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We analyze the $G$-skein theory invariants of the 3-torus $T^3$ and the two-torus $T^2$, for the groups $G = GL_N, SL_N$ and for generic quantum parameter. We obtain formulas for the dimension of the skein module of $T^3$, and we describe the algebraic structure of the skein category of $T^2$ -- namely of the $n$-point relative skein algebras. The case $n=N$ (the Schur-Weyl case) is special in our analysis. We construct an isomorphism between the $N$-point relative skein algebra and the double affine Hecke algebra at specialized parameters. As a consequence, we prove that all tangles in the relative $N$-point skein algebra are in fact equivalent to linear combinations of braids, modulo skein relations. More generally for $n$ an integer multiple of $N$, we construct a surjective homomorphism from an appropriate DAHA to the $n$-point relative skein algebra. In the case $G=SL_2$ corresponding to the Kauffman bracket we give proofs directly using skein relations. Our analysis of skein categories in higher rank hinges instead on the combinatorics of multisegment representations when restricting from DAHA to AHA and nonvanishing properties of parabolic sign idempotents upon them.

Over the last decade, the Kubernetes container orchestration platform has become essential to many scientific workflows. Despite its popularity, deploying a production-ready Kubernetes cluster on-premises can be challenging for system administrators. Many of the proprietary integrations that application developers take for granted in commercial cloud environments must be replaced with alternatives when deployed locally. This article will compare three popular deployment strategies for sites deploying Kubernetes on-premise: Kubeadm with Kubespray, OpenShift / OKD and Rancher via K3S/RKE2.

Optically addressable spin-based quantum sensors enable nanoscale measurements of temperature, magnetic field, pH, and other physical properties of a system. Advancing the sensors beyond proof-of-principle demonstrations in living cells and multicellular organisms towards reliable, damage-free quantum sensing poses three distinct technical challenges. First, spin-based quantum sensing requires optical accessibility and microwave delivery. Second, any microelectronics must be biocompatible and designed for imaging living specimens. Third, efficient microwave delivery and temperature control are essential to reduce unwanted heating and to maintain an optimal biological environment. Here, we present the Quantum Biosensing Chip (Q-BiC), which facilitates microfluidic-compatible microwave delivery and includes on-chip temperature control. We demonstrate the use of Q-BiC in conjunction with nanodiamonds containing nitrogen vacancy centers to perform optically detected magnetic resonance in living systems. We quantify the biocompatibility of microwave excitation required for optically detected magnetic resonance both in vitro in HeLa cells and in vivo in the nematode Caenorhabditis elegans for temperature measurements and determine the microwave-exposure range allowed before detrimental effects are observed. In addition, we show that nanoscale quantum thermometry can be performed in immobilised but non-anaesthetised adult nematodes with minimal stress. These results enable the use of spin-based quantum sensors without damaging the biological system under study, facilitating the investigation of the local thermodynamic and viscoelastic properties of intracellular processes.

We determine the $\sigma$-derivations of quantum tori and quantum affine spaces for a toric automorphism $\sigma$. By standard results, every toric automorphism $\sigma$ of a quantum affine space $\mathcal{A}$ and every $\sigma$-derivation of $\mathcal{A}$ extend uniquely to the corresponding quantum torus $\mathcal{T}$. We shall see that, for a toric automorphism $\sigma$, every $\sigma$-derivation of $\mathcal{T}$ is a unique sum of an inner $\sigma$-derivation and a $\sigma$-derivation that is conjugate to a derivation and that the latter is non-zero only if $\sigma$ is an inner automorphism of $\mathcal{T}$. This is applied to determine the $\sigma$-derivations of $\mathcal{A}$ for a toric automorphism $\sigma$, generalizing results of Alev and Chamarie for the derivations of quantum affine spaces and of Almulhem and Brzezi\'{n}ski for $\sigma$-derivations of the quantum plane. We apply the results to iterated Ore extensions $A$ of the base field for which all the defining endomorphisms are automorphisms and each of the adjoined indeterminates is an eigenvector for all the subsequent defining automorphisms. We present an algorithm which, in characteristic zero, will, for such an algebra $A$, either construct a quantum torus between $A$ and its quotient division algebra or show that no such quantum torus exists. Also included is a general section on skew derivations which become inner on localization at the powers of a normal element which is an eigenvector for the relevant automorphism. This section explores a connection between such skew derivations and normalizing sequences of length two. This connection is illustrated by known examples of skew derivations and by the construction of a family of skew derivations for the parametric family of subalgebras of the Weyl algebra that has been studied in three papers by Benkart, Lopes and Ondrus.

How do we capture the breadth of behavior in animal movement, from rapid body twitches to aging? Using high-resolution videos of the nematode worm $C. elegans$, we show that a single dynamics connects posture-scale fluctuations with trajectory diffusion, and longer-lived behavioral states. We take short posture sequences as an instantaneous behavioral measure, fixing the sequence length for maximal prediction. Within the space of posture sequences we construct a fine-scale, maximum entropy partition so that transitions among microstates define a high-fidelity Markov model, which we also use as a means of principled coarse-graining. We translate these dynamics into movement using resistive force theory, capturing the statistical properties of foraging trajectories. Predictive across scales, we leverage the longest-lived eigenvectors of the inferred Markov chain to perform a top-down subdivision of the worm's foraging behavior, revealing both "runs-and-pirouettes" as well as previously uncharacterized finer-scale behaviors. We use our model to investigate the relevance of these fine-scale behaviors for foraging success, recovering a trade-off between local and global search strategies., Comment: 28 pages, 15 figures

In this survey article for the Encyclopedia of Mathematical Physics, 2nd Edition, I give an introduction to quantum character varieties and quantum character stacks, with an emphasis on the unification between four different approaches to their construction.

We develop a $\mathtt{q}$-analogue of the theory of conjugation equivariant $\mathcal D$-modules on a complex reductive group $G$. In particular, we define quantum Hotta-Kashiwara modules and compute their endomorphism algebras. We use the Schur-Weyl functor of the second author, and develop tools from the corresponding double affine Hecke algebra to study this category in the cases $G=GL_N$ and $SL_N$. Our results also have an interpretation in skein theory (explored further in a sequel paper), namely a computation of the $GL_N$ and $SL_N$-skein algebra of the 2-torus., Comment: 51 pages, comments welcome!

How do the same mechanisms that faithfully regenerate complex developmental programs in spite of environmental and genetic perturbations also permit responsiveness to environmental signals, adaptation, and genetic evolution? Using the nematode Caenorhabditis elegans as a model, we explore the phenotypic space of growth and development in various genetic and environmental contexts. Our data are growth curves and developmental parameters obtained by automated microscopy. Using these, we show that among the traits that make up the developmental space, correlations within a particular context are predictive of correlations among different contexts. Further we find that the developmental variability of this animal can be captured on a relatively low dimensional phenoptypic manifold and that on this manifold, genetic and environmental contributions to plasticity can be deconvolved independently. Our perspective offers a new way of understanding the relationship between robustness and flexibility in complex systems, suggesting that projection and concentration of dimension can naturally align these forces as complementary rather than competing.

I introduce new Langlands duality conjectures concerning skein modules of 3-manifolds, which we have made recently with David Ben-Zvi, Sam Gunningham, and Pavel Safronov. I recount some historical motivation and some recent special cases where the conjecture is confirmed. The proofs in these cases combine the representation theory of double affine Hecke algebras and a new 1-form symmetry structure on skein modules related to electric-magnetic duality. This note is an expansion of my talk given at String Math 2022 in Warsaw, and is submitted to the String Math 2022 Proceedings publication., Comment: Comments welcome!

Isolating slower dynamics from fast fluctuations has proven remarkably powerful, but how do we proceed from partial observations of dynamical systems for which we lack underlying equations? Here, we construct maximally-predictive states by concatenating measurements in time, partitioning the resulting sequences using maximum entropy, and choosing the sequence length to maximize short-time predictive information. Transitions between these states yield a simple approximation of the transfer operator, which we use to reveal timescale separation and long-lived collective modes through the operator spectrum. Applicable to both deterministic and stochastic processes, we illustrate our approach through partial observations of the Lorenz system and the stochastic dynamics of a particle in a double-well potential. We use our transfer operator approach to provide a new estimator of the Kolmogorov-Sinai entropy, which we demonstrate in discrete and continuous-time systems, as well as the movement behavior of the nematode worm $C. elegans$., Comment: 22 pages, 14 figures

We initiate the study of decorated character stacks and their quantizations using the framework of stratified factorization homology. We thereby extend the construction by Fock and Goncharov of (quantum) decorated character varieties to encompass also the stacky points, in a way that is both compatible with cutting and gluing and equivariant with respect to canonical actions of the modular group of the surface. In the cases $G=SL_2,PGL_2$ we construct a system of categorical charts and flips on the quantum decorated character stacks which generalize the well--known cluster structures on the Fock--Goncharov moduli spaces.

We prove that a finite braided tensor category A is invertible in the Morita 4-category BrTens of braided tensor categories if, and only if, it is non-degenerate. This includes the case of semisimple modular tensor categories, but also non-semisimple examples such as categories of representations of the small quantum group at good roots of unity. Via the cobordism hypothesis, we obtain new invertible 4-dimensional framed topological field theories, which we regard as a non-semisimple framed version of the Crane-Yetter-Kauffman invariants, after Freed--Teleman and Walker's construction in the semisimple case. More generally, we characterize invertibility for E_1- and E_2-algebras in an arbitrary symmetric monoidal oo-category, and we conjecture a similar characterization of invertible E_n-algebras for any n. Finally, we propose the Picard group of BrTens as a generalization of the Witt group of non-degenerate braided fusion categories, and pose a number of open questions about it.

We study biochemical reaction networks capable of product discrimination inspired by biological proofreading mechanisms. At equilibrium, product discrimination, the selective formation of a "correct" product with respect to an "incorrect product", is fundamentally limited by the free energy difference between the two products. However, biological systems often far exceed this limit, by using discriminatory networks that expend free energy to maintain non-equilibrium steady states. Non-equilibrium systems are notoriously difficult to analyze and no systematic methods exist for determining parameter regimes which maximize discrimination. Here we introduce a measure that can be computed directly from the biochemical rate constants and which provides a necessary condition for proofreading. Non-equilibrium systems can discriminate using binding energy (energetic) differences or using activation energy (kinetic) differences, but in both cases, proofreading is optimal when dissipation is maximized, so long as the necessary condition we introduce is satisfied. We show that these constraints can be in opposition and that this explains why the error rate is a non-monotonic function of the irreversible drive in the original proofreading scheme of Hopfield and Ninio. Finally, we introduce mixed networks, in which one product is favored energetically and the other kinetically. In such networks, sensitive product switching can be achieved simply by spending free energy. Biologically, this corresponds to the ability to select between products by driving a single reaction without network fine tuning. This may be used to explore alternate product spaces in challenging environments.

We give a new, algebraically computable formula for skein modules of closed 3-manifolds via Heegaard splittings. As an application, we prove that skein modules of closed 3-manifolds are finite-dimensional, resolving in the affirmative a conjecture of Witten., Comment: 39 pages; final published version

We compute the center and Azumaya locus in the simplest non-abelian examples of quantized multiplicative quiver varieties at a root of unity: quantum Weyl algebras of rank $N$, and quantum differential operators on the quantum group $\mathrm{GL}_2$. These examples illustrate in elementary terms much more general phenomena explored further in [Ganev-Jordan-Safronov 2019]., Comment: 14 pages

We prove that quantized multiplicative quiver varieties and quantum character varieties define sheaves of Azumaya algebras over the corresponding classical moduli spaces, and we prove that the Azumaya locus of the Kauffman bracket skein algebras contains the smooth locus, proving a strong form of the Unicity Conjecture of Bonahon and Wong. The proofs exploit a strong compatibility between quantum Hamiltonian reduction and the quantum Frobenius homomorphism as it arises in each setting. We therefore introduce the concepts of Frobenius quantum moment maps and their Hamiltonian reduction, and of Frobenius Poisson orders. We use these tools to construct canonical central subalgebras of quantum algebras, and explicitly compute the resulting Azumaya loci we encounter, using a natural nondegeneracy assumption., Comment: 64 pages. v4: some proofs in Section 4.4 are rewritten

We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively 2, 3 and 4-dimensional framed local topological field theories. In particular, we produce a framed 3-dimensional local TFT attached to the category of representations of a quantum group at any value of $q$., Comment: Minor updates and edits; final version

We give simple formulas for the elements $c_k$ appearing in a quantum Cayley-Hamilton formula for the reflection equation algebra (REA) associated to the quantum group $U_q(\mathfrak{gl}_N)$, answering a question of Kolb and Stokman. The $c_k$'s are certain canonical generators of the center of the REA, and hence of $U_q(\mathfrak{gl}_N)$ itself; they have been described by Reshetikhin using graphical calculus, by Nazarov-Tarasov using quantum Yangians, and by Gurevich, Pyatov and Saponov using quantum Schur functions; however no explicit formulas for these elements were previously known. As byproducts, we prove a quantum Girard-Newton identity relating the $c_k$'s to the so-called quantum power traces, and we give a new presentation for the quantum group $U_q(\mathfrak{gl}_N)$, as a localization of the REA along certain principal minors.

Given a module $M$ for the algebra $\mathcal{D}_{\mathtt{q}}(G)$ of quantum differential operators on $G$, and a positive integer $n$, we may equip the space $F_n^G(M)$ of invariant tensors in $V^{\otimes n}\otimes M$, with an action of the double affine Hecke algebra of type $A_{n-1}$. Here $G= SL_N$ or $GL_N$, and $V$ is the $N$-dimensional defining representation of $G$. In this paper we take $M$ to be the basic $\mathcal{D}_{\mathtt{q}}(G)$-module, i.e. the quantized coordinate algebra $M= \mathcal{O}_{\mathtt{q}}(G)$. We describe a weight basis for $F_n^G(\mathcal{O}_{\mathtt{q}}(G))$ combinatorially in terms of walks in the type $A$ weight lattice, and standard periodic tableaux, and subsequently identify $F_n^G(\mathcal{O}_{\mathtt{q}}(G))$ with the irreducible "rectangular representation" of height $N$ of the double affine Hecke algebra., Comment: 37 pages, 14 figures; Several missing references added with discussion that includes proper citation to them

We consider algebras of quantum differential operators, for appropriate bicharacters on a polynomial algebra in one indeterminate and for the coordinate algebra of quantum $n$-space for $n\geq 3$. In the former case a set of generators for the quantum differential operators was identified in work by the first author and T. C. McCune but it was not known whether the algebra is Noetherian. We answer this question affirmatively, setting it in a more general context involving the behaviour Noetherian condition under localization at the powers of a single element. In the latter case we determine the algebra of quantum differential operators as a skew group algebra of the group $\mathbb{Z}^n$ over a quantized Weyl algebra. It follows from this description that this algebra is a simple right and left Noetherian domain.

We determine criteria for the prime spectrum of an ambiskew polynomial algebra $R$ over an algebraically closed field $K$ to be akin to those of two of the principal examples of such an algebra, namely the universal enveloping algebra $U(sl_2)$ (in characteristic $0$) and its quantization $U_q(sl_2)$ (when $q$ is not a root of unity). More precisely, we aim to determine when the prime spectrum of $R$ consists of $0$, the ideals $(z-\lambda)R$ for some central element $z$ of $R$ and all $\lambda\in K$, and, for some positive integer $d$ and each positive integer $m$, $d$ height two prime ideals with Goldie rank $m$. New applications are to certain ambiskew polynomial rings over coordinate rings of quantum tori which arise, as localizations of connected quantized Weyl algebras., Comment: Corrections and amendments following referee's comments

For an algebraically closed field $K$, we investigate a class of noncommutative $K$-algebras called connected quantized Weyl algebras. Such an algebra has a PBW basis for a set of generators $\{x_1,\dots,x_n\}$ such that each pair satisfies a relation of the form $x_ix_j=q_{ij}x_jx_i+r_{ij}$, where $q_{ij}\in K^*$ and $r_{ij}\in K$, with, in some sense, sufficiently many pairs for which $r_{ij}\neq 0$. We classify connected quantized Weyl algebras, showing that there are two types, linear and cyclic, each depending on a single parameter $q$. When $q$ is not a root of unity we determine the prime spectra for each type. In the linear case all prime ideals are completely prime but in the cyclic case, which can only occur if $n$ is odd, there are prime ideals for which the factors have arbitrarily large Goldie rank. We apply connected quantized Weyl algebras to obtain presentations of the quantum cluster algebras for two classes of quiver, namely, for $m$ even, the Dynkin quiver of type $A_m$ and the quiver $P_m^{(1)}$ identified by Fordy and Marsh in their analysis of periodic quiver mutation. We establish Poisson analogues of the results on prime ideals and quantum cluster algebras., Comment: Minor corrections to previous version, this version to appear in the Journal of Pure and Applied Algebra

We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants $\int_S\mathcal A$ of a surface $S$, determined by the choice of a braided tensor category $\mathcal A$, and computed via factorization homology. We identify the algebraic data governing marked points and boundary components with the notion of a {\em braided module category} for $\mathcal A$, and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called {\em quantum moment maps}. We then show that the quantum character variety of a decorated surface is obtained from that of the corresponding punctured surface as a quantum Hamiltonian reduction. Characters of braided $\mathcal A$-modules are objects of the torus category $\int_{T^2}\mathcal A$. We initiate a theory of character sheaves for quantum groups by identifying the torus integral of $\mathcal A=\operatorname{Rep_q} G$ with the category $\mathcal D_q(G/G)-\operatorname{mod}$ of equivariant quantum $\mathcal D$-modules. When $G=GL_n$, we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra (DAHA) $\mathbb{SH}_{q,t}$., Comment: 33 pages, 5 figures. Final version, to appear in Sel. Math. New Ser

We construct an explicit Harish-Chandra isomorphism, from the quantum Hamiltonian reduction of the algebra D_q(GL_2) of quantum differential operators on GL_2, to the spherical double affine Hecke algebra associated to GL2. The isomorphism holds for all deformation parameters non-zero q, t, such that t does not equal to +/-i, and q is not a non-trivial root of unity. We also discuss its extension to the root of unity case.

We apply the mechanism of factorization homology to construct and compute category-valued two-dimensional topological field theories associated to braided tensor categories, generalizing the $(0,1,2)$-dimensional part of Crane-Yetter-Kauffman 4D TFTs associated to modular categories. Starting from modules for the Drinfeld-Jimbo quantum group $U_q(\mathfrak g)$ we obtain in this way an aspect of topologically twisted 4-dimensional ${\mathcal N}=4$ super Yang-Mills theory, the setting introduced by Kapustin-Witten for the geometric Langlands program. For punctured surfaces, in particular, we produce explicit categories which quantize character varieties (moduli of $G$-local systems) on the surface; these give uniform constructions of a variety of well-known algebras in quantum group theory. From the annulus, we recover the reflection equation algebra associated to $U_q(\mathfrak g)$, and from the punctured torus we recover the algebra of quantum differential operators associated to $U_q(\mathfrak g)$. From an arbitrary surface we recover Alekseev's moduli algebras. Our construction gives an intrinsically topological explanation for well-known mapping class group symmetries and braid group actions associated to these algebras, in particular the elliptic modular symmetry (difference Fourier transform) of quantum $\mathcal D$-modules., Comment: 57 page, 5 figures. Final version, to appear in J. Top

We construct a certain cross product of two copies of the braided dual $\tilde H$ of a quasitriangular Hopf algebra $H$, which we call the elliptic double $E_H$, and which we use to construct representations of the punctured elliptic braid group extending the well-known representations of the planar braid group attached to $H$. We show that the elliptic double is the universal source of such representations. We recover the representations of the punctured torus braid group obtained in arXiv:0805.2766, and hence construct a homomorphism to the Heisenberg double $D_H$, which is an isomorphism if $H$ is factorizable. The universal property of $E_H$ endows it with an action by algebra automorphisms of the mapping class group $\widetilde{SL_2(\mathbb{Z})}$ of the punctured torus. One such automorphism we call the quantum Fourier transform; we show that when $H=U_q(\mathfrak{g})$, the quantum Fourier transform degenerates to the classical Fourier transform on $D(\mathfrak{g})$ as $q\to 1$., Comment: 12 pages, 1 figure. Final version, to appear in Quantum Topology

The lower central series invariants M_k of an associative algebra A are the two-sided ideals generated by k-fold iterated commutators; the M_k provide a filtration of A. We study the relationship between the geometry of X = Spec A_ab and the associated graded components N_k of this filtration. We show that the N_k form coherent sheaves on a certain nilpotent thickening of X, and that Zariski localization on X coincides with noncommutative localization of A. Under certain freeness assumptions on A, we give an alternative construction of N_k purely in terms of the geometry of X (and in particular, independent of A). Applying a construction of Kapranov, we exhibit the N_k as natural vector bundles on the category of smooth schemes.

We determine simplicity criteria in characteristics 0 and $p$ for a ubiquitous class of iterated skew polynomial rings in two indeterminates over a base ring. One obstruction to simplicity is the possible existence of a canonical normal element $z$. In the case where this element exists we give simplicity criteria for the rings obtained by inverting $z$ and the rings obtained by factoring out the ideal generated by $z$. The results are illustrated by numerous examples including higher quantized Weyl algebras and generalizations of some low-dimensional symplectic reflection algebras.

Poisson brackets on the polynomial algebra C[x,y,z] are studied. A description of all such brackets is given and, for a significant class of Poisson brackets, the Poisson prime ideals and Poisson primitive ideals are determined. The results are illustrated by numerous examples., Comment: includes minor corrections to published version

A significant class of Poisson brackets on the polynomial algebra $\C[x_1,x_2,..., x_n]$ is studied and, for this class of Poisson brackets, the Poisson prime ideals and Poisson primitive ideals are determined. Moreover it is established that these Poisson algebras satisfy the Poisson Dixmier-Moeglin equivalence.

For a derivation d of a commutative Noetherian complex algebra A, a homeomorphism is established between the prime spectrum of the Ore extension A[z;d] and the Poisson prime spectrum of the polynomial algebra A[z] endowed with the Poisson bracket such that {A,A}=0 and {z,a}=d(a) for all a in A. Several illustrative examples are discussed including some where A is known to be d-simple.

We construct a long exact sequence computing the obstruction space, pi_1(BrPic(C_0)), to G-graded extensions of a fusion category C_0. The other terms in the sequence can be computed directly from the fusion ring of C_0. We apply our result to several examples coming from small index subfactors, thereby constructing several new fusion categories as G-extensions. The most striking of these is a Z/2Z-extension of one of the Asaeda-Haagerup fusion categories, which is one of only two known 3-supertransitive fusion categories outside the ADE series. In another direction, we show that our long exact sequence appears in exactly the way one expects: it is part of a long exact sequence of homotopy groups associated to a naturally occuring fibration. This motivates our constructions, and gives another example of the increasing interplay between fusion categories and algebraic topology., Comment: 18 pages, 1 figure

Consider the free algebra A_n generated over Q by n generators x_1, ..., x_n. Interesting objects attached to A = A_n are members of its lower central series, L_i = L_i(A), defined inductively by L_1 = A, L_{i+1} = [A,L_{i}], and their associated graded components B_i = B_i(A) defined as B_i=L_i/L_{i+1}. These quotients B_i, for i at least 2, as well as the reduced quotient \bar{B}_1=A/(L_2+A L_3), exhibit a rich geometric structure, as shown by Feigin and Shoikhet and later authors, (Dobrovolska-Kim-Ma,Dobrovolska-Etingof,Arbesfeld-Jordan,Bapat-Jordan). We study the same problem over the integers Z and finite fields F_p. New phenomena arise, namely, torsion in B_i over Z, and jumps in dimension over F_p. We describe the torsion in the reduced quotient RB_1 and B_2 geometrically in terms of the De Rham cohomology of Z^n. As a corollary we obtain a complete description of \bar{B}_1(A_n(Z)) and \bar{B}_1(A_n(F_p)), as well as of B_2(A_n(Z[1/2])) and B_2(A_n(F_p)), p>2. We also give theoretical and experimental results for B_i with i>2, formulating a number of conjectures and questions based on them. Finally, we discuss the supercase, when some of the generators are odd (fermionic) and some are even (bosonic), and provide some theoretical results and experimental data in this case.

We study Poisson traces of the structure algebra A of an affine Poisson variety X defined over a field of characteristic p. According to arXiv:0908.3868v4, the dual space HP_0(A) to the space of Poisson traces arises as the space of coinvariants associated to a certain D-module M(X) on X. If X has finitely many symplectic leaves and the ground field has characteristic zero, then M(X) is holonomic, and thus HP_0(A) is finite dimensional. However, in characteristic p, the dimension of HP_0(A) is typically infinite. Our main results are complete computations of HP_0(A) for sufficiently large p when X is 1) a quasi-homogeneous isolated surface singularity in the three-dimensional space, 2) a quotient singularity V/G, for a symplectic vector space V by a finite subgroup G in Sp(V), and 3) a symmetric power of a symplectic vector space or a Kleinian singularity. In each case, there is a finite nonnegative grading, and we compute explicitly the Hilbert series. The proofs are based on the theory of D-modules in positive characteristic.

We study the lower central series filtration L_k for a symplectic quotient A=A_{2n}/ of the free algebra A_{2n} on 2n generators, where w=\sum [x_i,x_{i+n}]. We construct an action of the Lie algebra H_{2n} of Hamiltonian vector fields on the associated graded components of the filtration, and use this action to give a complete description of the reduced first component \bar{B}_1(A)= A/(L_2 + AL_3) and the second component B_2=L_2/L_3, and we conjecture a description for the third component B_3=L_3/L_4., Comment: Corrected formulas for singular vectors, implemented referee suggestions

We describe a physics simulation software framework, MAGE, that is based on the GEANT4 simulation toolkit. MAGE is used to simulate the response of ultra-low radioactive background radiation detectors to ionizing radiation, specifically the MAJORANA and GERDA neutrinoless double-beta decay experiments. MAJORANA and GERDA use high-purity germanium detectors to search for the neutrinoless double-beta decay of 76Ge, and MAGE is jointly developed between these two collaborations. The MAGE framework contains the geometry models of common objects, prototypes, test stands, and the actual experiments. It also implements customized event generators, GEANT4 physics lists, and output formats. All of these features are available as class libraries that are typically compiled into a single executable. The user selects the particular experimental setup implementation at run-time via macros. The combination of all these common classes into one framework reduces duplication of efforts, eases comparison between simulated data and experiment, and simplifies the addition of new detectors to be simulated. This paper focuses on the software framework, custom event generators, and physics lists., Comment: 12 pages, 6 figures

Beginning with the data of a quiver Q, and its dimension vector d, we construct an algebra D_q=D_q(Mat_d(Q)), which is a flat q-deformation of the algebra of differential operators on the affine space Mat_d(Q). The algebra D_q is equivariant for an action by a product of quantum general linear groups, acting by conjugation at each vertex. We construct a quantum moment map for this action, and subsequently define the Hamiltonian reduction A^lambda_d(Q) of D_q with moment parameter \lambda. We show that A^\lambda_d(Q) is a flat formal deformation of Lusztig's quiver varieties, and their multiplicative counterparts, for all dimension vectors satisfying a flatness condition of Crawley-Boevey: indeed the product on A^\lambda_d(Q) yields a Fedosov quantization the of symplectic structure on multiplicative quiver varieties. As an application, we give a description of the category of representations of the spherical double affine Hecke algebra of type A_{n-1}, and its generalization constructed by Etingof, Oblomkov, and Rains, in terms of a quotient of the category of equivariant D_q-modules by a Serre sub-category of aspherical modules., Comment: Re-written introduction, improvements to exposition

We continue the study of the lower central series of a free associative algebra, initiated by B. Feigin and B. Shoikhet (arXiv:math/0610410). We generalize via Schur functors the constructions of the lower central series to any symmetric tensor category; specifically we compute the modified first quotient \bar{B}_1, and second and third quotients B_2, and B_3 of the series for a free algebra T(V) in any symmetric tensor category, generalizing the main results of (arXiv:math/0610410) and (arXiv:0902.4899). In the case A_{m|n}:=T(\CC^{m|n}), we use these results to compute the explicit Hilbert series. Finally, we prove a result relating the lower central series to the corresponding filtration by two-sided associative ideals, confirming a conjecture from (arXiv:0805.1909), and another one from (arXiv:0902.4899), as corollaries., Comment: Corrected an error in the proof of Lemma 6.1 pointed out by Alexei Krasilnikov

We build representations of the affine and double affine braid groups and Hecke algebras of type $C^\vee C_n$, based upon the theory of quantum symmetric pairs $(U,B)$. In the case $U=U_q(gl_N)$, our constructions provide a quantization of the representations constructed by Etingof, Freund and Ma in arXiv:0801.1530, and also a type $BC$ generalization of the results in arXiv:0805.2766., Comment: Final version, to appear in Selecta Mathematica

We continue the study of the lower central series and its associated graded components for a free associative algebra with n generators, as initiated by B. Feigin and B. Shoikhet. We establish a linear bound on the degree of tensor field modules appearing in the Jordan-Hoelder series of each graded component, which is conjecturally tight. We also bound the leading coefficient of the Hilbert polynomial of each graded component. As applications, we confirm conjectures of P. Etingof and B. Shoikhet concerning the structure of the third graded component.

We advance the classification of fusion categories in two directions. Firstly, we completely classify integral fusion categories -- and consequently, semi-simple Hopf algebras -- of dimension $pq^2$, where $p$ and $q$ are distinct primes. This case is especially interesting because it is the simplest class of dimensions where not all integral fusion categories are group-theoretical. Secondly, we classify a certain family of $\ZZ/3\ZZ$-graded fusion categories, which are generalizations of the $\ZZ/2\ZZ$-graded Tambara-Yamagami categories. Our proofs are based on the recently developed theory of extensions of fusion categories.

We build representations of the elliptic braid group from the data of a quantum D-module M over a ribbon Hopf algebra U. The construction is modelled on, and generalizes, similar constructions by Lyubashenko and Majid, and also certain geometric constructions of Calaque, Enriquez, and Etingof concerning trigonometric Cherednik algebras. In this context, the former construction is the special case where M is the basic representation, while the latter construction can be recovered as a quasi-classical limit of U=U_t(sl_N), as t limits 1. In the latter case, we produce representations of the double affine Hecke algebra of type A_{n-1}, for each n.

A Monte Carlo framework, MaGe, has been developed based on the Geant4 simulation toolkit. Its purpose is to simulate physics processes in low-energy and low-background radiation detectors, specifically for the Majorana and Gerda $^{76}$Ge neutrinoless double-beta decay experiments. This jointly-developed tool is also used to verify the simulation of physics processes relevant to other low-background experiments in Geant4. The MaGe framework contains simulations of prototype experiments and test stands, and is easily extended to incorporate new geometries and configurations while still using the same verified physics processes, tunings, and code framework. This reduces duplication of efforts and improves the robustness of and confidence in the simulation output.

We prove a result that can be applied to determine the finite-dimensional simple Poisson modules over a Poisson algebra and apply it to numerous examples. In the discussion of the examples, the emphasis is on the correspondence with the finite-dimensional simple modules over deformations and on the behaviour of finite-dimensional simple Poisson modules on the passage from a Poisson algebra to the Poisson subalgebra of invariants for the action of a finite group of Poisson automorphisms., Comment: Minor corrections and some reorganisation of material

A skew Laurent polynomial ring R[x^{\pm 1};\alpha] is reversible if it has a reversing automorphism, that is, an automorphism of period two that transposes x and x^{-1} and restricts to an automorphism of R. We study invariants for reversing automorphisms and apply our methods to determine the rings of invariants of reversing automorphisms of two simple skew Laurent polynomial rings, namely a localization of the enveloping algebra of the two-dimensional non-abelian solvable Lie algebra and the coordinate ring of the quantum torus. These two skew Laurent polynomial rings are deformations of Poisson algebras and we interpret their reversing automorphisms and their invariants as deformations of Poisson automorphisms and their invariants.

In the early morning of January 15, 2006, the Stardust Sample Return Capsule (SRC) successfully delivered its precious cargo of cometary particles to the awaiting recovery team at the Utah Test and Training Range (UTTR). As the SRC entered at 12.8 km/s, the fastest manmade object to traverse the atmosphere, a team of researchers imaged the event aboard the NASA DC-8 airborne observatory. At SRC entry, the airplane was at an altitude of 11.9 km positioned within 6.4 km of the prescribed, preferred target view location. The incoming SRC was first acquired approximately 18 seconds (s) after atmospheric interface and tracked for approximately 60 s, an observation period that is roughly centered in time around predicted peak heating.