### No Seminar - Fall Break

- Series
- Geometry Topology Seminar
- Time
- Monday, October 14, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker

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- Series
- Geometry Topology Seminar
- Time
- Monday, October 14, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker

- Series
- Geometry Topology Seminar
- Time
- Monday, October 7, 2019 - 15:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Duncan McCoy – UQAM

I will explain how null-homologous twisting operations can be used to obtain bounds on the topological slice genus. In particular, I will discuss how one can obtain upper bounds on the topological slice genera of torus knots and satellite knots using these operations.

- Series
- Geometry Topology Seminar
- Time
- Monday, October 7, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Paul Melvin – Bryn Mawr College

It is a remarkable fact that some compact topological 4-manifolds X admit infinitely many exotic smooth structures, a phenomenon unique to dimension four. Indeed a fundamental open problem in the subject is to give a meaningful description of the set of all such structures on any given X. This talk will describe one approach to this problem when X is simply-connected, via cork twisting. First we'll sketch an argument to show that any finite list of smooth manifolds homeomorphic to X can be obtained by removing a single compact contractible submanifold (or cork) from X, and then regluing it by powers of a boundary diffeomorphism. In fact, allowing the cork to be noncompact, the collection of all smooth manifolds homeomorphic to X can be obtained in this way. If time permits, we will also indicate how to construct a single universal noncompact cork whose twists yield all smooth closed simply-connected 4-manifolds. This is joint work with Hannah Schwartz.

- Series
- Geometry Topology Seminar
- Time
- Monday, September 30, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Corey Bregman – Brandeis University – cbregman@brandeis.edu

Kodaira, and independently Atiyah, gave the first examples of surface bundles over surfaces whose signature does not vanish, demonstrating that signature need not be multiplicative. These examples, called Kodaira fibrations, are in fact complex projective surfaces admitting a holomorphic submersion onto a complex curve, whose fibers have nonconstant moduli. After reviewing the Atiyah-Kodaira construction, we consider Kodaira fibrations with nontrivial holomorphic invariants in degree one. When the dimension of the invariants is at most two, we show that the total space admits a branched covering over a product of curves.

- Series
- Geometry Topology Seminar
- Time
- Monday, September 23, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Michael Dougherty – Colby College – michael.dougherty@colby.edu

The space of degree-n complex polynomials with distinct roots appears frequently and naturally throughout mathematics, but its shape and structure could be better understood. In recent and ongoing joint work with Jon McCammond, we present a deformation retraction of this space onto a simplicial complex with rich structure given by the combinatorics of noncrossing partitions. In this talk, I will describe the deformation retraction and the resulting combinatorial data associated to each generic complex polynomial. We will also discuss a helpful comment from Daan Krammer which connects our work with the ideas of Bill Thurston and his collaborators.

- Series
- Geometry Topology Seminar
- Time
- Monday, September 16, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Nir Gadish – Massachusetts Institute of Technology – ngadish@mit.edu

As countless examples show, sequences of complicated objects should be studied all at once via the formalism of generating functions. We apply this point of view to the homology and combinatorics of (orbit-)configuration spaces: using the notion of twisted commutative algebras, which categorify exponential generating functions. With this idea the configuration space “generating function” factors into an infinite product, whose terms are surprisingly easy to understand. Beyond the intrinsic aesthetic of this decomposition and its quantitative consequences, it also gives rise to representation stability - a notion of homological stability for sequences of representations of differing groups.

- Series
- Geometry Topology Seminar
- Time
- Monday, September 9, 2019 - 14:00 for
- Location
- Speaker
- Miriam Kuzbary – Georgia Tech – kuzbary@gatech.edu

This is a general audience Geometry-Topology talk where I will give a broad overview of my research interests and techniques I use in my work. My research concerns the study of link concordance using groups, both extracting concordance data from group theoretic invariants and determining the properties of group structures on links modulo concordance. Milnor's invariants are one of the more fundamental link concordance invariants; they are thought of as higher order linking numbers and can be computed using both Massey products (due to Turaev and Porter) and higher order intersections (due to Cochran). In my work, I have generalized Milnor's invariants to knots inside a closed, oriented 3-manifold M. I call this the Dwyer number of a knot and show methods to compute it for null-homologous knots inside a family of 3-manifolds with free fundamental group. I further show Dwyer number provides the weight of the first non-vanishing Massey product in the knot complement in the ambient manifold. Additionally, I proved the Dwyer number detects knots K in M bounding smoothly embedded disks in specific 4-manifolds with boundary M which are not concordant to the unknot in M x I. This result further motivates my definition of a new link concordance group in joint work with Matthew Hedden using the knotification construction of Ozsv'ath and Szab'o. Finally, I will briefly discuss my recent result that the string link concordance group modulo its pure braid subgroup is non-abelian.

- Series
- Geometry Topology Seminar
- Time
- Monday, September 2, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker

- Series
- Geometry Topology Seminar
- Time
- Monday, August 26, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Jasmine Powell – University of Michigan – jtpowell@umich.edu

In complex dynamics, the main objects of study are rational maps on the Riemann sphere. For some large subset of such maps, there is a way to associate to each map a marked torus. Moving around in the space of these maps, we can then track the associated tori and get induced mapping classes. In this talk, we will explore what sorts of mapping classes arise in this way and use this to say something about the topology of the original space of maps.

- Series
- Geometry Topology Seminar
- Time
- Tuesday, August 13, 2019 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skile 114
- Speaker
- Jonathan Beardsley – Georgia Tech

**Please Note:** Please note different day and room.

In this talk, I will describe joint work with Maximilien Péroux on understanding Koszul duality in ∞-topoi. An ∞-topos is a particularly well behaved higher category that behaves like the category of compactly generated spaces. Particularly interesting examples of ∞-topoi are categories of simplicial sheaves on Grothendieck topologies. The main theorem of this work is that given a group object G of an ∞-topos, there is an equivalence of categories between the category of G-modules in that topos and the category of BG-comodules, where BG is the classifying object for G-torsors. In particular, given any pointed space X, the space of loops on X, denoted ΩX, can be lifted to a group object of any ∞-topos, so if X is in addition a connected space, there is an equivalence between objects of any ∞-topos with an ΩX-action, and objects with an X-coaction (where X is a coalgebra via the usual diagonal map). This is a generalization of the classical equivalence between G-spaces and spaces over BG for G a topological group.