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''Author(s)'': F. Morel and V. Voevodsky
''Journal'': Publications Mathematiques de l'IHES, no. 90, pp. 45-143
''Links'': [[Journal|http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1999__90__45_0]] [[Preprint|http://www.math.uiuc.edu/K-theory/0305/]]
''Abstract'': In this paper we begin to develop a machinery which we call ${\mathbb A}^1$-homotopy theory of schemes. All our constructions are based on the intuitive feeling that if the category of algebraic varieties is in any way similar to the category of topological spaces then there should exist a homotopy theory of algebraic varieties where the affine line plays the role of the unit interval.
''Notes:'' A long collection of notes to this paper is available at the [[motivic homotopy theory resource page|motivichomotopy.html#%5B%5BA%C2%B9-homotopy%20theory%20of%20schemes%5D%5D]]
* Additional Information: //Preprint v. 8 May 2008//
* Availability: [[DVI|papers/exoticIIb-contractibleFinal.dvi]],[[PDF|papers/exoticIIb-contractibleFinal.pdf]]
* ''Abstract'': We study the ${\mathbb A}^1$-homotopy types of smooth complex varieties $X$ such that $X({\mathbb C})$ is contractible as a topological space; we refer to such varieties as topologically contractible. Our study has two goals: i) to compare the ${\mathbb A}^1$-homotopy type and (Voevodsky's) motive of a smooth variety, and ii) to test some forms of the Hodge conjecture. While there exist arbitrary dimensional moduli of topologically contractible smooth complex surfaces, our first result shows that the only such variety contractible from the standpoint of ${\mathbb A}^1$-homotopy theory is ${\mathbb A}^2$. Even more, the sheaf of ${\mathbb A}^1$-connected components, which, roughly speaking, encodes information about (spaces of) morphisms ${\mathbb A}^1 \to X$, is often (conjecturally always) a sufficiently refined invariant to distinguish topologically contractible smooth surfaces from ${\mathbb A}^2$. Next, we study motives of topologically contractible smooth complex surfaces. In particular, while such surfaces are not necessarily ${\mathbb A}^1$-contractible, we show their motives are always isomorphic to that of ${\mathrm{Spec}} {\mathbb C}$. This serves two distinct purposes: we illustrate the stark difference between the unstable ${\mathbb A}^1$-homotopy category and Voevodsky's derived category of integral motives, and we provide evidence that the Hodge realization functor (from Voevodsky's triangulated category of rational geometric motives to an appropriate category of Hodge structures) is conservative.
* co-author(s): Brent Doran
* Additional Information: //Preprint v. 18 Dec 2007//
* Availability: [[DVI|papers/toricA1final.dvi]],[[PDF|papers/toricA1final.pdf]]
* ''Abstract'': We study some properties of ${\mathbb A}^1$-homotopy groups: geometric interpretations of connectivity, excision results, and a re-interpretation of geometric quotients by solvable groups in terms of covering spaces in the sense of ${\mathbb A}^1$-homotopy theory. These concepts and results are well-suited to the study of certain quotients via geometric invariant theory. As a case study in geometry of solvable group quotients, we focus on ${\mathbb A}^1$-homotopy groups of smooth toric varieties. We give simple combinatorial conditions (in terms of fans) guaranteeing vanishing of low degree ${\mathbb A}^1$-homotopy groups of smooth (proper) toric varieties. Finally, in certain cases, we can actually compute the ``next" non-vanishing A1-homotopy group (beyond the ${\mathbb A}^1$-fundamental group) of a smooth toric variety. From this point of view, ${\mathbb A}^1$-homotopy theory, even with its exquisite sensitivity to algebro-geometric structure, is almost ``as tractable" (in low degrees) as ordinary homotopy for large classes of interesting varieties.
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* co-author(s): Fabien Morel
* Additional Information: //Preprint v. 25 Jun 2008//
* Availability: In preparation; coming soon...
* ''Abstract'': (Tentative) We study the problem of classifying smooth proper schemes over an algebraically closed field from the standpoint of ${\mathbb A}^1$-homotopy theory. Specifically, in low dimensions, we compare the isomorphism classification and the classification up to ${\mathbb A}^1$-weak equivalence, at least with some homotopical connectness conditions, namely ${\mathbb A}^1$-connectedness, in place. We provide a concrete link between ${\mathbb A}^1$-connectedness and stable rationality (in the sense of birational geometry). Along the way, we discuss the general problem of classifying smooth schemes in a given ${\mathbb A}^1$-homotopy type.
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//And a foolish rowdy-dow over Geometers' Abstraction that cannot ever be seen.// Thomas Pynchon Mason & Dixon p. 56
* co-author(s): Brent Doran and Frances Kirwan
* Additional Information:
* Availability: In preparation
* ''Abstract'': This paper will provide complete proofs for the results stated in [[Yang- Mills theory and Tamagawa Numbers]]. In particular, we define a Borel-style equivariant motivic cohomology theory and prove properties of this theory analogous to those of [[ordinary equivariant cohomology|http://en.wikipedia.org/wiki/Equivariant_cohomology]]. We then show how the stratification studied in the thesis of the third author can be used to study equivariant motivic cohomology of quotients.
* Additional Information: //[[Pure and Applied Mathematics Quarterly v.2 no. 4|http://qjpam.henu.edu.cn/onlineinfo.jsp?id=21]], pp. 1085-1102, 2006//
* Availability: [[ArXiv|http://arxiv.org/abs/math/0604344]],[[DVI|papers/asok-equivvb.dvi]],[[PDF|papers/asok-equivvb.pdf]],[[Journal|http://qjpam.henu.edu.cn/articleinfo.jsp?articleID=135]],[[MathSciNet|http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&s4=Asok&s5=&s6=&s7=&s8=All&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=2&mx-pid=2282414]]
* ''Abstract'': We give a concrete description of the category of $G$-equivariant vector bundles on certain affine $G$-varieties (where $G$ is a reductive linear algebraic group) in terms of linear algebra data.
* co-author(s): James Parson
* Additional Information:
* Availability:
* ''Abstract'':
* co-author(s): Brent Doran
* Additional Information: ////Preprint v. 28 Mar 2008////
* Availability: [[DVI|papers/QuadricsRev2.dvi]], [[PDF|papers/QuadricsRev2.pdf]]
* ''Abstract'': We produce explicit smooth affine quadric hypersurfaces, smooth over ${\mathrm{Spec}} {\mathbb Z}$, that are [[motivic spheres|motivichomotopy.html#%5B%5BMotivic%20sphere%5D%5D]], i.e., spheres from the standpoint of ${\mathbb A}^1$-homotopy theory. Along the way, we compute various standard invariants for such quadrics (e.g., Chow groups, K-theory, or Hermitian K-theory).
!!! History and Motivation
In 1935, J.H.C. Whitehead constructed, as a counterexample to his ``proof" of the $3$-dimensional Poincare conjecture, the first example of an open (i.e., non-compact and without boundary) contractible manifold not homeomorphic to a ball (see [[Whitehead]]). Subsequently, D. R. McMillan produced infinitely many pairwise non-homeomorphic open contractible smooth $3$-manifolds (see [[McMillan]]). Slightly earlier, Mazur and Poenaru had provided examples of contractible open $4$-manifolds (Mazur's examples can be constructed as smooth manifolds, see [[Mazur]] or [[Poenaru]]). Generalizing these constructions, Curtis and Kwun (see [[CurtisKwun]]) showed that there exist infinitely many pairwise non-homeomorphic, contractible, open $n$-manifolds for every $n \geq 5$, and Glaser (see [[Glaser]]) showed that the same result held in dimension $4$.
Roughly contemporaneously, geometric topologists began to explore the possibility of ``exotic" $PL$ and smooth structures compatible with the usual topology on ${\mathbb R}^n$. Stallings proved (see [[Stallings]]) that if $M$ is an open contractible $n$-manifold of dimension $n \geq 5$, [[simply connected at infinity|Simply connected at infinity]], then $M$ is $PL$-isomorphic to ${\mathbb R}^n$; if further $M$ is smooth, then $M$ is in fact diffeomorphic to ${\mathbb R}^n$ with its usual smooth structure. (This result also follows from the $h$-cobordism theorem if $n \geq 6$, see [[Milnor]] Section 9 Proposition A). In other words, for any integer $n \geq 5$, ${\mathbb R}^n$ admits, up to the appropriate notion of isomorphism, unique $PL$ and smooth structures.
Surprisingly, a simple shift in perspective allows one to construct, at least in principle, all contractible manifolds; this will be the motivating theme of this paper. It follows from results of McMillan, Zeeman (see [[McMillanZeeman]]) and Stallings (see [[Stallings]]) that {\em all} of the examples just discussed can be realized as quotients of free ${\mathbb R}^k$ actions on ${\mathbb R}^{n+k}$ (for appropriate $n$ and $k$). In fact, Stallings (see \\[[loc. cit.|Stallings]]\\ Section 4, Section 5 and Proposition 2.2) shows that any open contractible (PL or smooth) $3$-manifold can be constructed as a quotient of ${\mathbb R}^5$ by a free (PL or smooth) ${\mathbb R}^2$-action and, more generally, for any $n \geq 4$ any open contractible (PL or smooth) $n$-manifold [[can be constructed as a quotient|Stallings' theorem]] of ${\mathbb R}^{n+1}$ by a free (PL or smooth) ${\mathbb R}$-action.
The naive algebro-geometric analog of this question is whether all smooth contractible complex algebraic varieties can be constructed as quotients of $\mathbb{C}^n$ by the free algebraic action of a unipotent group. All of the constructions just mentioned are inherently ``topological" so it perhaps came as a shock that a smooth contractible variety, apart from affine space, even exists. The [[first example]] was given by Ramanujam in his landmark paper [[Ramanujam]].\footnote{Ramanujam proved much more: any smooth, complex algebraic surface whose underlying analytic space is contractible and simply connected at infinity is necessarily {\em algebraically} isomorphic to ${\mathbb A}^2$. Ramanujam's example, being non-simply connected at infinity, was necessarily not homeomorphic to ${\mathbb R}^4$.} It, together with the fact that Zariski cancellation holds in dimension $2$ (see [[Fujita]]), provides a counter-example to this analog of the question.
Ramanujam's example was only the tip of the iceberg. Other authors showed that there exist many examples of contractible smooth algebraic varieties in every (complex) dimension $\geq 2$ (see the beautiful survey [[Zaidenberg]] for an overview and many more references). In this paper, we begin a study of contractible algebraic varieties from the standpoint of motivic homotopy theory. Rather, since topological contractibility only makes sense for varieties defined over fields that are embeddable in the complex numbers, we have to reformulate the notion of contractibility appropriately.
Following Morel and Voevodsky (see [[MV]]) we view the category of smooth schemes as analogous to the category of topological spaces with the affine line playing the role of the unit interval in ordinary topology. Morel and Voevodsky replace the category of (e.g. locally contractible) topological spaces by the category of (simplicial) Nisnevich sheaves on ${\mathcal Sm}/k$ (the category of smooth manifolds is replaced by the Nisnevich sheaves corresponding to smooth schemes), the notion of homeomorphism is replaced by isomorphism of smooth schemes and finally, the usual topological homotopy category is replaced by the Morel-Voevodsky ${{\mathbb A}^1}$-homotopy or ``motivic homotopy" category. These analogies are, of course, not perfect (as we shall explain), but hopefully
serve to guide intuition.
Our goal is to study ${{\mathbb A}^1}$-contractible smooth algebraic varieties, i.e. those varieties that are ${{\mathbb A}^1}$-weakly equivalent to ${\mathrm{Spec}} k$. Essentially by construction ${\mathbb A}^n$ is ${{\mathbb A}^1}$-contractible. However, we will see that there are many examples of smooth algebraic varieties, not isomorphic to affine space, that are ${{\mathbb A}^1}$-contractible. Henceforth, we call such a variety an {\em exotic ${{\mathbb A}^1}$-contractible variety}. We suggest that this notion gives the ``correct" algebro-geometric analog of our thematic question, namely:
!!!! Question
Does every smooth $\mathbb{A}^1$-contractible variety arise as a quotient of affine space by the free action of a unipotent group?
First, however, one needs to produce examples of interesting $\mathbb{A}^1$-contractible varieties. We prove, in this spirit, the following results.
!!!! Theorem (reference)
For every integer $m \geq 4$, there exists a denumerably infinite collection of pairwise non-isomorphic $m$-dimensional exotic ${{\mathbb A}^1}$-contractible varieties, each admitting an embedding into a smooth affine variety with pure codimension $2$ smooth boundary.
!!!! Theorem (reference)
For every integer $m \geq 6$ and every $n \geq 0$:
* there exists a connected $n$-dimensional scheme $S$ and a smooth morphism $f: X \to S$ of relative dimension $m$, whose fibers over $k$ points are ${{\mathbb A}^1}$-contractible, quasi-affine, not affine, and pairwise non-isomorphic.
* The morphism $f: X \to S$ admits a partial compactification to a flat family $\bar{f}: \overline{X} \to S$ whose fibers over $k$-points are smooth affine varieties. Furthermore, for any $k$-point $t \in S$, the map $X_t \rightarrow \bar{X}_t$ is an open immersion with a smooth complement of codimension $\geq 2$.
In other words, there exist arbitrary dimensional {\em moduli} of ${{\mathbb A}^1}$-contractible smooth varieties in dimension $\geq 6$. We stress that these examples are completely explicit and non-pathological. The families arise in a simple geometric manner: as ${\mathbb G}_{{\bf a}}$-quotients of families of ${\mathbb G}_{{\bf a}}$-invariant hypersurfaces in a fixed linear ${\mathbb G}_{{\bf a}}$-representation $W$. The resulting quotient varieties are complements of smooth codimension $2$ subvarieties in smooth hypersurfaces in ${\mathrm{Spec}} k[W]^{{\mathbb G}_{{\bf a}}}$. The simplest case, for example, is the complement in an affine quadric four-fold (defined by the vanishing of $x_1 x_4 - x_2 x_3 - x_5(x_5+1)$ in $\mathbb{A}^5$) of an explicit embedded copy of $\mathbb{A}^2$ (defined by $x_1 = x_2 = 0, x_5 = -1$); see [[Remark \ref{rem:Winkelex}]] for details.
When $k = {\mathbb C}$, we prove that it is impossible to construct exotic ${{\mathbb A}^1}$-contractible varieties of dimension $\leq 2$ by our method (see Claims [[ref{claim:dimension1}]] and [[ref{claim:dimension2}]]). Indeed, there exists a unique up to isomorphism smooth ${{\mathbb A}^1}$-contractible variety of dimension $1$, namely ${{\mathbb A}^1}$. It follows from results of several authors, that all the examples of ${{\mathbb A}^1}$-contractible smooth surfaces we produce are necessarily isomorphic to the affine plane. Therefore, only dimension $3$ seems mysterious. In analogy with the topological setting, one may need to use explicit $({\mathbb G}_{{\bf a}})^2$ actions to study dimension $3$.
The motivic homotopy category of schemes over ${\mathrm{Spec}} {\mathbb C}$, admits a ``topological realization functor" to the usual homotopy category of topological spaces. This realization functor takes ${{\mathbb A}^1}$-weak equivalences of smooth schemes to ordinary weak equivalences and, in particular, the topological realization of an ${{\mathbb A}^1}$-contractible smooth variety is a contractible smooth manifold. We are unable to produce examples of contractible algebraic varieties that are provably not ${{\mathbb A}^1}$-contractible. Topological intuition encourages us to believe that such varieties exist; however, ``motivic" intuition related to the Hodge conjecture imposes very strong topological restrictions on any such examples. Summarizing the above discussion, we make the following conjecture.\footnote{Note added in proof: we can now produce examples of smooth affine surfaces over ${\mathbb C}$ which are topologically contractible but {\em not} ${{\mathbb A}^1}$-contractible.}
!!!! Conjecture
For every $m \geq 3$, and every $n \geq 0$, there exists a connected $n$-dimensional scheme $S$ and a smooth morphism $f: X \to S$ of relative dimension $m$, whose fibers are ${{\mathbb A}^1}$-contractible and for a fixed field $k$, the fibers of $f$ over $k$-points of $S$ are all non-isomorphic.
Finally, we claim that the topological characterization ``at infinity" of the $PL$ or smooth structure of ${\mathbb R}^n$ for $n \geq 5$ gives rise to a natural question: can one give a motivic topological characterization of affine space? As a first step in this direction, one can try to define a notion of motivic homology at infinity. One such notion was introduced by Wildeshaus in his paper ``Basic properties of the boundary motive" (see [[Wildeshaus]]). All exotic ${{\mathbb A}^1}$-contractibles have, via Poincar\'e duality, motivic homology at infinity (in Wildeshaus' sense) that of a motivic sphere of appropriate dimension (see Lemma [[ref{lem:homatinfinity}]]). A natural question to ask is whether there exists a good notion of an ``${{\mathbb A}^1}$-singular chain complex at infinity" and of an ``${{\mathbb A}^1}$-fundamental group at infinity," analogous to the usual singular chain complex at infinity or the fundamental group at infinity, that one might use to characterize when an ${{\mathbb A}^1}$-contractible smooth variety is exotic. In this direction, Morel (see [[MorelICM]]) dreams of a ``motivic $s$-cobordism theorem:" a characterization of affine space as a smooth scheme should be a related consequence.
!!! Contents
As the techniques used in this paper have been introduced fairly recently, we have endeavored to make the paper as self-contained as possible. We begin, in Section [[ref{s:contractibility}]], by making a brief review of ${{\mathbb A}^1}$-contractibility. In particular, we state the main criterion we use to check that a morphism is an ${{\mathbb A}^1}$-weak equivalence (see Lemma [[lem:extendedhominv]]). In addition, we give the simplest examples of (singular) ${{\mathbb A}^1}$-contractible algebraic varieties. In Section [[ref{s:unipotents}]], we discuss the relevant elements from geometric invariant theory for non-reductive group actions as developed in [[DoranKirwan]]. In particular, after reviewing some basic facts about unipotent groups, we discuss a condition characterizing the existence of principal bundle quotients by unipotent group actions (see Theorems [[ref{thm:stablequotient}]] and Theorem [[ref{thm:affinequotient})]]. To keep this section self-contained, we have given complete proofs of all the main results; the focus here is on quotients of everywhere stable quasi-affine schemes and the material is essentially orthogonal to that contained in [[DoranKirwan]].
In Section \ref{s:hypersurfaces}, we study the simplest class of unipotent group actions: ${\mathbb G}_{{\bf a}}$-actions. The Jacobsen-Morozov theorem (see Theorem [[ref{thm:jacobsenmorozov}]]) essentially allows us to reduce the study of ${\mathbb G}_{{\bf a}}$-actions to $SL_2$-actions. We completely resolve the question of when the (principal bundle) quotient of a ${\mathbb G}_{{\bf a}}$-invariant hypersurface in a linear ${\mathbb G}_{{\bf a}}$-representation is an affine variety, a strictly quasi-affine variety, or not even a scheme. In particular, we give two explicit characterizations (see Theorems [[ref{thm:geomchar1}]] and [[ref{thm:algchar})]] which are used to produce all the examples discussed in Section [[ref{s:examples}]]. One curious consequence is a natural decomposition of {\em any} ${\mathbb G}_{{\bf a}}$-invariant function into a sum of an $SL_2$-invariant function and a ${\mathbb G}_{{\bf a}}$-invariant function of a very particular sort; the authors are not aware of a classical version of this statement in invariant theory (see Theorem [[ref{thm:algchar}]]).
In Section [[ref{s:examples}]], we prove the two theorems stated in the introduction by using a very simple class of ${\mathbb G}_{{\bf a}}$-equivariant linear embeddings of affine space (see Theorems [[ref{thm:dimension4}]] and [[ref{thm:moduli})]]. We also make a detailed study of strictly quasi-affine quotients in small dimensions. In Section [[ref{s:conjectures}]], we discuss various consequences of and conjectures related to the notion of ${{\mathbb A}^1}$-contractibility. We emphasize here that the very existence of the motivic homotopy category allows us to make very strong statements about the motivic topology of exotic ${{\mathbb A}^1}$-contractible varieties. In particular, we discuss briefly the idea of motivic topology at infinity. We close in the Appendix (Section [[ref{s:appendix}]]) with a summary of the main tools of the technique of faithfully flat descent, its application to Borel transfer, and the proof of the quite general and formal Theorem [[ref{thm:affinequotient}]]. Consequences of descent and its applications are utilized throughout the paper; rather than interrupt the main discussion with technical sidelights, we have compiled the relevant facts there.
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''[[A¹-homotopy theory|motivichomotopy.html]]''
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!!! Links
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!!! Acknowledgements
As it stands, the [[LaTeX|http://www.latex-project.org/]] functionality on this webpage is an amalgam of a few very nice pieces of (freely available!) technology. I use the [[LaTeX plugin|http://bob.mcelrath.org/tiddlyjsmath.html]] for [[TiddlyWiki|http://www.tiddlywiki.com]] developed by [[Bob McElrath|http://bob.mcelrath.org/]]. This, in turn uses the [[jsMath|http://www.math.union.edu/~dpvc/jsMath/]] javascript based [[LaTeX|http://www.latex-project.org/]] rendered developed by [[David Cervone|http://www.math.union.edu/~dpvc/]]. I also use [[Your equations|http://www.yourequations.com/docs/]], which produces small images (off-site) for [[LaTeX|http://www.latex-project.org]] code embedded in appropriate html tags. This second tool has the benefit of greater functionality than the [[LaTeX plugin|http://bob.mcelrath.org/tiddlyjsmath.html]], thought the disadvantage that it makes the Tiddlywiki less self-contained.
''Author(s)'': Y. Andre, B. Kahn, P. O'Sullivan
''Journal'': Rendiconti del Seminario Matematico della Universita di Padova 108 pp. 107-291
Links: [[Journal]] [[Preprint|http://arxiv.org/abs/math/0203273]]
''Abstract'': For $K$ a field, a Wedderburn $K$-linear category is a $K$-linear category ${\mathcal A}$ whose radical ${\mathcal R}$ is locally nilpotent and such that $\overline{{\mathcal A}} := {\mathcal A}/{\mathcal R}$ is semisimple and remains so after any extension of scalars. We prove existence and uniqueness results for sections of the projection ${\mathcal A} \to \overline{{\mathcal A}}$, in the vein of the theorems of Wedderburn. There are two such results: one in the general case and one when ${\mathcal A}$ has a monoidal structure for which ${\mathcal R}$ is a monoidal ideal. The latter applies notably to Tannakian categories over a field of characteristic zero, and we get a generalisation of the Jacobson-Morozov theorem: the existence of a proreductive envelope ${}^p{\mathrm{Red}}(G)$ associated with any affine group scheme $G$ over $K$ (${}^p{\mathrm{Red}}({\mathbb G}_a) = SL_2$, and ${}^p{\mathrm Red}(G)$ is infinite-dimensional for any bigger unipotent group). Other applications are given in this paper as well as in the note [[Construction inconditionnelle de groupes de Galois motiviques]] on motives.
''Notes'': We use the discussion of the Jacobsen- Morozov theorem given in this paper (Chapter 19 Theorem 19.5.1) in our paper [[On unipotent quotients and some A¹- contractible smooth schemes]]
* co-author(s): Brent Doran
* Additional Information:
* Availability: In preparation
* ''Abstract'':
* co-author(s): Brent Doran
* Additional Information: //v. 05 Mar 2007 - International Mathematics Research Papers 2007 V.2 Art. ID rpm005, 51 pages//
* Availability: [[ArXiv|http://arxiv.org/abs/math/0703137]],[[DVI|papers/exoticaffinesFinal.dvi]],[[PDF|papers/exoticaffinesFinal.pdf]],[[Journal|http://imrp.oxfordjournals.org/cgi/content/abstract/2007/rpm005/rpm005]],[[MathSciNet|http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&s4=Asok&s5=&s6=&s7=&s8=All&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1&mx-pid=2335246]]
* ''Abstract'': We study quotients of quasi-affine schemes by unipotent groups over fields of characteristic 0. To do this, we introduce a notion of stability which allows us to characterize exactly when a principal bundle quotient exists and, together with a cohomological vanishing criterion, to characterize whether or not the resulting quasi-affine quotient scheme is affine. We completely analyze the case of ${\mathbb G}_{{\bf a}}$-invariant hypersurfaces in a linear ${\mathbb G}_{{\bf a}}$-representation $W$; here the above characterizations admit simple geometric and algebraic interpretations. As an application, we produce arbitrary dimensional families of non-isomorphic smooth quasi-affine but not affine $n$-dimensional varieties $n \geq 6$ that are contractible in the sense of ${\mathbb A}^1$[[-homotopy theory|motivichomotopy.html#%5B%5BUnstable%20A%C2%B9-homotopy%20theory%5D%5D]]. Indeed, existence follows without any computation; yet explicit defining equations for the varieties depend only on knowing some linear ${\mathbb G}_{{\bf a}}$- and $SL_2$- invariants, which, for a sufficiently large class, we provide. Similarly, we produce infinitely many non-isomorphic examples in dimensions 4 and 5. Over ${\mathbb C}$, the analytic spaces underlying these varieties are non-isomorphic, non-Stein, topologically contractible and often diffeomorphic to ${\mathbb C}^n$.
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/***
|Name|Plugin: jsMath|
|Created by|BobMcElrath|
|Email|my first name at my last name dot org|
|Location|http://bob.mcelrath.org/tiddlyjsmath.html|
|Version|1.5.1|
|Requires|[[TiddlyWiki|http://www.tiddlywiki.com]] ≥ 2.0.3, [[jsMath|http://www.math.union.edu/~dpvc/jsMath/]] ≥ 3.0|
!Description
LaTeX is the world standard for specifying, typesetting, and communicating mathematics among scientists, engineers, and mathematicians. For more information about LaTeX itself, visit the [[LaTeX Project|http://www.latex-project.org/]]. This plugin typesets math using [[jsMath|http://www.math.union.edu/~dpvc/jsMath/]], which is an implementation of the TeX math rules and typesetting in javascript, for your browser. Notice the small button in the lower right corner which opens its control panel.
!Installation
In addition to this plugin, you must also [[install jsMath|http://www.math.union.edu/~dpvc/jsMath/download/jsMath.html]] on the same server as your TiddlyWiki html file. If you're using TiddlyWiki without a web server, then the jsMath directory must be placed in the same location as the TiddlyWiki html file.
I also recommend modifying your StyleSheet use serif fonts that are slightly larger than normal, so that the math matches surrounding text, and \\small fonts are not unreadable (as in exponents and subscripts).
{{{
.viewer {
line-height: 125%;
font-family: serif;
font-size: 12pt;
}
}}}
If you had used a previous version of [[Plugin: jsMath]], it is no longer necessary to edit the main tiddlywiki.html file to add the jsMath <script> tag. [[Plugin: jsMath]] now uses ajax to load jsMath.
!History
* 11-Nov-05, version 1.0, Initial release
* 22-Jan-06, version 1.1, updated for ~TW2.0, tested with jsMath 3.1, editing tiddlywiki.html by hand is no longer necessary.
* 24-Jan-06, version 1.2, fixes for Safari, Konqueror
* 27-Jan-06, version 1.3, improved error handling, detect if ajax was already defined (used by ZiddlyWiki)
* 12-Jul-06, version 1.4, fixed problem with not finding image fonts
* 26-Feb-07, version 1.5, fixed problem with Mozilla "unterminated character class".
* 27-Feb-07, version 1.5.1, Runs compatibly with TW 2.1.0+, by Bram Chen
!Examples
|!Source|!Output|h
|{{{The variable $x$ is real.}}}|The variable $x$ is real.|
|{{{The variable \(y\) is complex.}}}|The variable \(y\) is complex.|
|{{{This \[\int_a^b x = \frac{1}{2}(b^2-a^2)\] is an easy integral.}}}|This \[\int_a^b x = \frac{1}{2}(b^2-a^2)\] is an easy integral.|
|{{{This <IMG src="http://www.yourequations.com/eq.latex?\int_a^b \sin x = -(\cos b - \cos a)" alt="\int_a^b \sin x = -(\cos b - \cos a)" class="eq_latex" align="middle" border="0"> is another easy integral.}}}|This <IMG src="http://www.yourequations.com/eq.latex?\int_a^b \sin x = -(\cos b - \cos a)" alt="\int_a^b \sin x = -(\cos b - \cos a)" class="eq_latex" align="middle" border="0"> is another easy integral.|
|{{{Block formatted equations may also use the 'equation' environment \begin{equation} \int \tan x = -\ln \cos x \end{equation} }}}|Block formatted equations may also use the 'equation' environment \begin{equation} \int \tan x = -\ln \cos x \end{equation}|
|{{{Equation arrays are also supported \begin{eqnarray} a &=& b \\ c &=& d \end{eqnarray} }}}|Equation arrays are also supported \begin{eqnarray} a &=& b \\ c &=& d \end{eqnarray} |
|{{{I spent \$7.38 on lunch.}}}|I spent \$7.38 on lunch.|
|{{{I had to insert a backslash (\\) into my document}}}|I had to insert a backslash (\\) into my document|
!Code
***/
//{{{
// AJAX code adapted from http://timmorgan.org/mini
// This is already loaded by ziddlywiki...
if(typeof(window["ajax"]) == "undefined") {
ajax = {
x: function(){try{return new ActiveXObject('Msxml2.XMLHTTP')}catch(e){try{return new ActiveXObject('Microsoft.XMLHTTP')}catch(e){return new XMLHttpRequest()}}},
gets: function(url){var x=ajax.x();x.open('GET',url,false);x.send(null);return x.responseText}
}
}
// Load jsMath
jsMath = {
Setup: {inited: 1}, // don't run jsMath.Setup.Body() yet
Autoload: {root: new String(document.location).replace(/[^\/]*$/,'jsMath/')} // URL to jsMath directory, change if necessary
};
var jsMathstr;
try {
jsMathstr = ajax.gets(jsMath.Autoload.root+"jsMath.js");
} catch(e) {
alert("jsMath was not found: you must place the 'jsMath' directory in the same place as this file. "
+"The error was:\n"+e.name+": "+e.message);
throw(e); // abort eval
}
try {
window.eval(jsMathstr);
} catch(e) {
alert("jsMath failed to load. The error was:\n"+e.name + ": " + e.message + " on line " + e.lineNumber);
}
jsMath.Setup.inited=0; // allow jsMath.Setup.Body() to run again
// Define wikifers for latex
config.formatterHelpers.mathFormatHelper = function(w) {
var e = document.createElement(this.element);
e.className = this.className;
var endRegExp = new RegExp(this.terminator, "mg");
endRegExp.lastIndex = w.matchStart+w.matchLength;
var matched = endRegExp.exec(w.source);
if(matched) {
var txt = w.source.substr(w.matchStart+w.matchLength,
matched.index-w.matchStart-w.matchLength);
if(this.keepdelim) {
txt = w.source.substr(w.matchStart, matched.index+matched[0].length-w.matchStart);
}
e.appendChild(document.createTextNode(txt));
w.output.appendChild(e);
w.nextMatch = endRegExp.lastIndex;
}
}
config.formatters.push({
name: "displayMath1",
match: "\\\$\\\$",
terminator: "\\\$\\\$\\n?", // 2.0 compatability
termRegExp: "\\\$\\\$\\n?",
element: "div",
className: "math",
handler: config.formatterHelpers.mathFormatHelper
});
config.formatters.push({
name: "inlineMath1",
match: "\\\$",
terminator: "\\\$", // 2.0 compatability
termRegExp: "\\\$",
element: "span",
className: "math",
handler: config.formatterHelpers.mathFormatHelper
});
var backslashformatters = new Array(0);
backslashformatters.push({
name: "inlineMath2",
match: "\\\\\\\(",
terminator: "\\\\\\\)", // 2.0 compatability
termRegExp: "\\\\\\\)",
element: "span",
className: "math",
handler: config.formatterHelpers.mathFormatHelper
});
backslashformatters.push({
name: "displayMath2",
match: "\\\\\\\[",
terminator: "\\\\\\\]\\n?", // 2.0 compatability
termRegExp: "\\\\\\\]\\n?",
element: "div",
className: "math",
handler: config.formatterHelpers.mathFormatHelper
});
backslashformatters.push({
name: "displayMath3",
match: "\\\\begin\\{equation\\}",
terminator: "\\\\end\\{equation\\}\\n?", // 2.0 compatability
termRegExp: "\\\\end\\{equation\\}\\n?",
element: "div",
className: "math",
handler: config.formatterHelpers.mathFormatHelper
});
// These can be nested. e.g. \begin{equation} \begin{array}{ccc} \begin{array}{ccc} ...
backslashformatters.push({
name: "displayMath4",
match: "\\\\begin\\{eqnarray\\}",
terminator: "\\\\end\\{eqnarray\\}\\n?", // 2.0 compatability
termRegExp: "\\\\end\\{eqnarray\\}\\n?",
element: "div",
className: "math",
keepdelim: true,
handler: config.formatterHelpers.mathFormatHelper
});
// The escape must come between backslash formatters and regular ones.
// So any latex-like \commands must be added to the beginning of
// backslashformatters here.
backslashformatters.push({
name: "escape",
match: "\\\\.",
handler: function(w) {
w.output.appendChild(document.createTextNode(w.source.substr(w.matchStart+1,1)));
w.nextMatch = w.matchStart+2;
}
});
config.formatters=backslashformatters.concat(config.formatters);
window.wikify = function(source,output,highlightRegExp,tiddler)
{
if(source && source != "") {
if(version.major == 2 && version.minor > 0) {
var wikifier = new Wikifier(source,getParser(tiddler),highlightRegExp,tiddler);
wikifier.subWikifyUnterm(output);
} else {
var wikifier = new Wikifier(source,formatter,highlightRegExp,tiddler);
wikifier.subWikify(output,null);
}
jsMath.ProcessBeforeShowing();
}
}
//}}}
* Additional Information: ////Preprint v. 28 Mar 2008////
* Availability: In prepration
* ''Abstract'': (Tentative) Using the explicit geometric models of motivic spheres produced by Asok and Doran, we revisit the theory of polynomial maps of affine quadric hypersurfaces from the standpoint of ${\mathbb A}^1$-homotopy theory. One can produce motivic analogues of a host of classical unstable homotopy theoretic constructions. We modify the classical Hopf construction for bilinear forms to produce {\em smooth integral} Hopf morphisms. We produce a motivic version of the Hopf invariant. We formulate and solve a precise motivic analogue of the classical Hopf invariant $1$ problem. This result has the purely algebro-geometric consequence that over a field of characteristc not $2$, only smooth affine (homogeneous) quadric hypersurfaces of dimensions $0,1,3$ and $7$ can admit unital polynomial compositions.
!$\large {\mathbb A}^1$-homotopy types
In this group of papers, we study the problem described by Voevodsky in his [[1998 ICM address|http://www.mathematik.uni-bielefeld.de/documenta/xvol-icm/00/Voevodsky.MAN.html]] (see Section 7) on ${\mathbb A}^1$-homotopy theory: give an algebro-combinatorial description of ${\mathbb A}^1$-homotopy types. The natural candidates for the ``algebro-combinatorial" invariants in question are the ${\mathbb A}^1$-[[homotopy (sheaves of) groups|motivichomotopy.html#%5B%5BA%C2%B9-homotopy%20groups%5D%5D%20Overview%20%5B%5BOutlines%20of%20the%20theory%5D%5D%20%5B%5BSurveys%20on%20foundational%20material%5D%5D]]. The main goals of the papers below are, first, to compute these invariants in various cases, and second, to use these invariants to study basic ``geometric" questions in algebraic geometry. One key problem is: how can one distinguish the smooth varieties in a given ${\mathbb A}^1$-homotopy type? For smooth affine varieties, this problem seems tractable as many analogs of basic invariants of [[surgery theory|http://en.wikipedia.org/wiki/Surgery_theory]] are now available in the algebro-geometric setting.
* __[[On affine unipotent quotients]]__, with B. Doran - In preparation
* __[[Constructing A¹-weak equivalences by algebraic h-cobordism]]__, with F. Morel (//In preparation, draft v. 25 Jun 2008//)
* __[[Polynomial maps of smooth affine quadrics revisited]]__ (//In preparation, draft v. 28 Mar 2008//)
* __[[Homotopy theory of smooth affine quadrics revisited]]__, with. B. Doran - [[DVI|http://www.math.washington.edu/~asok/papers/QuadricsRev2.dvi]], [[PDF|http://www.math.washington.edu/~asok/papers/QuadricsRev2.pdf]] (//Preprint v. 28 Mar 2008//)
* __[[A¹-homotopy types and contractible smooth surfaces]]__ - [[DVI|http://www.math.washington.edu/~asok/papers/exoticIIb-contractibleFinal.dvi]],[[PDF|http://www.math.washington.edu/~asok/papers/exoticIIb-contractibleFinal.pdf]] (//Preprint v. 8 May 2008//)
* __[[A¹-homotopy types, excision, and solvable quotients]]__, with B. Doran - [[DVI|http://www.math.washington.edu/~asok/papers/toricA1final.dvi]],[[PDF|http://www.math.washington.edu/~asok/papers/toricA1final.pdf]] (//Preprint v. 18 Dec 2007//)
* __[[Vector bundles on contractible smooth schemes]]__, with B. Doran - [[ArXiv|http://arxiv.org/abs/0710.3607]],[[DVI|http://www.math.washington.edu/~asok/papers/exoticaffinesIIIfinal.dvi]],[[PDF|http://www.math.washington.edu/~asok/papers/exoticaffinesIIIfinal.pdf]] (//v. 15 Oct 2007 - Duke Math. Jour. 2008, 143 (3), pp. 513-530//)
* __[[On unipotent quotients and some A¹- contractible smooth schemes]]__, with B. Doran - [[ArXiv|http://arxiv.org/abs/math/0703137]],[[DVI|http://www.math.washington.edu/~asok/papers/exoticaffinesFinal.dvi]],[[PDF|http://www.math.washington.edu/~asok/papers/exoticaffinesFinal.pdf]](//v. 05 Mar 2007 - Int. Math. Res. Pap. 2007 V.2 Art. ID rpm005, 51 pages//)
!Cohomology of Quotients
In this group of papers, we extend the ideas of Kirwan's thesis to study motivic cohomology (in the sense of Voevodsky) of varieties constructed as quotients by means of geometric invariant theory.
* __[[Equivariant Motivic Cohomology and GIT Quotients]]__, with B. Doran and F. Kirwan- //In Preparation//
* __[[Yang- Mills theory and Tamagawa Numbers]]__, with B. Doran and F. Kirwan - [[ArXiv|http://arxiv.org/abs/0801.4733]],[[DVI|http://www.math.washington.edu/~asok/papers/TamagawaFinal2.dvi]],[[PDF|http://www.math.washington.edu/~asok/papers/TamagawaFinal2.pdf]] (//To Appear, Bulletin of the LMS//)
!Equivariant Vector Bundles
The key goal of this class of papers is to give ``concrete" descriptions of the category of equivariant vector bundles on certain varieties with group actions. The main example to keep in mind is [[Klyachko's description|http://arxiv.org/abs/math/0205311]] of the category of $T$-equivariant vector bundles on a smooth projective [[toric|http://en.wikipedia.org/wiki/Toric_variety]] $T$-variety.
* __[[Equivariant sheaves on spherical varieties]]__, with J. Parson - In preparation
* __[[Equivariant Vector Bundles on Certain Affine G- Varieties]]__ - [[ArXiv|http://arxiv.org/abs/math/0604344]],[[DVI|http://www.math.washington.edu/~asok/papers/asok-equivvb.dvi]],[[PDF|http://www.math.washington.edu/~asok/papers/asok-equivvb.pdf]] (//Pure and Applied Mathematics Quarterly v.2 no. 4, pp. 1085-1102, 2006//)
* Thesis - __Geometry of Simple G- Varieties__ (//Princeton, November 2004, Not fit for human consumption.//)
!Notes for talks
Here are some informal notes (transparencies) for some talks I've given.
* __A ``homotopic" view of affine lines on varieties__ - UCLA Colloqium, 1/31/08, ([[.PDF|papers/UCLATransparencies.pdf]])
* __Unipotent groups and some A¹-contractible smooth schemes__ - U Chicago Alg. Geom Sem, 3/7/07 ([[.PDF|papers/UChicagotransparencies.pdf]])
* Artist [[Lun-Yi Tsai|http://www.lunyitsai.com]]'s show [[Transition Gadgets]] (4/3/08-4/30/08)
* Workshop on Rational Curves on Quasi- Projective manifolds, [[MPIM Bonn|http://]] (5/5/08-5/10/08) - [[Workshop information|http://www.mpim-bonn.mpg.de/Events/This+Year+and+Prospect/]]
* Artist [[Lun-Yi Tsai|http://www.lunyitsai.com]]'s [[show|http://www.lunyitsai.com/berlin/index.htm]] at the Karl Hofer Gesellschaft in Berlin (2008)
* Jumbo Algebraic Geometry Program, MSRI Berkeley (1/12/09-5/22/09) - [[Organizers' webpage|http://www.math.washington.edu/~msri2009/]]
My research centers on geometry and topology of [[algebraic varieties|http://en.wikipedia.org/wiki/Algebraic_variety]]. Very roughly speaking, an algebraic variety is the set of solutions of some collection of [[polynomial|http://en.wikipedia.org/wiki/Polynomial]] equations over a [[field|http://en.wikipedia.org/wiki/Field_(mathematics)]], e.g, the $2$-dimensional sphere is given by the equation $x^2 + y^2 + z^2 = 1$. For algebraic varieties defined over the field ${\mathbb C}$ of [[complex numbers|http://en.wikipedia.org/wiki/Complex_number]], one can use tools from [[algebraic topology|http://en.wikipedia.org/wiki/Algebraic_topology]] to study the structure of the underlying point sets. For example, one can count the number of [[holes (or their higher dimensional analogs)|http://en.wikipedia.org/wiki/Betti_number]] of such a point set. For algebraic varieties over fields of [[less inherently geometric nature|http://en.wikipedia.org/wiki/Characteristic_(algebra)]], one needs different techniques. [[Alexander Grothendieck|http://en.wikipedia.org/wiki/Grothendieck]] introduced techniques (e.g., [[étale cohomology|http://en.wikipedia.org/wiki/%C3%89tale_cohomology]]) to study the structure of such algebraic varieties. One of my main interests is motivic or ${\mathbb A}^1$[[-homotopy theory|http://en.wikipedia.org/wiki/A%C2%B9_homotopy_theory]], which is a theory developed by [[Fabien Morel|http://www.mathematik.uni-muenchen.de/personen/morel.php]] and [[Vladimir Voevodsky|http://en.wikipedia.org/wiki/Vladimir_Voevodsky]]. The main goal of ${\mathbb A}^1$-homotopy theory is to import the full power of [[homotopy theory|http://en.wikipedia.org/wiki/Homotopy_theory]] into the study of algebraic varieties, and, in doing so, to unify the study of their [[arithmetic, geometric|http://en.wikipedia.org/wiki/Motivic_Galois_group]] and topological properties.
For more information see:
* ''[[Preprints, Publications and other notes|Preprints and Publications]]''
* ''[[Motivic or A¹-homotopy theory resource page|http://www.math.washington.edu/~asok/motivichomotopy.html]]''
<html><iframe src="http://www.google.com/calendar/embed?src=aravind.asok%40gmail.com&ctz=America/Los_Angeles" style="border: 0" width="800" height="600" frameborder="0" scrolling="no"></iframe></html>
[>img[Aravind Asok|webjpgs/selfpicsmall.jpg]]
''Employment and Education''
Spring 2009 - MSRI Postdoctoral Fellow ([[Mathematical Sciences Research Institute|http://www.msri.org]], Berkeley, CA)
2008- Hedrick Assistant Professor ([[University of California at Los Angeles|http://www.math.ucla.edu]], Los Angeles, CA)
2005-2008 - VIGRE Acting Assistant Professor ([[University of Washington|http://www.math.washington.edu]], Seattle, WA)
2004-2005 - EPSRC Postdoctoral Fellow ([[University of Oxford|http://www.maths.ox.ac.uk]], Oxford, England)
2004 - Ph.D. in Mathematics ([[Princeton University|http://www.math.princeton.edu/]], Princeton, NJ) Dissertation supervised by [[Bob MacPherson|http://www.math.ias.edu/pages/people/faculty/robert-d.-macpherson.php]]
Thesis entitled: //Geometry of simple $G$-varieties//
''Research Interests''
Group actions on algebraic varieties, topology of algebraic varieties, representation theory, ${\mathbb A}^1$-homotopy theory
''Collaborators''
[[Brent Doran|http://www.math.ias.edu/~brdoran]], [[Frances Kirwan|http://www.maths.ox.ac.uk/contact/details/kirwan]], [[Fabien Morel|http://www.mathematik.uni-muenchen.de/~morel/webpage.html]], James Parson
!!! Definition
An open manifold $M$ is said to be simply connected at infinity if for every compact subset $C$ of $M$, there is a larger compact subset $D$ of $M$ containing $C$ such that $M \setminus D$ is connected and simply connected.
VIGRE Acting Assistant Professor
http://www.www.math.washington.edu/~asok
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[[Math 326 - Summer 2008|http://www.math.washington.edu/~asok/326.html]]
[[Math 126 - Fall 2007|http://www.math.washington.edu/~asok/126.html]]
[[Math 308 - Fall 2007|http://www.math.washington.edu/~asok/308.html]]
[[Math 402 - Summer 2007|http://www.math.washington.edu/~asok/402.html]]
[[Math 407 - Summer 2007|http://www.math.washington.edu/~asok/407.html]]
|~ViewToolbar|closeTiddler closeOthers +editTiddler > fields syncing permalink references jump|
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<html><center><IMG SRC = "webjpgs/Gadget_01.jpg" WIDTH = 580></center></html>
<html><center><U>Classification of orientable surfaces</U>, <b><a href = "http://www.lunyitsai.com">Lun-Yi Tsai</a></b> (Miami, FL)</center></html>
<html><center><IMG SRC = "webjpgs/Gadget_10.jpg" HEIGHT = 500></center></html>
<html><center><u>Pinched tori and two banana spaces</u>, <b><a href = "http://www.lunyitsai.com">Lun-Yi Tsai</a></b> (Miami, FL)</center></html>
<html><center><IMG SRC = "webjpgs/Gadget_11.jpg" WIDTH = 580></center></html>
<html><center><u>Aravind's picture</u>, <b><a href = "http://www.lunyitsai.com">Lun-Yi Tsai</a></b> (Miami, FL)</center></html>
<html><center><IMG SRC = "webjpgs/Gadget_12.jpg" HEIGHT = 500></center></html>
<html><center><u>Hopf fibration</u>, <b><a href = "http://www.lunyitsai.com">Lun-Yi Tsai</a></b> (Miami, FL)</center></html>
<html><center><IMG SRC = "webjpgs/Gadget_13.jpg" WIDTH = 580></center></html>
<html><center><u>Hopf fibration</u>, <b><a href = "http://www.lunyitsai.com">Lun-Yi Tsai</a></b> (Miami, FL)</center></html>
<html><center><IMG SRC = "webjpgs/Gadget_14.jpg" WIDTH = 580></center></html>
<html><center><u>Elliptic lattice</u>, <b><a href = "http://www.lunyitsai.com">Lun-Yi Tsai</a></b> (Miami, FL)</center></html>
<html><center><IMG SRC = "webjpgs/Gadget_15.jpg" WIDTH = 580></center></html>
<html><center><u>Mayer-Vietoris exactness</u>, <b><a href = "http://www.lunyitsai.com">Lun-Yi Tsai</a></b> (Miami, FL)</center></html>
<html><center><IMG SRC = "webjpgs/Gadget_16.jpg" WIDTH = 580></center></html>
<html><center><u>Towards a universal gadget</u>, <b><a href = "http://www.lunyitsai.com">Lun-Yi Tsai</a></b> (Miami, FL)</center></html>
<html><center><IMG SRC = "webjpgs/Gadget_02.jpg" WIDTH = 580></center></html>
<html><center><u>The winding number</u>, <b><a href = "http://www.lunyitsai.com">Lun-Yi Tsai</a></b> (Miami, FL)</center></html>
<html><center><IMG SRC = "webjpgs/Gadget_03.jpg" WIDTH = 610></center></html>
<html><center><u>Hopf fibration</u>, <b><a href = "http://www.lunyitsai.com">Lun-Yi Tsai</a></b> (Miami, FL)</center></html>
<html><center><IMG SRC = "webjpgs/Gadget_04.jpg" WIDTH = 580></center></html>
<html><center><u>Hopf fibration</u>, <b><a href = "http://www.lunyitsai.com">Lun-Yi Tsai</a></b> (Miami, FL)</center></html>
<html><center><IMG SRC = "webjpgs/Gadget_05.jpg" WIDTH = 580></center></html>
<html><center><u>Hopf fibration</u>, <b><a href = "http://www.lunyitsai.com">Lun-Yi Tsai</a></b> (Miami, FL)</center></html>
<html><center><IMG SRC = "webjpgs/Gadget_06.jpg" WIDTH = 580></center></html>
<html><center><u>Homage a J.J. or stereographic projection</u>, <b><a href = "http://www.lunyitsai.com">Lun-Yi Tsai</a></b> (Miami, FL)</center></html>
<html><center><IMG SRC = "webjpgs/Gadget_07.jpg" WIDTH = 580></center></html>
<html><center><u>Hopf fibration</u>, <b><a href = "http://www.lunyitsai.com">Lun-Yi Tsai</a></b> (Miami, FL)</center></html>
<html><center><IMG SRC = "webjpgs/Gadget_08.jpg" WIDTH = 580></center></html>
<html><center><u>Monodromy</u>, <b><a href = "http://www.lunyitsai.com">Lun-Yi Tsai</a></b> (Miami, FL)</center></html>
<html><center><IMG SRC = "webjpgs/Gadget_09.jpg" WIDTH = 580></center></html>
<html><center><u>Monodromy</u>, <b><a href = "http://www.lunyitsai.com">Lun-Yi Tsai</a></b> (Miami, FL)</center></html>
!!! The show
Lun- Yi Tsai's show at Shoreline community college was entitled //Transition Gadgets//. For more of his writings on art, life, etc., see his [[blog|http://dimostrazioni.blogspot.com/]]. The show consisted of sixteen [[encaustic paintings|http://en.wikipedia.org/wiki/Encaustic_painting]] based on conversations pertaining to mathematics. I wrote two paragraphs for the [[press release|http://www.lunyitsai.com/pdfs/Tsai_Lun-Yi_ShorelineArtGallery_PR.pdf]] for this show.
``Lun- Yi Tsai’s most recent art explores the process of mathematics. To many people, this process consists of the formal manipulation of algebraic equations or the writing of proofs in plane geometry. While mathematicians often do write proofs, study geometry and perform formal manipulations, they don’t usually study ideas that are explained in a few lines. On the other hand, mathematicians often do use elementary pictures to guide their investigations and motivate their proofs. Lun- Yi’s latest work is inspired (in part) by our discussions about the evolution of the mathematical conception of space, and by the simple pictures I drew in an attempt to illustrate the concepts at hand.
``The classic Chinese work ‘[[The Nine Chapters on the Mathematical Art|http://en.wikipedia.org/wiki/The_Nine_Chapters_on_the_Mathematical_Art]]’ (1000-200 BCE) explains nine mathematical concepts in the form: (i) statement of the problem, (ii) statement of the solution, and finally (iii) explanation of the solution. Perhaps mirroring this construction, these new pieces begin with a ‘problem,’ the paper transcription of a mathematical lecture, on top of which Lun- Yi’s beautiful encaustic ‘solutions’ are superimposed. What remains is for the observer to provide the explanation of these ‘Transition Gadgets.’ One might call them ‘Sixteen Encaustics on the Mathematical Process.’"
!!! The images
Here are images of the actual encaustics (presented with permission of the artist). Contact [[Lun-Yi|http://www.lunyitsai.com/contact.htm]] for more information.
[img[webjpgs/Gadget_01thumb.jpg][Transition Gadget #1]] [img[webjpgs/Gadget_02thumb.jpg][Transition Gadget #2]] [img[webjpgs/Gadget_03thumb.jpg][Transition Gadget #3]] [img[webjpgs/Gadget_04thumb.jpg][Transition Gadget #4]] [img[webjpgs/Gadget_05thumb.jpg][Transition Gadget #5]] [img[webjpgs/Gadget_06thumb.jpg][Transition Gadget #6]] [img[webjpgs/Gadget_07thumb.jpg][Transition Gadget #7]] [img[webjpgs/Gadget_08thumb.jpg][Transition Gadget #8]]
[img[webjpgs/Gadget_09thumb.jpg][Transition Gadget #9]] [img[webjpgs/Gadget_10thumb.jpg][Transition Gadget #10]] [img[webjpgs/Gadget_11thumb.jpg][Transition Gadget #11]] [img[webjpgs/Gadget_12thumb.jpg][Transition Gadget #12]] [img[webjpgs/Gadget_13thumb.jpg][Transition Gadget #13]] [img[webjpgs/Gadget_14thumb.jpg][Transition Gadget #14]] [img[webjpgs/Gadget_15thumb.jpg][Transition Gadget #15]] [img[webjpgs/Gadget_16thumb.jpg][Transition Gadget #16]]
* co-author(s): Brent Doran
* Additional Information: //v. 15 Oct 2007 - Duke Mathematical Journal 2008, 143 (3), pp. 513-530 //
* Availability: [[ArXiv|http://arxiv.org/abs/0710.3607]],[[DVI|papers/exoticaffinesIIIfinal.dvi]],[[PDF|papers/exoticaffinesIIIfinal.pdf]] [[Journal|http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1212500465]]
* ''Abstract'': We discuss algebraic vector bundles on smooth $k$-schemes $X$ [[contractible from the standpoint of A¹-homotopy theory|On unipotent quotients and some A¹- contractible smooth schemes]]; when $k = {\mathbb C}$, the smooth manifolds $X({\mathbb C})$ are contractible as topological spaces. The integral algebraic K-theory and integral motivic cohomology of such schemes are that of ${\mathrm{Spec}} k$. One might hope that furthermore, and in analogy with the classification of topological vector bundles on manifolds, algebraic vector bundles on such schemes are all isomorphic to trivial bundles; [[this is almost certainly true when the scheme is affine|motivichomotopy.html#%5B%5BUnstable%20A%C2%B9-homotopy%20theory%5D%5D]]. However, in the non-affine case this is false: we show that (essentially) every smooth ${\mathbb A}^1$-contractible strictly quasi-affine scheme that admits a $U$-torsor whose total space is affine, for $U$ a unipotent group, possesses a non-trivial vector bundle. Indeed we produce explicit arbitrary dimensional families of non-isomorphic such schemes, with each scheme in the family equipped with ``as many" (i.e., arbitrary dimensional moduli of) non-isomorphic vector bundles, of every suffciently large rank $n$, as one desires; neither the schemes nor the vector bundles on them are distinguishable by algebraic K-theory. We also discuss the triviality of vector bundles for certain smooth complex affine varieties whose underlying complex manifolds are contractible, but that are not necessarily ${\mathbb A}^1$-contractible.
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* co-author(s): Brent Doran and Frances Kirwan
* Additional Information: //To appear, Bulletin of the LMS//
* Availability: [[ArXiv|http://arxiv.org/abs/0801.4733]],[[DVI|papers/TamagawaFinal2.dvi]],[[PDF|papers/TamagawaFinal2.pdf]], [[Journal|http://blms.oxfordjournals.org/cgi/content/abstract/bdn036]]
* ''Abstract'': Atiyah and Bott used [[equivariant Morse theory applied to the Yang–Mills functional|http://www.math.harvard.edu/history/bott/bottbio/node22.html]] to calculate the Betti numbers of moduli spaces of vector bundles over a [[Riemann surface|http://en.wikipedia.org/wiki/Riemann_surface]], rederiving inductive formulae obtained from an arithmetic approach which involved the [[Tamagawa number|http://en.wikipedia.org/wiki/Tamagawa_number]] of $SL_n$. This article attempts to survey and extend our understanding of this link between Yang–Mills theory and Tamagawa numbers, and to explain how methods used over the last three decades to study the singular cohomology of moduli spaces of bundles on a smooth projective curve over C can be adapted to the setting of ${\mathbb A}^1$[[-homotopy theory|motivichomotopy.html]] to study the [[motivic cohomology|http://en.wikipedia.org/wiki/Motivic_cohomology]] of these moduli spaces over an algebraically closed field.
Strictly speaking, Stallings shows that the product of the manifold with $\mathbb{R}^1$ (or $\mathbb{R}^2$ when $n=3$) is PL-isomorphic or diffeomorphic to $\mathbb{R}^n$; however, for contractible real manifolds, any $\mathbb{R}^k$-bundle is trivial, so our statement is equivalent to his, and generalizes well to our algebraic setting.}