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We discuss the extension of the faithfulness question for the Burau representation of braid groups to the case of Artin--Tits groups. We prove that the Burau representation is not faithful in affine type $\tilde{A_3}$, and not faithful over several finite rings in type $D_4$, using an algorithmic approach based on categorical methods that generalize Bigelow's curve strategy outside of type $A$., Comment: Comments welcome!

We recover Reidemeister's theorem using smooth functions and transversality., Comment: My goal here was to give an easily readable proof for the Reidemeister theorem based on transversality techniques. I hope it can prove useful when learning or teaching knot theory, or as a gentle introduction to the use of transversality for knots and knotted surfaces. The paper will appear in the Bulletin of the Australian Mathematical Society

These are lecture notes from a lecture series given at CIRM in the Fall 2023. They give a down-to-earth introduction to Khovanov and Seidel's categorical representation of Artin-Tits groups, emphasizing the fact that it is all explicitly computable. Several prospective applications to (geometric) group theory are mentioned., Comment: Lecture notes from the CIRM conference "Current trends in representation theory, cluster algebras and geometry", November 2023. Comments welcome!

Since their introduction in 1997, the Hardy spaces of Dirichlet series have been broadly and deeply studied. The increasing interest sparked by these Banach spaces of Dirichlet series motivated the introduction of new such spaces, as the Bergman spaces of Dirichlet series $\mathcal{A}^p_{\mu}$ here considered, where $\mu$ is a probability measure on $(0,\infty)$. Similarly, recent lines of research have focused their attention on the study of some classical operators acting on these spaces, as it is the case of the Volterra operator $T_g$. In this work, we introduce a new family of Bloch spaces of Dirichlet series, the $\text{Bloch}_{\mu}$-spaces, and study some of its most essential properties. Using these spaces we are able to provide a sufficient condition for the Volterra operator $T_g$ to act boundedly on the Bergman spaces $\mathcal{A}^p_{\mu}$. We also establish a necessary condition for a specific choice of the probability measures $\mu$. Sufficient and necessary conditions for compactness are also proven. The membership in Schatten classes is studied as well. Eventually, a radicality result is established for Bloch spaces of Dirichlet series., Comment: Main changes in Section 3 and Section 7. Some results have been improved and some proofs have been written more clearly

The theory of Banach spaces of Dirichlet series has drawn an increasing attention in the recent 25 years. One of the main interest of this new theory is that of defining analogues of the classical spaces of analytic functions on the unit disc. In this sense, Bergman spaces were introduced several years ago contributing to broaden the picture of this theory. In the paper presenting these new family of spaces, some of its most essential questions were considered. Among them, some partial estimates of the norm of the pointwise evaluation functional were given. In this work, we introduce a version of the Riemann-Liouville semigroup acting on these spaces, and, with this new tool, we are able to estimate the norm of this functional., Comment: Somo general corrections in order to ease the reading and the statements of the results

We give a complete characterization of the sequences $\beta = (\beta_n)$ of positive numbers for which all composition operators on $H^2 (\beta)$ are bounded, where $H^2 (\beta)$ is the space of analytic functions $f$ on the unit disk ${\mathbb D}$ such that $\sum_{n = 0}^\infty |a_n|^2 \beta_n < + \infty$ if $f (z) = \sum_{n = 0}^\infty a_n z^n$. We prove that all composition operators are bounded on $H^2 (\beta)$ if and only if $\beta$ is essentially decreasing and slowly oscillating. We also prove that every automorphism of the unit disk induces a bounded composition operator on $H^2 (\beta)$ if and only if $\beta$ is slowly oscillating. We give applications of our results., Comment: arXiv admin note: substantial text overlap with arXiv:2011.14928

We first consider some questions raised by N. Zorboska in her thesis. In particular she asked for which sequences $\beta$ every symbol $\varphi \colon \mathbb{D} \to \mathbb{D}$ with $\varphi \in H^2 (\beta)$ induces a bounded composition operator $C_\phi$ on the weighted Hardy space $H^2 (\beta)$. We give partial answers and investigate when $H^2 (\beta)$ is an algebra. We answer negatively another question in showing that there are a sequence $\beta$ and $\varphi \in H^2 (\beta)$ such that $\| \varphi \|_\infty < 1$ and the composition operator $C_\varphi$ is not bounded on $H^2 (\beta)$. In a second part, we show that for $p \neq 2$, no automorphism of $\mathbb{D}$, except those that fix $0$, induces a bounded composition operator on the Beurling-Sobolev space $\ell^p_A$, and even on any weighted version of this space.

This paper is a complement to our previous paper [21]. It surveys the works on the Furstenberg set $S=\{2^{m}3^{n}: n\ge 0, m\ge 0\}$ and its random version $T$. We also present some new results. For example, it is proved that $T$ almost surely contains a subset of positive lower density which is $\frac{4}{3}$-Rider. It is also proved that a class of random sets of integers are Sidon sets when Bourgain's condition is not satisfied; this generalizes a result of Kahane-Katznelson. Some open questions about $S$ and $T$ are listed at the end of the paper.

We show that the symbol of a bounded composition operator on the Wiener algebra of Dirichlet series does not need to belong to this algebra. Our example even gives an absolutely summing (hence compact) composition operator.

We prove full functoriality of Khovanov homology for tangled framed gl2 webs. We use this functoriality result to prove a strong positivity result for (orientable) surface skein algebras. The argument goes categorical and consists in proving that so-called linear complexes are stable under superposition., Comment: 71 pages. Preliminary version, comments welcome!

We use multijet transversality techniques to give a presentation by generators and relations of the categories of framed tangled webs and foams., Comment: 88 pages, many figures

We study some number-theoretic, ergodic and harmonic analysis properties of the Furstenberg set of integers $S=\{2^{m}3^{n}\}$ and compare them to those of its random analogue $T$. In this half-expository work, we show for example that $S$ is "Khinchin distributed", is far from being Hartman-distributed while $T$ is, and that $S$ is a $\Lambda(p)$ set for all $2

We characterize the symbols $\Phi$ for which there exists a weight w such that the weighted composition operator M w C $\Phi$ is compact on the weighted Bergman space B 2 $\alpha$. We also characterize the symbols for which there exists a weight w such that M w C $\Phi$ is bounded but not compact. We also investigate when there exists w such that M w C $\Phi$ is Hilbert-Schmidt on B 2 $\alpha$.

The definition of the Jones polynomial in the 80's gave rise to a large family of so-called quantum link invariants, based on quantum groups. These quantum invariants are all controlled by the same two-variable invariant (the HOMFLY-PT polynomial), which also specializes to the older Alexander polynomial. Building upon quantum Schur--Weyl duality and variants of this phenomenon, I will explain an algebraic setup that allows for global definitions of these quantum polynomials, and discuss extensions of these quantum objects designed to encompass all of the mentioned invariants, including the HOMFLY-PT polynomial., Comment: Lecture notes from Winter Braids IX conference, to appear in the Winter Braids Lecture Notes

We characterize the (essentially) decreasing sequences of positive numbers $\beta$ = ($\beta$ n) for which all composition operators on H 2 ($\beta$) are bounded, where H 2 ($\beta$) is the space of analytic functions f in the unit disk such that $\infty$ n=0 |c n | 2 $\beta$ n < $\infty$ if f (z) = $\infty$ n=0 c n z n. We also give conditions for the boundedness when $\beta$ is not assumed essentially decreasing.

An automatic sequence is a letter-to-letter coding of a fixed point of a uniform morphism. More generally, we have morphic sequences, which are letter-to-letter codings of fixed points of arbitrary morphisms. There are many examples where an, a priori, morphic sequence with a \emph{non-uniform} morphism happens to be an automatic sequence. An example is the Lys\"enok morphism $a \to aca$, $b \to d$, $c \to b$, $d \to c$, the fixed point of which is also a 2-automatic sequence. Such an identification is useful for the description of the dynamical systems generated by the fixed point. We give several ways to uncover such hidden automatic sequences, and present many examples. We focus in particular on morphisms associated with Grigorchuk(-like) groups.

We study a variant of the multi-agent path finding problem (MAPF) in which agents are required to remain connected to each other and to a designated base. This problem has applications in search and rescue missions where the entire execution must be monitored by a human operator. We re-visit the conflict-based search algorithm known for MAPF, and define a variant where conflicts arise from disconnections rather than collisions. We study optimizations, and give experimental results in which we compare our algorithms with the literature.

We prove that the norm of the Riesz projection from $L^\infty(\Bbb{T}^n)$ to $L^p(\Bbb{T}^n)$ is $1$ for all $n\ge 1$ only if $p\le 2$, thus solving a problem posed by Marzo and Seip in 2011. This shows that $H^p(\Bbb{T}^{\infty})$ does not contain the dual space of $H^1(\Bbb{T}^{\infty})$ for any $p>2$. We then note that the dual of $H^1(\Bbb{T}^{\infty})$ contains, via the Bohr lift, the space of Dirichlet series in $\operatorname{BMOA}$ of the right half-plane. We give several conditions showing how this $\operatorname{BMOA}$ space relates to other spaces of Dirichlet series. Finally, relating the partial sum operator for Dirichlet series to Riesz projection on $\Bbb{T}$, we compute its $L^p$ norm when $1

We compare the rate of decay of singular numbers of a given composition operator acting on various Hilbert spaces of analytic functions on the unit disk $\D$. We show that for the Hardy and Bergman spaces, our results are sharp. We also give lower and upper estimates of the singular numbers of the composition operator with symbol the ``cusp map'' and the lens maps, acting on weighted Dirichlet spaces.

We study when multiplication by a weight can turn a non-compact composition operator on H 2 into a compact operator, and when it can be in Schatten classes. The q-summing case in H p is considered. We also study when this multiplication can turn a compact composition operator into a non-compact one. MSC 2010 primary: 47B33 ; secondary: 46B28

Bernstein's classical inequality asserts that given a trigonometric polynomial $T$ of degree $n\geq1$, the sup-norm of the derivative of $T$ does not exceed $n$ times the sup-norm of $T$. We present various approaches to prove this inequality and some of its natural extensions/variants, especially when it comes to replacing the sup-norm with the $L^p-norm$.

Motivated by the increasing appeal of robots in information-gathering missions, we study multi-agent path planning problems in which the agents must remain interconnected. We model an area by a topological graph specifying the movement and the connectivity constraints of the agents. We study the theoretical complexity of the reachability and the coverage problems of a fleet of connected agents on various classes of topological graphs. We establish the complexity of these problems on known classes, and introduce a new class called sight-moveable graphs which admit efficient algorithms.

We construct an analytic self-map $\Phi$ of the bidisk ${\mathbb D}^2$ whose image touches the distinguished boundary, but whose approximation numbers of the associated composition operator on $H^2 ({\mathbb D}^2)$ are small in the sense that $\limsup_{n \to \infty} [a_{n^2} (C_\Phi)]^{1 / n} < 1$.

We give estimates of the entropy numbers of composition operators on the Hardy space of the disk and of the polydisk.

We give a new proof of the existence of a surjective symbol whose associated composition operator on H 2 (D) is in all Schatten classes, with the improvement that its approximation numbers can be, in some sense, arbitrarily small. We show, as an application, that, contrary to the 1-dimensional case, for N $\ge$ 2, the behavior of the approximation numbers a n = a n (C $\Phi$), or rather of $\beta$ -- N = lim inf n$\rightarrow$$\infty$ [a n ] 1/n 1/N or $\beta$ + N = lim sup n$\rightarrow$$\infty$ [a n ] 1/n 1/N , of composition operators on H 2 (D N) cannot be determined by the image of the symbol. MSC 2010 Primary: 47B33 Secondary: 32A35 ; 46B28

For suitable bounded hyperconvex sets $\Omega$ in $\mathbb{C}^N$, in particular the ball or the polydisk, we give estimates for the approximation numbers of composition operators $C_\phi \colon H^2 (\Omega) \to H^2 (\Omega)$ when $\phi (\Omega)$ is relatively compact in $\Omega$, involving the Monge-Amp\`ere capacity of $\phi (\Omega)$.

For every oriented surface of finite type, we construct a functorial Khovanov homology for links in a thickening of the surface, which takes values in a categorification of the corresponding gl(2) skein module. The latter is a mild refinement of the Kauffman bracket skein algebra, and its categorification is constructed using a category of gl(2) foams that admits an interesting non-negative grading. We expect that the natural algebra structure on the gl(2) skein module can be categorified by a tensor product that makes the surface link homology functor monoidal. We construct a candidate bifunctor on the target category and conjecture that it extends to a monoidal structure. This would give rise to a canonical basis of the associated gl(2) skein algebra and verify an analogue of a positivity conjecture of Fock--Goncharov and Thurston. We provide evidence towards the monoidality conjecture by checking several instances of a categorified Frohman-Gelca formula for the skein algebra of the torus. Finally, we recover a variant of the Asaeda--Przytycki--Sikora surface link homologies and prove that surface embeddings give rise to spectral sequences between them., Comment: 56 pages

In previous work, we have constructed diagrammatic idempotents in an affine extension of the Temperley-Lieb category, which describe extremal weight projectors for sl(2), and which categorify Chebyshev polynomials of the first kind. In this paper, we generalize the construction of extremal weight projectors to the case of gl(N) for N greater than or equal to 2, with a view towards categorifying the corresponding torus skein algebras via Khovanov-Rozansky link homology. As by-products, we obtain compatible diagrammatic presentations of the representation categories of gl(N) and its Cartan subalgebra, and a categorification of power-sum symmetric polynomials., Comment: 32 pages

We use categorical annular evaluation to give a uniform construction of both $\mathfrak{sl}_n$ and HOMFLYPT Khovanov-Rozansky link homology, as well as annular versions of these theories. Variations on our construction yield $\mathfrak{gl}_{-n}$ link homology, i.e. a link homology theory associated to the Lie superalgebra $\mathfrak{gl}_{0|n}$, both for links in $S^3$ and in the thickened annulus. In the $n=2$ case, this produces a categorification of the Jones polynomial that we show is distinct from Khovanov homology, and gives a finite-dimensional categorification of the colored Jones polynomial. This behavior persists for general $n$. Our approach yields simple constructions of spectral sequences relating these theories, and emphasizes the roles of super vector spaces, categorical traces, and current algebras in link homology., Comment: 47 pages, many figures

By using the Schur test, we give some upper and lower estimates on the norm of a composition operator on $\mathcal{H}^2$, the space of Dirichlet series with square summable coefficients, for the inducing symbol $\varphi(s)=c_1+c_{q}q^{-s}$ where $q\geq 2$ is a fixed integer. We also give an estimate on the approximation numbers of such an operator., Comment: 12 pages, one figure; To appear in Integral Equations and Operator Theory

We give examples of composition operators $C\_\Phi$ on $H^2 (\D^2)$ showing that the condition $\|\Phi \|\_\infty = 1$ is not sufficient for their approximation numbers $a\_n (C\_\Phi)$ to satisfy $\lim\_{n \to \infty} [a\_n (C\_\Phi) ]^{1/\sqrt{n}} = 1$, contrary to the $1$-dimensional case. We also give a situation where this implication holds. We make a link with the Monge-Amp\`ere capacity of the image of $\Phi$.

We study Artin-Tits braid groups $\mathbb{B}_W$ of type ADE via the action of $\mathbb{B}_W$ on the homotopy category $\mathcal{K}$ of graded projective zigzag modules (which categorifies the action of the Weyl group $W$ on the root lattice). Following Brav-Thomas, we define a metric on $\mathbb{B}_W$ induced by the canonical $t$-structure on $\mathcal{K}$, and prove that this metric on $\mathbb{B}_W$ agrees with the word-length metric in the canonical generators of the standard positive monoid $\mathbb{B}_W^+$ of the braid group. We also define, for each choice of a Coxeter element $c$ in $W$, a baric structure on $\mathcal{K}$. We use these baric structures to define metrics on the braid group, and we identify these metrics with the word-length metrics in the Birman-Ko-Lee/Bessis dual generators of the associated dual positive monoid $\mathbb{B}_{W.c}^\vee$. As consequences, we give new proofs that the standard and dual positive monoids inject into the group, give linear-algebraic solutions to the membership problem in the standard and dual positive monoids, and provide new proofs of the faithfulness of the action of $\mathbb{B}_W$ on $\mathcal{K}$. Finally, we use the compatibility of the baric and $t$-structures on $\mathcal{K}$ to prove a conjecture of Digne and Gobet regarding the canonical word-length of the dual simple generators of ADE braid groups., Comment: 40 pages

We study the composition operators of the Hardy space on D $\infty$ $\cap$ {\ell} 1 , the {\ell} 1 part of the infinite polydisk, and the behavior of their approximation numbers.

We introduce a simple diagrammatic 2-category $\mathscr{A}$ that categorifies the image of the Fock space representation of the Heisenberg algebra and the basic representation of $\mathfrak{sl}_\infty$. We show that $\mathscr{A}$ is equivalent to a truncation of the Khovanov--Lauda categorified quantum group $\mathscr{U}$ of type $A_\infty$, and also to a truncation of Khovanov's Heisenberg 2-category $\mathscr{H}$. This equivalence is a categorification of the principal realization of the basic representation of $\mathfrak{sl}_\infty$. As a result of the categorical equivalences described above, certain actions of $\mathscr{H}$ induce actions of $\mathscr{U}$, and vice versa. In particular, we obtain an explicit action of $\mathscr{U}$ on representations of symmetric groups. We also explicitly compute the Grothendieck group of the truncation of $\mathscr{H}$. The 2-category $\mathscr{A}$ can be viewed as a graphical calculus describing the functors of $i$-induction and $i$-restriction for symmetric groups, together with the natural transformations between their compositions. The resulting computational tool is used to give simple diagrammatic proofs of (apparently new) representation theoretic identities., Comment: 36 pages. v2: Minor corrections, published version

We introduce a quotient of the affine Temperley-Lieb category that encodes all weight-preserving linear maps between finite-dimensional sl(2)-representations. We study the diagrammatic idempotents that correspond to projections onto extremal weight spaces and find that they satisfy similar properties as Jones-Wenzl projectors, and that they categorify the Chebyshev polynomials of the first kind. This gives a categorification of the Kauffman bracket skein algebra of the annulus, which is well adapted to the task of categorifying the multiplication on the Kauffman bracket skein module of the torus., Comment: 16 pages

We study the approximation numbers of weighted composition operators $f\mapsto w\cdot(f\circ\varphi)$ on the Hardy space $H^2$ on the unit disc. For general classes of such operators, upper and lower bounds on their approximation numbers are derived. For the special class of weighted lens map composition operators with specific weights, we show how much the weight $w$ can improve the decay rate of the approximation numbers, and give sharp upper and lower bounds. These examples are motivated from applications to the analysis of relative commutants of special inclusions of von Neumann algebras appearing in quantum field theory (Borchers triples)., Comment: 35 pages, no figures. Some typos removed, minor improvements in presentation, updated references

We introduce an sl(n) homology theory for knots and links in the thickened annulus. To do so, we first give a fresh perspective on sutured annular Khovanov homology, showing that its definition follows naturally from trace decategorifications of enhanced sl(2) foams and categorified quantum gl(m), via classical skew Howe duality. This framework then extends to give our annular sl(n) link homology theory, which we call sutured annular Khovanov-Rozansky homology. We show that the sl(n) sutured annular Khovanov-Rozansky homology of an annular link carries an action of the Lie algebra sl(n), which in the n=2 case recovers a result of Grigsby-Licata-Wehrli., Comment: 29 pages, many figures

We present a short and unified representation-theoretical treatment of type A link invariants (that is, the HOMFLY-PT polynomials, the Jones polynomial, the Alexander polynomial and, more generally, the gl(m|n) quantum invariants) as link invariants with values in the quantized oriented Brauer category.

We give general estimates for the approximation numbers of composition operators on the Hardy space on the ball $B\_d$ and the polydisk $D^d$ .

We define a ribbon category $\mathsf{Sp}(\beta)$, depending on a parameter $\beta$, which encompasses Cautis, Kamnitzer and Morrison's spider category, and describes for $\beta=m-n$ the monoidal category of representations of $U_q(\mathfrak{gl}_{m|n})$ generated by exterior powers of the vector representation and their duals. We identify this category $\mathsf{Sp}(\beta)$ with a direct limit of quotients of a dual idempotented quantum group $\dot{\mathsf{U}}_q(\mathfrak{gl}_{r+s})$, proving a mixed version of skew Howe duality in which exterior powers and their duals appear at the same time. We show that the category $\mathsf{Sp}(\beta)$ gives a unified natural setting for defining the colored $\mathfrak{gl}_{m|n}$ link invariant (for $\beta=m-n$) and the colored HOMFLY-PT polynomial (for $\beta$ generic)., Comment: Corrected incomplete argument in Section 4

We give estimates for the approximation numbers of composition operators on the Hp spaces, 1 $\le$ p \textless{} $\infty$.

We define a generalization of the idempotented Schur algebra which gives a unified setting for a quantum group presentation of the Homfly-Pt polynomial, together with its specializations to the Alexander polynomial and the slm Reshetikhin-Turaev invariants., Comment: 13 pages

For approximation numbers $a_n (C_\phi)$ of composition operators $C_\phi$ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol $\phi$ of uniform norm $< 1$, we prove that $\lim_{n \to \infty} [a_n (C_\phi)]^{1/n} = \e^{- 1/ \capa [\phi (\D)]}$, where $\capa [\phi (\D)]$ is the Green capacity of $\phi (\D)$ in $\D$. This formula holds also for $H^p$ with $1 \leq p < \infty$., Comment: 25 pages