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Victor Petrov

Publications and source records attributed to Victor Petrov.

At least 19 recordsLinked to original sources
arXiv2026

Counter-examples to a conjecture of Karpenko via truncated Brown-Peterson cohomology

Let $G$ be a split semisimple linear algebraic group and let $X$ denote the generically twisted variety of Borel subgroups in $G$. Nikita Karpenko conjectured that the map from the Chow ring of $X$ to the associated graded ring of the topological filtration on the Grothendieck ring of $X$ is an isomorphism. After having been verified for many $G$, the conjecture was disproved by Nobuaki Yagita for some spinor groups. Later, other counter-examples were constructed by Baek-Karpenko and Baek-Devyatov. We present a new method for constructing counter-examples that is based on the connection of the truncated Brown-Peterson cohomology with the connective K-theory. Using this method, we disprove the conjecture for new groups, including $\mathrm{Spin}_{15}$, which is now the smallest known spinor group for which the conjecture fails.

math.AG↗
arXiv2024

Morava $J$-invariant

We compute the co-multiplication of the algebraic Morava K-theory for split orthogonal groups. This allows us to compute the decomposition of the Morava motives of generic maximal orthogonal Grassmannians and to compute a Morava K-theory analogue of the $J$-invariant in terms of the ordinary (Chow) $J$-invariant.

math.KT↗
arXiv2024

Tits construction and Rost invariant

We show a Springer type theorem for the variety of parabolic subgroups of type $1,2,6$ for all groups of type $E_6$. As far as we know this gives the first example for the validity of the Springer theorem for projective homogeneous varieties of type $^2E_6$ different from varieties of Borel subgroups. The proof combines several topics, notably the Rost invariant, a Tits construction, Cartan's symmetric spaces and indirectly the structure of the Chow motives of projective homogeneous varieties of exceptional type.

math.AG↗
arXiv2024

Morava K-theory of orthogonal groups and motives of projective quadrics

We compute the algebraic Morava K-theory ring of split special orthogonal and spin groups. In particular, we establish certain stabilization results for the Morava K-theory of special orthogonal and spin groups. Besides, we apply these results to study Morava motivic decompositions of orthogonal Grassmannians. For instance, we determine all indecomposable summands of the Morava motives of a generic quadric.

math.KT↗
arXiv2024

High-fidelity velocity and concentration measurements of turbulent buoyant jets

Accurate models of turbulent buoyant flows are essential for the design of nuclear reactors thermal hydraulics and passive safety systems. However, available models fail to fully capture the physics of turbulent mixing when buoyancy becomes predominant with respect to momentum. Therefore, high-fidelity experiments of well-controlled fundamental flows are needed to develop and validate more accurate models. We analyze experiments of positive and negative turbulent buoyant jets, both in uniform and stratified environments, with the aim of understanding the thermal hydraulics of turbulent mixing with variable density and providing high-fidelity data for the development and validation of turbulence models. Non-intrusive, simultaneous Particle Image Velocimetry and Laser Induced Fluorescence measurements were carried out to acquire instantaneous velocity and concentration fields on a vertical section parallel to the axis of a jet in the self-similar region. The Refractive Index Matching method was applied to measure high-resolution buoyant jets with up to 8.6% density difference. These data are free of the typical errors that characterize optical measurements of buoyancy driven flows (e.g., natural and mixed convection) where the refractive index of the fluid is inhomogeneous throughout the measurement domain. Turbulent statistics and entrainment of buoyant jets in uniform and stratified environments are presented. These data are compared with non-buoyant jets in uniform environment, as a reference to investigate the effects of buoyancy and stratification on turbulent mixing. The results will be used for the assessment of current turbulence models and as basis for the development of a new one that captures turbulent mixing.

physics.flu-dyn↗
arXiv2023

Morava K-theory and Rost invariant

We prove that inner forms of a variety of Borel subgroups have isomorphic motives with respect to the second Morava K-theory if and only if the corresponding Tits algebras and Rost invariants coincide. This extends Panin's results on interrelationship of K-theory with Tits algebras to the case of cohomological invariants of degree 3.

math.KT↗
arXiv2023

RANS Simulations of Turbulent Round Jets in the Presence of Density Difference and Comparison with High-Resolution Experimental Data

In this paper, the novel experimental data reported by Qin et al. [1] are used to assess the predictive capability of the Realizable k-epsilon (RKE) model and Reynolds stress transport (RST) model for buoyant jets and understand the reasons for discrepancies. In particular, we present the comparison between simulation results of a turbulent buoyant jet flow in the self-similar region with high-resolution experimental data obtained for a jet injected from a 2 mm nozzle into a 300x300x300 $mm^3$ tank, with nominal Reynolds number equal to 10,000. Results show that streamwise velocity profiles predicted by the RST model had good agreement with experimental data, while the larger spreading rate was predicted by the RKE model. For turbulent statistics, turbulent kinetic energy witnessed a discrepancy in the center region, with shear stress well predicted for both models. Comparison of the turbulent kinetic energy production term with experimental data revealed reasons for the discrepancy and also showed that the gradient of the streamwise velocity in the crosswise direction contributes the most to the turbulent kinetic energy production. Investigation of model coefficients of the turbulent dissipation equation for the RKE model has revealed that $C_{\varepsilon2}$ is critical in model accuracy.

physics.flu-dyn↗
arXiv2021

The Allison-Faulkner construction of $E_8$

We show that the Tits index $E_8^{133}$ cannot be obtained by means of the Tits construction over a field with no odd degree extensions. We construct two cohomological invariants, in degrees 6 and 8, of the Tits construction and the more symmetric Allison-Faulkner construction of Lie algebras of type $E_8$ and show that these invariants can be used to detect the isotropy rank.

math.RA↗
arXiv2020

Hopf-theoretic approach to motives of twisted flag varieties

Let $G$ be a split semisimple algebraic group over a field and let $A^*$ be an oriented cohomology theory in the sense of Levine--Morel. We provide a uniform approach to the $A^*$-motives of geometrically cellular smooth projective $G$-varieties based on the Hopf algebra structure of $A^*(G)$. Using this approach we provide various applications to the structure of motives of twisted flag varieties.

math.AG↗
arXiv2016

Motivic decompositions of twisted flag varieties and representations of Hecke-type algebras

Let G be a split semisimple linear algebraic group over a field k0. Let E be a G-torsor over a field extension k of k0. Let h be an algebraic oriented cohomology theory in the sense of Levine-Morel. Consider a twisted form E/B of the variety of Borel subgroups G/B over k. Following the Kostant-Kumar results on equivariant cohomology of flag varieties we establish an isomorphism between the Grothendieck groups of the h-motivic subcategory generated by E/B and the category of finitely generated projective modules of certain Hecke-type algebra H which depends on the root datum of G, on the torsor E and on the formal group law of the theory h. In particular, taking h to be the Chow groups with finite coefficients Fp and E to be a generic G-torsor we prove that all indecomposable submodules of an affine nil-Hecke algebra H of G with coefficients in Fp are isomorphic to each other and correspond to the (non-graded) generalized Rost-Voevodsky motive for (G,p).

math.AG↗
arXiv2015

A rational construction of Lie algebras of type E_7

We give an explicit construction of Lie algebras of type $E_7$ out of a Lie algebra of type $D_6$ with some restrictions. Up to odd degree extensions, every Lie algebra of type $E_7$ arises this way. For Lie algebras that admit a $56$-dimensional representation we provide a more symmetric construction based on an observation of Manivel; the input is seven quaternion algebras subject to some relations.

math.RA↗
arXiv2015

Rost motives, affine varieties, and classifying spaces

In the present article we investigate ordinary and equivariant Rost motives. We provide an equivariant motivic decomposition of the variety X of full flags of a split semisimple algebraic group over a smooth base scheme, study torsion subgroup of the Chow group of twisted forms of X, define some equivariant Rost motives over a field and ordinary Rost motives over a general base scheme, and relate equivariant Rost motives with classifying spaces of some algebraic groups.

math.AG↗
arXiv2014

Morava K-theory of twisted flag varieties

In the present article we prove some results about the Morava K-theory. In particular, we construct an operation from the Morava K-theory to the Chow theory analogous to the second Chern class for Grothendieck's K0-theory. Furthermore, we investigate ordinary and equivariant oriented cohomology theories in the sense of Levine-Morel of projective quadrics, and discuss the Rost motives.

math.AG↗
arXiv2013

Effective Lagrangian for the Polyakov line on a lattice

We formulate a method for computing the effective Lagrangian of the Polyakov line on the lattice. Using mean field approximation we calculate the effective potential for high temperatures. The result agrees with recent lattice simulations. We reveal a new type of ultraviolet divergence (coming from longitudinal gluons) which dominates the effective potential and explains the discrepancy of the lattice simulations and standard perturbative calculations performed in covariant gauges.

hep-lat↗
arXiv2013

A theory of baryon resonances at large N_c

At large number of colors, N_c quarks in baryons are in a mean field of definite space and flavor symmetry. We write down the general Lorentz and flavor structure of the mean field, and derive the Dirac equation for quarks in that field. The resulting baryon resonances exhibit an hierarchy of scales: The crude mass is O(N_c), the intrinsic quark excitations are O(1), and each intrinsic quark state entails a finite band of collective excitations that are split as O(1/N_c). We build a (new) theory of those collective excitations, where full dynamics is represented by only a few constants. In a limiting (but unrealistic) case when the mean field is spherically-and flavor-symmetric, our classification of resonances reduces to the SU(6) classification of the old non-relativistic quark model. Although in the real world N_c is only three, we obtain a good accordance with the observed resonance spectrum up to 2 GeV.

hep-ph↗
arXiv2012

Rationally isotropic exceptional projective homogeneous varieties are locally isotropic

Assume that R is a local regular ring containing an infinite perfect field, or that R is the local ring of a point on a smooth scheme over an infinite field. Let K be the field of fractions of R and the characteristic of K is not 2. Let X be an exceptional projective homogeneous scheme over R. We prove that in most cases the condition that X has a K-point implies that X has an R-point.

math.AG↗
arXiv2012

Baryon resonances at large Nc, or Quark Nuclear Physics

We suggest a new point of view according to which baryon resonances can be understood as collective excitations about intrinsic one-quark excitations in a mean field of definite symmetry. This approach is justified in the limit of large number of colours Nc, and is similar to the physics of large-A nuclei, hence "quark nuclear physics". Although in the real world Nc is only three, we obtain a good agreement with the observed resonance spectrum of light baryons up to 2 GeV, and of lowest charmed baryon multiplets. A by-product of the scheme is the prediction of new exotic charmed (and bottom) baryons that may be stable against strong decays.

hep-ph↗