Day/Time: Tuesdays from 3:30pm–4:30pm
Location: Phillips 385
A seminar focusing on algebraic, geometric, and quantum topology, as well as related topics. This seminar is also the UNC home of the Triangle Topology Seminar (TTS), a joint topology seminar between UNC, Duke, and NC State. The TTS talks may occur on non-Tuesdays, to accommodate the schedules of a majority of Triangle topologists.
Schedule:
| Date | Speaker(s) | Title | Info |
| Friday 9/13 | Isabella Khan Lisa Piccirillo |
Koszul duality for partial Heegaard diagrams Small exotic 4-manifolds |
TTS at NCSU at 3pm in 4201 SAS Hall |
| Tuesday 9/24 | Dave Rose | Spin link homology | |
| Tuesday 10/8 | Joshua Turner | Haiman ideals, link homology, and affine Springer fibers | |
| Monday 10/21 | Joshua Wang | TBD | TTS at UNC |
| Monday 11/11 | Joshua Greene | TBD | TTS at Duke |
Abstracts:
Josh Turner (10/8): Haiman ideals, link homology, and affine Springer fibers
Abstract: We will discuss a class of ideals in a polynomial ring studied by Mark Haiman in his work on the Hilbert scheme of points, and ask some purely algebraic questions about them. It turns out that these questions are very closely tied to homology of affine Springer fibers, Khovanov-Rozansky homology of links, and to the ORS conjecture. We will discuss which cases are known and unknown, and compute some simple examples.
Dave Rose (9/24): Spin link homology
Abstract: For each finite-dimensional simple complex Lie algebra, Reshetikhin-Turaev define a quantum invariant of knots/links in the 3-sphere with components colored by finite-dimensional representations. These invariants generalize the Jones polynomial, which corresponds to the case of sl(2) and the vector representation. One categorical level higher, Khovanov and Khovanov-Rozansky construct sl(n) link homology theories, explicit and computable homological invariants of knots/links that “categorify” the sl(n) Reshetikhin-Turaev invariants. Extending this theory, Webster defines link homologies associated to arbitrary finite-dimensional simple complex Lie algebras. While sl(n) link homologies have been widely studied and have found spectacular applications in 3- and 4-dimensional topology, essentially nothing is known about link homologies for non type A Lie algebras, beyond their existence.
In this talk, we will present a new categorification of the spin-colored so(2n+1) link invariant, which arises from a novel involution on n-colored sl(2n) Khovanov-Rozansky homology. Our approach involves categorical representation theory, and highlights a subtle connection between link invariants in types A and B that is hidden at the decategorified level. Further, our construction is explicit and computable, thus should be amenable to topological applications. (This is joint work with Elijah Bodish and Ben Elias.)
Isabella Khan (9/13): Koszul duality for partial Heegaard diagrams
Abstract: By slicing a Heegaard diagram for a knot K in $S^3$, it is possible to retrieve the knot Floer homology of K as a tensor product of bimodules over an $\A_{\infty}$ algebra corresponding to the slice. The first step in this process is to assign an $\mathcal{A}_{\infty}$ algebra to this slice, which can also be written as a planar graph. In this talk, we construct a pair of Koszul dual $\mathcal{A}_{\infty}$ algebras corresponding to planar graphs arising as slices of these Heegaard diagrams, and discuss how to verify the Koszul duality relation.
Lisa Piccirillo (9/13): New constructions and invariants of exotic 4-manifolds
Abstract: Dimension four is the lowest dimension where smooth and topological manifolds can differ; any difference between these categories is known as exotica. In particular, a smooth 4-manifold is exotic if there is another smooth 4-manifold which is homeomorphic but not diffeomorphic to it. There is a wealth of literature, mostly written between 1983 and 2008, on producing exotic manifolds, but the techniques pioneered in this era can be hard to use in practice. I will discuss joint work with triangle locals Adam Levine and Tye Lidman in which we give some new techniques for producing exotic 4-manifolds.