases of p/q, before going on to present joint work with Kyle Hayden and Lisa Piccirillo in which we prove the conjecture using rational RBG links and the zeroth HOMFLYPT polynomial.
Institution: UT Austin
Website: https://sites.google.com/view/laurawakelin
Spring Break
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Speaker: Kyle Hayden
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Institution: Rutgers University-Newark
Website: https://sites.google.com/view/kylehayden/home
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