arXiv subjects

Richard Gonzales

Publications and source records attributed to Richard Gonzales.

8 recordsLinked to original sources

arXiv2025

Development of a High-Doppler Shift Optical Simulator to Test Femtosecond-Level Optical Time-Frequency Transfer

We have developed an optical simulator to test optical two-way time-frequency transfer (O-TWTFT) at the femtosecond level capable of simulating relative motion between two linked optical clock nodes up to Mach 1.8 with no moving parts. The technique is enabled by artificially Doppler shifting femtosecond pulses from auxiliary stabilized optical frequency combs. These pulses are exchanged between the nodes to simulate the Doppler shifts observed from a changing optical path length. We can continuously scan the simulated velocity from 14 to 620 m/s while simultaneously measuring velocity-dependent clock shifts at much higher velocities than has been previously recorded. This system provides an effective testbed that allows us to explore issues and solutions to enable femtosecond-level optical time transfer at high velocity.

physics.optics↗

arXiv2021

Geometry of horospherical varieties of Picard rank one

We study the geometry of non-homogeneous horospherical varieties. These have been classified by Pasquier and include the well-known odd symplectic Grassmannians. We focus our study on quantum cohomology, with a view towards Dubrovin's conjecture. In particular, we describe the cohomology groups of these varieties as well as a Chevalley formula, and prove that many Gromov-Witten invariants are enumerative. This enables us to prove that in many cases the quantum cohomology is semisimple. We give a presentation of the quantum cohomology ring for odd symplectic Grassmannians. The final section is devoted to the derived categories of coherent sheaves on horospherical varieties. We first discuss a general construction of exceptional bundles on these varieties. We then study in detail the case of the horospherical variety associated to the exceptional group $G_2$, and construct a full rectangular Lefschetz exceptional collection in the derived category.

math.AG↗

arXiv2020

Equivariant Grothendieck-Riemann-Roch and localization in operational K-theory

We produce a Grothendieck transformation from bivariant operational $K$-theory to Chow, with a Riemann-Roch formula that generalizes classical Grothendieck-Verdier-Riemann-Roch. We also produce Grothendieck transformations and Riemann-Roch formulas that generalize the classical Adams-Riemann-Roch and equivariant localization theorems. As applications, we exhibit a projective toric variety $X$ whose equivariant $K$-theory of vector bundles does not surject onto its ordinary $K$-theory, and describe the operational $K$-theory of spherical varieties in terms of fixed-point data. In an appendix, Vezzosi studies operational $K$-theory of derived schemes and constructs a Grothendieck transformation from bivariant algebraic $K$-theory of relatively perfect complexes to bivariant operational $K$-theory.

math.AG↗

arXiv2015

Algebraic rational cells and equivariant intersection theory

We provide a notion of algebraic rational cell with applications to intersection theory on singular varieties with torus action. Based on this notion, we study the algebraic analogue of $\mathbb{Q}$-filtrable varieties: algebraic varieties where a torus acts with isolated fixed points, such that the associated Bialynicki-Birula decomposition consists of algebraic rational cells. We show that the rational equivariant Chow group of any $\mathbb{Q}$-filtrable variety is freely generated by the cell closures. We apply this result to group embeddings, and more generally to spherical varieties. This paper is an extension of arxiv.org/abs/1112.0365 to equivariant Chow groups.

math.AG↗

arXiv2015

Equivariant Cohomology of Rationally Smooth Group Embeddings

We describe the equivariant cohomology ring of rationally smooth projective embeddings of reductive groups. These embeddings are the projectivizations of reductive monoids. Our main result describes their equivariant cohomology in terms of roots, idempotents, and underlying monoid data. Also, we characterize those embeddings whose equivariant cohomology ring is obtained via restriction to the associated toric variety. Such characterization is given in terms of the closed orbits.

math.AG↗

arXiv2015

Equivariant operational Chow rings of T-linear schemes

We study $T$-linear schemes, a class of objects that includes spherical and Schubert varieties. We provide a localization theorem for the equivariant Chow cohomology of these schemes that does not depend on resolution of singularities. Furthermore, we give an explicit presentation of the equivariant Chow cohomology of possibly singular complete spherical varieties admitting a smooth equivariant envelope (e.g. group embeddings).

math.AG↗

arXiv2014

Localization in equivariant operational K-theory and the Chang-Skjelbred property

We establish a localization theorem of Borel-Atiyah-Segal type for the equivariant operational K-theory of Anderson and Payne. Inspired by the work of Chang-Skjelbred and Goresky-Kottwitz-MacPherson, we establish a general form of GKM theory in this setting, applicable to singular schemes with torus action. Our results are deduced from those in the smooth case via Gillet-Kimura's technique of cohomological descent for equivariant envelopes. As an application, we extend Uma's description of the equivariant K-theory of smooth compactifications of reductive groups to the equivariant operational K-theory of all, possibly singular, projective group embeddings.

math.AG↗

arXiv2012

Rational smoothness, cellular decompositions and GKM theory

We introduce the notion of Q-filtrable varieties: projective varieties with a torus action and a finite number of fixed points, such that the cells of the associated Bialynicki-Birula decomposition are all rationally smooth. Our main results develop GKM theory in this setting. We also supply a method for building nice combinatorial bases on the equivariant cohomology of any Q-filtrable GKM variety. Applications to the theory of group embeddings are provided.

math.AG↗