L6

It has long been appreciated that in QCD-like theories without fundamental matter, confinement can be given a sharp characterization in terms of symmetry. More recently, such symmetries have been identified as 1-form symmetries, which fit into the broader category of generalized global symmetries.  In this talk I will discuss obstructions to the existence of a 1-form symmetry in large N QCD, where confinement is a sharp notion. I give general arguments for this disconnect between 1-form symmetries and confinement, and use 2d scalar QCD on the lattice as an explicit example.  

Khovanov-Rozansky homology is a link invariant that categorifies the HOMFLY-PT polynomial. I will describe a geometric model for this invariant, living in the monodromic Hecke category. I will also explain how it allows to identify objects representing graded pieces of Khovanov-Rozansky homology, using a remarkable family of character sheaves. Based on joint works with Roman Bezrukavnikov.

Fibres coming from the Springer resolution on the nilpotent cone are incredibly rich algebraic varieties that have many applications in representation theory and combinatorics. Though their geometry can be very difficult to describe in general, in type A at least, their irreducible components can be described using standard Young tableaux, and this can help describe their geometry in small dimensions. In this talk, I will report on recent and ongoing work with Lewis Topley and separately Daniele Rosso on geometrical and combinatorial applications of the classical ‘type A’ Springer fibres and the ‘exotic’ type C Springer fibres coming from Kato’s exotic Springer correspondence.

In this talk we will present some work in progress generalising Lubin-Tate formal group laws to higher dimensions. There have been some other generalisations, but ours is different in that the ring over which the formal group law is defined changes as the dimension increases. We will state some conjectures about these formal group laws, including their relationship to the Drinfeld tower over the p-adic upper half plane, and provide supporting evidence for these conjectures.

Let F be a p-adic field, and k an algebraically closed field of characteristic l different from p. In this talk, we will first give a category decomposition of Rep_k(SL_n(F)), the category of smooth k-representations of SL_n(F), with respect to the GL_n(F)-equivalent supercuspidal classes of SL_n(F), which is not always a block decomposition in general. We then give a block decomposition of the supercuspidal subcategory, by introducing a partition on each GL_n(F)-equivalent supercuspidal class through type theory, and we interpret this partition by the sense of l-blocks of finite groups. We give an example where a block of Rep_k(SL_2(F)) is defined with several SL_2(F)-equivalent supercuspidal classes, which is different from the case where l is zero. We end this talk by giving a prediction on the block decomposition of Rep_k(A) for a general p-adic group A.

We outline Dirac cohomology for Lie algebras and Vogan’s conjecture. We then cover some basic material on Schur-Weyl duality and Arakawa-Suzuki functors. Finishing with current efforts and results on generalising Vogan’s conjecture to a Schur-Weyl duality setting. This would relate the centre of a Lie algebra with the centre of the relevant tantaliser algebra. We finish by considering a unitary module X and giving a bound on the action of the tantalizer algebra.

I will discuss algebraic conditions that for a given group guarantee or characterize the vanishing of cohomology in a given degree with coefficients in any unitary representation. These conditions will be expressed in terms positivity of certain elements over group algebras, where positivity is meant as being a sum of hermitian squares. I will explain how conditions like this can be used to give computer-assisted proofs of vanishing of cohomology. 

The work of Kraft-Procesi classifies closures of nilpotent orbits that are normal in the cases of classical complex Lie algebras. Subsequent work of Ranee Brylinsky combines this work with the Theta correspondence as defined by Howe to attach a representation of the corresponding complex group. It provides a quantization of the closure of a nilpotent orbit. In joint work with Daniel Wong, we carry out a detailed analysis of these representations viewed as (\g,K)-modules of the complex group viewed as a real group. One consequence is a "representation theoretic" proof of the classification of Kraft-Procesi.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

During growth and development, tissue dynamics, such as tissue folding, cell intercalations and oriented cell divisions, are critical for shaping tissues and organs. However, less is known about how tissues regulate their dynamics during tissue homeostasis and repair, to maintain their shape after development. In this talk, we will discuss how differential growth rates can generate precise folds in tissues. We will also discuss how tissues respond to mechanical perturbations, such as stretching or wounding, by altering their actomyosin contractile structures, to change tissue dynamics, and thus preserve tissue shape and patterning. We combine genetics, biophysics and computational modelling to study these processes.