56 results on '"Ivan Panin"'
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2. Effective Design for Sobol Indices Estimation Based on Polynomial Chaos Expansions.
- Author
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Evgeny Burnaev, Ivan Panin, and Bruno Sudret
- Published
- 2016
- Full Text
- View/download PDF
3. Efficient design of experiments for sensitivity analysis based on polynomial chaos expansions.
- Author
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Evgeny Burnaev, Ivan Panin, and Bruno Sudret
- Published
- 2017
- Full Text
- View/download PDF
4. Adaptive Design of Experiments for Sobol Indices Estimation Based on Quadratic Metamodel.
- Author
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Evgeny Burnaev and Ivan Panin
- Published
- 2015
- Full Text
- View/download PDF
5. Notes on a Grothendieck–Serre Conjecture in Mixed Characteristic Case
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Ivan Panin
- Subjects
Statistics and Probability ,Pure mathematics ,Zariski topology ,Conjecture ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Principal (computer security) ,01 natural sciences ,Discrete valuation ring ,010305 fluids & plasmas ,Generic point ,Simple (abstract algebra) ,Residue field ,Group scheme ,0103 physical sciences ,0101 mathematics ,Mathematics - Abstract
Let R be a discrete valuation ring with infinite residue field and X a smooth projective curve over R. Let G be a simple simply-connected group scheme over R and E a principal G-bundle over X. It is proved that E is trivial locally for the Zariski topology on X providing E is trivial over the generic point of X. The main aim of the present paper is to develop a method rather than to get a very strong concrete result.
- Published
- 2021
6. Two purity theorems and the Grothendieck-Serre conjecture concerning principal -bundles
- Author
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Ivan Panin
- Subjects
Pure mathematics ,Algebra and Number Theory ,Conjecture ,Principal (computer security) ,Mathematics - Abstract
The main results of the paper are two purity theorems for reductive group schemes over regular local rings containing a field. Using these two theorems a well-known Grothendieck-Serre conjecture on principal bundles is reduced to the simply-connected case. We point out that the mentioned reduction is one of the major steps in the proof of the conjecture that the author published in another work. Bibliography: 25 titles.
- Published
- 2020
7. A Short Proof of a Theorem Due to O. Gabber
- Author
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Ivan Panin
- Subjects
Statistics and Probability ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Field (mathematics) ,Regular local ring ,Reductive group ,01 natural sciences ,010305 fluids & plasmas ,Finite field ,Scheme (mathematics) ,0103 physical sciences ,Fraction (mathematics) ,0101 mathematics ,Mathematics - Abstract
A very short proof of an unpublished result due to O. Gabber is given. More exactly, let R be a regular local ring containing a finite field k. Let G be a simply-connected reductive group scheme over k. It is proved that a principal G-bundle over R is trivial if it is trivial over the fraction field of R. This is the mentioned unpublished result due to O. Gabber. In this paper, this result is derived from a purely geometric one, proved in another paper of the author and stated in the Introduction.
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- 2020
8. Homotopy invariant presheaves with framed transfers
- Author
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Grigory Garkusha and Ivan Panin
- Subjects
Mathematics - Algebraic Geometry ,Pure mathematics ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Homotopy ,FOS: Mathematics ,Presheaf ,Sheaf ,Base field ,Abelian group ,Invariant (mathematics) ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
The category of framed correspondences $Fr_*(k)$, framed presheaves and framed sheaves were invented by Voevodsky in his unpublished notes [12]. Based on the theory, framed motives are introduced and studied in [7]. The main aim of this paper is to prove that for any $\mathbb A^1$-invariant quasi-stable radditive framed presheaf of Abelian groups $\mathcal F$, the associated Nisnevich sheaf $\mathcal F_{nis}$ is $\mathbb A^1$-invariant whenever the base field $k$ is infinite of characteristic different from 2. Moreover, if the base field $k$ is infinite perfect of characteristic different from 2, then every $\mathbb A^1$-invariant quasi-stable Nisnevich framed sheaf of Abelian groups is strictly $\mathbb A^1$-invariant and quasi-stable. Furthermore, the same statements are true in characteristic 2 if we also assume that the $\mathbb A^1$-invariant quasi-stable radditive framed presheaf of Abelian groups $\mathcal F$ is a presheaf of $\mathbb Z[1/2]$-modules. This result and the paper are inspired by Voevodsky's paper [13].
- Published
- 2020
9. Lectures on Russian Literature: Pushkin, Gogol, Turgenef, Tolstoy
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Ivan Panin and Ivan Panin
- Published
- 2010
10. On the motivic commutative ring spectrum $\mathbf {BO}$
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C. Walter and Ivan Panin
- Subjects
Ring (mathematics) ,Pure mathematics ,Algebra and Number Theory ,Homotopy category ,Applied Mathematics ,010102 general mathematics ,Commutative ring ,01 natural sciences ,Spectrum (topology) ,Cohomology ,Weak equivalence ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Scheme (mathematics) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Commutative property ,Analysis ,Mathematics - Abstract
We construct an algebraic commutative ring T -spectrum BO which is stably fibrant and (8, 4)-periodic and such that on SmOp/S the cohomology theory (X, U) 7→ BO(X+/U+) and Schlichting’s hermitian K-theory functor (X, U) 7→ KO [q] 2q−p(X, U) are canonically isomorphic. We use the motivic weak equivalence Z×HGr ∼ −→ KSp relating the infinite quaternionic Grassmannian to symplectic K-theory to equip BO with the structure of a commutative monoid in the motivic stable homotopy category. When the base scheme is SpecZ[ 1 2 ], this monoid structure and the induced ring structure on the cohomology theory BO are the unique structures compatible with the products KO [2m] 0 (X)× KO [2n] 0 (Y ) → KO [2m+2n] 0 (X × Y ). on Grothendieck-Witt groups induced by the tensor product of symmetric chain complexes. The cohomology theory is bigraded commutative with the switch map acting on BO(T∧T ) in the same way as multiplication by the Grothendieck-Witt class of the symmetric bilinear space 〈−1〉.
- Published
- 2019
11. A short exact sequence
- Author
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Ivan Panin
- Subjects
Statistics and Probability ,Exact sequence ,Conjecture ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Dedekind domain ,Field (mathematics) ,K-Theory and Homology (math.KT) ,01 natural sciences ,Injective function ,010305 fluids & plasmas ,Combinatorics ,Mathematics - Algebraic Geometry ,Morphism ,Scheme (mathematics) ,Mathematics - K-Theory and Homology ,0103 physical sciences ,Bibliography ,FOS: Mathematics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
Let R be a regular semi-local integral domain containing a field and K be its fraction field. Let mu: G --> T be an R-group schemes morphism between reductive R-group schemes, which is smooth as a scheme morphism. Suppose that T is an R-torus.Then the map T(R)/mu(G(R)) --> T(K)/mu(G(K)) is injective and certain purity theorem is true.These and other results are derived from an extended form of Grothendieck--Serre conjecture proven in the present paper for rings R as above., arXiv admin note: text overlap with arXiv:1707.01763, arXiv:1406.1129
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- 2021
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12. Risk of estimators for Sobol’ sensitivity indices based on metamodels
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Ivan Panin
- Subjects
FOS: Computer and information sciences ,Statistics and Probability ,65T40 ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,62J10, 62J05, 65T40 ,Statistics - Computation ,62J05 ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,Sensitivity (control systems) ,Legendre polynomials ,Computation (stat.CO) ,Mathematics ,polynomial chaos approximation ,Statistics::Applications ,Nonparametric statistics ,Estimator ,Sobol sequence ,Function (mathematics) ,Statistics::Computation ,Metamodeling ,Global sensitivity analysis ,62J10 ,Statistics, Probability and Uncertainty ,Sobol’ indices - Abstract
Sobol' sensitivity indices allow to quantify the respective effects of random input variables and their combinations on the variance of mathematical model output. We focus on the problem of Sobol' indices estimation via a metamodeling approach where we replace the true mathematical model with a sample-based approximation to compute sensitivity indices. We propose a new method for indices quality control and obtain asymptotic and non-asymptotic risk bounds for Sobol' indices estimates based on a general class of metamodels. Our analysis is closely connected with the problem of nonparametric function fitting using the orthogonal system of functions in the random design setting. It considers the relation between the metamodel quality and the error of the corresponding estimator for Sobol' indices and shows the possibility of fast convergence rates in the case of noiseless observations. The theoretical results are complemented with numerical experiments for the approximations based on multivariate Legendre and Trigonometric polynomials.
- Published
- 2021
13. Cancellation theorem for framed motives of algebraic varieties
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Grigory Garkusha, Ivan Panin, and Alexey Ananyevskiy
- Subjects
Group (mathematics) ,General Mathematics ,010102 general mathematics ,K-Theory and Homology (math.KT) ,Algebraic variety ,14F42, 14F05 ,01 natural sciences ,Suspension (topology) ,Combinatorics ,Mathematics - Algebraic Geometry ,Mathematics - K-Theory and Homology ,0103 physical sciences ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
The machinery of framed (pre)sheaves was developed by Voevodsky [V1]. Based on the theory, framed motives of algebraic varieties are introduced and studied in [GP1]. An analog of Voevodsky's Cancellation Theorem [V1] is proved in this paper for framed motives stating that a natural map of framed $S^1$-spectra $$M_{fr}(X)(n)\to\underline{\textrm{Hom}}(\mathbb G,M_{fr}(X)(n+1)),\quad n\geq 0,$$ is a schemewise stable equivalence, where $M_{fr}(X)(n)$ is the $n$th twisted framed motive of $X$. This result is also necessary for the proof of the main theorem of [GP1] computing fibrant resolutions of suspension $\mathbb P^1$-spectra $\Sigma^\infty_{\mathbb P^1}X_+$ with $X$ a smooth algebraic variety. The Cancellation Theorem for framed motives is reduced to the Cancellation Theorem for linear framed motives stating that the natural map of complexes of abelian groups \[ \mathbb ZF(\Delta^\bullet \times X,Y) \to \mathbb ZF((\Delta^\bullet \times X)\wedge (\mathbb G_m,1),Y\wedge (\mathbb G_m,1)),\quad X,Y\in Sm/k, \] is a quasi-isomorphism, where $\mathbb ZF(X,Y)$ is the group of stable linear framed correspondences in the sense of [GP1]., Comment: This is the final revised version; accepted by Advances Math
- Published
- 2021
14. Witt sheaves and the η-inverted sphere spectrum
- Author
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Alexey Ananyevskiy, Ivan Panin, and Marc Levine
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Pure mathematics ,Mathematics::K-Theory and Homology ,Computation ,010102 general mathematics ,0103 physical sciences ,Sphere spectrum ,Torsion (algebra) ,Field (mathematics) ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Ananyevskiy has recently computed the stable operations and cooperations of rational Witt theory. These computations enable us to show a motivic analog of Serre's finiteness result. Theorem. Let k be a field of characteristic different from two. Then πnA1(Sk−)∗ is torsion for n>0. As an application, we define a category of Witt motives and show that rationally this category is equivalent to the minus part of SH(k)Q.
- Published
- 2017
15. On the Grothendieck–Serre Conjecture Concerning Principal G-Bundles Over Semilocal Dedekind Domains
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Anastasia Stavrova and Ivan Panin
- Subjects
Statistics and Probability ,Conjecture ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Field of fractions ,Dedekind domain ,01 natural sciences ,Prime (order theory) ,Combinatorics ,Kernel (algebra) ,Scheme (mathematics) ,0103 physical sciences ,Simply connected space ,Dedekind cut ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Let R be a semilocal Dedekind domain, and let K be the field of fractions of R. Let G be a reductive semisimple simply connected R-group scheme such that every semisimple normal R-subgroup scheme of G contains a split R-torus $$ {\mathbb{G}}_{m,R} $$ . It is proved that the kernel of the map $$ {H}_{\overset{\prime }{e}t}^1\left(R,\kern0.5em G\right)\to {H}_{\overset{\prime }{e}t}^1\left(K,\kern0.5em G\right) $$ induced by the inclusion of R into K is trivial. This result partially extends the Nisnevich theorem.
- Published
- 2017
16. A moving lemma for motivic spaces
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Ivan Panin
- Subjects
Lemma (mathematics) ,Pure mathematics ,Algebra and Number Theory ,Applied Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Analysis ,Mathematics - Published
- 2018
17. ON GROTHENDIECK–SERRE CONJECTURE CONCERNING PRINCIPAL BUNDLES
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Ivan Panin
- Subjects
Pure mathematics ,Conjecture ,Principal (computer security) ,Mathematics - Published
- 2019
18. Algebraic Cobordism and Projective Homogeneous Varieties
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Alexander Vishik, Marc Levine, Ivan Panin, and Stefan Gille
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Discrete mathematics ,Pure mathematics ,Algebraic cobordism ,Algebraic geometry of projective spaces ,Complex projective space ,Projective space ,Cobordism ,Algebraic variety ,General Medicine ,Projective variety ,Twisted cubic ,Mathematics - Published
- 2016
19. ON THE MOTIVIC SPECTRAL SEQUENCE
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Ivan Panin and Grigory Garkusha
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,K-Theory and Homology (math.KT) ,Algebraic variety ,Mathematics::Algebraic Topology ,01 natural sciences ,Tower (mathematics) ,Spectrum (topology) ,Motivic cohomology ,Mathematics - Algebraic Geometry ,Mathematics::K-Theory and Homology ,Algebraic K-theory ,Mathematics - K-Theory and Homology ,19E08, 55T99 ,0103 physical sciences ,Spectral sequence ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,010307 mathematical physics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
It is shown that the Grayson tower for $K$-theory of smooth algebraic varieties is isomorphic to the slice tower of $S^1$-spectra. We also extend the Grayson tower to bispectra and show that the Grayson motivic spectral sequence is isomorphic to the motivic spectral sequence produced by the Voevodsky slice tower for the motivic $K$-theory spectrum $KGL$. This solves Suslin's problem for these two spectral sequences in the affirmative., Comment: This is the final revised version
- Published
- 2015
20. In Memoriam: Andrei Suslin
- Author
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Christian Haesemeyer, Alexander Beilinson, Marc Levine, Alexander Merkurjev, Raman Parimala, Ivan Panin, Christophe Soulé, Charles A. Weibel, Serge Yagunov, and Eric M. Friedlander
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General Mathematics - Published
- 2020
21. A Variant of the Levine–Morel Moving Lemma
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Ivan Panin and K. I. Pimenov
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Statistics and Probability ,Lemma (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Geometric Topology ,Mathematics::Algebraic Topology ,01 natural sciences ,Combinatorics ,Mathematics::K-Theory and Homology ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
A version of the lemma proved by M. Levine and F. Morel in their book “Algebraic cobordisms,” is reformulated in the Chow group context. The obtained statement turns out to be valid in any characteristic and its proof is substantially shortened.
- Published
- 2016
22. Goldie ranks of primitive ideals and indexes of equivariant Azumaya algebras
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Ivan Panin and Ivan Losev
- Subjects
Rank (linear algebra) ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,17B35, 16A16 ,Mathematics - Rings and Algebras ,16. Peace & justice ,01 natural sciences ,Primitive ideal ,Combinatorics ,03 medical and health sciences ,0302 clinical medicine ,Rings and Algebras (math.RA) ,Azumaya algebra ,Irreducible representation ,Homogeneous space ,FOS: Mathematics ,Equivariant map ,030212 general & internal medicine ,Representation Theory (math.RT) ,0101 mathematics ,Semisimple Lie algebra ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
Let $\mathfrak{g}$ be a semisimple Lie algebra. We establish a new relation between the Goldie rank of a primitive ideal $\mathcal{J}\subset U(\mathfrak{g})$ and the dimension of the corresponding irreducible representation $V$ of an appropriate finite W-algebra. Namely, we show that $\operatorname{Grk}(\mathcal{J}) \leqslant \dim V/d_V$, where $d_V$ is the index of a suitable equivariant Azumaya algebra on a homogeneous space. We also compute $d_V$ in representation theoretic terms., 13 pages; v2 15 pages, improved exposition, accepted version
- Published
- 2018
23. The Last Twelve Verses of Mark - Their Genuineness Established
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Ivan Panin and Ivan Panin
- Abstract
'The Last Times and the Great Consummation'is a detailed treatise on the numeric patterns in the last twelve verses of the Gospel according to Mark, which have been widely omitted from Scripture by numerous scholars, editors, and translators. Ivan Nikolayevitsh Panin (12 December 1855 - 30 October 1942) was a Russian-born emigrant to the United States who became famous for his discover of various numeric patterns hidden in the Greek and Hebrew Bible, as well as for his extensive work based on his related research. This fascinating volume by the'father of Bible numerics'will appeal to those with an interest in numerology and Christian Scripture. Many vintage books such as this are becoming increasingly scarce and expensive. We are republishing this volume now in an affordable, modern, high-quality edition complete with the original text and artwork.
- Published
- 2018
24. A proof of the Grothendieck–Serre conjecture on principal bundles over regular local rings containing infinite fields
- Author
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Roman Fedorov and Ivan Panin
- Subjects
General Mathematics ,Local ring ,Field (mathematics) ,Regular local ring ,Reductive group ,Cohomology ,Combinatorics ,Mathematics - Algebraic Geometry ,Kernel (algebra) ,Number theory ,Scheme (mathematics) ,FOS: Mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
Let R be a regular local ring, containing an infinite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R., Section "Formal loops and affine Grassmannians" is removed as this is now covered in arXiv:1308.3078. Exposition is improved and slightly restructured. Some minor corrections
- Published
- 2015
25. Nice triples and Grothendieck--Serre's conjecture concerning principal G-bundles over reductive group schemes
- Author
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Ivan Panin
- Subjects
Pure mathematics ,Conjecture ,Reduction (recursion theory) ,Statement (logic) ,Grothendieck–Serre’s conjecture ,General Mathematics ,010102 general mathematics ,Principal (computer security) ,20G35 ,reductive group schemes ,Reductive group ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics::K-Theory and Homology ,Scheme (mathematics) ,0103 physical sciences ,Line (geometry) ,FOS: Mathematics ,010307 mathematical physics ,Affine transformation ,20G10 ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics ,principal G-bundles - Abstract
In a series of papers [Pan0], [Pan1], [Pan2], [Pan3] we give a detailed and better structured proof of the Grothendieck--Serre's conjecture for semi-local regular rings containing a finite field. The outline of the proof is the same as in [P1],[P2],[P3]. If the semi-local regular ring contains an infinite field, then the conjecture is proved in [FP]. Thus the conjecture is true for regular local rings containing a field. The present paper is the one [Pan1] in that new series. Theorem 1.1 is one of the main result of the paper. It is also one of the key steps in the proof of the Grothendieck--Serre's conjecture for semi-local regular rings containing a field (see [Pan3]). The proof of Theorem 1.1 is completely geometric., Comment: arXiv admin note: text overlap with arXiv:1406.0241
- Published
- 2017
- Full Text
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26. The triangulated category of K-motives
- Author
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Grigory Garkusha and Ivan Panin
- Subjects
Combinatorics ,Algebra and Number Theory ,Triangulated category ,Algebraic K-theory ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Perfect field ,Geometry and Topology ,GeneralLiterature_REFERENCE(e.g.,dictionaries,encyclopedias,glossaries) ,Mathematics - Abstract
For any perfect field k a triangulated category of K-motives is constructed in the style of Voevodsky's construction of the category . To each smooth k-variety X the K-motive is associated in the category andwhere pt = Spec(k) and K(X) is Quillen's K-theory of X.
- Published
- 2014
27. Effective Design for Sobol Indices Estimation Based on Polynomial Chaos Expansions
- Author
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Ivan Panin, Evgeny Burnaev, and Bruno Sudret
- Subjects
0209 industrial biotechnology ,Mathematical optimization ,Dependency (UML) ,Polynomial chaos ,Design of experiments ,020101 civil engineering ,Sobol sequence ,02 engineering and technology ,Measure (mathematics) ,0201 civil engineering ,Set (abstract data type) ,020901 industrial engineering & automation ,Metric (mathematics) ,Applied mathematics ,Sensitivity (control systems) ,Mathematics - Abstract
Sobol' indices are a common metric of dependency in sensitivity analysis. It is used as a measure of confidence of input variables influence on the output of the analyzed mathematical model. We consider a problem of selection of experimental design points for Sobol' indices estimation. Based on the concept of D-optimality, we propose a method for constructing an adaptive design of experiments, effective for the calculation of Sobol' indices from Polynomial Chaos Expansions. We provide a set of applications that demonstrate the efficiency of the proposed approach.
- Published
- 2016
28. Framed motives of relative motivic spheres
- Author
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Alexander Neshitov, Grigory Garkusha, and Ivan Panin
- Subjects
Sequence ,Applied Mathematics ,General Mathematics ,Homotopy ,010102 general mathematics ,K-Theory and Homology (math.KT) ,Base field ,Homology (mathematics) ,01 natural sciences ,Weak equivalence ,Combinatorics ,Mathematics - Algebraic Geometry ,Nisnevich topology ,Scheme (mathematics) ,Mathematics - K-Theory and Homology ,FOS: Mathematics ,Algebraic Topology (math.AT) ,SPHERES ,Mathematics - Algebraic Topology ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
The category of framed correspondences $Fr_*(k)$ and framed sheaves were invented by Voevodsky in his unpublished notes [V2]. Based on the theory, framed motives are introduced and studied in [GP1]. These are Nisnivich sheaves of $S^1$-spectra and the major computational tool of [GP1]. The aim of this paper is to show the following result which is essential in proving the main theorem of [GP1]: given an infinite perfect base field $k$, any $k$-smooth scheme $X$ and any $n\geq 1$, the map of simplicial pointed Nisnevich sheaves $(-,\mathbb{A}^1//\mathbb G_m)^{\wedge n}_+\to T^n$ induces a Nisnevich local level weak equivalence of $S^1$-spectra $$M_{fr}(X\times (\mathbb{A}^1// \mathbb G_m)^{\wedge n})\to M_{fr}(X\times T^n).$$ Moreover, it is proven that the sequence of $S^1$-spectra $$M_{fr}(X \times T^n \times \mathbb G_m) \to M_{fr}(X \times T^n \times\mathbb A^1) \to M_{fr}(X \times T^{n+1})$$ is locally a homotopy cofiber sequence in the Nisnevich topology. Another important result of this paper shows that homology of framed motives is computed as linear framed motives in the sense of [GP1]. This computation is crucial for the whole machinery of framed motives [GP1]., Comment: This is the final revised version
- Published
- 2016
- Full Text
- View/download PDF
29. Duality theorem for motives
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Ivan Panin and Serge Yagunov
- Subjects
Pure mathematics ,Algebra and Number Theory ,Triangulated category ,Betti number ,Applied Mathematics ,Duality (mathematics) ,Abstract nonsense ,Algebraic geometry ,Homology (mathematics) ,Cohomology ,symbols.namesake ,Mathematics::Category Theory ,symbols ,Analysis ,Poincaré duality ,Mathematics - Abstract
A general duality theorem for the category of motives is established, with a short, simple, and self-contained proof. Introduction Recently, due to the active study of cohomological invariants in algebraic geometry, “transplantation” of classical topological constructions to the algebraic-geometrical “soil” seems to be rather important. In particular, it is very interesting to study topological properties of the category of motives. The concept of a motive was introduced by Alexander Grothendieck in 1964 in order to formalize the notion of universal (co-)homology theory (see the detailed exposition of Grothendieck’s ideas in [5]). For us, the principal example of this type is the category of motives DM−, constructed by Voevodsky [12] for algebraic varieties. The Poincare duality is a classical and fundamental result in algebraic topology that initially appeared in Poincare’s first topological memoir “Analysis Situs” [9] (as a part of the Betti numbers symmetry theorem proof). The proof of the general duality theorem for extraordinary cohomology theories apparently belongs to Adams [1]. Our purpose in this paper is to establish a general duality theorem for the category of motives. Essentially, we extend the main statement of [8] to this category. Many known results can easily be interpreted in these terms. In particular, we get a generalization of the Friedlander–Voevodsky duality theorem [4] to the case of the ground field of arbitrary characteristic. The proof of this fact, involving the main result of [8], was kindly conveyed to the authors by Andrěi Suslin in a private communication. Being inspired by his work and Dold–Puppe’s category approach [2] to the duality phenomenon in topology, we decided to present a short, simple, and self-contained proof of a similar result for the category of motives. Our result might be viewed as a purely abstract theorem and rewritten in the spirit of “abstract nonsense” as a statement about some category with a distinguished class of morphisms. Essentially, what is required for the proof is the existence of finite fiber products and the terminal object in the category of varieties, a small part of the tensor triangulated category structure for motives, and finally, the existence of transfers for the class of morphisms generated by graphs of a special type (of projective morphisms). However, rather, we preferred to formulate all statements for motives of algebraic varieties in order to clarify the geometric nature of the construction and make possible applications easier. This led, in particular, to the appearence of the second (co)homology index responsible for twist with the Tate object Z(1) (see Voevodsky [12]). The only exception is the classical Example 2. 2000 Mathematics Subject Classification. Primary 14F42.
- Published
- 2010
30. Oriented cohomology theories of algebraic varieties II
- Author
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Ivan Panin
- Subjects
14F43 ,14F42 ,Group cohomology ,Cobordism ,oriented cohomology theories ,Mathematics::Algebraic Topology ,Cohomology ,algebraic cobordism ,Motivic cohomology ,Algebra ,Mathematics (miscellaneous) ,Mathematics::K-Theory and Homology ,Cup product ,55N22 ,De Rham cohomology ,Equivariant cohomology ,Complex cobordism ,Mathematics - Abstract
The concept of oriented cohomology theory is well-known in topology. Examples of these kinds of theories are complex cobordism, complex $K$-theory, usual singular cohomology, and Morava $K$-theories. A specific feature of these cohomology theories is the existence of trace operators (or Thom-Gysin operators, or push-forwards) for morphisms of compact complex manifolds. The main aim of the present article is to develop an algebraic version of the concept. Bijective correspondences between orientations, Chern structures, Thom structures and trace structures on a given ring cohomology theory are constructed. The theory is illustrated by singular cohomology, motivic cohomology, algebraic $K$-theory, the algebraic cobordism of Voevodsky and by other examples.
- Published
- 2009
31. Gersten resolution with support
- Author
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Ivan Panin and Kirill Zainoulline
- Subjects
Lemma (mathematics) ,Pure mathematics ,Number theory ,Functor ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,General Mathematics ,Calculus ,Algebraic geometry ,Mathematics::Algebraic Topology ,Mathematics - Abstract
In the present paper, we generalize the Quillen presentation lemma. As an application, for a given functor with transfers, we prove the exactness of its Gersten complex with support.
- Published
- 2008
32. Rationally isotropic quadratic spaces are locally isotropic
- Author
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Ivan Panin
- Subjects
Discrete mathematics ,Pure mathematics ,Lemma (mathematics) ,Quadratic equation ,General Mathematics ,Isotropy ,Zero (complex analysis) ,Field of fractions ,Field (mathematics) ,Regular local ring ,Isotropic quadratic form ,Mathematics - Abstract
Let R be a regular local ring, K its field of fractions and (V,ϕ) a quadratic space over R. Assume that R contains a field of characteristic zero we show that if (V,ϕ)⊗ R K is isotropic over K, then (V,ϕ) is isotropic over R. This solves the characteristic zero case of a question raised by J.-L. Colliot-Thelene in [3]. The proof is based on a variant of a moving lemma from [7]. A purity theorem for quadratic spaces is proved as well. It generalizes in the charactersitic zero case the main purity result from [9] and it is used to prove the main result in [2].
- Published
- 2008
33. On the relation of Voevodsky’s algebraic cobordism to Quillen’s K-theory
- Author
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Ivan Panin, Oliver Röndigs, and Konstantin Pimenov
- Subjects
Pure mathematics ,Ring (mathematics) ,Algebraic cobordism ,General Mathematics ,14F05 ,K-Theory and Homology (math.KT) ,K-theory ,Mathematics::Algebraic Topology ,Cohomology ,Ground field ,Mathematics - Algebraic Geometry ,symbols.namesake ,55P43 ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Euler characteristic ,Mathematics - K-Theory and Homology ,55N22 ,FOS: Mathematics ,symbols ,Sheaf ,Complex cobordism ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
Quillen's algebraic K-theory is reconstructed via Voevodsky's algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P^1-spectrum MGL of Voevodsky is considered as a commutative P^1-ring spectrum. There is a unique ring morphism MGL^{2*,*}(k)--> Z which sends the class [X]_{MGL} of a smooth projective k-variety X to the Euler characteristic of the structure sheaf of X. Our main result states that there is a canonical grade preserving isomorphism of ring cohomology theories MGL^{*,*}(X,U) \tensor_{MGL^{2*,*}(k)} Z --> K^{TT}_{- *}(X,U) = K'_{- *}(X-U)} on the category of smooth k-varieties, where K^{TT}_* is Thomason-Trobaugh K-theory and K'_* is Quillen's K'-theory. In particular, the left hand side is a ring cohomology theory. Moreover both theories are oriented and the isomorphism above respects the orientations. The result is an algebraic version of a theorem due to Conner and Floyd. That theorem reconstructs complex K-theory via complex cobordism., LaTeX, 18 pages, uses XY-pic
- Published
- 2008
34. Purity of -torsors
- Author
-
Vladimir Chernousov and Ivan Panin
- Subjects
Combinatorics ,Functor ,Mathematics::K-Theory and Homology ,Simple (abstract algebra) ,Algebraic group ,Zero (complex analysis) ,Local ring ,Field (mathematics) ,General Medicine ,Algebraic number ,Group theory ,Mathematics - Abstract
Let k be a field of characteristic zero, and let G be a split simple algebraic group of type G2 over k. We prove that the functor R↦He´t1(R,G) of G-torsors satisfies purity for regular local rings containing k. To cite this article: V. Chernousov, I. Panin, C. R. Acad. Sci. Paris, Ser. I 345 (2007).
- Published
- 2007
35. Adaptive Design of Experiments for Sobol Indices Estimation Based on Quadratic Metamodel
- Author
-
Ivan Panin and Evgeny Burnaev
- Subjects
Mathematical optimization ,Quadratic equation ,Statistics::Applications ,Computer science ,Active learning (machine learning) ,Design of experiments ,Monte Carlo method ,Sobol sequence ,Sensitivity (control systems) ,Focus (optics) ,Statistics::Computation ,Metamodeling - Abstract
Sensitivity analysis aims to identify which input parameters of a given mathematical model are the most important. One of the well-known sensitivity metrics is the Sobol sensitivity index. There is a number of approaches to Sobol indices estimation. In general, these approaches can be divided into two groups: Monte Carlo methods and methods based on metamodeling. Monte Carlo methods have well-established mathematical apparatus and statistical properties. However, they require a lot of model runs. Methods based on metamodeling allow to reduce a required number of model runs, but may be difficult for analysis. In this work, we focus on metamodeling approach for Sobol indices estimation, and particularly, on the initial step of this approach — design of experiments. Based on the concept of D-optimality, we propose a method for construction of an adaptive experimental design, effective for calculation of Sobol indices from a quadratic metamodel. Comparison of the proposed design of experiments with other methods is performed.
- Published
- 2015
36. To the anniversary of Sergei Vladimirovich Vostokov
- Author
-
Mikhail V. Bondarko, A. I. Generalov, N. L. Gordeev, Nikolai Vavilov, G. A. Leonov, I. B. Zhukov, Ivan Panin, A. L. Smirnov, A. V. Yakovlev, Maxim Vsemirnov, B. B. Lurie, I. B. Fesenko, and D. G. Benois
- Subjects
Algebra and Number Theory ,Applied Mathematics ,Art history ,Analysis ,Mathematics - Published
- 2016
37. [Untitled]
- Author
-
A. L. Smirnov and Ivan Panin
- Subjects
Statistics and Probability ,Discrete mathematics ,Zariski topology ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Image (category theory) ,Mathematics::Algebraic Geometry ,Morphism ,Mathematics::Category Theory ,Bounded function ,Bibliography ,Sheaf ,Variety (universal algebra) ,Constant (mathematics) ,Mathematics - Abstract
A smooth projective morphism p : T → S to a smooth variety S is considered. In particular, the following result is proved. The total direct image Rp*(ℤ/nℤ) of the constant etale sheaf ℤ/nℤ is locally (in Zariski topology) quasiisomorphic to a bounded complex \(\mathcal{L}\) on S that consists of locally constant, constructible etale sheaves of ℤ/nℤ-modules. Bibliography: 2 titles.
- Published
- 2003
38. On the relation of symplectic algebraic cobordism to hermitian K-theory
- Author
-
Charles Walter, Ivan Panin, St. Petersburg Department of V.A. Steklov Mathematical Institute (PDMI RAS), Steklov Mathematical Institute [Moscow] (SMI), Russian Academy of Sciences [Moscow] (RAS)-Russian Academy of Sciences [Moscow] (RAS), Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), and Ivan Panin a bénéficié d'un poste de professeur invité de l'Université de NIce pendant le mois de juin 2009.
- Subjects
Algebraic cobordism ,Astrophysics::High Energy Astrophysical Phenomena ,Commutative ring ,01 natural sciences ,Mathematics::Algebraic Topology ,K-théorie hermitienne ,Combinatorics ,Mathematics - Algebraic Geometry ,Mathematics (miscellaneous) ,classes de Thom ,Mathematics::K-Theory and Homology ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Physics ,MSC 14F42, 19G38, 19E20, 19E08 ,Homotopy category ,010102 general mathematics ,Cobordism ,K-Theory and Homology (math.KT) ,changement de coefficients ,K-theory ,classes de Pontryagin ,Cohomology ,Computer Science::Sound ,Mathematics - K-Theory and Homology ,[MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT] ,14F42, 19G38, 19E20, 19E08 ,cobordisme symplectique algébrique ,010307 mathematical physics ,Isomorphism ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Symplectic geometry - Abstract
We reconstruct hermitian K-theory via algebraic symplectic cobordism. In the motivic stable homotopy category SH(S), there is a unique morphism ϕ: MSp → BO of commutative ring T-spectra which sends the Thom class thMSp to the Thom class thBO. Using ϕ we construct an isomorphism of bigraded ring cohomology theories on the category $${\mathop{\rm Sm}\nolimits} {\mathcal O}p/S,\bar \varphi :{{\mathop{\rm MSp}\nolimits} ^{*,*}}(X,U){ \otimes _{{\rm{MS}}{{\rm{p}}^{4*,0*}}({\rm{pt}})}}{\rm{B}}{{\rm{O}}^{4*,2*}}({\rm{pt}}) \cong {\rm{B}}{{\rm{O}}^{*,*}}(X,U)$$. The result is an algebraic version of the theorem of Conner and Floyd reconstructing real K-theory using symplectic cobordism. Rewriting the bigrading as MSpp,q = MSp1[q], we have an isomorphism $$\bar \varphi :{{\mathop{\rm MSp}\nolimits} _*}^{[*]}(X,U){ \otimes _{{\rm{MSp}}_0^{[2*]}({\rm{pt}})}}{\rm{KO}}_0^{[2*]}({\rm{pt}}) \cong {\rm{K}}{{\rm{O}}_*}^{[*]}(X,U)$$, where the KOi[](X,U) are Schlichting’s hermitian K-theory groups.
- Published
- 2010
39. Rigidity for orientable functors
- Author
-
Serge Yagunov and Ivan Panin
- Subjects
Discrete mathematics ,Pure mathematics ,Algebra and Number Theory ,Algebraic cobordism ,Group cohomology ,Étale cohomology ,Mathematics::Algebraic Topology ,Cohomology ,Motivic cohomology ,Grothendieck topology ,Mathematics::K-Theory and Homology ,Ext functor ,Equivariant cohomology ,Mathematical Physics and Mathematics ,Mathematics - Abstract
In the following paper we introduce the notion of orientable functor (orientable cohomology theory) on the category of projective smooth schemes and define a family of transfer maps. Applying this technique, we prove that with finite coefficients orientable cohomology of a projective variety is invariant with respect to the base-change given by an extension of algebraically closed fields. This statement generalizes the classical result of Suslin, concerning algebraic K -theory of algebraically closed fields. Besides K -theory, we treat such examples of orientable functors as etale cohomology, motivic cohomology, algebraic cobordism. We also demonstrate a method to endow algebraic cobordism with multiplicative structure and Chern classes.
- Published
- 2002
40. The Gersten conjecture for Witt groups in the equicharacteristic case
- Author
-
Paul Balmer, Stefan Gille, Ivan Panin, and Charles Walter
- Subjects
General Mathematics - Published
- 2002
41. Framed motives of algebraic varieties (after V. Voevodsky)
- Author
-
Grigory Garkusha and Ivan Panin
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Homotopy ,Zero (complex analysis) ,Algebraic variety ,K-Theory and Homology (math.KT) ,Type (model theory) ,Suspension (topology) ,Stable homotopy theory ,Mathematics - Algebraic Geometry ,Mathematics - K-Theory and Homology ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Smooth scheme ,Mathematics - Algebraic Topology ,Algebraically closed field ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
A new approach to stable motivic homotopy theory is given. It is based on Voevodsky’s theory of framed correspondences. Using the theory, framed motives of algebraic varieties are introduced and studied in the paper. They are the major computational tool for constructing an explicit quasi-fibrant motivic replacement of the suspension P 1 \mathbb P^1 -spectrum of any smooth scheme X ∈ S m / k X\in Sm/k . Moreover, it is shown that the bispectrum ( M f r ( X ) , M f r ( X ) ( 1 ) , M f r ( X ) ( 2 ) , … ) , \begin{equation*} (M_{fr}(X),M_{fr}(X)(1),M_{fr}(X)(2),\ldots ), \end{equation*} each term of which is a twisted framed motive of X X , has the motivic homotopy type of the suspension bispectrum of X X . Furthermore, an explicit computation of infinite P 1 \mathbb P^1 -loop motivic spaces is given in terms of spaces with framed correspondences. We also introduce big framed motives of bispectra and show that they convert the classical Morel–Voevodsky motivic stable homotopy theory into an equivalent local theory of framed bispectra. As a topological application, it is proved that the framed motive M f r ( p t ) ( p t ) M_{fr}(pt)(pt) of the point p t = Spec k pt=\operatorname {Spec} k evaluated at p t pt is a quasi-fibrant model of the classical sphere spectrum whenever the base field k k is algebraically closed of characteristic zero.
- Published
- 2014
42. Rationally trivial hermitian spaces are locally trivial
- Author
-
Ivan Panin and Manuel Ojanguren
- Subjects
regular local ring ,Symmetric algebra ,Discrete mathematics ,Pure mathematics ,reductive group scheme ,General Mathematics ,Group algebra ,trace form ,Reductive group ,Grothendieck's conjecture ,Filtered algebra ,$epsilon$-hermitian space ,Azumaya algebra with involution ,Azumaya algebra ,Algebra representation ,Cellular algebra ,Composition algebra ,essentially smooth algebra ,Mathematics - Abstract
Keywords: regular local ring ; Azumaya algebra with involution ; essentially smooth algebra ; $epsilon$-hermitian space ; reductive group scheme ; Grothendieck's conjecture ; trace form Reference CMA-ARTICLE-2001-001doi:10.1007/PL00004859 Record created on 2008-12-16, modified on 2016-08-08
- Published
- 2001
43. A purity theorem for the witt group
- Author
-
Ivan Panin and Manuel Ojanguren
- Subjects
regular local ring ,Discrete mathematics ,Exact sequence ,Functor ,Witt group ,Mathematics::Commutative Algebra ,General Mathematics ,Local ring ,Field of fractions ,Field (mathematics) ,Regular local ring ,trace form ,integral scheme ,Prime (order theory) ,Combinatorics ,essentially smooth algebra ,Mathematics - Abstract
Let A be a regular local ring and K its field of fractions. We denote by W the Witt group functor that classifies quadratic spaces. We say that purity holds for A if W(A) is the intersection of all W(A p ) ⊂ W(K), as p runs over the height-one prime ideals of A. We prove purity for every regular local ring containing a field of characteristic ≠ 2. The question of purity and of the injectivity of W(A) into W(K) for arbitrary regular local rings is still open.
- Published
- 1999
44. Index Reduction Formulas for Twisted Flag Varieties, II
- Author
-
Ivan Panin, Alexander Merkurjev, and Adrian R. Wadsworth
- Subjects
Discrete mathematics ,Index (economics) ,General Mathematics ,Algebraic group ,Generalized flag variety ,Integration by reduction formulae ,Central simple algebra ,Mathematics ,Flag (geometry) - Published
- 1998
45. Purity for Pfister forms and F4-torsors with trivial g3 invariant
- Author
-
Ivan Panin and Vladimir Chernousov
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Invariant (mathematics) ,Mathematics - Published
- 2013
46. $K$-motives of algebraic varieties
- Author
-
Grigory Garkusha and Ivan Panin
- Subjects
Pure mathematics ,spectral category ,Triangulated category ,19E08 ,Motivic homotopy theory ,01 natural sciences ,010104 statistics & probability ,Mathematics - Algebraic Geometry ,Mathematics (miscellaneous) ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Point (geometry) ,Mathematics - Algebraic Topology ,0101 mathematics ,Algebra over a field ,Algebraic number ,55U35 ,Algebraic Geometry (math.AG) ,Mathematics ,14F42 ,010102 general mathematics ,Algebraic variety ,K-Theory and Homology (math.KT) ,Motivic cohomology ,Algebraic K-theory ,Spectral sequence ,Mathematics - K-Theory and Homology ,algebraic $K$-theory - Abstract
A kind of motivic algebra of spectral categories and modules over them is developed to introduce K-motives of algebraic varieties. As an application, bivariant algebraic K-theory as well as bivariant motivic kohomology groups are defined and studied. We use Grayson's machinery to produce the Grayson motivic spectral sequence connecting bivariant K-theory to bivariant motivic kohomology. It is shown that the spectral sequence is naturally realized in the triangulated category of K-motives constructed in the paper. It is also shown that ordinary algebraic K-theory is represented by the K-motive of the point., Comment: This is the final version; to appear Homology, Homotopy and Applications. arXiv admin note: text overlap with arXiv:math/0108143 by other authors
- Published
- 2012
47. A splitting principle and the algebraic K-theory of some homogeneous varieties
- Author
-
Ivan Panin
- Subjects
Statistics and Probability ,Discrete mathematics ,Pure mathematics ,Function field of an algebraic variety ,Applied Mathematics ,General Mathematics ,Algebraic extension ,Algebraic closure ,Algebraic element ,Algebraic cycle ,Real algebraic geometry ,Algebraic geometry and analytic geometry ,Splitting principle ,Mathematics - Published
- 1993
48. Rationally Isotropic Quadratic Spaces Are Locally Isotropic: II
- Author
-
Ivan Panin and Konstantin Pimenov
- Published
- 2010
49. On Voevodsky's Algebraic K-Theory Spectrum
- Author
-
Konstantin Pimenov, Ivan Panin, and Oliver Roendigs
- Subjects
Discrete mathematics ,Mathematics::Commutative Algebra ,Homotopy category ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Smash product ,Algebraic K-theory ,Noetherian scheme ,A¹ homotopy theory ,Krull dimension ,Algebraic number ,Weak equivalence ,Mathematics - Abstract
Under a certain normalization assumption we prove that the P1-spectrum BGL of Voevodsky which represents algebraic K-theory is unique over Spec.(Z). Following an idea of Voevodsky, we equip the P1-spectrum BGL with the structure of a commutative P1-ring spectrum in the motivic stable homotopy category. Furthermore, we prove that under a certain normalization assumption this ring structure is unique over Spec.(Z). For an arbitrary Noetherian scheme S of finite Krull dimension we pull this structure back to obtain a distinguished monoidal structure on BGL. This monoidal structure is relevant for our proof of the motivic Conner–Floyd theorem (Panin et al., Invent Math 175:435–451, 2008). It has also been used to obtain a motivic version of Snaith’s theorem (Gepner and Snaith, arXiv:0712.2817v1 [math.AG]).
- Published
- 2009
50. T-spectra and Poincaré duality
- Author
-
Serge Yagunov and Ivan Panin
- Subjects
symbols.namesake ,Applied Mathematics ,General Mathematics ,symbols ,Algebraic geometry ,Spectral line ,Poincaré duality ,Mathematics ,Mathematical physics - Published
- 2008
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