/*** SET BUTTON'S FOLDER HERE ***/ var buttonFolder = "buttons/"; /*** SET BUTTONS' FILENAMES HERE ***/ upSources = new Array("button1up.png","button2up.png","button3up.png","button4up.png","button5up.png","button6up.png","button7up.png","button8up.png","button9up.png","button10up.png","button11up.png","button12up.png","button13up.png","button14up.png","button15up.png","button16up.png","button17up.png","button18up.png","button19up.png","button20up.png","button21up.png","button22up.png","button23up.png","button24up.png","button25up.png","button26up.png","button27up.png","button28up.png","button29up.png","button30up.png","button31up.png","button32up.png","button33up.png","button34up.png","button35up.png","button36up.png","button37up.png","button38up.png","button39up.png","button40up.png","button41up.png"); overSources = new Array("button1over.png","button2over.png","button3over.png","button4over.png","button5over.png","button6over.png","button7over.png","button8over.png","button9over.png","button10over.png","button11over.png","button12over.png","button13over.png","button14over.png","button15over.png","button16over.png","button17over.png","button18over.png","button19over.png","button20over.png","button21over.png","button22over.png","button23over.png","button24over.png","button25over.png","button26over.png","button27over.png","button28over.png","button29over.png","button30over.png","button31over.png","button32over.png","button33over.png","button34over.png","button35over.png","button36over.png","button37over.png","button38over.png","button39over.png","button40over.png","button41over.png"); // SUB MENUS DECLARATION, YOU DONT NEED TO EDIT THIS subInfo = new Array(); subInfo[1] = new Array(); subInfo[2] = new Array(); subInfo[3] = new Array(); subInfo[4] = new Array(); subInfo[5] = new Array(); subInfo[6] = new Array(); subInfo[7] = new Array(); subInfo[8] = new Array(); subInfo[9] = new Array(); subInfo[10] = new Array(); subInfo[11] = new Array(); subInfo[12] = new Array(); subInfo[13] = new Array(); subInfo[14] = new Array(); subInfo[15] = new Array(); subInfo[16] = new Array(); subInfo[17] = new Array(); subInfo[18] = new Array(); subInfo[19] = new Array(); subInfo[20] = new Array(); subInfo[21] = new Array(); subInfo[22] = new Array(); subInfo[23] = new Array(); subInfo[24] = new Array(); subInfo[25] = new Array(); subInfo[26] = new Array(); subInfo[27] = new Array(); subInfo[28] = new Array(); subInfo[29] = new Array(); subInfo[30] = new Array(); subInfo[31] = new Array(); subInfo[32] = new Array(); subInfo[33] = new Array(); subInfo[34] = new Array(); subInfo[35] = new Array(); subInfo[36] = new Array(); subInfo[37] = new Array(); subInfo[38] = new Array(); subInfo[39] = new Array(); subInfo[40] = new Array(); subInfo[41] = new Array(); //*** SET SUB MENUS TEXT LINKS AND TARGETS HERE ***// subInfo[1][1] = new Array("Citytech","teaching.html#Citytech",""); subInfo[1][2] = new Array("Graduate Center","teaching.html#Gradcenter",""); subInfo[2][1] = new Array("Articles","articles.html",""); subInfo[2][2] = new Array("Preprints","preprints.html",""); subInfo[2][3] = new Array("Other","otherpublications.html",""); subInfo[3][1] = new Array("Talks","talks.html",""); subInfo[3][2] = new Array("CV","PDF/CV2005.pdf",""); subInfo[3][3] = new Array("Grants","cv.html",""); subInfo[3][4] = new Array("Research interests","cv.html",""); subInfo[4][1] = new Array("Seminars","links.html#seminars",""); subInfo[4][2] = new Array("Math links","links.html#mathlinks",""); subInfo[4][3] = new Array("Other home pages","links.html#otherlinks",""); subInfo[4][4] = new Array("Picture galery","links.html#pictures",""); subInfo[6][1] = new Array("empty","articles.html",""); subInfo[7][1] = new Array("Let $K$ be an algebraically closed field endowed with a complete non-archimedean norm. Let $f: Y\\to X$ be a map of $K$-affinoid varieties. We prove that for each point $x\\in X$, either $f$ is flat at $x$, or there exists, at least locally around $x$, a maximal locally closed analytic subvariety $Z\\subset X$ containing $x$, such that the base change $f^{-1}(Z)\\to Z$ is flat at $x$, and, moreover, $g^{-1}(Z)$ has again this property in any point of the fiber of $x$ after base change over an arbitrary map $g: X'\\to X$ of affinoid varieties. If we take the local blowing up $\\pi:\\tilde X\\to X$ with this centre $Z$, then the fiber with respect to the strict transform $\\tilde f$ of $f$ under $\\pi$, of any point of $\\tilde X$ lying above $x$, has grown strictly smaller. Among the corollaries to these results we quote, that flatness in rigid analytic geometry is local in the source; that flatness over a reduced quasi-compact rigid analytic variety can be tested by surjective families; that an inclusion of affinoid domains is flat in a point, if it is unramified in that point. ","articles.html",""); subInfo[8][1] = new Array("The class of all Artinian local rings of length at most $l$ is $\\forall_2$-elementary, axiomatised by a finite set of axioms $ART_l$. We show that its existentially closed models are Gorenstein, of length exactly $l$ and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form. The theory $GOR_l$ of all Artinian local Gorenstein rings of length $l$ with algebraically closed residue field is model complete and the theory $ART_l$ is companionable, with model-companion $GOR_l$.","articles.html",""); subInfo[9][1] = new Array("We give a rigid analytic version of Hironaka's Embedded Resolution of Singularities over an algebraically closed field of characteristic zero, complete with respect to a non-archimedean norm. This resolution is local with respect to the Grothendieck topology. The proof uses Hironaka's original result, together with an application of our analytization functor.","articles.html",""); subInfo[10][1] = new Array("We introduce a measure of complexity for affine algebras and their finitely generated modules, in terms of the degrees of the polynomials used in their description. We then study how various cohomological operations and numerical invariants are uniformly bounded with respect to these complexities. We apply this to give first order characterisations of certain algebraic-geometric properties. This enables us to apply the Lefschetz Principle to transfer properties between various characteristics. As an application, we obtain the following version of the Zariski-Lipman Conjecture in positive characteristic: let $R$ be the local ring of a point $P$ on a hypersurface over an algebraically closed field $K$ such that the module of $K$-invariant derivatives on $R$ is free, then $P$ is a non-singular point, provided the characteristic is larger than some bound only depending on the degree of the hypersurface.","articles.html",""); subInfo[11][1] = new Array("We show the existence of a first order theory $CMM_{d,e}$ whose Noetherian models are precisely the local Cohen-Macaulay rings of dimension $d$ and multiplicity $e$. The completion of a model of $CMM_{d,e}$ is again a model and is moreover Noetherian. If $R$ is an equicharacteristic local Gorenstein ring of dimension $d$ and multiplicity $e$ with algebraically closed residue field and if the Artin Approximation Property holds for $R$, then $R$ is an existentially closed model in the subclass of all Noetherian models of $CMM_{d,e}$. In case $R$ is moreover excellent, Spivakovski proved that the weaker Henselian assumption implies the Artin Approximation. This suggests an alternative, model theoretic strategy for proving Artin Approximation under the additional assumptions that $R$ is Gorenstein, equicharacteristic and has algebraically closed residue field.","articles.html",""); subInfo[12][1] = new Array("Let $K$ be an algebraically closed field endowed with a complete non-archimedean norm with valuation ring $R$. Let $f: Y\\to X$ be a map of $K$-affinoid varieties. In this paper we study the analytic structure of the image $f(Y)\\subset X$; such an image is a typical example of a subanalytic set. Using Embedded Resolution of Singularities, we derive in the zero characteristic case a Uniformization Theorem for subanalytic sets: after finitely many local blowing ups with smooth centers, a subanalytic set becomes semi-analytic. To prove this we establish a Flattening Theorem for affinoid varieties in the style of Hironaka, which allows a reduction to the study of subanalytic sets arising from flat maps. Specifically we show that a map of affinoid varieties can be rendered flat by using only finitely many local blowing ups. The case of an image under a flat map is then dealt with by a small extension of a result of Raynaud. Our result can be conveniently stated as a Quantifier Elimination theorem for the valuation ring $R$ in an analytic expansion of the language of valued fields. This formulation is in the style of Denef and van den Dries.","articles.html",""); subInfo[13][1] = new Array("In this survey article, we introduce various measures of complexity for algebraic constructions in polynomial rings over fields and show how they are often uniformly bounded by the complexity of the starting data. In problems which have a linear nature, the degree of the polynomials provide a sufficient notion of complexity. However, in the non-linear case, the more sophisticated measure of etale complexity is needed. These bounds lead often to the constructible nature of geometric problems, where in the non-linear case, one should work in the etale site rather than in the Zariski site. As another application of the existence of these bounds we mention the possibility of transferring results from one characteristic to another by means of the Lefschetz Principle. We will give some examples of new results as well as some new proofs to old results.","articles.html",""); subInfo[14][1] = new Array("If $A\\to B$ is a faithfully flat ring homomorphism of Noetherian rings and $(x_n|n)$ is a sequence of elements in $A$ satisfying a linear recursion relation with coefficients in $B$, then this sequence already satisfies such a recursion relation (of the same length) with coefficients in $A$. As a corollary, we obtain that, if $A$ and $B$ are moreover normal domains, then any power series over $A$ which is rational over $B$, is already rational over $A$.","articles.html",""); subInfo[15][1] = new Array("Let $K$ be an algebraically closed field endowed with a complete non-archimedean norm. Let $f:Y\\to X$ be a map of $K$-affinoid varieties. In this paper we study the analytic structure of the image $f(Y)\\subset X$; such an image is a typical example of a subanalytic set. We show that the subanalytic sets are precisely the $D$-semianalytic sets, where $D$ is the truncated division function first introduced by Denef and van den Dries. To prove this we establish a Flattening Theorem for affinoid varieties in the style of Hironaka, which allows a reduction to the study of subanalytic sets arising from flat maps. More precisely, we show that a map of affinoid varieties can be rendered flat by using only finitely many local blowing ups. The case of a flat map is then dealt with by a small extension of a result of Raynaud and Gruson showing that the image of a flat map of affinoid varieties is open in the Grothendieck topology. Using Embedded Resolution of Singularities, we derive in the zero characteristic case a Uniformization Theorem for subanalytic sets: a subanalytic set can be rendered semianalytic using only finitely many local blowing ups with smooth centers. As a corollary we obtain that any subanalytic set in the plane is semianalytic.","articles.html",""); subInfo[16][1] = new Array("We show how Resolution of Singularities in characteristic $p$ implies the decidability of the existential theory of $F_p[[t]]$ in the language of discrete valuation rings, where $t$ is a single variable and $F_p$ the $p$-element field.","articles.html",""); subInfo[17][1] = new Array("Using a tight closure argument in characteristic $p$ and then lifting the argument to characteristic zero with aid of ultraproducts, I present an elementary proof of the Briancon-Skoda Theorem: for an $m$-generated ideal $I$ of $C[[X_1,\\dots,X_n]]$, the $m$-th power of its integral closure is contained in $I$. It is well-known that as a corollary, one gets a solution to the following classical problem. Let $f$ be a convergent power series in $n$ variables over $C$ which vanishes at the origin. Then $f^n$ lies in the ideal generated by the partial derivatives of $f$.","articles.html",""); subInfo[18][1] = new Array("There is an algorithm which computes the minimal number of generators of the ideal of a reduced curve $C$ in affine $n$-space over an algebraically closed field $K$, provided $C$ is not a local complete intersection. The existence of such an algorithm follows from the fact that given $d$, there exists a bound $d'$, such that if $I$ is a height $n-1$ radical ideal in $K[X]$ with $X$ an $n$-tuple of variables, generated by polynomials of degree at most $d$, then $I$ admits a set of generators of minimal cardinality, with each generator having degree at most $d'$, except possibly when $K[X]/I$ is an (unmixed) local complete intersection.","articles.html",""); subInfo[19][1] = new Array("For a Noetherian local ring $R$, if $R/I$ is Cohen-Macaulay, then the ideal $I$ can be generated by at most $(e-2)(v-d-1)+2$ elements, where $v$ is the embedding dimension of $R$ and where $d$ and $e\\geq 3$ are the dimension and the multiplicity of $R/I$ respectively. This bound is in general much sharper than the bounds given by Sally or Boratynski-Eisenbud-Rees in case $I$ has height bigger than two. Moreover, no Cohen-Macaulay assumption on $R$ is required. ","articles.html",""); subInfo[20][1] = new Array("Let $R$ be a locally finitely generated algebra over a discrete valuation ring $V$ of mixed characteristic. For any of the homological properties, the Direct Summand Theorem, the Monomial Theorem, the Improved New Intersection Theorem, the Vanishing of Maps of Tors and the Hochster-Roberts Theorem, we show that it holds for $R$ and possibly some other data defined over $R$, provided the residual characteristic of $V$ is sufficiently large in terms of the complexity of the data, where the complexity is primarily given in terms of the degrees of the polynomials over $V$ that define the data, but possibly also by some additional invariants.","articles.html",""); subInfo[21][1] = new Array("For a Noetherian local ring, the prime ideals in the singular locus completely determine the category of finitely generated modules up to direct summand, extensions and syzygies. From this some simple homological criteria are derived for testing whether an arbitrary module has finite projective dimension.","articles.html",""); subInfo[22][1] = new Array("In this paper, non-standard tight closure is proposed as an alternative for classical tight closure on finitely generated algebras over $C$. It has the advantage that it admits a functional definition, similar to the characteristic $p$ definition of tight closure, where instead of the characteristic $p$ Frobenius, we use now its ultraproduct, the non-standard Frobenius. This new closure operation $cl(I)$ on ideals $I$ of $A$, has the same properties as classical tight closure, to wit, (1) if $A$ is regular, then $I=cl(I)$; (2) if $A\\subset B$ is an integral extension of domains, then $cl(IB)\\cap A\\subset cl(I)$; (3) if $A$ is local and $(x_1,\\dots,x_n)$ is a system of parameters, then $((x_1,\\dots,x_i)A:x_{i+1})$ is contained in $cl((x_1,\\dots,x_i)A)$ (Colon-Capturing); (4) if $I$ is generated by $m$ elements, then $cl(I)$ is contained in the integral closure of $I$ and contains the integral closure of $I^m$ (Briancon-Skoda).","articles.html",""); subInfo[23][1] = new Array("In this paper, an alternative proof is presented of the following result on symbolic powers due to Ein-Lazarsfeld-Smith (for the affine case over $C$) and to Hochster-Huneke (for the general case). Let $A$ be a regular ring containing a field $K$. Let $I$ be a radical ideal of $A$ and let $h$ be the maximum of the heights of its minimal primes. Then for all $n$, we have an inclusion $I^{(hn)}\\subset I^n$, where the first ideal denotes the $hn$-th symbolic power of $I$. In prime characteristic, this result admits an easy tight closure proof due to Hochster-Huneke. In this paper, the characteristic zero version is obtained from this by an application of the Lefschetz Principle. The paper is entirely self-contained.","articles.html",""); subInfo[24][1] = new Array("We give a canonical construction of a balanced big Cohen-Macaulay algebra for a domain of finite type over $C$ by taking ultraproducts of absolute integral closures in positive characteristic. This yields a new tight closure characterization of rational singularities in characteristic zero.","articles.html",""); subInfo[25][1] = new Array("Let $R$ be an excellent local domain of positive characteristic with residue field $k$. This paper investigates properties of $R$ in case $Tor^R_1(R^+,k)$ vanishes, where $R^+$ denotes the absolute integral closure of $R$. Such a ring is F-rational and F-pure. If $R$ has at most an isolated singularity or has dimension at most two, then $R$ is regular.","articles.html",""); subInfo[26][1] = new Array("Generalizing work of Smith and Hara, we give a new characterization of log-terminal singularities for finitely generated algebras over $C$, in terms of purity properties of ultraproducts of characteristic $p$ Frobenii. As a first application we obtain a Boutot-type theorem for log-terminal singularities: given a pure morphism $Y \\to X$ between affine $Q$-Gorenstein varieties of finite type over $C$, if $Y$ has at most a log-terminal singularities, then so does $X$. The second application is the Vanishing for Maps of Tor for log-terminal singularities: if $A\\subset R$ is a Noether Normalization of a finitely generated $C$-algebra $R$ and $S$ is an $R$-algebra of finite type with log-terminal singularities, then the natural morphism $Tor^A_i(M,R) \\to Tor^A_i(M,S)$ is zero, for every $A$-module $M$ and every $i\\geq 1$. The final application is Kawamata-Viehweg Vanishing for a connected projective variety $X$ of finite type over $C$ whose affine cone has a log-terminal vertex (for some choice of polarization). As a corollary, we obtain a proof of the following conjecture of Smith: if $G$ is the complexification of a real Lie group acting algebraically on a projective smooth Fano variety $X$, then for any numerically effective line bundle $L$ on any GIT quotient $Y$, each cohomology module $H^i(Y,L)$ vanishes for $i\\geq 1$, and, if $L$ is moreover big, then $H^i(Y,L^{-1})$ vanishes for $i\\leq dim Y-1$","articles.html",""); subInfo[27][1] = new Array("We prove a generalization of the Hochster-Roberts-Boutot-Kawamata Theorem: let $R\\to S$ be a pure homomorphism of equicharacteristic zero Noetherian local rings. If $S$ is regular, then $R$ is pseudo-rational, and if $R$ is moreover $Q$-Gorenstein, then it is pseudo-log-terminal.","articles.html",""); subInfo[28][1] = new Array("For a Noetherian local domain $R$, there exists an upperbound $N_t=N_t(R)$ on the minimal number of generators of any height two ideal $I$ for which $R/I$ is Cohen-Macaulay of type $t$. If $R$ contains an infinite field, one can take $N_t=(t+1)e$, where $e$ is the homological multiplicity of $R$.","articles.html",""); subInfo[29][1] = new Array("In this paper, various Homological Conjectures are studied for local rings which are locally finitely generated over a discrete valuation ring $V$ of mixed characteristic. Typically, we can only conclude that a particular Conjecture holds for such a ring provided the residual characteristic of $V$ is sufficiently large in terms of the complexity of the data, where the complexity is primarily given in terms of the degrees of the polynomials over $V$ that define the data, but possibly also by some additional invariants such as (homological) multiplicity. Thus asymptotic versions of the Improved New Intersection Theorem, the Monomial Conjecture, the Direct Summand Conjecture, the Hochster-Roberts Theorem and the Vanishing of Maps of Tors Conjecture are given. That the results only hold asymptotically, is due to the fact that non-standard arguments are used, relying on the Ax-Kochen-Ershov Principle, to infer their validity from their positive characteristic counterparts. A key role in this transfer is played by the Hochster-Huneke canonical construction of big Cohen-Macaulay algebras in positive characteristic via absolute integral closures.","articles.html",""); subInfo[30][1] = new Array("We prove various extensions of the Local Flatness Criterion over a Noetherian local ring $R$ with residue field $k$. For instance, if $M$ is a complete $R$-module of finite projective dimension, then $M$ is flat if and only if $Tor^R_n(M,k)=0$ for all $n=1,...,depth(R)$. In low dimensions, we have the following criteria. If $R$ is one-dimensional and reduced, then $M$ is flat if and only if $Tor^R_1(M,k)=0$. If $R$ is two-dimensional, then in order for $M$ to be flat, it suffices that it is separated, that its projective dimension is finite and that $Tor^R_1(M,k)=0$. Many of these criteria have global counterparts and in particular, it is shown that the $I$-adic completion of a flat module of finite projective dimension over an arbitrary Noetherian ring is again flat.","articles.html",""); subInfo[31][1] = new Array("Let $R$ be a mixed characteristic Artinian local ring of length $l$ and let $X$ be an $n$-tuple of variables. We prove that several algebraic constructions in the ring $ R[X]$ admit uniform bounds on the degrees of their output in terms of $l$, $n$ and the degrees of the input. For instance, if $I$ is an ideal in $R[X]$ generated by polynomials $g_i$ of degree at most $d$ and if $f$ is a polynomial of degree at most $d$ belonging to $I$, then $f=q_1f_1+\dots+q_sf_s$, with $q_i$ of degree bounded in terms of $d$, $l$ and $n$ only. Similarly, the module of syzygies of $I$ is generated by tuples all of whose entries have degree bounded in terms of $d$, $l$ and $n$ only.","articles.html",""); subInfo[32][1] = new Array("We associate to every equicharacteristic zero Noetherian local ring $R$ a faithfully flat ring extension which is an ultraproduct of rings of various prime characteristics, in a weakly functorial way. Since such ultraproducts carry naturally a non-standard Frobenius, we can define a new tight closure operation on $R$ by mimicking the positive characteristic functional definition of tight closure. This approach avoids the use of generalized Neron Desingularization and only relies on Rotthaus' result on Artin Approximation in characteristics zero. If $R$ is moreover equidimensional and universally catenary, then we can also associate to it in a canonical, weakly functorial way a balanced big Cohen-Macaulay algebra","articles.html",""); subInfo[33][1] = new Array("In this paper, a local invariant is a map $\omega$ which assigns to a local ring $R$ a natural number $\omega(R)$. It induces on any scheme $X$ a partition given by the sets consisting of all points $x$ of $X$ for which $\omega(\mathcal O_{X,x})$ is constant. Criteria are given for this partition to be constructible, in case $X$ is a scheme of finite type over a field. It follows that if the partition is constructible, then it is finite, so that the invariant takes only finitely many different values on $X$. Examples of local invariants to which these results apply, are the regularity defect, the Cohen-Macaulay defect, the Gorenstein defect, the complete intersection defect, the Betti numbers and the (twisted) Bass numbers. As an application, we obtain that for an affine scheme $X$ of finite type over a field $K$, there is a number $\delta(X)$, such that for any $n$ and any closed immersion $X\subset\mathbb A_K^n$, we can realize $X$ as the scheme-theoretic intersection of $\delta(X)+n$ hypersurfaces. Moreover, this bound $\delta(X)$ is uniform in families.","articles.html",""); subInfo[34][1] = new Array("An overview of the author's work on the use of ultraproducts to lift characteristic $p$ methods, such as tight closure and absolute integral closure/big CM algebras, to characterisitic zero","articles.html",""); subInfo[35][1] = new Array("Abstract","articles.html",""); subInfo[36][1] = new Array("Abstract","articles.html",""); subInfo[37][1] = new Array("Abstract","articles.html",""); subInfo[38][1] = new Array("Abstract","articles.html",""); subInfo[39][1] = new Array("Abstract","articles.html",""); subInfo[40][1] = new Array("Abstract","articles.html",""); subInfo[41][1] = new Array("Abstract","articles.html",""); //*** SET SUB MENU POSITION ( RELATIVE TO BUTTON ) ***// var xSubOffset = 64; var ySubOffset = 22; //*** NO MORE SETTINGS BEYOND THIS POINT ***// var overSub = false; var delay = 10; totalButtons = upSources.length; // GENERATE SUB MENUS for ( x=0; x'); // SET DIV FOR BUTTONS WITH SUBMENU } else { document.write(''); } //*** MAIN BUTTONS FUNCTIONS ***// // PRELOAD MAIN MENU BUTTON IMAGES function preload() { for ( x=0; x