Mon, 16 Nov 2015

12:00 - 13:00
L3

Energy Gaps and Casimir Energies in Holographic CFTs

Andrew Hickling
(Imperial College)
Abstract

Two interesting properties of static curved space QFTs are Casimir Energies, and the Energy Gaps of fluctuations. We investigate what AdS/CFT has to say about these properties by examining holographic CFTs defined on curved but static spatially closed spacetimes. Being holographic, these CFTs have a dual gravitational description under Gauge/Gravity duality, and these properties of the CFT are reflected in the geometry of the dual bulk.  We can turn this on its head and ask, what does the existence of the gravitational bulk dual imply about these properties of the CFTs? In this talk we will consider holographic CFTs where the dual vacuum state is described by pure Einstein gravity with negative cosmological constant.  We will argue using the bulk geometry first, that if the CFT spacetime's spatial scalar curvature is positive there is a lower bound on the gap for scalar fluctuations, controlled by the minimum value of the boundary Ricci scalar. In fact, we will show that it is precisely the same bound as is satisfied by free scalar CFTs, suggesting that this bound might be something that applies more generally than just in a Holographic context. We will then show, in the case of 2+1 dimensional CFTs, that the Casimir energy is non-positive, and is in fact negative unless the CFT's scalar curvature is constant. In this case, there is no restriction on the boundary scalar curvature, and we can even allow singularities in the bulk, so long as they are 'good' singularities. If time permits, we will also describe some new results about the Hawking-Page transition in this context. 

 
 
Mon, 09 Nov 2015

12:00 - 13:00
L3

Yang-Mills origin of gravitational symmetries

Mike Duff
(Imperial College)
Abstract

By regarding gravity as the convolution of left and right Yang-Mills theories together with a spectator scalar field in the bi-adjoint representation, we derive in linearised approximation the gravitational symmetries of general covariance, p-form gauge invariance, local Lorentz invariance and local supersymmetry from the flat space Yang-Mills symmetries of local gauge invariance and global super-Poincare. As a concrete example we focus on the new-minimal (12+12) off-shell version of simple four-dimensional supergravity obtained by tensoring the off-shell Yang-Mills multiplets (4+4,NL =1)and(3+0,NR =0). 

 
Mon, 26 Oct 2015

12:00 - 13:00
L5

Generalising Calabi-Yau for generic flux backgrounds

Anthony Ashmore
(Imperial College)
Abstract

Calabi-Yau manifolds without flux are perhaps the best-known
supergravity backgrounds that leave some supersymmetry unbroken. The
supersymmetry conditions on such spaces can be rephrased as the
existence and integrability of a particular geometric structure. When
fluxes are allowed, the conditions are more complicated and the
analogue of the geometric structure is not well understood.

In this talk, I will define the analogue of Calabi-Yau geometry for
generic D=4, N=2 backgrounds with flux in both type II and
eleven-dimensional supergravity. The geometry is characterised by a
pair of G-structures in 'exceptional generalised geometry' that
interpolate between complex, symplectic and hyper-Kahler geometry.
Supersymmetry is then equivalent to integrability of the structures,
which appears as moment maps for diffeomorphisms and gauge
transformations. Similar structures also appear in D=5 and D=6
backgrounds with eight supercharges.

As a simple application, I will discuss the case of AdS5 backgrounds
in type IIB, where deformations of these geometric structures give
exactly marginal deformations of the dual field theories.

 
 
Thu, 29 Oct 2015

16:00 - 17:00
L5

Arthur's multiplicity formula for automorphic representations of certain inner forms of special orthogonal and symplectic groups

Olivier Taibi
(Imperial College)
Abstract

I will explain the formulation and proof of Arthur's multiplicity formula for automorphic representations of special orthogonal groups and certain inner forms of symplectic groups $G$ over a number field $F$. I work under an assumption that substantially simplifies the use of the stabilisation of the trace formula, namely that there exists a non-empty set $S$ of real places of $F$ such that $G$ has discrete series at places in $S$ and is quasi-split at places outside $S$, and restricting to automorphic representations of $G(A_{F})$ which have algebraic regular infinitesimal character at the places in $S$. In particular, this proves the general multiplicity formula for groups $G$ such that $F$ is totally real, $G$ is compact at all real places of $F$ and quasi-split at all finite places of $F$. Crucially, the formulation of Arthur's multiplicity formula is made possible by Kaletha's recent work on local and global Galois
gerbes and their application to the normalisation of Kottwitz-Langlands-Shelstad transfer factors.

Mon, 11 May 2015
14:15
L4

New G2 holonomy cones and exotic nearly Kähler structures on compact 6-manifolds

Mark Haskins
(Imperial College)
Abstract

A long-standing problem in almost complex geometry has been the question of existence of (complete) inhomogeneous nearly Kahler 6-manifolds. One of the main motivations for this question comes from $G_2$ geometry: the Riemannian cone over a nearly Kahler 6-manifold is a singular space with holonomy $G_2$.

Viewing Euclidean 7-space as the cone over the round 6-sphere, the induced nearly Kahler structure is the standard $G_2$-invariant almost complex structure on the 6-sphere induced by octonionic multiplication. We resolve this problem by proving the existence of exotic (inhomogeneous) nearly Kahler metrics on the 6-sphere and also on the product of two 3-spheres. This is joint work with Lorenzo Foscolo, Stony Brook.

Thu, 30 Apr 2015

16:00 - 17:00
L4

Utility-Risk Portfolio Selection

Dr Harry Zheng
(Imperial College)
Abstract

In this talk we discuss a utility-risk portfolio selection problem. By considering the first order condition for the objective function, we derive a primitive static problem, called Nonlinear Moment Problem, subject to a set of constraints involving nonlinear functions of “mean-field terms”, to completely characterize the optimal terminal wealth. Under a mild assumption on utility, we establish the existence of the optimal solutions for both utility-downside-risk and utility-strictly-convex-risk problems, their positive answers have long been missing in the literature. In particular, the existence result in utility-downside-risk problem is in contrast with that of mean-downside-risk problem considered in Jin-Yan-Zhou (2005) in which they prove the non-existence of optimal solution instead and we can show the same non-existence result via the corresponding Nonlinear Moment Problem. This is joint work with K.C. Wong (University of Hong Kong) and S.C.P. Yam (Chinese University of Hong Kong).

Thu, 28 May 2015

12:00 - 13:00
L6

Can we compute everything?

Jonathan Ben-Artzi
(Imperial College)
Abstract
It is often desirable to solve mathematical problems as a limit of simpler problems. However, are such techniques always guaranteed to work? For instance, the problem of finding roots of polynomials of degree higher than three starting from some initial guess and then iterating was only solved in the 1980s (Newton's method isn't guaranteed to converge): Doyle and McMullen showed that this is only possible if one allows for multiple independent limits to be taken, not just one. They called such structures "towers of algorithms". In this talk I will apply this idea to other problems (such as computational quantum mechanics, inverse problems, spectral analysis), show that towers of algorithms are a necessary tool, and introduce the Solvability Complexity Index. An important consequence is that solutions to some problems can never be obtained as a limit of finite dimensional approximations (and hence can never be solved numerically). If time permits, I will mention connections with analogous notions in logic and theoretical computer science.
 

Joint work with Anders Hansen (Cambridge), Olavi Nevalinna (Aalto) and Markus Seidel (Zwickau).             

 
Thu, 11 Jun 2015

16:00 - 17:00
L6

Moduli stacks of potentially Barsotti-Tate Galois representations

Toby Gee
(Imperial College)
Abstract

I will discuss joint work with Ana Caraiani, Matthew Emerton and David Savitt, in which we construct moduli stacks of two-dimensional potentially Barsotti-Tate Galois representations, and study the relationship of their geometry to the weight part of Serre's conjecture.

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