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Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in an earlier paper, three of the present authors derived representations for certain sums of products of trigonometric functions as double series of Bessel functions [8] . These series are generalized in the present paper by introducing the novel notion of balanced derivatives, leading to further theorems. As we will see below, the regions of convergence in the unbalanced case are entirely different than those in the balanced case. From this viewpoint, it is remarkable that Ramanujan had the intuition to formulate entries that are, in our new terminology, “balanced”. If x denotes the number of products of the trigonometric functions appearing in our sums, in addition to proving the identities mentioned above, theorems and conjectures for upper and lower bounds for the sums as x → ∞ are established.

The Langlands functoriality conjecture, as reformulated in the “beyond endoscopy” program, predicts comparisons between the (stable) trace formulas of different groups G 1 , G 2 for every morphism G 1 L → L G 2 between their L-groups. This conjecture can be seen as a special case of a more general conjecture, which replaces reductive groups by spherical varieties and the trace formula by its generalization, the relative trace formula. The goal of this article and its precursor [11] is to demonstrate, by example, the existence of “transfer operators” between relative trace formulas, which generalize the scalar transfer factors of endoscopy. These transfer operators have all properties that one could expect from a trace formula comparison: matching, fundamental lemma for the Hecke algebra, transfer of (relative) characters. Most importantly, and quite surprisingly, they appear to be of abelian nature (at least, in the low-rank examples considered in this paper), even though they encompass functoriality relations of non-abelian harmonic analysis. Thus, they are amenable to application of the Poisson summation formula in order to perform the global comparison. Moreover, we show that these abelian transforms have some structure — which presently escapes our understanding in its entirety — as deformations of well-understood operators when the spaces under consideration are replaced by their “asymptotic cones”. In this second paper we use Rankin–Selberg theory to prove the local transfer behind Rudnick's 1990 thesis (comparing the stable trace formula for SL 2 with the Kuznetsov formula) and Venkatesh's 2002 thesis (providing a “beyond endoscopy” proof of functorial transfer from tori to GL 2 ). As it turns out, the latter is not completely disjoint from endoscopic transfer — in fact, our proof “factors” through endoscopic transfer. We also study the functional equation of the symmetric-square L-function for GL 2 , and show that it is governed by an explicit “Hankel operator” at the level of the Kuznetsov formula, which is also of abelian nature. A similar theory for the standard L-function was previously developed (in a different language) by Jacquet.

A concept of emergence was recently introduced in the paper [Berger] in order to quantify the richness of possible statistical behaviors of orbits of a given dynamical system. In this paper, we develop this concept and provide several new definitions, results, and examples. We introduce the notion of topological emergence of a dynamical system, which essentially evaluates how big the set of all its ergodic probability measures is. On the other hand, the metric emergence of a particular reference measure (usually Lebesgue) quantifies how non-ergodic this measure is. We prove fundamental properties of these two emergences, relating them with classical concepts such as Kolmogorov's $\epsilon$-entropy of metric spaces and quantization of measures. We also relate the two types of emergences by means of a variational principle. Furthermore, we provide several examples of dynamics with high emergence. First, we show that the topological emergence of some standard classes of hyperbolic dynamical systems is essentially the maximal one allowed by the ambient. Secondly, we construct examples of smooth area-preserving diffeomorphisms that are extremely non-ergodic in the sense that the metric emergence of the Lebesgue measure is essentially maximal. These examples confirm that super-polynomial emergence indeed exists, as conjectured in the paper [Berger]. Finally, we prove that such examples are locally generic among smooth diffeomorphisms., Comment: v3: Final version; to appear in Advances in Mathematics

The groups mentioned in the title are certain matrix groups of infinite size over a finite field $\mathbb F_q$. They are built from finite classical groups and at the same time they are similar to reductive $p$-adic Lie groups. In the present paper, we initiate the study of invariant measures for the coadjoint action of these infinite-dimensional groups. We examine first the group $\mathbb{GLB}$, a topological completion of the inductive limit group $\varinjlim GL(n, \mathbb F_q)$. As was shown by Gorin, Kerov, and Vershik [arXiv:1209.4945], the traceable factor representations of $\mathbb{GLB}$ admit a complete classification, achieved in terms of harmonic functions on the Young graph $\mathbb Y$. We show that there exists a parallel theory for ergodic coadjoint-invariant measures, which is linked with a deformed version of harmonic functions on $\mathbb Y$. Here the deformation means that the edges of $\mathbb Y$ are endowed with certain formal multiplicities coming from the simplest version of Pieri rule (multiplication by the first power sum $p_1$) for the Hall-Littlewood (HL) symmetric functions with parameter $t:=q^{-1}$. This fact serves as a prelude to our main results, which concern topological completions of two inductive limit groups built from finite unitary groups. We show that in this case, coadjoint-invariant measures are linked to some new branching graphs. The latter are still related to the HL functions, but the novelty is that now the formal edge multiplicities come from the multiplication by $p_2$ (not $p_1$) and the HL parameter $t$ turns out to be negative (as in Ennola's duality)., 46 pages; v2: minor modifications, shortened abstract, journal version, to appear in Adv. Math

Fa\`a di Bruno's formula gives an expression for the derivatives of the composition of two real-valued functions. In this paper we prove a multivariate and synthesized version of Fa\`a di Bruno's formula in higher dimensions, providing a combinatorial expression for the derivatives of chain compositions $F^{(1)} \circ \ldots \circ F^{(m)}$ of functions $F^{(l)} : \mathbb{R}^N \to \mathbb{R}^N$ in terms of sums over labelled trees. We give several applications of this formula, including a new involution formula for the inversion of multivariate power series. We use this framework to outline a combinatorial approach to studying the invertibility of polynomial mappings, giving a purely combinatorial restatement of the Jacobian conjecture. Our methods extend naturally to the non-commutative case, where we prove a free version of Fa\`a di Bruno's formula for multivariate power series in free indeterminates, and use this formula as a tool for obtaining a new inversion formula for free power series., Comment: 38 pages, 11 figures

In this paper we study the dependence of geometric properties of Radon measures, such as Hausdorff dimension and rectifiability of singular sets, on the wavefront set. This is achieved by adapting the method of Brummelhuis to the non-analytic case. As an application we obtain a general form of uncertainty principle for measures on the complex sphere which subsumes certain classical results about pluriharmonic measures., Submitted

This paper presents various functional-geometric aspects of the logarithmic Sobolev capacity Cap log ⁡ , γ , p - a capacity generated by the logarithmic Sobolev space W p log ⁡ , γ , where ( γ , p ) ∈ ( 0 , ∞ ) × [ 1 , ∞ ) . This new nonlinear-inhomogeneous-nontrivial capacity is closely related to the logarithmic Hausdorff capacity. Not only continuously sharp embedding property of W p log ⁡ , γ into certain logarithmic weighted Lebesgue space and the tracing inequalities are established, but also the logarithmic perimeter and the Lebesgue measure are utilized to: reformulate Cap log ⁡ , γ , p ; find the first variation of the logarithmic perimeter via the corresponding logarithmic mean curvature; characterize the hypersurface of a constant logarithmic mean curvature.

We examine an unstudied manuscript of N.~S.~Koshliakov over $150$ pages long and containing the theory of two interesting generalizations $\zeta_p(s)$ and $\eta_p(s)$ of the Riemann zeta function $\zeta(s)$, which we call \emph{Koshliakov zeta functions}. His theory has its genesis in a problem in the analytical theory of heat distribution which was analyzed by him. In this paper, we further build upon his theory and obtain two new modular relations in the setting of Koshliakov zeta functions, each of which gives an infinite family of identities, one for each $p\in\mathbb{R^{+}}$. The first one is a generalization of Ramanujan's famous formula for $\zeta(2m+1)$ and the second is an elegant extension of a modular relation on page $220$ of Ramanujan's Lost Notebook. Several interesting corollaries and applications of these modular relations are obtained including a new representation for $\zeta(4m+3)$., Comment: 34 pages, submitted for publication; comments are welcome

For any $k\geq 1$, this paper studies the number of polynomials having $k$ irreducible factors (counted with or without multiplicities) in $\mathbf{F}_q[t]$ among different arithmetic progressions. We obtain asymptotic formulas for the difference of counting functions uniformly for $k$ in a certain range. In the generic case, the bias dissipates as the degree of the modulus or $k$ gets large, but there are cases when the bias is extreme. In contrast to the case of products of $k$ prime numbers, we show the existence of complete biases in the function field setting, that is the difference function may have constant sign. Several examples illustrate this new phenomenon., The exposition has been improved, we now present the case of the number of irreducible factors both counting and not counting multiplicities. We also add some results on the possible values of the bias

This paper is devoted to interior estimates for eigenfunctions of the restricted fractional Laplacian on a bounded domain in R d . We prove that the eigenfunctions satisfy the expected L p bounds analogous to the classical results by Sogge [24] . As the fractional Laplacian is nonlocal, the standard method for Laplacian eigenfunction estimates can no longer work here. In the proof, we mainly reduce L p bounds to a kind of L 2 commutator estimates, which can be handled by the explicit integral expression of the restricted fractional Laplacian and its heat kernel estimates.

Let $G$ be connected reductive algebraic group defined over an algebraically closed field of characteristic $p > 0$ and suppose that $p$ is a good prime for the root system of $G$, the derived subgroup of $G$ is simply connected and the Lie algebra $\mathfrak{g} = \operatorname{Lie}(G)$ admits a non-degenerate Ad$(G)$-invariant symmetric bilinear form. Given a linear function $\chi$ on $\mathfrak{g}$ we denote by $U_\chi(\mathfrak{g})$ the reduced enveloping algebra of $\mathfrak{g}$ associated with $\chi$. By the Kac-Weisfeiler conjecture (now a theorem), any irreducible $U_\chi(\mathfrak{g})$-module has dimension divisible by $p^{d(\chi)}$ where $2d(\chi)$ is the dimension of the coadjoint $G$-orbit containing $\chi$. In this paper we give a positive answer to the natural question raised in the 1990s by Kac, Humphreys and the first-named author and show that any algebra $U_\chi(\mathfrak{g})$ admits a module of dimension $p^{d(\chi)}$., Comment: The present version is accepted for publication

In this paper, we show that one can naturally associate a limiting dynamical system $F: T\longrightarrow T$ on an $\R$-tree to any degenerating sequence of rational maps $f_n: \hat\C \longrightarrow \hat\C$ of fixed degree. The construction of $F$ is in $2$ steps: first we use barycentric extension to get $\E f_n : \Hyp^3 \longrightarrow \Hyp^3$; second, we take appropriate limit on rescalings of hyperbolic space. An important ingredient we prove here is that the Lipschitz constant depends only on the degree of the rational map. We show that the dynamics of $F$ records the limiting length spectra of the sequence $f_n$., Comment: 41 pages, 5 figure. Final version

In previous work, we used an $\infty$-categorical version of ultraproducts to show that, for a fixed height $n$, the symmetric monoidal $\infty$-categories of $E_{n,p}$-local spectra are asymptotically algebraic in the prime $p$. In this paper, we prove the analogous result for the symmetric monoidal $\infty$-categories of $K_{p}(n)$-local spectra, where $K_{p}(n)$ is Morava $K$-theory at height $n$ and the prime $p$. This requires $\infty$-categorical tools suitable for working with compactly generated symmetric monoidal $\infty$-categories with non-compact unit. The equivalences that we produce here are compatible with the equivalences for the $E_{n,p}$-local $\infty$-categories., Comment: 33 pages

Let G be a connected reductive complex algebraic group with a maximal torus T. We denote by Λ the coweight lattice of T. Let Λ + ⊂ Λ be the submonoid of dominant coweights. For λ ∈ Λ + , μ ∈ Λ , μ ⩽ λ , in [7] , authors defined a generalized transversal slice W ‾ μ λ . This is an algebraic variety of the dimension 〈 2 ρ ∨ , λ − μ 〉 , where 2 ρ ∨ is the sum of positive roots of G. In this paper, we construct an isomorphism W ‾ μ λ ≃ W ‾ μ + λ × A 〈 2 ρ ∨ , μ + − μ 〉 for μ ∈ Λ such that 〈 α ∨ , μ 〉 ⩾ − 1 for any positive root α ∨ , here μ + ∈ W μ is the dominant representative in the Weyl group orbit of μ. We consider the example when λ is minuscule, μ ∈ W λ and describe natural coordinates, Poisson structure on W ‾ μ λ ≃ A 〈 2 ρ ∨ , λ − μ 〉 and its T × C × -character. We apply these results to compute T × C × -characters of tangent spaces at fixed points of convolution diagrams W ˜ μ λ _ with minuscule λ i . We also apply our results to construct open coverings by affine spaces of convolution diagrams W ˜ μ λ _ over slices with μ such that 〈 α ∨ , μ 〉 ⩾ − 1 for any positive root α ∨ and minuscule λ i and to compute Poincare polynomials of such convolution diagrams W ˜ μ λ _ .

In this paper, we consider the nonlinear Schrodinger equation, i ∂ t u + Δ u = μ | u | p u , ( t , x ) ∈ R d + 1 , with μ = ± 1 , p > 0 . In this work, we consider the mass-subcritical cases, that is, p ∈ ( 0 , 4 d ) . We prove that under some restrictions on d , p , any radial initial data in the critical space H ˙ s c ( R d ) with compact support, implies global well-posedness.

In this paper, we will introduce Quantum Toric Varieties which are (non-commutative) generalizations of ordinary toric varieties where all the tori of the classical theory are replaced by quantum tori. Quantum toric geometry is the non-commutative version of the classical theory; it generalizes non-trivially most of the theorems and properties of toric geometry. By considering quantum toric varieties as (non-algebraic) stacks, we show that their category is equivalent to a certain category of quantum fans. We develop a Quantum Geometric Invariant Theory (QGIT) type construction of Quantum Toric Varieties. Unlike classical toric varieties, quantum toric varieties admit moduli and we define their moduli spaces, prove that these spaces are orbifolds and, in favorable cases, up to homotopy, they admit a complex structure.

We develop algebraic geometry for general Segal's Γ-rings and show that this new theory unifies two approaches we had considered earlier on (for a geometry under Spec Z ). The starting observation is that the category obtained by gluing together the category of commutative rings and that of pointed commutative monoids, that we used in [3] to define F 1 -schemes, is naturally a full subcategory of the category of Segal's Γ-rings (equivalently of S -algebras). In this paper we develop the affine case of this general algebraic geometry: one distinctive feature is that the spectrum Spec ( A ) of an S -algebra is in general a Grothendieck site rather than a point set endowed with a topology. Two striking features of this new geometry are that it is the natural domain for cyclic homology and for homological algebra as in [20] , and that new operations, which do not make sense in ordinary algebraic geometry, are here available. For instance, in this new context, the quotient of a ring by a multiplicative subgroup is still an S -algebra to which our general theory applies. Thus the adele class space gives rise naturally to an S -algebra.

In this paper we introduce new symmetrization with respect to mixed volume or anisotropic curvature integral, which generalizes the one with respect to quermassintegral due to Talenti and Tso. We show a P\'olya-Szego type principle for such symmetrization -- it diminishes the anisotropic Hessian integral for quasi-convex functions. We achieve this by a systematic study of invariants on non-symmetric matrices with real eigenvalues and the higher order anisotropic mean curvatures of level sets, which may be of independent interest. As applications, we establish a comparison principle for anisotropic Hessian equations and sharp anisotropic Sobolev inequalities., Comment: accepted by Adv. Math

Fano orbifold lines are classified by the Dynkin diagrams of type A , D , and E. It is known that the corresponding total descendant potential is a tau-function of an appropriate Kac–Wakimoto hierarchy. It is also known that in the A-case the Kac–Wakimoto hierarchies admit an extension and that the total descendant potential is a tau-function of an extended Kac–Wakimoto hierarchy. The goal of this paper is to prove that in the D-case the total descendent potential is also a tau-function of an extended Kac–Wakimoto hierarchy.

In this paper we develop an explicit method for studying the distribution of rational points near manifolds. As a consequence we obtain optimal lower bounds on the number of rational points of bounded height lying at a given distance from an arbitrary non-degenerate curve in R n . This generalises previous results for analytic non-degenerate curves. Furthermore, the main results are proved in the inhomogeneous setting. For n ≥ 3 , the inhomogeneous aspect is new even under the additional assumption of analyticity. Applications of the main distribution theorem also include the inhomogeneous Khintchine-Jarnik type theorem for divergence for arbitrary non-degenerate curves in R n .

This paper begins a series studying D-modules on the Feigin-Frenkel semi-infinite flag variety from the perspective of the Beilinson-Drinfeld factorization (or chiral) theory. Here we calculate Whittaker-twisted cohomology groups of Zastava spaces, which are certain finite-dimensional subvarieties of the affine Grassmannian. We show that such cohomology groups realize the nilradical of a Borel subalgebra for the Langlands dual group in a precise sense, following earlier work of Feigin-Finkelberg-Kuznetsov-Mirkovic and Braverman-Gaitsgory. Moreover, we compare this geometric realization of the Langlands dual group to the standard one provided by (factorizable) geometric Satake.