High Energy Physics

There have been many attempts to define quasilocal mass for a spacelike 2-surface in a spacetime by the Hamilton-Jacobi method. The essential difficulty in this approach is the choice of the background configuration to be subtracted from the physical Hamiltonian. The quasilocal mass should be positive for general surfaces, but on the other hand should be zero for surfaces in the flat spacetime. In this talk, I shall describe how to use isometric embeddings into the Minkowski space to overcome this difficulty and propose a new definition of gauge-independent quasi-local mass that has the desired properties, in addition to other natural requirements for a mass. This talk is based on a joint work with Shing-Tung Yau at Harvard.

We introduce a projective hypersurface ''normal form'' for a class of K3 surfaces which generalizes the classical Weierstrass normal form for complex elliptic curves. A geometric two-isogeny relates these K3 surfaces to the Kummer K3 surfaces of principally polarized abelian surfaces, with the normal form coefficients naturally identifying with the Igusa basis of Siegel modular forms of degree two. These results are reinterpreted through the lens of the Kuga-Satake Hodge Conjecture, and seen as a prediction coming from mirror symmetry.

The topological recursion of Eynard and Orantin has found many applications in various areas of mathematics. In this talk I will discuss the recursion from the point of view of Hurwitz numbers and local mirror symmetry. I will explain the mathematics underlying the recursion, its relation with the cut-and-join equation, and explore first steps towards proving (and understanding geometrically) the appearance of the recursion in local mirror symmetry.

The Planck-weak hierarchy is investigated in an extradimensional, soft-wall model originally proposed by Batell and Gherghetta. In this model the soft-wall is dynamically generated by background fields that, in the Einstein frame, cause the metric factor to deviate from anti-de Sitter by a power-law of the conformal coordinate. This talk will demonstrate that in order to achieve the appropriate Planck-weak hierarchy, the power of the conformal coordinate must be less than one. This in turn implies that the gravitational sector contains scalar fields that act like unparticles without a mass gap.

We will discuss the notion of categorical Lie algebra actions, as introduced by Rouquier and Khovanov-Lauda. In particular, we will give examples of categorical Lie algebra actions on derived categories of coherent sheaves. We will show that such categorical Lie algebra actions lead to actions of braid groups.

There appear to be only two essentially distinct ways to understand intersection numbers on moduli spaces of curves --- via Hurwitz numbers or symplectic volumes. In this talk, we will consider polynomials defined by Norbury which bridge the gap between these two pictures. They appear in the enumeration of lattice points in moduli spaces of curves and it appears that their coefficients store interesting information. We will also describe a connection between these polynomials and the topological recursion defined by Eynard and Orantin.

An enormous effort is underway to search for the Higgs boson at the LHC. One new development of the past couple of years is to look into the kinematic region where the Higgs boosted, which has led to the possibility to observe the dominant b-bar decay mode as a single "fat jet" when the Higgs is light. I'll discuss how this technique has great promise not only within the Standard Model, but potentially has even greater promise to find a light Higgs in new physics models such as supersymmetry.

The Petrov classification of the Weyl tensor is an important tool in the study of exact solutions of the Einstein equation in 4d. For example, the Kerr solution was discovered in a study of spacetimes with algebraically special Weyl tensors. Algebraic classification of the Weyl tensor has been extended to higher dimensions. I shall review this classification and describe known families of algebraically special solutions. Recent progress towards obtaining a higher dimensional generalization of the Goldberg-Sachs theorem will be described.

I will discuss recent joint work with A. Ionescu and S. Klainerman on the black hole uniqueness problem. A classical result of Hawking (building on earlier work of Carter and Robinson) asserts that any vacuum, stationary black hole exterior region must be isometric to the Kerr exterior, under the restrictive assumption that the space-time metric shouldbe analytic in the entire exterior region. We prove that Hawking's theorem remains valid without the assumption of analyticity, for black hole exteriors which are apriori assumed to be ''close'' to the Kerr exterior solution in a very precise sense. Our method of proof relies on certain geometric Carleman-type estimates for the wave operator. Time permitting, some more recent developments will also be surveyed.

The seminar is devoted to the solution of the AdS/CFT spectral problem, both for infinite and finite volume cases, using integrability. The basic constructions will be explained using an analogy with the relativistic O(4) sigma model. We devote a special attention to the study of the so called dressing factor. This is a scalar factor of the scattering matrix fixed using discrete crossing symmetry. We will show how to explicitly solve the equations dictated by the crossing symmetry, discuss essential properties of the dressing factor and give a shorthand and suggestive notation for the Bethe Ansatz of AdS/CFT. Then we will sketch a shorthand way to pass from the Bethe Ansatz to the Y-system.

A systematic method to construct 4d N=2 supersymmetric theories by compactifying M5-branes on a Riemann surface was found by Gaiotto last year. This suggests that any physical quantity of the 4d theory should be reflected in another physical quantity of the 2d theory living on the Riemann surface. Indeed, one finds that the instanton partition function of the 4d theories equals the conformal blocks of the 2d theory. I would like to illustrate this construction through explicit examples.