The generalised Mukai conjecture (GMC) concerns the characterisation of powers of projective space among Fano varieties. In this talk, we will prove the GMC for spherical varieties. These varieties generalise toric varieties, and their geometry may be interpreted combinatorially via a 'spherical dictionary'. If time permits, we will explain how this result can be extended to prove a purely combinatorial smoothness criterion for spherical varieties.
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In this talk we will discuss the notion of a quiver and its applications to algebraic geometry. In particular we will construct moduli spaces of quiver representations and stability conditions on quivers. We will show how to construct some simple GIT quotients from this data. Along the way Gabriel's theorem and Mumford's criterion will be covered.
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(Co)homology theories are ubiquitous throughout pure mathematics. Can we find some objects which `capture’ the (co)homological behaviour of our spaces of interest? In algebraic geometry, the answer is the theory of motives. They allow one to use topological methods in the context of algebraic geometry, and can be thought of as a `universal cohomology theory’ for varieties. In this talk I will give a brief introduction to the theory, highlighting the analogies with topology, and I’ll also discuss some applications.
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TBA
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TBA
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TBA
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