I will claim (and maybe show) that a lot of problems in differential geometry can be reformulated in terms of non-linear elliptic differential operators. After reviewing the theory of linear elliptic operators, I will show what can be said about the non-linear setting.
I will explain how moduli spaces of quadratic differentials on Riemann surfaces can be interpreted as spaces of stability conditions for certain 3-Calabi-Yau triangulated categories. These categories are defined via quivers with potentials, but can also be interpreted as Fukaya categories. This work (joint with Ivan Smith) was inspired by the papers of Gaiotto, Moore and Neitzke, but connections with hyperkahler metrics, Fock-Goncharov coordinates etc. will not be covered in this talk.
We discuss some recent developments on the following long-standing problem known as Ryser's
conjecture. Let $H$ be an $r$-partite $r$-uniform hypergraph. A matching in $H$ is a set of disjoint
edges, and we denote by $\nu(H)$ the maximum size of a matching in $H$. A cover of $H$ is a set of
vertices that intersects every edge of $H$. It is clear that there exists a cover of $H$ of size at
most $r\nu(H)$, but it is conjectured that there is always a cover of size at most $(r-1)\nu(H)$.
Random geometric graphs have been well studied over the last 50 years or so. These are graphs that
are formed between points randomly allocated on a Euclidean space and any two of them are joined if
they are close enough. However, all this theory has been developed when the underlying space is
equipped with the Euclidean metric. But, what if the underlying space is curved?
The aim of this talk is to initiate the study of such random graphs and lead to the development of
their theory. Our focus will be on the case where the underlying space is a hyperbolic space. We
will discuss some typical structural features of these random graphs as well as some applications,
related to their potential as a model for networks that emerge in social life or in biological
sciences.
A dot product representation of a graph assigns to each vertex $s$ a vector $v(s)$ in ${\bf R}^k$ in such a way that $v(s)^T v(t)$ is greater than $1$ if and only $st$ is an edge. Similarly, in a distance representation $|v(s)-v(t)|$ is less than $1$ if and only if $st$ is an edge.
I will discuss the solution of some open problems by Spinrad, Breu and Kirkpatrick and others on these and related geometric representations of graphs. The proofs make use of a connection to oriented pseudoline arrangements.
(Joint work with Colin McDiarmid and Ross Kang)
In this talk, I will focus on an infinite class of 3d N=4 gauge theories
which can be constructed from a certain set of ordered pairs of integer
partitions. These theories can be elegantly realised on brane intervals in
string theory. I will give an elementary review on such brane constructions
and introduce to the audience a symmetry, known as mirror symmetry, which
exchanges two different phases (namely the Higgs and Coulomb phases) of such
theories. Using mirror symmetry as a tool, I will discuss a certain
geometrical aspect of the vacuum moduli spaces of such theories in the
Coulomb phase. It turns out that there are certain infinite subclasses of
such spaces which are special and rather simple to study; they are complete intersections. I will mention some details and many interesting features of these spaces.
For a fixed n we will investigate homomorphisms Out(F_n) to
Out(F_m) (i.e. free representations) and Out(F_n) to
GL_m(K) (i.e. K-linear representations). We will
completely classify both kinds of representations (at least for suitable
fields K) for a range of values $m$.
Rado introduced the following `lion and man' game in the 1930's: two players (the lion and the man) are in the closed unit disc and they can run at the same speed. The lion would like to catch the man and the man would like to avoid being captured.
This game has a chequered history with several false `winning strategies' before Besicovitch finally gave a genuine winning strategy.
We ask the surprising question: can both players win?
Abstract:
Motivated by the correlation functions-Wilson loop correspondence in
maximally supersymmetric Yang-Mills theory, we will investigate a
conjecture of Alday, Buchbinder, and Tseytlin regarding correlators of
null polygonal Wilson loops with local operators in general position.
By translating the problem to twistor space, we can show that such
correlators arise by taking null limits of correlation functions in the
gauge theory, thereby providing a proof for the conjecture.
Additionally, twistor methods allow us to derive a recursive formula for
computing these correlators, akin to the BCFW recursion for scattering
amplitudes.
Abstract: In this talk, I will discuss the peeling behaviour of the Weyl tensor near null infinity for asymptotically flat higher dimensional spacetimes. The result is qualitatively different from the peeling property in 4d. Also, I will discuss the rewriting of the Bondi energy flux in terms of "Newman-Penrose" Weyl components.