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R. P. Thomas
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Beginning with the theorems of Beilinson and Orlov on derived categories, we show how these lead naturally to Kuznetsovs beautiful theory of Homological Projective Duality. We then survey some examples.
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The Katz-Klemm-Vafa conjecture expresses the Gromov-Witten theory of K3 surfaces (and K3-fibred 3-folds in fibre classes) in terms of modular forms. Its recent proof gives the first non-toric geometry in dimension greater than 1 where Gromov-Witten theory is exactly solved in all genera. We survey the various steps in the proof. The MNOP correspondence and a new Pairs/Noether-Lefschetz correspondence for K3-fibred 3-folds transform the Gromov-Witten problem into a calculation of the full stable pairs theory of a local K3-fibred 3-fold. The stable pairs calculation is then carried out via degeneration, localisation, vanishing results, and new multiple cover formulae.
We review a combinatoric approach to the Hodge Conjecture for Fermat Varieties and announce new cases where the conjecture is true.
We provide some corrections and clarifications to the paper [Gr3] of the title. In particular, we clarify the left/right conventions on complex reflection groups and their braid groups. Most importantly, we fill in a gap related to the treatment of cuts in the Picard-Lefschetz theory part of the argument. The statements of the main results are not affected.
This is an expository paper. Its purpose is to explain the linear algebra that underlies Donaldson-Thomas theory and the geometry of Riemannian manifolds with holonomy in $G_2$ and ${rm Spin}(7)$.
The representation problem of finite-dimensional Markov matrices in Markov semigroups is revisited, with emphasis on concrete criteria for matrix subclasses of theoretical or practical relevance, such as equal-input, circulant, symmetric or doubly stochastic matrices. Here, we pay special attention to various algebraic properties of the embedding problem, and discuss the connection with the centraliser of a Markov matrix.
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