Dual Perspectives Meetings

Abstract: N=1 supergravity theory in d=4 dimensions is the simplest locally supersymmetric extension of Einstein's general theory of relativity. A short overview of its many remarkable properties will be given. In particular I will introduce the notion of complex quaternionic exterior differential forms and discuss their relevance for simple supergravity.

Abstract: The spectral properties of natural self-adjoint operators such as the Hodge Laplacian and the Dirac operator play a fundamental role in understanding many geometric and physical problems. I will discuss some aspects of the theory in the very rich context of closed hyperbolic three-manifolds, and highlight connections with geometry, low-dimensional topology, and number theory.

Abstract: Non-Lorentzian geometry refers to a differential geometric framework for space-times with a degenerate metric structure and a local causal structure that differs from the one of Lorentzian geometry. It has recently found new applications, e.g., in the study of field and gravitational theories in non- and ultra-relativistic regimes. In the first of these two lectures, I will provide an introduction to non-Lorentzian geometry, focusing on the examples of Galilean and Newton-Cartan geometry that describe non-relativistic space-times. I will discuss their metric structure and metric-compatible affine connections with and without torsion in a frame formulation. I will furthermore comment on the physical interpretation of these structures and outline differences with Lorentzian geometry. The second lecture will focus on the appearance of non-Lorentzian geometry in non-relativistic string theory, a consistent and UV-complete string theory whose excitations exhibit non-relativistic dispersion relations and Newtonian gravitational interactions. After an introduction to non-relativistic string theory, I will argue that its target space-time geometry is given by an extension of Newton-Cartan geometry, called string Newton-Cartan geometry. I will discuss the structure of string Newton-Cartan geometry and show how it can be viewed as a particular limit of Lorentzian geometry equipped with an extra two-form Kalb-Ramond gauge field. If time permits, I will outline how this limit can be used to obtain effective gravitational field equations for non-relativistic string theory and comment on T-duality in non-relativistic string theory.

Abstract: Smooth Fano 3-folds are classified in 105 families (Iskovskikh, Mori, Mukai). For the description of these families, see https://www.fanography.info . We know which deformation families have K-polystable (Kahler-Einstein) members and which do not (Araujo, Castravet, Cheltsov, Fujita, Kaloghiros, Martinez-Garcia, Shramov, Suess, Viswanathan). Since K-polystable Fano threefolds form good moduli spaces, it would be interesting to describe K-moduli of smoothable Fano 3-folds (moduli that parametrize K-polystable smooth members of a given deformation family and their K-polystable limits). This is a very active area of research, but the problem has only been solved for the following 52 deformation families: zero-dimensional families (47 families), two one-dimensional families (families 2-24 and 2-25), cubic 3-folds (Liu, Xu), complete intersection of two quadrics (Spotti, Sun), quartic double solids (Ascher, DeVleming, Liu). In this talk, I will speak about K-moduli of Fano 3-folds in the family 3-10, see https://www.fanography.info/3-10 . This is a two-dimensional family whose smooth members can be obtained by blowing up a smooth quadric 3-fold along two disjoint conics. We know that a general member of this family is K-stable (Kahler-Einstein and finite automorphism groups), but some smooth members are not K-polystable (not Kahler-Einstein), and some members have infinite automorphism group (Cheltsov, Przyjalkowski, Shramov). In the talk, I will give explicit classification of all smooth members of the family 3-10 (normal forms), explain which smooth Fano 3-folds in this family are K-polystable and which are not (Araujo, Castravet, Cheltsov, Fujita, Kaloghiros, Martinez-Garcia, Shramov, Suess, Viswanathan), and describe all singular K-polystable members of this family (work in progress with Alan Thompson from Loughborough). If time permits, I will explain how to prove K-polystability of one singular and very symmetric member of this deformation family.

Abstract:In the first lecture I will present the geometric construction of a wide class of integrable many-body systems by using the Hamiltonian and Poisson reduction techniques. I will demonstrate that in some cases the corresponding reduction scheme pertains to quantisation where it shows up as a special factorisation problem in quantum algebras. In the second lecture I will discuss an advanced application of the reduction technique to the construction of the hyperbolic Ruijsenaars-Schneider model with spin. I will show that the model enjoys the Poisson-Lie symmetry which explains its superintegrability. For the quantum model without spin I will present the L-operator algebra and integrals of motion in the formalism of the quantum inverse scattering method.

Abstract

Abstract

Abstract