Feb 10
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Tim Riley (Cornell): What calculating in the trivial group has to do with RNA-folding, the design of liquid crystals, and counting fixed points
Abstract: I will give polynomial-time dynamic-programming algorithms finding the "areas" of words in certain presentations of the trivial group. I will explain how this relates to RNA-folding, the design of liquid crystals, and counting fixed points of self-maps of compact surfaces.
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Mar 3
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Johanna Mangahas (Buffalo): Normal right-angled Artin groups in mapping class groups
Abstract: Free normal subgroups of mapping class groups abound, by the result of Dahmani, Guirardel, and Osin that the normal closure of high powers of pseudo-Anosovs is free. At the other extreme, if a normal subgroup contains a mapping class supported on too small a subsurface, it can never be isomorphic to a right-angled Artin group, by work of Brendle and Margalit. I will talk about a case right in between: a family of normal subgroups isomorphic to non-free right-angled Artin groups. We also recover, expand, and make constructive the result of Dahmani, Guirardel, and Osin about free normal subgroups. We do this by creating a version of their "windmill" construction tailor-made for the projection complexes introduced by Bestvina, Bromberg, and Fujiwara. This is joint work with Matt Clay and Dan Margalit.
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Mar 10
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Marco Varisco (SUNY-Albany): On Whitehead groups of infinite groups with torsion
Abstract: Given a group G (finite or infinite, abelian or not), its Whitehead group Wh(G) is an abelian group defined using elementary linear algebra. Remarkably, as I will review, Whitehead groups play an important role in the classification of high-dimensional manifolds. Unfortunately, Whitehead groups are very hard to compute. I will overview what is known and what is conjectured about the structure of Whitehead groups. Then I will present joint work with Wolfgang Lück, Holger Reich, and John Rognes [Adv. Math. 304 (2017), 930–1020, arXiv:1504.03674] and with Ross Geoghegan [arXiv:1401.0357] about Whitehead groups of certain infinite groups with torsion, including outer automorphism groups of free groups, and Thompson’s group T.
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Apr 14
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Tolga Etgü (Koç/Princeton): Fukaya categories of plumbings and multiplicative preprojective algebras
Abstract: I will talk about the wrapped Fukaya category of the Weinstein manifold obtained by plumbing copies of cotangent bundles of surfaces. A symplectic handlebody decomposition of this manifold will be described. As objects of the Fukaya category, the cocores of the 2-handles provide a generating set whose endomorphism algebra turns out to be quasi-isomorphic to the derived multiplicative preprojective algebra associated to the plumbing graph. This talk is based on joint work with Yanki Lekili.
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Apr 21
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Abigail Thompson (UC Davis): An invariant of trisected 4-manifolds
Abstract: A closed, orientable 3-manifold M always has a Heegaard splitting, that is, $M^3$ can be described as the union of two handlebodies glued together along their boundaries. Gay and Kirby extended this idea to 4-manifolds, showing that any closed orientable 4-manifold $M^4$ can be described as the union of three 4-dimensional handlebodies, glued together (carefully) along their boundaries. They called this a trisection of $M^4$. I’ll discuss their result, and describe a natural 4-manifold invariant, $L(M^4)$, that arises from this decomposition. This is joint work with D. Gay and R. Kirby.
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Apr 28
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Jonah Gaster (Boston College): Combinatorial properties of curve graphs
Abstract: The curve graph of a closed oriented surface of genus $g$ has vertices given by simple closed curves, and edges that correspond to curves that can be realized disjointly. Inquiry into the large scale geometry of these graphs has borne considerable fruit, and lead to the resolution of some of Thurston's conjectures. We will take a more naive perspective and explore instead combinatorial properties of this graph. For instance, what is its chromatic number (finite due to work of Bestvina-Bromberg-Fujiwara)? What are its induced subgraphs? Though precise answers to these questions are currently beyond reach, we will present progress that informs them.
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May 5
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Sam Taylor (Yale): Veering triangulations and fibered faces of 3-manifolds
Abstract: From a pseudo-Anosov homeomorphism of a surface, Agol’s veering triangulation gives a canonical ideal triangulation of the associated mapping torus punctured along singular fibers. By recent work of Gueritaud, this triangulation can be directly obtained from the stable and unstable laminations of the monodromy. We study the way in which these triangulations interact with the arc complexes of fibers and their subsurfaces. In particular, we find that the veering triangulation records the hierarchy of subsurface projections associated to each fiber in a fibered face of the Thurston norm ball. These projections are in fact visible as embedded subcomplexes of the veering triangulation itself. Through this structure, we obtained explicit control over the size and nature of subsurface projections occurring over a fixed fibered face. This is joint work with Yair Minsky.
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May 12
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Steven Sivek (Bonn): SU(2)-cyclic surgeries and the pillowcase
Abstract: The cyclic surgery theorem of Culler, Gordon, Luecke, and Shalen implies that any knot in S^3 other than a torus knot has at most two nontrivial cyclic surgeries. In this talk, we investigate the weaker notion of SU(2)-cyclic surgeries on a knot, meaning surgeries whose fundamental groups only admit SU(2) representations with cyclic image. By studying the image of the SU(2) character variety of a knot in the “pillowcase”, we will show that if it has infinitely many SU(2)-cyclic surgeries, then the corresponding slopes (viewed as a subset of RP^1) have a unique limit point, which is a finite, rational number, and that this limit is a boundary slope for the knot. As a corollary, it follows that for any nontrivial knot, the set of SU(2)-cyclic surgery slopes is bounded. This is joint work with Raphael Zentner.
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