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: The (unique) maximally extended D=11 supermembrane theory stands out as a candidate for the non-perturbative unification of superstring theory. In this talk I will review some basic features, in particular the light-cone gauge reformulation of the theory as the N-->∞ limit of the maximally supersymmetric SU(N) matrix model, and present new evidence for the existence of the N-->∞ limit, using a path integral formulation. I will also touch on several open issues, such as the construction of supermembrane vertex operators.

Abstract: In the first part of the talk, I will give a quick introduction to D-modules, local cohomology, s-parametric annihilators and Bernstein-Sato polynomials. In the second part, I will consider holonomic D-modules associated to a hypersurface with non-isolated singularities. I will show a method for computing holonomic D-modules and microlocal b-functions. The keys of the method are the use of a Poincare-Birkhoff-Witt algebra and local cohomology. I will discuss, as applications, relations between the structure of holonomic D-module and the vertical monodromy introduced by Siersma and Le cycles introduced by Massey.

Abstract: Supersymmetric extensions of Einstein geometric theory of general relativity have the remarkable feature that they can give rise to unexpected global symmetries, known as hidden symmetries. In recent years, there has been progress in phrasing them in a geometric manner through so-called generalised or exceptional geometry. In the first part, I will review these structures and how they give rise to the symmetries of supergravity. I will sketch how they can be used to formulate and study deformations of supergravity in a systematic way. In the second part, I will apply these ideas to a particularly interesting case, both mathematically and physically. This is the case of two-dimensional dilaton-gravity, where work since the 1970s on solution generating techniques and integrable system has been active and is related to an underlying affine Kac-Moody symmetry. I will review recent work on constructing the associated exceptional field theory and use it to obtain new dilaton-gravity models that allow for AdS-type solutions that are conjectured to be holographically dual to matrix models, such as the one associated to D0-branes.

Abstract: In this first general talk I will explain what "symplectic" means, and sketch a proof of Gromov's non-squeezing theorem and of Gromov's 2-ball theorem. These basic symplectic rigidity results already have applications to problems in dynamics, such as short-time super-recurrence and the non-existence of local attractors of certain Hamiltonian PDEs. For the second part, write $B^4(a)$ for the ball of capacity $a=\pi r^2$, and $Z^4$ for the symplectic cylinder $D^2(1) \times \RR^2$ where $D^2(1)$ is the disc of area 1. Going beyond Gromov's non-squeezing theorem, Sackel, Song, Varolgunes, and Zhu recently showed that for $a>1$ the complement $B^4(a) - S$ of a subset $S$ in the ball cannot be embedded symplectically into $Z^4$ if the Minkowski dimension of $S$ is $<2$. They also found that this result is sharp provided that $a<2$, and then Brendel extended this to $a<3$. In joint work with Emmanuel Opshtein, we find in any ball $B^4(a)$ a finite union of planar Lagrangian discs $S$ such that $B^4(a) \setminus S$ symplectically embeds into $Z^4$. Among the applications are: capacity killing; non-displaceability of the Clifford torus $T(1/d,1/d)$ from $S$ in $B^4(d)$; and the existence of very short Reeb chords from a Legendrian knot back to itself or to $S$.