University of Lancaster

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Given a non-orientable Lagrangian surface L in a symplectic 4-manifold, how far
can the cohomology class of the symplectic form be deformed before there is no
longer a Lagrangian isotopic to L? I will properly introduce this and a
related question, both of which are less interesting for orientable
Lagrangians due to topological conditions. The majority of this talk will be
an exposition on Evans' 2020 work in which he solves this deformation
question for a particular Klein bottle. The proof employs the heavy machinery
of symplectic field theory and more classical pseudoholomorphic
curve theory to severely constrain the topology and intersection properties of
the limits of certain pseudoholomorphic curves under a process called
neck-stretching. The treatment of SFT-related material will be light and focus
mainly on how one can use the compactness theorem to prove interesting things.

I will discuss various definitions of quantum or noncommutative graphs that have appeared in the literature, along with motivating examples.  One definition is due to Weaver, where examples arise from quantum channels and the study of quantum zero-error communication.  This definition works for any von Neumann algebra, and is "spatial": an operator system satisfying a certain operator bimodule condition.  Another definition, first due to Musto, Reutter, and Verdon, involves a generalisation of the concept of an adjacency matrix, coming from the study of (simple, undirected) graphs.  Here we study finite-dimensional C*-algebras with a given faithful state; examples are perhaps less obvious.  I will discuss generalisations of the latter framework when the state is not tracial, and discuss various notions of a "morphism" of the resulting objects

Hirschman-Widder densities may be viewed as the probability density functions of positive linear combinations of independent and identically distributed exponential random variables. They also arise naturally in the study of Pólya frequency functions, which are integrable functions that give rise to totally positive Toeplitz kernels. This talk will introduce the class of Hirschman-Widder densities and discuss some of its properties. We will demonstrate connections to Schur polynomials and to orbital integrals. We will conclude by describing the rigidity of this class under composition with polynomial functions.

 This is joint work with Dominique Guillot (University of Delaware), Apoorva Khare (Indian Institute of Science, Bangalore) and Mihai Putinar (University of California at Santa Barbara and Newcastle University).

(Joint with Y. Lekili) If someone gives you a variety with a singular point, you can try and get some understanding of what the singularity looks like by taking its “link”, that is you take the boundary of a neighbourhood of the singular point. For example, the link of the complex plane curve with a cusp $y^2 = x^3$ is a trefoil knot in the 3-sphere. I want to talk about the links of a class of 3-fold singularities which come up in Mori theory: the compound Du Val (cDV) singularities. These links are 5-dimensional manifolds. It turns out that many cDV singularities have the same 5-manifold as their link, and to tell them apart you need to keep track of some extra structure (a contact structure). We use symplectic cohomology to distinguish the contact structures on many of these links.

Planar random growth processes occur widely in the physical world. Examples include diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. One approach to mathematically modelling such processes is to represent the randomly growing clusters as compositions of conformal mappings. In 1998, Hastings and Levitov proposed one such family of models, which includes versions of the physical processes described above. An intriguing property of their model is a conjectured phase transition between models that converge to growing disks, and 'turbulent' non-disk like models. In this talk I will describe a natural generalisation of the Hastings-Levitov family in which the location of each successive particle is distributed according to the density of harmonic measure on the cluster boundary, raised to some power. In recent joint work with Norris and Silvestri, we show that when this power lies within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability. This phase transition in fluctuations can be interpreted as the beginnings of a macroscopic phase transition from disks to non-disks analogous to that present in the Hastings-Levitov family.
 

It is well-known that nilpotent orbits in $\mathfrak{sl}_n(\mathbb C)$ correspond bijectively with the set of partitions of $n$, such that the closure (partial) ordering on orbits is sent to the dominance order on partitions. Taking dual partitions simply turns this poset upside down, so in type $A$ there is an order-reversing involution on the poset of nilpotent orbits. More generally, if $\mathfrak g$ is any simple Lie algebra over $\mathbb C$ then Lusztig-Spaltenstein duality is an order-reversing bijection from the set of special nilpotent orbits in $\mathfrak g$ to the set of special nilpotent orbits in the Langlands dual Lie algebra $\mathfrak g^L$.
It was observed by Kraft and Procesi that the duality in type $A$ is manifested in the geometry of the nullcone. In particular, if two orbits $\mathcal O_1<\mathcal O_2$ are adjacent in the partial order then so are their duals $\mathcal O_1^t>\mathcal O_2^t$, and the isolated singularity attached to the pair $(\mathcal O_1,\mathcal O_2)$ is dual to the singularity attached to $(\mathcal O_2^t,\mathcal O_1^t)$: a Kleinian singularity of type $A_k$ is swapped with the minimal nilpotent orbit closure in $\mathfrak{sl}_{k+1}$ (and vice-versa). Subsequent work of Kraft-Procesi determined singularities associated to such pairs in the remaining classical Lie algebras, but did not specifically touch on duality for pairs of special orbits.
In this talk, I will explain some recent joint research with Fu, Juteau and Sommers on singularities associated to pairs $\mathcal O_1<\mathcal O_2$ of (special) orbits in exceptional Lie algebras. In particular, we (almost always) observe a generalized form of duality for such singularities in any simple Lie algebra.