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.

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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.

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In their celebrated work, Gordon and Luecke proved that knots in the three-dimensional sphere are determined by their complements. Subsequently, Boileau asked whether the same result holds for null-homotopic knots in arbitrary 3-manifolds. In this talk, I will discuss a program to answer this question. In particular, I will explain how one can give an affirmative answer to Boileau's question for arbitrary knots in some families of 3-manifolds including any connected sum of Brieskorn homology spheres. This is joint work with Tye Lidman.

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To participate in this seminar, please register HERE.

Heegaard-Floer homology was originally introduced by Ozsvath-Szabo as a variation of Lagrangian-Floer homology of a space associated to a Heegaard diagram. It was then generalized to compact three manifolds with boundary, where we associated to the boundary a strand algebra, the three manifold with boundary a module over the strand algebra, and they should glue to our Heegaard-Floer homology. Auroux showed that the strand algebras associated to the bordered Floer homology, or more generally some algebras coming from an open Riemann surface, generate the Fukaya category of the space. In this talk I’ll introduce the construction of Heegaard-Floer, bordered Floer and Ozsvath-Szabo algebras, Fukaya category of the symmetric product of Riemann surfaces, and Auroux’ work on generation results. It’s a special case of my joint work in progress with Sukjoo Lee, Yin Li and Cheuk-Yu Mak.

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To participate in this seminar, please register HERE.

Ten years ago Gaiotto, Moore and Neitzke proposed a twistorial method of constructing Hyperkähler metrics on moduli spaces of Higgs bundles via solutions of a certain Riemann-Hilbert problem. Additionally they conjectured that a different construction, using Fock-Goncharov coordinates, exists for the specific case of weakly parabolic Higgs bundles. I will explain how it is possible to use recent results of Fredrickson, Mazzeo, Swoboda and Weiß to carry out this specific construction, how it relates to the general Riemann-Hilbert case and how it helps in comparing the L^2 metric and the semiflat metric on the moduli space.

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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.

An important principle in 4-dimesional topology, as discovered by Wall in the 1960s, states that all exotic phenomena are eliminated by sufficiently many stabilizations (i.e., taking connected sum with S2 xS2's). Since then, it has been a fundamental problem to search for exotic phenomena that survives one stabilization. In this talk, we will establish the first pair of orientable exotic surfaces (in a puctured K3) which are not smoothly isotopic even after one stabilization. A key ingredient in our argument is a vanishing theorem for the family Bauer-Furuta invariant, proved using equivariant stable homotopy theory. This theorem applies to a large family of spin 4-manifolds and has some interesting applications in Smale's conjecture (about exotic diffeomorphisms on S^4). In particular, it implies that the S^1-equivariant or non-equivariant family Bauer-Furuta invariant do not detect an exotic diffeomorphism on S^4 and it suggests that the Pin(2)-symmetry could be a game changer. This is a joint work with Jianfeng Lin. Also if time permits I will discuss exotically embedded 3-manifolds in 4-manifolds which is an upcoming work joint with Hokuto Konno and Masaki Taniguchi.

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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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.