Publications

A group of Gerry Schwarz’s colleagues and collaborators gathered at the Fields Institute in Toronto for a mathematical festschrift in honor of his 60th birthday. This volume is an outgrowth of that event, covering the wide range of mathematics to which Gerry Schwarz has either made fundamental contributions or stimulated others to pursue. The articles are a sampling of modern day algebraic geometry with associated group actions from its leading experts, with a particular focus on characteristic 0 and modular invariant theory.

This article studies the sequence of iterative degrees of a birational map of the plane. This sequence is known either to be bounded or to have a linear, quadratic or exponential growth. 
The classification elements of infinite order with a bounded sequence of degrees is achieved, the case of elements of finite order being already known. The coefficients of the linear and quadratic growth are then described, and related to geometrical properties of the map. The dynamical number of base-points is also studied. 
Applications of our results are the description of embeddings of the Baumslag-Solitar groups and GL(2,Q) into the Cremona group.

The present paper is concerned with the numerical solutionof a shape identification problem for the heat equation. The goal is to determine of the shape of a void or inclusion of zero temperature from measurements of the temperature and the heat flux at the exterior boundary. This nonlinear and ill-posed shape identification problem is reformulated in terms of three different shape optimization problems: (a) minimization of a least-squares energy variational functional,(b) tracking of the Dirichlet data, and (c) tracking of the Neumann data. The states and their adjoint equations are expressed as parabolic boundary integral equations and solved using a Nyström discretization and a space-time fast multipole method for the rapid evaluation of thermal potentials. Special quadrature rules are derived to handle singularities of the kernel and the solution. Numerical experiments are carried out to demonstrate and compare the different formulations.

We study the algebraic structure of the n-dimensional Cremona

group and show that it is not an algebraic group of infinite dimension (indgroup)

if n>1. We describe the obstruction to this, which is of a topological

nature.

By contrast, we show the existence of a Euclidean topology on the Cremona

group which extends that of its classical subgroups and makes it a topological

group.

We give a way to construct group of pseudo-automorphisms of

rational varieties of any dimension that fix pointwise the image of a cubic

hypersurface of p^n. These group are free products of involutions, and most of

their elements have dynamical degree > 1. Moreover, the Picard group of the

varieties obtained is not big, if the dimension is at least 3.

We also answer a question of E. Bedford on the existence of birational maps

of the plane that cannot be lifted to automorphisms of dynamical degree > 1,

even if we compose them with an automorphism of the plane.

In this paper, we give a sufficient condition for a set G of polynomials to be a Groebner basis with respect to a given term-order for the ideal I that it generates. Our criterion depends on the linkage pattern of the ideal I and of the ideal generated by the initial terms of the elements of G. We then apply this criterion to ideals generated by minors and pfaffians. More precisely, we consider large families of ideals generated by minors or pfaffians in a matrix or a ladder, where the size of the minors or pfaffians is allowed to vary in different regions of the matrix or the ladder. We use the sufficient condition that we established to prove that the minors or pfaffians form a reduced Groebner basis for the ideal that they generate, with respect to any diagonal or anti-diagonal term-order. We also show that the corresponding initial ideal is Cohen-Macaulay. Our proof relies on known results in liaison theory, combined with a simple Hilbert function computation. In particular, our arguments are completely algebraic.

In Rⁿ, n⩾2, we study the constructive and numerical solution of minimizing the energy relative to the Riesz kernel , where 1<α<n, for the Gauss variational problem, considered for finitely many compact, mutually disjoint, boundaryless (n−1)-dimensional C^∞-manifolds Γ_ℓ, ℓ∈L, each Γ_ℓ being charged with Borel measures with the sign α_ℓ≔±1 prescribed. We show that the Gauss variational problem over an affine cone of Borel measures can alternatively be formulated as a minimum problem over an affine cone of surface distributions belonging to the Sobolev–Slobodetski space H^−ε/2(Γ), where ε≔α−1 and . This allows the application of simple layer boundary integral operators on Γ and, hence, a penalty approximation. A corresponding numerical method is based on the Galerkin–Bubnov discretization with piecewise constant boundary elements. Wavelet matrix compression is applied to sparsify the system matrix. To the discretized problem, a gradient-projection method is applied. Numerical results are presented to illustrate the approach.

In this short note, we propose a unified approach, based on the variation-of-constants formula, for the numerical discretisation over long-time intervals of several stochastic oscillators. The proposed schemes permit the use of large step sizes and offer various geometric properties. These new numerical integrators can be viewed as a stochastic-generalisation of the trigonometric integrators for highly oscillatory deterministic problems.

We give a simple set of generators and relations for the Cre-

mona group of the plane. Namely, we show that the Cremona group is the

amalgamated product of the de Jonquieres group with the group of auto-

morphisms of the plane, divided by one relation .

We characterise smooth curves in P3 whose blow-up produces

a threefold with anticanonical divisor big and nef. These are

curves C of degree d and genus g lying on a smooth quartic, such that

(i) 4d 􀀀 30  g  14 or (g; d) = (19; 12), (ii) there is no 5-secant line,

9-secant conic, nor 13-secant twisted cubic to C. This generalises the

classical similar situation for the blow-up of points in P2.

We describe then Sarkisov links constructed from these blow-ups, and

are able to prove the existence of Sarkisov links which were previously

only known as numerical possibilities.

We develop technics of birational geometry to study automorphisms of affine surfaces

admitting many distinct rational fibrations, with a particular focus on the interactions between automorphisms

and these fibrations. In particular, we associate to each surface S of this type a graph

encoding equivalence classes of rational fibrations from which it is possible to decide for instance if the

automorphism group of S is generated by automorphisms preserving these fibrations.

We give the classication of elements { respectively cyclic subgroups {

of nite order of the Cremona group, up to conjugation. Natural parametri-

sations of conjugacy classes, related to xed curves of positive genus, are

provided.

In this article we study the transitivity of the group of automorphisms of real algebraic sur- faces. We characterize real algebraic surfaces with very transitive automorphism groups. We give applications to the classification of real algebraic models of compact surfaces: these applications yield new insight into the geometry of the real locus, proving several surprising facts on this geometry. This geometry can be thought of as a half-way point between the biregular and birational geometries.

The algebra of basic covers of a graph G, denoted by ¯A(G), was introduced by Jürgen
Herzog as a suitable quotient of the vertex cover algebra. In this paper we show that if
the graph is bipartite then ¯A(G) is a homogeneous algebra with straightening laws and
thus is Koszul. Furthermore, we compute the Krull dimension of ¯A(G) in terms of the
combinatorics of G. As a consequence we get new upper bounds on the arithmetical rank
of monomial ideals of pure codimension 2. Finally, we characterize the Cohen-Macaulay
property and the Castelnuovo-Mumford regularity of the edge ideal of a certain class of
graphs.

The present paper is dedicated to the analytical computation of shape derivatives in the polarizable continuum model. We derive expressions for the interaction energy's sensitivity withrespect to variations of the cavity's shape by means of the Hadamard representation of the shape gradient. In particular, by using the adjoint approach, the shape gradient depends only on two solutions of the underlying electrostatic problem. We further formulate boundary integral equations to compute the involved quantities.

Locally periodic, elliptic multiscale problems in a bounded Lipschitz domain D⊂Rⁿ with K ⩾ 2 separated scales are reduced to an elliptic system of K coupled, anisotropic elliptic one-scale problems in a cartesian product domain of total dimension Kn (e.g. [2], [3], [11] and [26]). In [23] and [31], it has been shown how these coupled elliptic problems could be solved by sparse tensor wavelet Finite Element Methods in log-linear complexity with respect to the number N of degrees of freedom required by multilevel solvers for elliptic one-scale problems in D with the same convergence rate. In the present paper, the high dimensional one-scale limiting problems are discretized by a sparse tensor product Finite Element Method (FEM) with standard, one-scale FE basis functions as used in engineering FE codes. Sparse tensorization and multilevel preconditioning is achieved by a BPX multilevel iteration.

We show that the resulting sparse tensor multilevel FEM resolves all physical length scales throughout the domain, with efficiency (i.e., accuracy versus work and memory) comparable to that of multigrid solvers for elliptic one-scale problems in the physical domain D. In particular, our sparse tensor FEM gives numerical approximations of the correct homogenized limit as well as compressed numerical representations of all first order correctors, throughout the physical domain with performance independent of the physical problem’s scale parameters. Numerical examples with standard FE shape functions and BPX multilevel preconditioners for elliptic problems with K = 2 separated, physical scales in spatial dimension n = 2 confirm the theoretical results. In particular, the present approach allows to avoid the construction of wavelet FE bases necessary in previous work [23] and [31] while achieving resolution of all scales throughout the physical domain in log-linear complexity, with the logarithmic exponent behaving linearly in the number K of scales.

The present paper is concerned with the rapid solution of a boundary integral equation for the apparent surface charge which arises from solvation continuum models. In order to apply the wavelet Galerkin scheme the molecular surface needs to be represented as a parametric surface consisting of smooth four-sided patches. We develop an algorithmwhich decomposes a solvent excluded surface into a set of globally continuous four-sided NURBS patches. Numerical experiments are carried out to demonstrate the feasibility and scope of the present approach.

The Cremona group is connected in any dimension and, endowed

with its topology, it is simple in dimension 2.

Given a finite Abelian subgroup of the Cremona group of the plane, we

provide a way to decide whether it is birationally conjugate to a group of

automorphisms of a minimal surface.

In particular, we prove that a finite cyclic group of birational transformations

of the plane is linearisable if and only if none of its non-trivial elements

fix a curve of positive genus. For finite Abelian groups, there exists only one

surprising exception, a group isomorphic to Z/2Z xZ/4Z, whose non-trivial

elements do not fix a curve of positive genus but which is not conjugate to

a group of automorphisms of a minimal rational surface.

We also give some descriptions of automorphisms (not necessarily of finite

order) of del Pezzo surfaces and conic bundles.

This article deals with the study of the birational transformations of the projective complex plane which leave invariant an irreducible algebraic curve. We try to describe the state of the art and provide some new results on this subject.

The paper starts with the following simple observation. Let V be a representation of a reductive group G, and let f₁,f₂,...,f_n be homogeneous invariant functions. Then the polarizations of f₁,f₂,...,f_n define the nullcone of k ≤ m copies of V if and only if every linear subspace L of the nullcone of V of dimension ≤ m is annhilated by a one-parameter subgroup (shortly a 1-PSG). (This means that there is a group homomorphism h: C* -> G such that lim_{t->0} h(t) x = 0 for all x in L.)

This is then applied to many examples. A surprising result is about the group SL(2,C) where almost all representations V have the property that all linear subspaces of the nullcone are annihilated. Again, this has interesting applications to the invariants on several copies. Another result concerns the n-qubits which appear in quantum computing. This is the representation of a product of n copies of SL(2,C) on the n-fold tensor product C² otimes C²otimes ... otimes C². Here we show just the opposite, namely that the polarizations never define the nullcone of several copies if n ≥ 3.

(An earlier version of this paper, distributed in 2002, was split into two parts; the first part with the title ``On the nullcone of representations of reductive groups'' is published in Pacific J. Math. 224 (2006), 119-140.)

We give a complete classification of maximal algebraic subgroups of the Cremona group Bir(P2) and provide algebraic varieties that parametrize the conjugacy classes.

For classes of symplectic and symmetric time-stepping methods- trigonometric integrators and the Stormer-Verlet or leapfrog method-applied to spectral semi-discretizations of semilinear wave equations in a weakly non-linear setting, it is shown that energy, momentum, and all harmonic actions are approximately preserved over long times. For the case of interest where the CFL number is not a small parameter, such results are outside the reach of standard backward error analysis. Here, they are instead obtained via a modulated Fourier expansion in time.

The authors T.Harima, J.C.Migliore, U.Nagel and J.Watanabe characterize
the Hilbert function of algebras with the Lefschetz property.
We extend this characterization to algebras with the Lefschetz
property m times. We also give upper bounds for the Betti numbers
of Artinian algebras with a given Hilbert function and with the Lefschetz
property m times and describe the cases in which these bounds
are reached

A modulated Fourier expansion in time is used to show long-time near-conservation of the harmonic actions associated with spatial Fourier modes along the solutions of nonlinear wave equations with small initial data. The result implies the long-time near-preservation of the Sobolev-type norm that specifies the smallness condition on the initial data.

We study the group of birational transformations of the plane that

fix (each point of) a curve of geometric genus 1.

A precise description of the finite elements is given; it is shown in particular

that the order is at most 6, and that if the group contains a non-trivial torsion,

the fixed curve is the image of a smooth cubic by a birational transformation

of the plane.

We show that for a smooth cubic, the group is generated by its elements

of degree 3, and prove that it contains a free product of Z/2Z, indexed by the

points of the curve.

We continue the study of G. Castelnuovo on the group of birational

transformation of the complex plane that fix each point of a curve of genus

> 1 ; we use adjoint linear system of the curve as Castelnuovo does.

We prove that these groups are abelian, and that these are either finite, of

order 2 or 3, or conjuguate to a subgroup of the de Jonqui`eres group. We show

also that these results do not generalise to curves of genus  1.

Keywords. Cremona transformations, birational transformations, fixed curves,

curves of big genus, adjoint linear system, de Jonqui`eres transformations.

Amorosointroduced a new class of number fields, the so-called class of $ PCM $-fields (proche d'un corps $CM$), having interesting arithmetic proprierties.

The main result of our work is the proof of the fact that no totally real field is $PCM$.

Moreover we show how our proof, in the special case of finite totally real abelian extensions of $mathbb{Q}$, implies a lower bound for the number of non-zero coefficients of polynomials whose certain zeros are roots of $1$.

Let R be a ring. The Special Automorphism Group SAut(R)R[x(1),..., x(n)] is the set of all automorphisms with determinant of the Jacobian equal to 1. It is shown that the canonical map of SAut(R[t]) R[t][x(1),...,x(n)] to SAutR(m) R-m[x(1),...,x(n)] where R-m := R[t]/(t(m)) and Q subset of R is surjective. This result is used to study a particular case of the following question: if A is a subring of a ring B and f is an element of A vertical bar n vertical bar is a coordinate over B does it imply that f is a coordinate over A? It is shown that if A = R[t(m), t(m+1),...] subset of R[t] = B then the answer to this question is ``yes``. Also, a question on the Venereau polynomial is settled, which indicates another ``coordinate-like property`` of this polynomial. (c) 2006 Elsevier B.V. All rights reserved.

We describe the conjugacy classes of affine automorphisms in the group

Aut(n,K) (respectively Bir(Kn)) of automorphisms (respectively of birational

maps) of Kn. From this we deduce also the classification of conjugacy classes of

automorphisms of Pn in the Cremona group Bir(Kn).

We give necessary and sufficient conditions on a commutative ring with unity R so that the following holds: any polynomial f in R[x, y] such that R[x, y]/(f) similar or equal to R-[1] (a line for short) is a variable (= alpha(x) for some automorphism alpha:R[x, y] -> R[x, y]). The main ingredient here is the construction of new bad lines (lines which are not variables). (c) 2006 Elsevier Inc. All rights reserved.

We study the geometry of the nullcone N(V^k) for several copies of a representation V of a reductive group G and its behavior for different k. We show that for large k there is a certain "stability" with respect to the irreducible components. In the case of the so-called theta-representations, this can be made more precise by using the combinatorics of the weight system as a subset of the root system. All this finally allows to calculate explicitly and in detail a number of important examples, e.g. the cases of 3- and 4-qubits which play a fundamental role in quantum computing.

We study algebraic actions of the additive group C+ on an affine threefold X and prove a smoothness property for the quotient morphism X -> X//C+. Then, following Shulim Kaliman, we give a proof of the conjecture that every free C+ action on affine 3-space C^3 is a translation.

We prove that an automorphism of order 5 of the Del Pezzo surface S5 of degree 5 is conjugate through a

birational map S 99K P2 to a linear automorphism of P2. This completes the classification of conjugacy classes of

elements of prime order in the Cremona group.

The Abhyankar-Sathaye Problem asks wether any biregular embedding
of C^k into Cⁿ can be rectified. Here we study
this problem for the embeddings of C³ into C⁴ whose image X is
given in C⁴ by an equation p = f(x; y)u+g(x; y; z) = 0. Under certain additional assumptions we show that, indeed,
the polynomial p is a variable of the polynomial ring C^[4] = C[x; y; z; u] (i.e., a
coordinate of a polynomial automorphism of C⁴). This is an analog of a theorem
due to Sathaye (1976) which concerns the case of embeddings of C² into C³.
Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial
p as above, a criterion for when X is isomorphic to C³.

An algebra A over a field k is FCR if every finite dimensional representation of A is completely reducible and the intersection of the kernels of these representations is zero. We give a useful characterization of FCR-algebras and apply this to C*-algebras and to localizations. Moreover, we show that ``small'' products and sums of FCR-algebras are again FCR.

In this article, we show that the multilevel Monte Carlo method for elliptic stochastic partial differential equations can be interpreted as a sparse grid approximation.By using this interpretation, the method can straightforwardly be generalized to any given quadrature rule for high dimensional integrals like the quasi Monte Carlo method or the polynomial chaos approach. Besides the multilevel quadrature for approximatingthe solution's expectation, a simple and efficient modification of the approach is proposed to compute the stochastic solution's variance. Numerical results are provided to demonstrate and quantify the approach.

A finite element heterogeneous multiscale method (FE-HMM) is proposed for the time dependent wave equation with highly oscillatory, albeit not necessarily periodic, coefficients. It is based on a finite element discretization of an effective wave equation at the macro scale, whose a-priori unknown effective coefficients are computed “on the fly” on sampling domains within each macro finite element at the micro scale ε > 0. Since the sampling domains scale in size with ε, which corresponds to the finest scales in the possibly highly heterogeneous medium, the computational work is independent of ε. In [1], we proved optimal error estimates in the energy norm and the L²-norm with respect to the micro and macro scale mesh parameters, h and H, and also convergence to the homogenized solution as ε → 0.

The pole placement problem belongs to the classical problems of linear systems theory. It is often assumed that the ground field is the real numbers R or the complex numbers C. The major result over the complex numbers derived in 1981 by Brockett and Byrnes states that arbitrary static pole placement is possible for a generic set of m-inputs, p-outputs and McMillan degree n system as soon as mp>=n. Moreover the number of solutions in the situation mp=n is an intersection number first computed by Hermann Schubert in the 19th century. In this paper we show that the same result with slightly different proofs holds over any algebraically closed field.