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In this talk we take for granted the existence of the Khovanov stable homotopy type and the Steenrod algebra action on Khovanov homology that arises from this construction. We will use basic properties of the behavior of Kh-stable homotopy type under connected sum and the behavior of Khovanov homology under ribbon concordances to show how to propagate non-trivial Sq^n actions on composite knots to prime knots.

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To participate in this seminar, please register HERE.

This will be an introduction to Witten's program to obtain topological information about knots from studying solutions of Kapustin-Witten equations. I will state the main conjecture and explain the difficulties involved in proving it. Then I will introduce the dimensional reduction of KW, explain the significance of its solutions to Witten's program and sketch a Kobayashi-Hitchin type correspondence for such solutions. Time permitting I will discuss my own work on the subject and future directions.

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In this talk, we will review Hitchin's self-duality equation and Hitchin fibration in the moduli space of Higgs bundles and the notion of limiting configuration. This is in some sense the limiting of a family of hermitian metrics with diverging Higgs field, which has application in the study of asymptotic geometry of the Hitchin moduli space.

We will review the gluing technique of Mazzeo, Swoboda, Weiss, Witt on limiting configuration of SL(2,C) Higgs bundles and discuss a new result for real Lie group SU(1,2). One of the main challenges is dealing with complications of considering singular Hitchin fiber and we will also mention an explicit description of spectral data for the SU(1,2) Higgs bundles.

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I will review the ideas going into a construction of global charts for Hamiltonian Floer trajectories. I will describe a few geometric constructions which, when combined with a soft variant of Abouzaid-McLean-Smith’s global charts and with Bai-Xu’s FOP perturbations, allows for a relatively efficient construction of the Floer complex over the integers. The goal will be primarily to help other symplectic topologists use these methods flexibly in their own work. These methods suffice to prove the Arnol’d Conjecture over the integers.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

The Atiyah-Singer index theorem is a landmark result which bridges analysis, geometry, and topology in a striking manner and has subsequently been used in many areas of contemporary mathematics. In this talk I will provide some insight into the modern developments in index theory which aim at broadening the scope of the index theorem. I will focus primarily on the newfound role of deformation spaces in index theory and sketch how one can derive new index theorems in settings where the old techniques cannot be applied.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

A knot is called slice if it bounds a smoothly embedded disk in the four-ball. Much of low-dimensional topology revolves around obstructing this property of a knot. In this talk, we will introduce a method of obstructing sliceness which takes into account the actions of different symmetries on the Floer homology group. Using this, we will show that the (2,1)-cable of the figure-eight knot is not slice, answering a 42-year-old question posed by Kawauchi. Historically, this knot gained particular interest since it was regarded as one of the simplest "potential" counter-example to the slice-ribbon conjecture. Our proof also shows that the branched double cover of the aforementioned knot is a cork. This is joint work with Dai, Kang, Park, and Stoffregen.

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I will show how to construct a global Kuranishi chart for Gromov--Witten moduli spaces (parametrizing closed holomorphic curves of some fixed genus in a symplectic manifold) which generalizes the genus 0 construction of Abouzaid--McLean--Smith. The global chart and its smoothness properties allow us to give a simple definition of virtual classes, identify the resulting classes with their algebro-geometric counterparts and prove a torus localization formula in the symplectic setting. This is based on joint work-in-progress with Amanda Hirschi.

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The work of Aganagic and her collaborators suggests a relation between the moduli spaces of monopoles and knot homology. I will give an explicit description of the spaces of SU(2) monopoles on R^3 (following Donaldson), discuss their relation to  Hilbert schemes of points, and review some facts about the geometry and topology of these spaces. Recently, Bullimore-Dimofte-Gaiotto and Braverman-Finkelberg-Nakajima related the spaces of monopoles and their cousins to Coulomb branches of supersymmetric gauge theories, these developments will be discussed as well.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

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By extending a result of Kronheimer-Mrowka to the family setting, we prove a gluing formula for the family Seiberg-Witten invariant. When the cutting 3-manifold is an L-space, this formula implies a relation between the family Seiberg-Witten invariant, the Seiberg-Witten invariant of the fiber and the index of the family Dirac operator. We use this relation to calculate the Seiberg-Witten invariant of families of 4-manifolds that arise when resolving an ADE singularity using a hyperkähler family of complex structures near the singularity.  Applications include a Z^\infty summand in the fundamental group of Diff(M) for many simply connected 4-manifolds.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

To participate in this seminar, please register HERE.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

To participate in this seminar, please register HERE.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.