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Geometrical Aspects of Gauge Configurations

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    Title: Introduction to knot theory. This course will give an introduction to polynomial invariants of knots and their categorifications Khovanov homology. Knot polynomials have a long history, dating back to work of Alexander in the 20's, but the modern history of the subject starts with the discovery the Jones polynomial in My goal is to define the invariants, describe some of their properties, and give some topological applications.

    I'll start by discussing the Alexander polynomial and its relation to cyclic coverings and the Seifert genus.

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    Knot Floer homology, which categorifies the Alexander polynomial, will be discussed in Hom's lectures. I'll discuss its definition via the Jones-Ocneanu trace, and its categorification, due to Khovanov and Rozansky. I'll describe the categorified analog spectral sequences and applications to the slice genus of knots.

    Course Outline:. Lecture 1: The Alexander polynomial.

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    Lecture 2: The Jones polynomial and Khovanov homology. Lecture 3: Khovanov homology for tangles. Lecture 5: Deformations and Spectral Sequences. Lecture 6: Colored homologies.

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    Prerequisites: Basic algebraic topology, including covering spaces and homology. Some experience with representation theory for either finite groups or Lie groups would be helpful for context, but is not strictly necessary. Suggested reading overlaps with the first two lectures :.

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    Higgs bundles provide a unifying structure for diverse branches of mathematics and physics. The Dolbeault Moduli space of Higgs bundles has a hyperkahler structure, and through different complex structures it can be understood as different moduli spaces: Via the non-abelian Hodge correspondence the moduli space is diffeomorphic as a real manifold to the De Rahm moduli space of flat connections. Via the Riemann-Hilbert correspondence there is an analytic correspondence with the Betti moduli space of conjugacy classes of surface group representations.

    We will begin by introducing Higgs bundles and their main properties Lecture 1 , and then we will discuss the Hitchin fibration and its different uses Lecture 2. The second half of the course will be dedicated to studying different types of subspaces branes of the moduli space of Higgs bundles, their appearances in terms of flat connections and representations Lecture 3 , as well as correspondences between them Lecture 4. Lecture 2: The Hitchin fibration and Langlands duality.

    Lecture 3: Branes in the moduli space of Higgs bundles. Prerequisites: The main prerequisite is some familiarity with complex and differential geometry, as well as basic Lie theory. Hitchin, The self-duality equations on a Riemann surface, Proc. LMS 55 3 Hitchin, Stable bundles and integrable systems, Duke Math. Hitchin, Langlands duality and G2 spectral curves, Quat. Math Hitchin, Higgs bundles and characteristic classes, arXiv Baraglia, L.