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In their 2012 paper, Bobadilla and Koll\'ar studied topological conditions which guarantee that a proper map of complex algebraic varieties is a topological or differentiable fibration. They also asked whether a certain finiteness property on the relative covering space can imply that a proper map is a fibration. In this paper, we answer positively the integral homology version of their question in the case of abelian varieties, and the rational homology version in the case of compact ball quotients. We also propose several conjectures in relation to the Singer-Hopf conjecture in the complex projective setting., Comment: published/final version

Author(s): Bamler, Richard H; Kleiner, Bruce | Abstract: In a recent paper, Brendle showed the uniqueness of the Bryant soliton among 3-dimensional $\kappa$-solutions. In this paper, we present an alternative proof for this fact and show that compact $\kappa$-solutions are rotational symmetric. Our proof arose from independent work relating to our Strong Stability Theorem for singular Ricci flows.

In 1980, Onishchik introduced the index of a Riemannian symmetric space as the minimal codimension of a (proper) totally geodesic submanifold. He calculated the index for symmetric spaces of rank less than or equal to 2, but for higher rank it was unclear how to tackle the problem. In earlier papers we developed several approaches to this problem, which allowed us to calculate the index for many symmetric spaces. Our systematic approach led to a conjecture for how to calculate the index. The purpose of this paper is to verify the conjecture., 33 pages; Table 1 corrected; to appear in Journal fuer die Reine und Angewandte Mathematik

As a continuation of a previous paper, we prove Perelman’s assertion, that is, for ancient solutions to the Ricci flow with bounded nonnegative curvature operator, uniformly bounded entropy is equivalent to κ-noncollapsing on all scales. We also establish an equality between the asymptotic entropy and the asymptotic reduced volume, which is a result similar to a paper by Xu (2017), where he assumes the Type I curvature bound.

The main result of this paper is the statement that the Hodge theoretic Donaldson–Thomas invariant for a quiver with zero potential and a generic stability condition agrees with the compactly supported intersection cohomology of the closure of the stable locus inside the associated coarse moduli space of semistable quiver representations. In fact, we prove an even stronger result relating the Donaldson–Thomas “function” to the intersection complex. The proof of our main result relies on a relative version of the integrality conjecture in Donaldson–Thomas theory. This will be the topic of the second part of the paper, where the relative integrality conjecture will be proven in the motivic context.

This is the second in a series of papers ([7] and [6] are the others) that studies the behavior of harmonic maps into the Weil–Petersson completion 𝒯 ¯ {\overline{\mathcal{T}}} of Teichmüller space. The boundary of 𝒯 ¯ {\overline{\mathcal{T}}} is stratified by lower-dimensional Teichmüller spaces and the normal space to each stratum is a product of copies of a singular space 𝐇 ¯ {\overline{\bf H}} called the model space. The significance of 𝐇 ¯ {\overline{\bf H}} is that it captures the singular behavior of the Weil–Petersson geometry of 𝒯 ¯ {\overline{\mathcal{T}}} . The main result of the paper is that certain subsets of 𝐇 ¯ {\overline{\bf H}} are essentially regular in the sense that harmonic maps to those spaces admit uniform approximation by affine functions. This is a modified version of the notion of essential regularity introduced by Gromov–Schoen in [12] for maps into Euclidean buildings and is one of the key ingredients in proving superrigidity. In the process, we introduce new coordinates on 𝐇 ¯ {\overline{\bf H}} and estimate the metric and its derivatives with respect to the new coordinates. These results form the technical core for studying the analytic behavior of harmonic maps into the completion of Teichmüller space and are utilized in our subsequent paper [6], where we prove the holomorphic rigidity of the Teichmüller space and several rigidity results for the mapping class group.

The classification of subfactors of small index revealed several new subfactors. The first subfactor above index 4, the Haagerup subfactor, is increasingly well understood and appears to lie in a (discrete) infinite family of subfactors where the ℤ \mathbb{Z} /3 ℤ \mathbb{Z} symmetry is replaced by other finite abelian groups. The goal of this paper is to give a similarly good description of the Asaeda–Haagerup subfactor which emerged from our study of its Brauer–Picard groupoid. More specifically, we construct a new subfactor 𝒮 {\mathcal{S}} which is a ℤ \mathbb{Z} /4 ℤ \mathbb{Z} × \times ℤ \mathbb{Z} /2 ℤ \mathbb{Z} analogue of the Haagerup subfactor and we show that the even parts of the Asaeda–Haagerup subfactor are higher Morita equivalent to an orbifold quotient of 𝒮 {\mathcal{S}} . This gives a new construction of the Asaeda–Haagerup subfactor which is much more symmetric and easier to work with than the original construction. As a consequence, we can settle many open questions about the Asaeda–Haagerup subfactor: calculating its Drinfeld center, classifying all extensions of the Asaeda–Haagerup fusion categories, finding the full higher Morita equivalence class of the Asaeda–Haagerup fusion categories, and finding intermediate subfactor lattices for subfactors coming from the Asaeda–Haagerup categories. The details of the applications will be given in subsequent papers.

We define a “period-ring-valued beta function” and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark’s conjecture implies that the exponentials of the derivatives at s = 0 s=0 of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler’s formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative proof of a weaker result by using the reciprocity law on the period-ring-valued beta function. In other words, the reciprocity law given in this paper is a refinement of the reciprocity law on cyclotomic units.

In hyperbolic 3-space $\mathbb{H}^3$ surfaces of constant mean curvature $H$ come in three types, corresponding to the cases $0 \leq H < 1$, $H = 1$, $H > 1$. Via the Lawson correspondence the latter two cases correspond to constant mean curvature surfaces in Euclidean 3-space $\mathbb{E}^3$ with H=0 and $H \neq 0$, respectively. These surface classes have been investigated intensively in the literature. For the case $0 \leq H < 1$ there is no Lawson correspondence in Euclidean space and there are relatively few publications. Examples have been difficult to construct. In this paper we present a generalized Weierstra{\ss} type representation for surfaces of constant mean curvature in $\mathbb{H}^3$ with particular emphasis on the case of mean curvature $0\leq H < 1$. In particular, the generalized Weierstra{\ss} type representation presented in this paper enables us to construct simultaneously minimal surfaces (H=0) and non-minimal constant mean curvature surfaces ($0, Comment: 37 pages, 4 figures. v3: Various typos fixed. v4: Proposition D.1 has been fixed

Following [`Gaussian densities and stability for some Ricci solitons', preprint 2004], in this paper we study the linear stability of Perelman's ν-entropy on Einstein manifolds with positive Ricci curvature. We observe the equivalence between the linear stability (also called ν-stability in this paper) restricted to the transversal traceless symmetric 2-tensors and the stability of Einstein manifolds with respect to the Hilbert action. As a main application, we give a full classification of linear stability of the ν-entropy on symmetric spaces of compact type. In particular, we exhibit many more ν-stable and ν-unstable examples than previously known and also the first ν-stable examples, other than the standard spheres, whose second variations are negative definite.

Suppose that W is a finite, unitary reflection group acting on the complex vector space V. Let 𝒜 = 𝒜(W) be the associated hyperplane arrangement of W. Terao [J. Fac. Sci. Univ. Tokyo 27 (1980), 293–320] has shown that each such reflection arrangement 𝒜 is free. There is the stronger notion of an inductively free arrangement. In 1992, Orlik and Terao [Arrangements of hyperplanes, Springer-Verlag, Berlin 1992, Conjecture 6.91] conjectured that each reflection arrangement is inductively free. It has been known for quite some time that the braid arrangement as well as the Coxeter arrangements of type B ℓ and type D ℓ are inductively free. Barakat and Cuntz [Adv. Math. 229 (2012), 691–709] completed this list only recently by showing that every Coxeter arrangement is inductively free. Nevertheless, Orlik and Terao's conjecture is false in general. In a paper which will appear in Tôhoku Math. J., we already gave two counterexamples to this conjecture among the exceptional complex reflection groups. In this paper we classify all inductively free reflection arrangements. In addition, we show that the notions of inductive freeness and that of hereditary inductive freeness coincide for reflection arrangements. As a consequence of our classification, we get an easy, purely combinatorial characterization of inductively free reflection arrangements 𝒜 in terms of exponents of the restrictions to any hyperplane of 𝒜.

The first goal of this paper is to study the question of finiteness of integral points on a cofinite non-compact complex two-dimensional ball quotient defined over a number field. Along the process we will also consider some negatively curved compact surfaces. Using some fundamental results of Faltings, the question is to reduce to a conjecture of Borel about existence of virtual holomorphic one-forms on cofinite non-cocompact complex ball quotients. The second goal of this paper is to study the conjecture on such non-compact surfaces.

In this paper we construct explicitly a square integrable residual automorphic representation of the special orthogonal group $SO_{2n}$, through Eisenstein series. We show that this representation comes from an elliptic Arthur parameter $\psi$ and appears in the space $L^2(SO_{2n}(\mathbb{Q})\backslash SO_{2n}(\mathbb{A}_{\mathbb{Q}}))$ with multiplicity one. Next, we consider parameters whose Hecke matrices, at the unramified places, have eigenvalues bigger (in absolute value), than those of the parameter constructed before. The main result is, that these parameters cannot be cuspidal. We establish bounds for the eigenvalues of Hecke operators, as consequences of Arthur's conjectures for $SO_{2n}$. Next, we calculate the character and the twisted characters for the representations that we constructed. Finally, we find the composition of the global and local Arthur's packets associated to our parameter $\psi$. All the results in this paper are true if we replace $\mathbb{Q}$ by any number field $F$., Comment: 39 pages

For each positive integer k, we investigate the L-function attached to the k-th symmetric power of the F-crystal associated to the family of cubic exponential sums of x 3 + ‚x where ‚ runs over Fp. We explore its rationality, fleld of deflnition, degree, trivial factors, functional equation, and Newton polygon. The paper is essentially self-contained, due to the remarkable and attractive nature of Dwork’s p-adic theory. A novel feature of this paper is an extension of Dwork’s efiective decomposition theory when k < p. This allows for explicit computations in the associated p-adic cohomology. In particular, the action of Frobenius on the (primitive) cohomology spaces may be explicitly studied.

The first part of this paper is a generalization of the Feix-Kaledin theorem on the existence of a hyperkahler metric on a neighbourhood of the zero section of the cotangent bundle of a Kahler manifold. We show that the problem of constructing a hyperkahler structure on a neighbourhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix-Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kahler type has a hyperkahler structure on a neighbourhood of its identity section. More generally, we reduce the existence of a hyperkahler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin's unobstructedness theorem., 20 pages. To appear in Journal fur die reine und angewandte Mathematik (Crelle's Journal)

Classical Serre-Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips the base of its universal deformation with a Frobenius lifting and canonical multiplicative coordinates. A variant of this theory has been obtained for ordinary K3 surfaces by Nygaard and Ogus., In this paper, we construct canonical liftings modulo p(2) of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Frobenius lifting on the moduli space of such varieties. The quite explicit construction uses Frobenius splittings and a relative version of Witt vectors of length two. If the variety has unobstructed deformations and bijective first higher Hasse-Witt operation, the Frobenius lifting gives rise to canonical coordinates. One of the key features of our liftings is that the crystalline Frobenius preserves the Hodge filtration., We also extend Nygaard's approach from K3 surfaces to higher dimensions, and show that no non-trivial families of such varieties exist over simply connected bases with no global one-forms.

The local Langlands correspondence is a conjectural connection between representations of groups G(k) for connected reductive groups G over a padic field k and certain homomorphisms (Langlands parameters) from the Galois (or Weil-Deligne group) of k into a complex Lie group G which is dual, in a certain sense, to G and which encodes the splitting structure of G over k. More introductory remarks on the local Langlands correspondence can be found in [21]. WhenG = GL1 this correspondence should reduce to local abelian class field theory. For G = GLn, the Langlands correspondence is uniquely determined by local factors [24] and was shown to exist in [23] and [25]. So far this correspondence is not completely explicit, but much progress has been made in this direction; see [9], [10], for example. For groups other than GLn or PGLn, the theory is much less advanced; new phenomena appear, arising on the arithmetic side from the difference between conjugacy and stable conjugacy and on the dual side from nontrivial monodromy of Langlands parameters. This means that a single Langlands parameter φ should determine not just one, but a finite set of representations Π(φ); these are the “L-packets” of the title. However, since local factors have not been defined in general, there is no precise characterization of an L-packet for general groups. One can, at present, only hope to define finite sets of representations Π(φ) attached to Langlands parameters φ, and show that they have properties expected (or perhaps unexpected) of L-packets. (See [14, chap. 3] for some of these properties.) One is thereby proposing a definition of local factors for the representations in the sets Π(φ) (cf. [4, chap.3]). This paper is a sequel to [14]. The aim of both papers is to verify, in an explicit and natural way, the local Langlands correspondence for the simplest kinds of non-abelian extensions of k, and the simplest kinds of

Let k be a field of characteristic zero, V a smooth, positive-dimensional, quasiprojective variety over k, and D a nonempty effective divisor on V. Let K be the function field of V, and A the semilocal ring of D in K. In this paper, we prove the Diophantine undecidability of: (1) A, in all cases; (2) K, when k is (formally) real and V has a real point; (3) K, when k is a subfield of a p-adic field, for some odd prime p. To achieve this, we use Denef's method: from an elliptic curve E over Q, without complex multiplication, one constructs a quadratic twist E' of E over Q(t), which has Mordell-Weil rank one. Most of the paper is devoted to proving (using a theorem of R. Noot) that one can choose f in K, vanishing at D, such that the group E'(K) deduced from the field extension K/Q(f)=Q(t) is equal to E'(Q(t)). Then we mimic the arguments of Denef (for the real case) and of Kim and Roush (for the p-adic case).

The aim of this paper is to introduce a notion of L-homology in the context of von Neumann algebras. Finding a suitable (co)homology theory for von Neumann algebras has been a dream for several generations (see [KR71a, KR71b, JKR72, SS95] and references therein). One’s hope is to have a powerful invariant to distinguish von Neumann algebras. Unfortunately, little positive is known about the Kadison-Ringrose cohomology H∗ b (M,M), except that it vanishes in many cases. Furthermore, there does not seem to be a good connection between the bounded cohomology theory of a group and of the bounded cohomology of its von Neumann algebra. Our interest in developing an L-cohomology theory was revived by the introduction of Lcohomology invariants in the field of ergodic equivalence relations in the paper of Gaboriau [Gab02]. His results in particular imply that L-Betti numbers β (2) i (Γ) of a discrete group are the same for measure-equivalent groups (i.e., for groups that can generate isomorphic ergodic measure-preserving equivalence relations). Parallels between the “worlds” of von Neumann algebras and measurable equivalence relations have been noted for a long time (starting with the parallel between the work of Murray and von Neumann [MvN] and that of H. Dye [Dy]). Thus there is hope that an invariant of a group that “survives” measure equivalence will survive also “von Neumann algebra equivalence”, i.e., will be an invariant of the von Neumann algebra of the group. The original motivation for our construction comes from the well understood analogy between the theory of II1-factors and that of discrete groups, based on the theory of correspondences [Con, Con94]. This analogy has been remarkably efficient to transpose analytic properties such as “amenability” or “property T” from the group context to the factor context [Con80] [CJ] and more recently in the breakthrough work of Popa [Popa] [Con03]. We use the theory of correspondences together with the algebraic description of L-Betti numbers given by Luck. His definition involves the computation of the algebraic group homology with coefficients in the group von Neumann algebra. Following the guiding idea that the category of bimodules over a von Neumann algebra is the analogue of the category of modules over a group, we are led to the following algebraic definition of L-homology of a von Neumann algebra M

For A|F a central simple algebra over a p-adic local field the group of units A× ∼= GLm(Dd) is a general linear group over a central division algebra Dd|F of index d. The product n = dm being fixed, the Abstract Matching Theorem (AMT) implies the existence of bijective maps JL between the sets of discrete series representations of the groups A× such that a character relation is preserved. In this paper we construct maximal level zero extended type components for every level zero discrete series representation of A×. Its maximal level zero extended type determines the discrete series representation uniquely (without any twist ambiguities as for the usual types) and, conversely, every level zero discrete series representation Π contains a maximal level zero extended type component Σ(Π) which is unique up to conjugacy. In order to determine how JL matches the extended types we find certain regular elliptic elements where the characters of Σ(Π) and Π are the same and we compute the character values at these elements by using a version of Shintani descent which we develop in Appendix B. Surprisingly, we find that AMT also implies explicit Shintani descent for irreducible characters of finite general linear groups which have cuspidal descent. named author Let F be a locally compact p-adic field, let Dd be a central division algebra of index d over F , and let A = Mm(Dd), the algebra of all m × m matrices with entries in Dd. Then A is a central simple F algebra of reduced degree n := dm and the group of units A× of A is the group of F points of a connected reductive algebraic group defined over F . The group A× is separable, totally disconnected, and unimodular and the harmonic analysis of A×, as is well known, is linked to numerous problems in arithmetic. It is interesting, and perhaps may be regarded as a version of the Abstract Matching Theorem (AMT) (see §0.2), that the representation theory of A× depends upon the parameters d,m associated to A, but not more specifically on the isomorphism class of A. We follow the convention of writing Dd for any central F division algebra of index d; we also write Dn for any one of the φ(n) isomorphism class representatives of central F division algebras of index n. 1991 Mathematics Subject Classification. 22E50. 1The work reported in this paper grew out of a “research in pairs” sojourn at the Mathematisches Institut, Oberwolfach. The authors wish to express their appreciation to the Institut for its support of their research as well as for an enjoyable stay at the Institut.

We give a full classification of all braided semisimple tensor categories whose Grothendieck semiring is the one of Rep(O(\infty) (formally), Rep(O(N), Rep(Sp(N) or of one of its associated fusion categories. If the braiding is not symmetric, they are completely determined by the eigenvalues of a certain braiding morphism, and we determine precisely which values can occur in the various cases. If the category allows a symmetric braiding, it is essentially determined by the dimension of the object corresponding to the vector representation. Note that the paper is followed by a brief erratum, which corrects a mistake, which does not affect the main results of the paper.

This paper introduces a space of nonabelian modular symbols 𝒮 ⁢ ( V ) {{\mathcal{S}}(V)} attached to any hyperbolic Riemann surface V, and applies it to obtain new results on polygonal billiards and holomorphic 1-forms. In particular, it shows the scarring behavior of periodic trajectories for billiards in a regular polygon is governed by a countable set of measures homeomorphic to ω ω + 1 {\omega^{\omega}+1} .

The Skew Mean Curvature Flow (SMCF) is a Schrödinger-type geometric flow canonically defined on a co-dimension two submanifold, which generalizes the famous vortex filament equation in fluid dynamics. In this paper, we prove the local existence and uniqueness of general-dimensional SMCF in Euclidean spaces.

Let 𝕍 {{\mathbb{V}}} be a polarized variation of integral Hodge structure on a smooth complex quasi-projective variety S. In this paper, we show that the union of the non-factor special subvarieties for ( S , 𝕍 ) {(S,{\mathbb{V}})} , which are of Shimura type with dominant period maps, is a finite union of special subvarieties of S. This generalizes previous results of Clozel and Ullmo (2005) and Ullmo (2007) on the distribution of the non-factor (in particular, strongly) special subvarieties in a Shimura variety to the non-classical setting and also answers positively the geometric part of a conjecture of Klingler on the André–Oort conjecture for variations of Hodge structures.

When a hypersurface t evolves with normal velocity equal to its mean curvature plus a forcing term g x t the generalized viscosity solution may be fattened at some moment when t is singular This phenomenon cor responds to nonuniqueness of codimension one solutions A speci c type of geometric singularity occurs if t includes two smooth pieces at the moment t when the two pieces touch each other If each piece is strictly convex at that moment and at that point then we show that fattening occurs at the rate t That is for small positive time the generalized solution contains a ball of IR of radius ct but its complement meets a ball of a larger radius t In this sense the sharp rate of fattening of the generalized solution is characterized We assume that the smooth evolution of the two pieces of t considered separately do not cross each other for small positive time Introduction Consider the problem of a hypersurface t in IRn which ows in time with normal velocity given by its mean curvature plus perhaps a continuous forcing term g x t When singularities develop in this problem the smooth solutions may cease to exist and the weak solutions may become nonunique This has been observed in a number of recent papers see BSS and BP A weak solution as de ned by Brakke B in particular is not unique see I However uniqueness holds for the generalized solution de ned as follows A real valued function u on IRn t T is constructed so that at the initial time t u t is positive on one side of the oriented initial hypersurface t and negative on the other side u x t is then required to be continuous and to satisfy the degenerate parabolic partial di erential equation u t jruj div ru jruj g x t in the viscosity sense with the the initial condition u x t The signi cance of this equation is as follows see ES if all level hypersurfaces of u x t were smooth then each of the level sets f x t u x t g for various real values of would evolve with normal velocity equal to its mean curvature plus g x t The level set for is a closed subset of IRn which evolves uniquely in time and does not depend on the choice of the initial function u x t This solution is known as the generalized solution to the problem since it need not be smooth need not have Hausdor codimension one and may even have a nonempty interior as a subset of IRn The phenomenon of an initially smooth hypersurface which later develops a nonempty interior is known as fattening or ballooning This phenomenon occurs precisely when Brakke s weak solution is nonunique I In Belletini and Paolini BP worked out some interesting examples of fattening in IR which involved two circles meeting externally at a certain time t In Koo K extended the results of BP and showed that their examples were manifestations of a general principle valid for hypersurfaces in IRn evolving by mean curvature plus a forcing term which guaranteed that the generalized solution begins to have positive Lebesgue measure as soon as two components t of an immersed solution touch from the outside at time t without crossing each other immediately before or after the critical time t An examination of the proof in K shows that the size of the fat level set grows at least as fast as p t i e at the rate suggested by parabolic scaling In the present paper we shall show that in fact the lower bound c p t on growth of the fat level set may be replaced by the much faster growth ct Theorem below This improves the estimate of K Moreover this estimate is sharp In fact with the additional assumption of strict convexity at the touching point Theorem below shows that the region outside t and outside the fat level set is at a distance at most t from the touching point for a larger constant More precisely combining Theorems and below we have the Theorem Let t jtj T be two smooth oriented hypersurfaces of IRn which move with normal velocity V H g x t for some continuous forcing term g IRn IR Suppose t and t are disjoint for t but that they meet at one point x at time t at which each is strictly convex Then there are c and so that for all t the region outside t and outside the generalized solution t includes some points at distance t from x but does not intersect the ball Bct x It should be observed that fattening of a speci c level set cannot happen in most circumstances More precisely if t is a disjoint one parameter family of generalized solutions evolving according to the same function of curvature then fat tening does not occur for almost all In fact at any time t the set of real numbers f j u t has positive measureg has measure zero in IR by the additivity of Lebesgue measure This observation is consistent with Koo s principle since Koo s result only applies to the rst time fattening occurs and requires touching to occur from the outside Assuming the family t is real analytic as might follow from parabolicity the set of to which Koo s principle applies is discrete The intuition behind the distinction between K and the present paper may be understood in the following way Koo s proof relies on comparison with a self similar solution of the degenerate parabolic partial di erential equation v t jrvj div rv jrvj and the parabolic scaling x pt follows from parabolic self similarity However the spatial aspect of self similarity is homothetic scaling Homothetic scaling is adapted to manifold like geometries such as Euclidean space and more generally to cone like geometries In the problem considered by BSS BP and K however the region exterior to is not a cone but a sort of cusp The region rescales in a small neighborhood of the touching point to small neighborhoods of a hyperplane In particular the homothetic scaling of K occurs independently of this cusp geometry and in a certain sense replaces it by a cone This replacement of the region given in the problem by a larger and very di erent region might lead one to suspect that the c p t estimate cannot be sharp Thus as Theorem shows for the analysis of behavior inside a cusp self similar solutions are not enough We conjecture that if the strict convexity of in Theorem is replaced by contact of order m then the generalized solution t grows like ct m An interface which moves by mean curvature plus a forcing term is a simple al though perhaps suggestive model for solidi cation of isotropic materials It would be of interest to understand the phenomena discussed in the present paper and analogous phenomena in the context of a more realistic system of equations in corporating temperature as a dependent variable along with one or more order parameters of the material Anisotropic materials would also be of interest We would like to acknowledge valuable discussions with Perry Leo Walter Littman Stephan Luckhaus and Juan Vel asquez This work was supported by the Max Planck Institute for Mathematics in the Sciences Leipzig Level set formulation of hypersurface ow In this section the forcing term g x t will depend on t alone When applied to our main results g x t will be estimated above or below by a function g t For a function r of one space variable y and of time we write r y t r y y t For x IRn we will use the potentially confusing notation x x x IR IRn We trust that in context the reader will be able to distinguish this use of the notation x x xn for a point x x xn IRn from the notation for the space derivative r y t of a function r y t of two variables Proposition Let fra y t g be a one parameter family of viscosity subsolutions to ra t r a r a n ra g t p r a satisfying ara y t Choose a continuous locally monotone function IR IR and let a function v be de ned on IRn by

In the present paper, we generalize the celebrated classical lemma of Birch and Heegner on quadratic twists of elliptic curves over $\mathbb{Q}$. We prove the existence of explicit infinite families of quadratic twists with analytic ranks $0$ and $1$ for a large class of elliptic curves, and use Heegner points to explicitly construct rational points of infinite order on the twists of rank $1$. In addition, we show that these families of quadratic twists satisfy the $2$-part of the Birch and Swinnerton-Dyer conjecture when the original curve does. We also prove a new result in the direction of the Goldfeld conjecture., Comment: 22 pages, to appear in Crelle's Journal

The Plateau–Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior Hölder regularity of solutions. Our results generalize corresponding results of Jost and Tomi-Tromba from the setting of Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality. The special case of a disc-type surface spanning a single Jordan curve corresponds to the classical problem of Plateau, in proper metric spaces recently solved by Lytchak and the second author.

In this paper we prove a result on the effective generation of pluri-canonical linear systems on foliated surfaces of general type. Fix a function {P:\mathbb{Z}_{\geq 0}\to\mathbb{Z}} , then there exists an integer {N>0} such that if {(X,{\mathcal{F}})} is a canonical or nef model of a foliation of general type with Hilbert polynomial {\chi(X,{\mathcal{O}}_{X}(mK_{\mathcal{F}}))=P(m)} for all {m\in\mathbb{Z}_{\geq 0}} , then {|mK_{\mathcal{F}}|} defines a birational map for all {m\geq N} . On the way, we also prove a Grauert–Riemenschneider-type vanishing theorem for foliated surfaces with canonical singularities.

In this paper we obtain rigidity results for a non-constant entire solution u of the Allen–Cahn equation in {\mathbb{R}^{n}} , whose level set {\{u=0\}} is contained in a half-space. If {n\leq 3} , we prove that the solution must be one-dimensional. In dimension {n\geq 4} , we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations to one phase free boundary problems are also obtained.

In recent years, a number of papers have been devoted to the study of roots of period polynomials of modular forms. Here, we study cohomological analogues of the Eichler-Shimura period polynomials corresponding to higher $L$-derivatives. We state general conjectures about the locations of the roots of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative "period polynomials" in the case of cusp forms. We prove a special case of this conjecture in the case of Eisenstein series., 21 pages

The first part of the paper discusses a second-order quasilinear parabolic equation in a vector bundle over a compact manifold $M$ with boundary $\partial M$. We establish a short-time existence theorem for this equation. The second part of the paper is devoted to the investigation of the Ricci flow on $M$. We propose a new boundary condition for the flow and prove two short-time existence results., 18 pages

This paper studies stable sheaves on abelian surfaces of Picard number one. Our main tools are semi-homogeneous sheaves and Fourier-Mukai transforms. We introduce the notion of semi-homogeneous presentation and investigate the behavior of stable sheaves under Fourier-Mukai transforms. As a consequence, an affirmative proof is given to the conjecture proposed by Mukai in the 1980s. This paper also includes an explicit description of the birational correspondence between the moduli spaces of stable sheaves and the Hilbert schemes.

An equifocal submanifold M of a symmetric space N of compact type induces a foliation with singular leaves on N. In this paper we will show how to reconstruct the equifocal foliation starting from one of the singular leaves, the so-called focal manifolds. To be more concrete: The equifocal submanifold is equal to a partial tube B around the focal manifold and we will show how to construct B in this paper. Moreover, we will find a geometrical characterization of focal manifolds., Comment: 23 pages

that is exact everywhere except at F0, where the cokernel is M . Such a resolution always exists. If R is a noetherian N-graded ring, with R0 a field, and M is a finitely generated Z-graded R-module, then a resolution may be constructed in a minimal way in which each Fi is graded and each differential is homogeneous of degree 0. The rank of the ith free module in a minimal free resolution of M is an invariant of M , called the ith Betti number of M and denoted β i (M). Likewise, β R ij(M), the number of elements of degree j in a minimal set of homogeneous generators of Fi, is also an invariant of M , called the (i, j)th graded Betti number of M . In this paper we study modules with maximal graded Betti numbers. More precisely, consider a set Π consisting of pairs (R,M) where R is a noetherian N-graded ring, with R0 a field, and M is a finitely generated graded R-module. Is there some (R ,M ) ∈ Π such that β ′ ij (M ) ≥ β ij(M) for every i and j and every (R,M) ∈ Π? There are trivial examples in which the answer is yes (e.g., Π has only one element) or no (e.g., Π consists of all such pairs). Theorems 1 and 2 below give affirmative answers for certain sets Π defined by conditions on the Hilbert series and the depths of R and M . Theorem 3 allows one to compute the maximal graded Betti numbers for the pairs in the sets considered in Theorems 1 and 2. Throughout this paper, k denotes a field and Q the polynomial ring k[x1, . . . , xn] with the usual N-grading given by degxi = 1. The Hilbert series of a finitely generated graded Q-module M is HM (s) = ∑ dimk Mis . We define d-lexicographic ideals and submodules in Section 2. The Poincare series of an R-module M is P M (s; t) = ∑ β ij(M)s t. We write 4 for coefficient-wise inequality of Laurent series with coefficients in Z.

S = {x ∈ gp(S) | mx ∈ S for some m > 0}. One calls S normal if S = S. For simplicity we will often assume that gp(S) = Z; this is harmless because we can replace Z by gp(S) if necessary. The rank of S is the rank of gp(S). We will only be interested in the case in which S ∩ (−S) = 0; such affine semigroups will be called positive. The positivity of S is equivalent to the pointedness of the cone C(S) = R+S generated by S in R; one has rankS = dimC(S). (All the cones appearing in this paper have their apex at the origin.) The normality of S can now be characterized geometrically: one obviously has S = C(S) ∩ Z, and so S is normal if and only if S = C(S) ∩ Z. Conversely, every convex, pointed, finitely generated rational cone C ⊂ R yields a normal affine semigroup S(C) = C ∩ Z (this semigroup is finitely generated by Gordan’s lemma). If we simply speak of a cone C in the following, then it is always assumed that C is convex, pointed, finitely generated, and rational. Since dimC(S) = rankS, we often call the dimension of a cone its rank. Each positive affine semigroup S can be embedded into Z+ , m = rankS (one uses m linearly independent integral linear forms representing support hyperplanes of C(S)). Therefore every element can be written as a sum of irreducible elements, and the set of irreducible elements of S is finite since S is finitely generated. For a cone C the set of irreducible elements of S(C) is often called the Hilbert basis Hilb(C) of C in the combinatorial literature, a convention we will follow. More generally, we also call the set of irreducible elements of a positive affine semigroup S its Hilbert basis and denote it by Hilb(S). This paper is devoted to a discussion of sufficient and potentially necessary conditions for the normality of S in terms of combinatorial properties of Hilb(S). As far as the necessity is concerned, these conditions can of course be formulated in terms of Hilbert bases of rational cones. The first property under consideration is the existence of a unimodular (Hilbert) cover

has cardinality r ≥ 1. Let G be the F-rational points of a simple Chevalley group defined over F. In his thesis, Matsumoto [5] gave a beautiful construction for the metaplectic cover G of G, a central extension of G by μr(F) whose existence is intimately connected with the deep properties of the r-th order Hilbert symbol (·, ·)F : F × × F → μr(F). Metaplectic groups figure prominently in the study of number theory, representation theory, and physics, arising naturally in the theory of theta functions, dual pair correspondences, Weil representations, and spin geometry. In this paper we study the class of central extensions of a simple Chevalley group over an arbitrary infinite field, of which the metaplectic groups form an important subclass. Metaplectic groups were constructed quite explicitly in Weil’s memoir [10] in the case that G is symplectic. In [3] and [4], Kubota gave the construction of the r-fold metaplectic cover of GL2(F). Moreover, he described an explicit 2-cocycle σK on GL2(F) that represents the second cohomology class of the extension (cf. §3 Corollary 8), which makes it possible to deal quite rapidly with many concrete problems in this setting. Steinberg [9] and Moore [7] considered the algebraic problem of determining the central extensions of a simple Chevelley group over an arbitrary field; they were also led to the metaplectic groups. This line of investigation was completed by Matsumoto [5], whose work forms the foundation of the present paper. To summarize our results, let F be an infinite field, G the F-rational points of a simple Chevalley group defined over F, A an abelian group, and c : F × F → A a Steinberg symbol that is bilinear if G is not symplectic (cf. §1). In this paper we describe an explicit 2-cocycle σG in Z (G;A) that represents the cohomology class in H(G;A) of the central extension G of G by A constructed by Matsumoto [5]

In this paper we are interested in computing the Brauer group of K3 surfaces. To an element of the Brauer–Grothendieck group Br(X) of a smooth projective variety X over a number field k class field theory associates the corresponding Brauer–Manin obstruction, which is a closed condition satisfied by k-points inside the topological space of adelic points of X, see [20, Ch. 5.2]. If such a condition is non-trivial, X is a counterexample to weak approximation, and if no adelic point satisfies this condition, X is a counterexample to the Hasse principle. The computation of Br(X) is thus a first step in the computation of the Brauer–Manin obstruction on X. Let k be an arbitrary field with a separable closure k, Γ = Gal(k/k). Recall that for a variety X over k the subgroup Br0(X) ⊂ Br(X) denotes the image of Br(k) in Br(X), and Br1(X) ⊂ Br(X) denotes the kernel of the natural map Br(X) → Br(X), where X = X ×k k. In [22] we showed that if X is a K3 surface over a field k finitely generated over Q, then Br(X)/Br0(X) is finite. No general approach to the computation of Br(X)/Br0(X) seems to be known; in fact until recently there was not a single K3 surface over a number field for which Br(X)/Br0(X) was known. One of the aims of this paper is to give examples of K3 surfaces X over Q such that Br(X) = Br(Q). We study a particular kind of K3 surfaces, namely Kummer surfaces X = Kum(A) constructed from abelian surfaces A. Let Br(X)n denote the n-torsion subgroup of Br(X). Section 1 is devoted to the geometry of Kummer surfaces. We show that there is a natural isomorphism of Γ-modules Br(X)−→Br(A) (Proposition 1.3). When A is a product of two elliptic curves, the algebraic Brauer group Br1(X) often coincides with Br(k), see Proposition 1.4. Section 2 starts with a general remark on the etale cohomology of abelian varieties which may be of independent interest (Proposition 2.2). It implies that if n is an odd integer, then for any abelian variety A the group Br(A)n/Br1(A)n is canonically isomorphic to the quotient of Het(A, μn) Γ by (NS (A)/n), where NS (A) is the Neron–Severi group (Corollary 2.3). For any n ≥ 1 we prove that Br(X)n/Br1(X)n is a subgroup of Br(A)n/Br1(A)n, and this inclusion is an equality for odd n, see

In this paper, we prove that, to any Hilbert cuspidal eigenform, one may attach a compatible system of Galois representations. This result extends the analogous results of Deligne and Deligne–Serre for elliptic modular forms. The principal work on this conjecture was carried out by Carayol and Taylor, but their results left one case remaining, which we complete in this paper. We also investigate the compatibility of our results with the local Langlands correspondence, and prove that whenever the local component of the automorphic representation is not special, then the results coincide.

In this paper, we study the structure of the pointed-Gromov-Hausdorff limits of sequences of Ricci shrinkers. We define a regular-singular decomposition following the work of Cheeger-Colding for manifolds with a uniform Ricci curvature lower bound, and prove that the regular part of any Ricci shrinker limit space is convex, inspired by Colding-Naber's original idea of parabolic smoothing of the distance functions., Comment: revised version, to appear in J. Reine Angew. Math

Consider the parabolic free boundary problem Δu – ∂ t u = 0 in {u > 0}, |∇u| = 1 on ∂{u > 0}. For a realistic class of solutions, containing for example all limits of the singular perturbation problem Δue – ∂ t ue = βe (ue ) as e → 0, we prove that one-sided flatness of the free boundary implies regularity. In particular, we show that the topological free boundary ∂{u > 0} can be decomposed into an open regular set (relative to ∂{u > 0}) which is locally a surface with Holder-continuous space normal, and a closed singular set. Our result extends the main theorem in the paper by H. W. Alt-L. A. Caffarelli (1981) to more general solutions as well as the time-dependent case. Our proof uses methods developed in H. W. Alt-L. A. Caffarelli (1981), however we replace the core of that paper, which relies on non-positive mean curvature at singular points, by an argument based on scaling discrepancies, which promises to be applicable to more general free boundary or free discontinuity problems.

In an earlier paper we proved some results concerning Gross's conjecture on tori. This conjecture, which we call the Minus Conjecture, is closely related to a conjecture of Burns, which is now known to hold generally in the absolutely abelian setting; however Burns' conjecture does not directly imply the Minus Conjecture. The result proved in the earlier paper was concerned with imaginary absolutely abelian extensions K/Q of the form K=FK+, with F imaginary quadratic and K+/Q being tame, l-elementary and ramified at most at two primes. In the present paper we complement these results by proving the Minus Conjecture for extensions K/Q as above but without any restriction on the number s of ramified primes. The price we have to pay for this generality is that our proof only works if the odd prime l>=3(s+1) and l does not divide hF.

How does the associative structure of the algebra A e¤ect the Lie structure of A? In particular, is it possible to describe Lie ideals of A through associative ideals of A? This and related questions such as the link between Lie ideals and conjugate-invariant subspaces of unital algebras, have been an active area of research for more than 50 years. They have been studied in pure algebra (see e.g. [A], [BFM], [H1], [H2], [H3], [JR], [LM], [MM1], [MM2], [Mu]) and, more or less independently, also in functional analysis, particularly in operator algebras (see e.g. [BM], [CY], [FM], [FR], [FMS], [FN], [HMS], [HP], [Ma], [MaMu], [MS], [Mi], [To]). One of the goals of this paper is to ‘‘glue’’ these two areas; that is, we will apply purely algebraic results, derived in the first part of the paper, to the second part dealing with Banach algebras, especially with W -algebras and C -algebras.

We develop a theory of quasi–coherent and constructible sheaves on algebraic stacks correcting a mistake in the recent book of Laumon and Moret-Bailly. We study basic cohomological properties of such sheaves, and prove stack–theoretic versions of Grothendieck’s Fundamental Theorem for proper morphisms, Grothendieck’s Existence Theorem, Zariski’s Connectedness Theorem, as well as finiteness Theorems for proper pushforwards of coherent and constructible sheaves. We also explain how to define a derived pullback functor which enables one to carry through the construction of a cotangent complex for a morphism of algebraic stacks due to Laumon and Moret–Bailly. 1.1. In the book ([LM-B]) the lisse-etale topos of an algebraic stack was introduced, and a theory of quasi–coherent and constructible sheaves in this topology was developed. Unfortunately, it was since observed by Gabber and Behrend (independently) that the lisse-etale topos is not functorial as asserted in (loc. cit.), and hence the development of the theory of sheaves in this book is not satisfactory “as is”. In addition, since the publication of the book ([LM-B]), several new results have been obtained such as finiteness of coherent and etale cohomology ([Fa], [Ol]) and various other consequences of Chow’s Lemma ([Ol]). The purpose of this paper is to explain how one can modify the arguments of ([LM-B]) to obtain good theories of quasi–coherent and constructible sheaves on algebraic stacks, and in addition we provide an account of the theory of sheaves which also includes the more recent results mentioned above. 1.2. The paper is organized as follows. In section 2 we recall some aspects of the theory of cohomological descent ([SGA4], V) which will be used in what follows. In section 3 we review the basic definitions of the lisse-etale site, cartesian sheaves over a sheaf of algebras, and verify some basic properties of such sheaves. In section 4 we relate the derived category of cartesian sheaves over some sheaf of rings to various derived categories of sheaves on the simplicial space obtained from a covering of the algebraic stack by an algebraic space. Loosely speaking the main result states that the cohomology of a complex with cartesian cohomology sheaves can be computed by restricting to the simplicial space obtained from a covering and computing cohomology on this simplicial space using the etale topology. In section 5 we generalize these results to comparisons between Ext–groups computed in the lisse-etale topos and Ext–groups computed using the etale topology on a hypercovering. In section 6 we specialize the discussion of sections 3-5 to quasi–coherent sheaves. We show that if X is an algebraic stack and OXlis-et denotes the structure sheaf of the lisse-etale topos, then the triangulated category D qcoh(X ) of bounded below complexes of OXlis-et–modules with quasi–coherent cohomology sheaves satisfies all the basic properties that one would expect from the theory for schemes. For example we show in this section that if f : X → Y is a quasi–compact morphism of algebraic stacks and M is a quasi–coherent sheaf on X Date: November 2, 2005. 1

In this paper, we prove the conjectural endoscopic character identities for tempered representations of metaplectic group $Mp_{2n}$ based on the formalism of endoscopy theory by J. Adams, D. Renard and W.W. Li., This is part of my thesis. Submitted

We prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne–Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne–Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes X h {X_{h}} . Boyarchenko’s two conjectures are on the maximality of X h {X_{h}} and on the behavior of the torus-eigenspaces of their cohomology. Both of these conjectures were known in full generality only for division algebras with Hasse invariant 1 / n {1/n} in the case h = 2 {h=2} (the “lowest level”) by the work of Boyarchenko–Weinstein on the cohomology of a special affinoid in the Lubin–Tate tower. We prove that the number of rational points of X h {X_{h}} attains its Weil–Deligne bound, so that the cohomology of X h {X_{h}} is pure in a very strong sense. We prove that the torus-eigenspaces of the cohomology group H c i ⁢ ( X h ) {H_{c}^{i}(X_{h})} are irreducible representations and are supported in exactly one cohomological degree. Finally, we give a complete description of the homology groups of the semi-infinite Deligne–Lusztig varieties attached to any division algebra, thus giving a geometric realization of a large class of supercuspidal representations of these groups. Moreover, the correspondence θ ↦ H c i ⁢ ( X h ) ⁢ [ θ ] {\theta\mapsto H_{c}^{i}(X_{h})[\theta]} agrees with local Langlands and Jacquet–Langlands correspondences. The techniques developed in this paper should be useful in studying these constructions for p-adic groups in general.

Kubota [19] showed how the theory of Eisenstein series on the higher metaplectic covers of SL2 (which he discovered) can be used to study the analytic properties of Dirichlet series formed with n-th order Gauss sums. In this paper we will prove a functional equation for such Dirichlet series in the precise form required by the companion paper [2]. Closely related results are in Eckhardt and Patterson [10]. The Kubota Dirichlet series are the entry point to a fascinating universe. Their residues, for example, are mysterious if n > 3, though there is tantalizing evidence that these residues exhibit a rich structure that can only be partially glimpsed at this time. When n = 4 the residues are the Fourier coefficients of the biquadratic theta function that were studied by Suzuki [23]. Suzuki found that he could only determine some of the coefficients. This failure to determine all the coefficients was explained in terms of the failure of uniqueness of Whittaker models for the generalized theta series by Deligne [9] and by Kazhdan and Patterson [15]. On the other hand, Patterson [22] conjectured that the mysterious coefficients are essentially square roots of Gauss sums. Evidence for Patterson’s conjecture is discussed in Bump and Hoffstein [6] and in Eckhardt and Patterson [10], where the conjecture is refined in light of numerical data. Partial proofs were given by Suzuki in [24] and [25]. Another set of conjectures relevant to the mysterious coefficients of n-th order theta functions were given by Bump and Hoffstein, who considered theta functions on the n-fold covers of GLr for arbitrary r. They are expressed as identities between Rankin-Selberg convolutions of generalized theta series and Whittaker coefficients of Eisenstein series on the metaplectic group, but they boil down to properties of the residues of Kubota Dirichlet series, and their higher rank generalizations. See Bump and Hoffstein [6], Bump [4] and Hoffstein [12]. These conjectures are different from the Patterson conjecture, and there are other considerations which suggest that there may be further unproved relations beyond those described in the conjectures of Patterson and Bump and Hoffstein.