In joint work with K. Zhang we construct some explicit canonical geometries on various classes of complex manifolds, following a general symmetry principle pioneered by Calabi in the 70's. Our focus is to allow edge type singularities (that are the natural higher-dimensional analogues of conical Riemann surfaces studied by Picard and others since the 19th century) and study Gromov-Hausdorff limits as the angle in the cone tends to zero.
I will explain how Feynman diagrams arise in purealgebra: how the computation of compositions of maps of a certain naturalclass, from one polynomial ring into another, naturally leads to a certaincomposition operation of quadratics and to Feynman diagrams.
I will also explain, with very little detail, how this is used inthe construction of some very well-behaved poly-time computable knotpolynomials, and then with better detail, why I care about having suchinvariants.
In geometric approach the starting point of the formulation of quantum theory is the cone of states. I'll explain this approach, the algebraic approach , the textbook formulation of quantum mechanics and the relations between these approaches. I'll show how the formulas for probabilities can be obtained from basic principles. I'll talk in more detail about the case when the cone of states is homogeneous and about the relation of homogeneous cones to Jordan algebras and homogeneous complex domains.
Mirror symmetry relates the algebraic and symplectic geometry of spaces which are related by dualizing a Lagrangian torus fibration. From the perspective of representation theory this is particularly interesting, with ties to geometric Langlands duality, in cases where the spaces are hyperkähler, and the Lagrangian tori are actually holomorphic Lagrangian. Such spaces, which arise as moduli spaces of four-dimensional field theories, include character varieties, multiplicative quiver varieties, and the "K-theoretic Coulomb branches" of Braverman-Finkelberg-Nakajima.