We consider BPS states in a large class of , field theories, obtained by reducing six-dimensional superconformal field theories on Riemann surfaces, with defect operators inserted at points of the Riemann surface. Further dimensional reduction on yields sigma models, whose target spaces are moduli spaces of Higgs bundles on Riemann surfaces with ramification. In the case where the Higgs bundles have rank 2, we construct canonical Darboux coordinate systems on their moduli spaces. These coordinate systems are related to one another by Poisson transformations associated to BPS states, and have well-controlled asymptotic behavior, obtained from the WKB approximation. The existence of these coordinates implies the Kontsevich–Soibelman wall-crossing formula for the BPS spectrum. This construction provides a concrete realization of a general physical explanation of the wall-crossing formula which was proposed in Gaiotto et al. . It also yields a new method for computing the spectrum using the combinatorics of triangulations of the Riemann surface.

We introduce the idea of (and several variations) for a sequence of representations of groups . A central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability. This makes it possible to extend classical theorems of homological stability to a much broader variety of examples. Representation stability also provides a framework in which to find and to predict patterns, from classical representation theory (Littlewood–Richardson and Murnaghan rules, stability of Schur functors), to cohomology of groups (pure braid, Torelli and congruence groups), to Lie algebras and their homology, to the (equivariant) cohomology of flag and Schubert varieties, to combinatorics (the conjecture). The majority of this paper is devoted to exposing this phenomenon through examples. In doing this we obtain applications, theorems and conjectures. Beyond the discovery of new phenomena, the viewpoint of representation stability can be useful in solving problems outside the theory. In addition to the applications given in this paper, it is applied by Church–Ellenberg–Farb (in preparation) to counting problems in number theory and finite group theory. Representation stability is also used by Church (2012) to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds.

We develop a variational calculus for a certain free energy functional on the space of all probability measures on a Kähler manifold . This functional can be seen as a generalization of Mabuchiʼs K-energy functional and its twisted versions to more singular situations. Applications to Monge–Ampère equations of mean field type, twisted Kähler–Einstein metrics and Moser–Trudinger type inequalities on Kähler manifolds are given. Tianʼs -invariant is generalized to singular measures, allowing in particular a proof of the existence of Kähler–Einstein metrics with positive Ricci curvature that are singular along a given anti-canonical divisor (which combined with very recent developments concerning Kähler metrics with conical singularities confirms a recent conjecture of Donaldson). As another application we show that if the Calabi flow in the (anti-)canonical class exists for all times then it converges to a Kähler–Einstein metric, when a unique one exists, which is in line with a well-known conjecture.

In this paper, we study the birational geometry of the Hilbert scheme P-2[n] of n-points on P-2. We discuss the stable base locus decomposition of the effective cone and the corresponding birational models. We give modular interpretations to the models in terms of moduli spaces of Bridgeland semi-stable objects. We construct these moduli spaces as moduli spaces of quiver representations using G.I.T. and thus show that they are projective. There is a precise correspondence between wall-crossings in the Bridgeland stability manifold and wall-crossings between Mori cones. For n <= 9, we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions. (C). 2012 Elsevier Inc. All rights reserved.

A skew Calabi–Yau algebra is a generalization of a Calabi–Yau algebra which allows for a non-trivial Nakayama automorphism. We prove three homological identities about the Nakayama automorphism and give several applications. The identities we prove show (i) how the Nakayama automorphism of a smash product algebra is related to the Nakayama automorphisms of a graded skew Calabi–Yau algebra and a finite-dimensional Hopf algebra that acts on it; (ii) how the Nakayama automorphism of a graded twist of is related to the Nakayama automorphism of ; and (iii) that the Nakayama automorphism of a skew Calabi–Yau algebra has trivial homological determinant in case is noetherian, connected graded, and Koszul.

Since the early 2000s physicists have developed an ingenious but non-rigorous formalism called the to put forward precise conjectures on phase transitions in random problems (Mézard et al., 2002 ). The cavity method predicts that the satisfiability threshold in the random -SAT problem is , with (Mertens et al., 2006 ). This paper contains a proof of the conjecture.

In this paper, we study the birational geometry of the Hilbert scheme of -points on . We discuss the stable base locus decomposition of the effective cone and the corresponding birational models. We give modular interpretations to the models in terms of moduli spaces of Bridgeland semi-stable objects. We construct these moduli spaces as moduli spaces of quiver representations using G.I.T. and thus show that they are projective. There is a precise correspondence between wall-crossings in the Bridgeland stability manifold and wall-crossings between Mori cones. For , we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions.

We initiate a new study of differential operators with symmetries and combine this with the study of branching laws for Verma modules of reductive Lie algebras. By the criterion for discretely decomposable and multiplicity-free restrictions of generalized Verma modules (T. Kobayashi (2012) ), we are brought to natural settings of parabolic geometries for which there exist unique equivariant differential operators to submanifolds. Then we apply a new method (F-method) relying on the Fourier transform to find singular vectors in generalized Verma modules, which significantly simplifies and generalizes many preceding works. In certain cases, it also determines the Jordan–Hölder series of the restriction for singular parameters. The F-method yields an explicit formula of such unique operators, for example, giving an intrinsic and new proof of Juhl's conformally invariant differential operators (Juhl (2009) ) and its generalizations to spinor bundles. This article is the first in the series, and the next ones include their extension to curved cases together with more applications of the F-method to various settings in parabolic geometries.

We consider equally-weighted Cantor measures arising from iterated function systems of the form , , where . We classify the so that they have infinitely many mutually orthogonal exponentials in . In particular, if divides , the measures have a complete orthogonal exponential system and hence spectral measures. Improving the construction by Dutkay et al. (2009) , we characterize all the maximal orthogonal sets when divides via a maximal mapping on the -adic tree in which all elements in are represented uniquely in finite -adic expansions and we can separate the maximal orthogonal sets into two types: regular and irregular sets. For a regular maximal orthogonal set, we show that its completeness in is crucially determined by the certain growth rate of non-zero digits in the tail of the -adic expansions of the elements. Furthermore, we exhibit complete orthogonal exponentials with zero Beurling dimensions. These examples show that the technical condition in Theorem 3.5 of Dutkay et al. (2011) cannot be removed. For an irregular maximal orthogonal set, we show that under some conditions, its completeness is equivalent to that of the corresponding regularized mapping.

In the paper, we establish a blow-up criterion in terms of the integrability of the density for strong solutions to the Cauchy problem of compressible isentropic Navier–Stokes equations in with vacuum, under the assumptions on the coefficients of viscosity: . This extends the corresponding results in Huang et al. (2011), Sun et al. (2011) where a blow-up criterion in terms of the upper bound of the density was obtained under the condition . As a byproduct, the restriction in Fan et al. (2010), Sun et al. (2011) is relaxed to for the full compressible Navier–Stokes equations by giving a new proof of Lemma 3.1. Besides, we get a blow-up criterion in terms of the upper bound of the density and the temperature for strong solutions to the Cauchy problem of the full compressible Navier–Stokes equations in . The appearance of vacuum could be allowed. This extends the corresponding results in Sun et al. (2011) where a blow-up criterion in terms of the upper bound of was obtained without vacuum. The effective viscous flux plays a very important role in the proofs.

We consider the inverse mean curvature flow in smooth Riemannian manifolds of the form with metric and non-positive radial sectional curvature. We prove, that for initial mean-convex graphs over the flow exists for all times and remains a graph over . Under weak further assumptions on the ambient manifold, we prove optimal decay of the gradient and that the flow leaves become umbilic exponentially fast. We prove optimal -estimates in case that the ambient pinching improves.

We study Artin algebras and commutative Noetherian complete local rings in connection with the following decomposition property of Gorenstein-projective modules: We show that the class of algebras enjoying ( ) coincides with the class of virtually Gorenstein algebras of finite Cohen–Macaulay type, introduced in Beligiannis and Reiten (2007) , Beligiannis (2005) . Thus we solve the problem stated in Chen (2008) . This is proved by characterizing when a resolving subcategory is of finite representation type in terms of decomposition properties of its closure under filtered colimits, thus generalizing a classical result of Auslander (1976) and Ringel and Tachikawa (1974) . In the commutative case, if admits a non-free finitely generated Gorenstein-projective module, then we show that is of finite Cohen–Macaulay type iff is Gorenstein and satisfies ( ). We also generalize a result of Yoshino (2005) by characterizing when finitely generated modules without extensions with the ring are Gorenstein-projective. Finally we study the (stable) relative Auslander algebra of a virtually Gorenstein algebra of finite Cohen–Macaulay type and, under the presence of a cluster tilting object, we give descriptions of the stable category of Gorenstein-projective modules in terms of the cluster category associated to the quiver of the stable relative Auslander algebra. In this setting we show that the cluster category is invariant under derived equivalences.

We find upper and lower bounds of the multiplicities of irreducible admissible representations of a semisimple Lie group occurring in the induced representations from irreducible representations of a closed subgroup . As corollaries, we establish geometric criteria for finiteness of the dimension of (induction) and of (restriction) by means of the real flag variety , and discover that uniform boundedness property of these multiplicities is independent of real forms and characterized by means of the complex flag variety.

We consider Liouville-type and partial regularity results for the nonlinear fourth-order problem where and . We give a complete classification of stable and finite Morse index solutions (whether positive or sign changing), in the full exponent range. We also compute an upper bound of the Hausdorff dimension of the singular set of extremal solutions. Our approach is motivated by Fleming's tangent cone analysis technique for minimal surfaces and Federer's dimension reduction principle in partial regularity theory. A key tool is the monotonicity formula for biharmonic equations.

We prove that the coefficients of certain weight harmonic Maass forms are “traces” of singular moduli for weak Maass forms. To prove this theorem, we construct a theta lift from spaces of weight harmonic weak Maass forms to spaces of weight vector-valued harmonic weak Maass forms on , a result which is of independent interest. We then prove a general theorem which guarantees (with bounded denominator) when such Maass singular moduli are algebraic. As an example of these results, we derive a formula for the partition function as a finite sum of algebraic numbers which lie in the usual discriminant ring class field.

This paper addresses the classification of locally conformally flat gradient Yamabe solitons. In the first part it is shown that locally conformally flat gradient Yamabe solitons with positive sectional curvature are rotationally symmetric. In the second part the classification of all radially symmetric gradient Yamabe solitons is given and their correspondence to smooth self-similar solutions of the fast diffusion equation on is shown. In the last section it is shown that any eternal solution to the Yamabe flow with positive Ricci curvature and with the scalar curvature attaining an interior space–time maximum must be a steady Yamabe soliton.

Let be a semisimple (so, finite dimensional) Hopf algebra over an algebraically closed field of characteristic zero and let be a commutative domain over . We show that if arises as an -module algebra via an inner faithful -action, then must be a group algebra. This answers a question of E. Kirkman and J. Kuzmanovich and partially answers a question of M. Cohen. The main results of this article extend to working over of positive characteristic. On the other hand, we obtain results on Hopf actions on Weyl algebras as a consequence of the main theorem.

We prove various Hardy-type and uncertainty inequalities on a stratified Lie group . In particular, we show that the operators , where is a homogeneous norm, , and is the sub-Laplacian, are bounded on the Lebesgue space . As consequences, we estimate the norms of these operators sufficiently precisely to be able to differentiate and prove a logarithmic uncertainty inequality. We also deduce a general version of the Heisenberg–Pauli–Weyl inequality, relating the norm of a function to the norm of and the norm of .

We consider some analogs of the quantum unique ergodicity conjecture for geodesics and “shrinking” families of sets. In particular, we prove the analog of the QUE conjecture for Eisenstein series restricted to the infinite geodesic connecting 0 and ∞ inside the modular surface.