In 2008, Juhasz published the following result, which was proved using sutured Floer homology.
Let $K$ be a prime, alternating knot. Let $a$ be the leading coefficient of the Alexander polynomial of $K$. If $|a|
We will present a physical motivation of the SYZ conjecture and try to understand the conjecture via calibrated geometry. We will define calibrated submanifolds, and also give sketch proofs of some properties of the moduli space of special Lagrangian submanifolds. The talk will be elementary and accessible to a broad audience.
We will consider the monodromy action on mod 2 cohomology for SL(2) Hitchin systems. We will study Copeland's approach to the subject and use his results to compute the monodromy action on mod 2 cohomology. An interpretation of our results in terms of geometric properties of fixed points of a natural involution on the moduli space is given.
Many interesting classes of projective varieties can be studied in terms of their graded rings. For weighted projective varieties, this has been done in the past in relatively low codimension.
Let $G$ be a simple and simply connected Lie group and $P$ be a parabolic subgroup of $G$, then homogeneous space $G/P$ is a projective subvariety of $\mathbb{P}(V)$ for some\\
$G$-representation $V$. I will describe weighted projective analogues of these spaces and give the corresponding Hilbert series formula for this construction. I will also show how one may use such spaces as ambient spaces to construct weighted projective varieties of higher codimension.
After reviewing the salient details from last week's seminar, I will construct an explicit example of a spectral curve, using co-Higgs bundles of rank 2. The role of the spectral curve in understanding the moduli space will be made clear by appealing to the Hitchin fibration, and from there inferences (some of them very concrete) can be made about the structure of the moduli space. I will make some conjectures about the higher-dimensional picture, and also try to show how spectral varieties might live in that picture.
PLEASE NOTE THE CHANGE OF TIME FOR THIS WEEK: 13.30 instead of 12.
In the first of two talks, I will simultaneously introduce the notion of a co-Higgs vector bundle and the notion of the spectral curve associated to a compact Riemann surface equipped with a vector bundle and some extra data. I will try to put these ideas into both a historical context and a contemporary one. As we delve deeper, the emphasis will be on using spectral curves to better understand a particular moduli space.
Physical space-time is a manifold with a Lorentzianmetric, but the more mathematical treatments of the theory usually prefer toreplace the metric with a positive - i.e. Riemannian - one. The passage fromLorentzian to Riemannian metrics is called 'Wick rotation'. In my talk I shallgive a precise description of what is involved, and shall explain some of itsimplications for physics.