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Zoom Link:  https://pitp.zoom.us/j/93581608531?pwd=d3NRQXRGNTNISkhuWmxLYkJMZllTUT09

Based on recent work arXiv:1902.08207 and arXiv:1911.02018 with E. Verlinde.

Over the last decade, the Effective Field Theory of Large Scale Structure (EFTofLSS) has emerged as a frontrunner in the effort to produce accurate models of cosmological statistics. Quantities such as power spectra can be fit with sub-percent precision, and there is a wealth of literature applying the formalism to more complex statistics. It is interesting to ask what lies ahead for the theory. Can it be used for cosmological parameter inference? And is it just for statistics based on the 3D density field?

 

In this talk, I will present a pedagogical introduction to the EFTofLSS, discussing its motivation and basic formalism, as well as a few of its theoretical challenges. To demonstrate the utility of the theory beyond the blackboard, I will discuss the analysis of galaxy power spectra. Using this model, competitive constraints can be placed on cosmology utilizing all the large-scale spatial information, not just the position of the Baryon Acoustic Oscillations. In particular, this yields the strongest CMB-independent measurement of H0. 

 

In answering the second question, I will introduce two less conventional applications of the EFTofLSS. Firstly, the marked density field. This is simply the matter field weighted by its local overdensity, and recent works have shown its power spectrum to be capable of placing strong constraints on the neutrino mass. I will discussing its perturbative modeling, highlighting its unusual features, and how the model allows us to shed light on the surprising constraining of the statistic. An additional statistic of interest is weak lensing. Creating an analytic model for this has its own complications, since it is sensitive to a large range of scales, requiring extensions to the usual EFTofLSS modeling. Crucial to this effort is the development of a matter power spectrum model which is accurate on all scales; I will present the results of recent modeling efforts within the ‘Effective Halo Model’, and discuss future applications to integrated statistics such as weak lensing.

Zoom Link: https://pitp.zoom.us/j/91697113596?pwd=OTh3ZjZ5SHd5Q09sTFdReUMyb0hpUT09

Hessenberg varieties are a distinguished family of projective varieties associated to a semisimple complex algebraic group. We use the formalism of perverse sheaves to study their cohomology rings. We give a partial characterization, in terms of the Springer correspondence, of the irreducible representations which appear in the action of the Weyl group on the cohomology ring of a regular semisimple Hessenberg variety. We also prove a support theorem for the universal family of regular Hessenberg varieties, and we deduce that its fibers, though not necessarily smooth, always have the "Kahler package". This is joint work with Peter Crooks.

Categorical representation theory is filled with graded additive categories (defined by generators and relations) whose Grothendieck groups are algebras over \mathbb{Z}[q,q^{-1}]. For example, Khovanov-Lauda-Rouquier (KLR) algebras categorify the quantum group, and the diagrammatic Hecke categories categorify Hecke algebras. Khovanov introduced Hopfological algebra in 2006 as a method to potentially categorify the specialization of these \mathbb{Z}[q,q^{-1}]-algebras at q = \zeta_n a root of unity. The schtick is this: one equips the category (e.g. the KLR algebra) with a derivation d of degree 2, which satisfies d^p = 0 after specialization to characteristic p, making this specialization into a p-dg algebra. The p-dg Grothendieck group of a p-dg algebra is automatically a module over \mathbb{Z}[\zeta_{2p}]... but it is NOT automatically the specialization of the ordinary Grothendieck group at a root of unity! Upgrading the categorification to a p-dg algebra was done for quantum groups by Qi-Khovanov and Qi-Elias. Recently, Qi-Elias accomplished the task for the diagrammatic Hecke algebra in type A, and ruled out the possibility for most other types. Now the question is: what IS the p-dg Grothendieck group? Do you get the quantum group/hecke algebra at a root of unity, or not? This is a really hard question, and currently the only techniques for establishing such a result involve explicit knowledge of all the important idempotents in the category. These techniques sufficed for quantum \mathfrak{sl}_n with n \le 3, but new techniques are required to make further progress. After reviewing the theory of p-dg algebras and their Grothendieck groups, we will present some new techniques and conjectures, which we hope will blow your mind. Everything is joint with You Qi.

For quantum groups at a root of unity, there is a web of theorems (due to Bezrukavnikov and Ostrik, and relying on work of Lusztig) connecting the following topics: (i) tilting modules; (ii) vector bundles on nilpotent orbits; and (iii) Kazhdan–Lusztig cells in the affine Weyl group. In this talk, I will review these results, and I will explain a (partly conjectural) analogous picture for reductive algebraic groups over fields of positive characteristic, inspired by a conjecture of Humphreys. This is joint work with W. Hardesty and S. Riche.

Koszul duality, as conceived by Beilinson-Ginzburg-Soergel, describes a remarkable symmetry in the representation theory of Langlands dual reductive groups. Geometrically, Koszul duality can be stated as an equivalence of categories of mixed (motivic) sheaves on flag varieties. In this talk, I will argue that there should be an an 'ungraded' version of Koszul duality between monodromic constructible sheaves and equivariant K-motives on flag varieties. For this, I will explain what K-motives are and present preliminary results.

Admissible representations of real reductive Lie groups are a key player in the world of unitary representation theory. The characters of irreducible admissible representations were described by Lustig—Vogan in the 80’s in terms of a geometrically-defined module over the associated Hecke algebra. In this talk, I’ll describe a categorification of this module using Soergel bimodules, with a focus on examples. This is work in progress.

Suzuki's functor relates the representation theory of the affine Lie algebra to the representation theory of the rational Cherednik algebra in type A. In this talk, we discuss an extension of this functor to the critical level, t=0 case. This case is special because the respective categories of representations have large centres. Our main result describes the relationship between these centres, and provides a partial geometric interpretation in terms of Calogero-Moser spaces and opers.

In this note we give an alternative presentation of the rational Cherednik algebra H_c corresponding to the permutation representation of S_n. As an application, we give an explicit combinatorial basis for all standard and simple modules if the denominator of c is at least n, and describe the action of H_c in this basis. We also give a basis for the irreducible quotient of the polynomial representation and compare it to the basis of fixed points in the homology of the parabolic Hilbert scheme of points on the plane curve singularity {x^n=y^m}. This is a joint work with José Simental and Monica Vazirani.

The study of compact objects in the strong field regime needs a thorough understanding of the initial value problem in general relativity at the resence of hydrodynamical or magnetohydrodynamical sources. This is a twofold problem that includes general relativistic solutions that represent realistic astrophysical systems at a given moment in time as well as their subsequent evolutions. In this talk I will present the fundamental principles of this endeavor as well as efforts in understanding a great variety of astrophysical systems from binary neutron stars to ergostars and black hole-disks through numerical relativity.

 

The category of Harish-Chandra bimodules is ubiquitous in representation theory. In this talk I will explain their relationship to the theory of dynamical R-matrices (going back to the works of Donin and Mudrov) and quantum moment maps. I will also relate the monoidal properties of the parabolic restriction functor for Harish-Chandra bimodules to the so-called standard dynamical R-matrix. This is a report on work in progress, joint with Artem Kalmykov.

The preprojective algebra of a quiver naturally appears when computing the cotangent to the quiver moduli, via the moment map. When considering the derived setting, it is replaced by its differential graded (dg) variant, introduced by Ginzburg. This construction can be generalized using potentials, so that one retrieves critical loci when considering moduli of perfect modules. Our idea is to consider some relative, or constrained critical loci, deformations of the above, and study Calabi--Yau structures on the underlying relative versions of Ginzburg's dg-algebras. It yields for instance some new lagrangian subvarieties of the Hilbert schemes of points on the plane. This reports a joint work with Damien Calaque and Sarah Scherotzke arxiv.org/abs/2006.01069