Yale University

The Steinberg scheme and the equivariant coherent sheaves on it play a very important role in Geometric Representation theory. In this talk we will discuss various t-structures on the equivariant derived category of the Steinberg of importance for Representation theory in positive characteristics. Based on arXiv:2302.05782.

The resolutions of Slodowy slices S͂_e are symplectic varieties that contain the Springer fiber (G/B)_e as a Lagrangian subvariety. In joint work with R. Bezrukavnikov, M. McBreen, and Z. Yun, we construct analogues of these spaces for homogeneous affine Springer fibers. We further understand the categories of microlocal sheaves in these symplectic spaces supported on the affine Springer fiber as some categories of coherent sheaves.

In this talk I will mostly focus on the case of the homogeneous element ts for s a regular semisimple element and will discuss some relations of these categories with the small quantum group providing a categorification of joint work with R.Bezrukavnikov, P. Shan and E. Vasserot.

Hydrodynamic excitations corresponding to sound and shear modes in fluids are characterized by gapless dispersion relations. In the hydrodynamic gradient expansion, their frequencies are represented by power series in spatial momenta. In this talk we will discuss the convergence properties of the hydrodynamic series by studying the associated spectral curve in the space of complexified frequency and complexified spatial momentum. For the N=4 supersymmetric Yang-Mills plasma at infinite 't Hooft coupling, we will use the holographic methods to demonstrate that the derivative expansions have finite non-zero radii of convergence. Obstruction to the convergence of hydrodynamic series arises from level-crossings in the quasinormal spectrum at complex momenta. We will discuss how finiteness of 't Hooft coupling affects the radius of convergence. We will show that the purely perturbative calculation in terms of inverse 't Hooft coupling gives the increasing radius of convergence when the coupling is decreasing. Applying the non-perturbative resummation techniques will make radius of convergence piecewise continuous function that decreases after the initial increase. Finally, we will provide arguments in favour of the non-perturbative approach and show that the presence of nonperturbative modes in the quasinormal spectrum can be indirectly inferred from the analysis of perturbative critical points.

In many data-driven applications, the data follows some geometric structure, and the goal is to recover this structure. In many cases, the observed data is noisy and the recovery task is even more challenging. A common assumption is that the data lies on a low dimensional manifold. Estimating a manifold from noisy samples has proven to be a challenging task. Indeed, even after decades of research, there was no (computationally tractable) algorithm that accurately estimates a manifold from noisy samples with a constant level of noise.

In this talk, we will present a method that estimates a manifold and its tangent. Moreover, we establish convergence rates, which are essentially as good as existing convergence rates for function estimation.

This is a joint work with Barak Sober.

The roots of a function represented by its Chebyshev expansion are known to be the eigenvalues of the so-called colleague matrix, which is a Hessenberg matrix that is the sum of a symmetric tridiagonal matrix and a rank 1 perturbation. The rootfinding problem is thus reformulated as an eigenproblem, making the computation of the eigenvalues of such matrices a subject of significant practical interest. To obtain the roots with the maximum possible accuracy, the eigensolver used must posess a somewhat subtle form of stability.

In this talk, I will discuss a recently constructed algorithm for the diagonalization of colleague matrices, satisfying the relevant stability requirements.  The scheme has CPU time requirements proportional to n^2, with n the dimensionality of the problem; the storage requirements are proportional to n. Furthermore, the actual CPU times (and storage requirements) of the procedure are quite acceptable, making it an approach of choice even for small-scale problems. I will illustrate the performance of the algorithm with several numerical examples.

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The geodesic flow on hyperbolic finite-volume hyperbolic manifolds is a particularly well-studied dynamical system; this is in part due to its connection to other important dynamical systems on the manifold, as well as orbital counting and other number-theoretic problems related to discrete subgroups of orthogonal groups. In recent years, there has been some interest in generalizing many of the properties of the geodesic flow on finite-volume manifolds to the infinite-volume setting. I will discuss joint work with Hee Oh in which we establish exponential mixing of the geodesic flow on infinite-volume geometrically finite hyperbolic manifolds with large enough critical exponent. Patterson-Sullivan densities and Burger-Roblin measures, the Lax-Phillips spectral gap for the Laplace operator on infinite volume geometrically finite hyperbolic manifolds, and complementary series representations are all involved in both the statement and proof of our result, and I will try to explain how these different objects are related in this setting.

There is a well-known correspondence between Weil-Petersson geodesic loops in the moduli space of a surface S and hyperbolic 3-manifolds fibering over the circle with fibre S. Much is unknown, however, about the detailed relationship between geometric features of the loops and those of the 3-manifolds.

In work with Leininger-Souto-Taylor we study the relation between WP length and 3-manifold volume, when the length (suitably normalized) is bounded and the fiber topology is unbounded. We obtain a WP analogue of a theorem proved by Farb-Leininger-Margalit for the Teichmuller metric. In work with Modami, we fix the fiber topology and study connections between the thick-thin decomposition of a geodesic loop and that of the corresponding 3-manifold. While these decompositions are often in direct correspondence, we exhibit examples where the correspondence breaks down. This leaves the full conjectural picture somewhat mysterious, and raises many questions. 

Shear Thickening fluids such as cornstarch and water show remarkable response under impact, which allows, for example, a person to run on the surface of the suspension. We perform constant velocity impact experiments along with imaging and particle tracking in a shear thickening fluid at velocities lower than 500 mm/s and suspension heights of a few cm. In this regime, where inertial effects are insignificant, we find that a solid-like dynamically jammed region with a propagating front is generated under impact. The suspension is able to support large stresses like a solid only when the front reaches the opposite boundary. These impact-activated fronts are generated only above a critical velocity. We construct a model by taking into account that sufficiently large stresses are generated when this solid like region spans to the opposite boundary and the work necessary to deform this solid like material dissipates the kinetic energy of the impacting object. The model shows quantitative agreement of the measured penetration depth using high speed video of a person running on cornstarch and water suspensions.