Search Talks

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

I will explain a generalized Albanese property for smooth curves, which implies Deligne's geometric class field theory with arbitrary ramification. The proof essentially reduces to some well-known Cartier duality statements. This is joint work with Andreas Hayash.

The Springer resolution and resulting Springer sheaf are key players in geometric representation theory. While one can construct the Springer sheaf geometrically, Hotta and Kashiwara gave it a purely algebraic reincarnation in the language of equivariant $D(\mathfrak{g})$-modules. For $G = GL_N$, the endomorphism algebra of the Springer sheaf, or equivalently of the associated $D$-module, is isomorphic to $\mathbb{C}[\mathcal{S}_n]$ the group algebra of the symmetric group. In this talk, I'll discuss a quantum analogue of this. In joint work with Sam Gunningham and David Jordan, we define quantum Hotta-Kashiwara $D$-modules $\mathrm{HK}_\chi$, and compute their endomorphism algebras. In particular $\mathrm{End}_{\mathcal{D}_q(G)}(\mathrm{HK}_0) \simeq \mathbb{C}[\mathcal{S}_n]$. This is part of a larger program to understand the category of strongly equivariant quantum $D$-modules. Our main tool to study this category is Jordan's elliptic Schur-Weyl duality functor to representations of the double affine Hecke algebra (DAHA). When we input $\mathrm{HK}_0$ into Jordan's functor, the endomorphism algebra over the DAHA of the output is $\mathbb{C}[\mathcal{S}_n]$ from which we deduce the result above. From studying the output of all the $\mathrm{HK}_\chi$, we are able to compute that for input a distinguished projective generator of the category the output is the DAHA module generated by the sign idempotent. This is joint work with Sam Gunningham and David Jordan.

I will give an overview of a joint project with Simon Riche and Laura Rider and another one with Dima Arinkin aimed at a modular version of the equivalence between two geometric realization of the affine Hecke algebra and derived Satake equivalence respectively. As a byproduct we obtain a proof of the Finkelberg-Mirkovic conjecture and a possible approach to understanding cohomology of higher Frobenius kernels with coefficients in a G-module.

The Beilinson-Drinfeld Grassmannian of a simple complex algebraic group admits a natural stratification into "global spherical Schubert varieties". In the case when the underlying curve is the affine line, we determine algebraically the global sections of the determinant line bundle over these global Schubert varieties as modules over the corresponding Lie algebra of currents. The resulting modules are the global Weyl modules (in the simply laced case) and generalizations thereof. This is a joint work with Ilya Dumanski and Evgeny Feigin.

Elliptic stable envelopes, introduced by Aganagic and Okounkov, are a key ingredient in the study of quantum integrable systems attached to a symplectic resolution. I will describe a relation between elliptic stable envelopes on a hypertoric variety and a certain 'loop space' of that variety. Joint with Artan Sheshmani and Shing-Tung Yau.

Given a quiver with potential, Kontsevich-Soibelman constructed a Hall algebra on the cohomology of the stack of representations of (Q,W). In particular cases, one recovers positive parts of Yangians as defined by Maulik-Okounkov. For general (Q,W), the Hall algebra has nice structure properties, for example Davison-Meinhardt proved a PBW theorem for it using the decomposition theorem. One can define a K-theoretic version of this algebra using certain categories of singularities that depend on the stack of representations of (Q,W). In particular cases, these Hall algebras are positive parts of quantum affine algebras. We show that some of the structure properties in cohomology, such as the PBW theorem, can be lifted to K-theory, replacing the use of the decomposition theorem with semi-orthogonal decompositions.

Since the 1980s, mathematicians have found connections between orbit closures in type A quiver representation varieties and Schubert varieties in type A flag varieties. For example, singularity types appearing in type A quiver orbit closures coincide with those appearing in Schubert varieties in type A flag varieties (Bobinski-Zwara); combinatorics of type A quiver orbit closure containment is governed by Bruhat order on the symmetric group (follows from work of Zelevinsky, Kinser-R); and multiple researchers have produced formulas for classes of type A quiver orbit closures in equivariant cohomology and K-theory in terms of Schubert polynomials, Grothendieck polynomials, and related objects. After recalling some of this type A story, I will discuss joint work with Ryan Kinser on type D quiver representation varieties. I will describe explicit embeddings which completes a circle of links between orbit closures in type D quiver representation varieties, B-orbit closures (for a Borel subgroup B of GL_n) in certain symmetric varieties GL_n/K, and B-orbit closures in double Grassmannians Gr(a, n) x Gr(b, n). I will end with some geometric and combinatorial consequences, as well as a brief discussion of joint work in progress with Zachary Hamaker and Ryan Kinser on formulas for classes of type D quiver orbit closures in equivariant cohomology.

Neutrinos are a key (although implicit) ingredient of the standard cosmological model, LambdaCDM. Firstly, neutrinos directly participate in neutron freeze out during BBN, and secondly, they represent 40% of the energy density of the Universe after electron positron annihilation up to almost matter radiation equality. The latter fact makes neutrinos a necessary element to understand CMB observations. 

In this talk, I will review the cosmological implications of neutrinos. I will explain how current cosmological observations can be used to constrain their masses, their abundances, and their properties -- such as their interaction rate with other species. In particular, I will highlight that the typically very stringent constraint on their masses can be substantially relaxed if neutrinos decay on cosmological timescales. I will illustrate the implications of neutrino decays in cosmology with a few well-motivated neutrino mass models in which neutrinos can decay. I will then show that Planck CMB observations are a powerful tool to constraint neutrino interactions with neutrinophilic bosons. In particular, I will demonstrate that Planck legacy constraints neutrinophilic bosons with couplings as small as 10^{-13} with neutrinos for boson masses in the 0.1 eV < m < 300 eV range. I will finish by reviewing the role neutrinos can play with regards to the outstanding Hubble tension. I will show that pseudogoldstone bosons (majorons) interacting with neutrinos right before recombination represent a well motivated possibility to ameliorate (and potentially solve) the Hubble tension.