arXiv subjects

Anders Karlsson

Publications and source records attributed to Anders Karlsson.

At least 19 recordsLinked to original sources

arXiv2026

Torsion subgroups and fixed-point rigidity in CAT(0) geometry

We develop new methods for studying groups acting on CAT(0) spaces, which lead to several general structural results. First, we prove that every torsion subgroup of a CAT(0) group is finite, resolving a question of Swenson from the 1990s. The proof is based on showing that random walks on any finitely generated torsion group with bounded exponent acting on a CAT(0) space have zero drift. This is then combined with the fixed-point rigidity that we develop. Second, we show that any finitely generated torsion group of bounded exponent has a global fixed point whenever it acts properly by isometries on a CAT(0) space of bounded geometry, or, without the properness assumption, by isometries on a finite-dimensional CAT(0) space. Third, we establish a Kazhdan-type rigidity principle that underlies many of our results: let $\Gamma$ be a finitely generated group such that every isometric action of $\Gamma$ on $\mathbb{R}^n$ has a fixed point. Then every fixed-point-free action of $\Gamma$ on a geodesically complete $n$-dimensional CAT(0) space of bounded geometry has joint minimal displacement uniformly bounded away from zero. In particular, almost fixed points imply a global fixed point. This applies in particular to groups with property (T), torsion groups, certain branch groups, and mapping class groups. Fourth, we establish the following alternative for any finitely generated amenable group: either every action on a finite-dimensional CAT(0) space has a global fixed point, or the group has non-vanishing virtual first Betti number. Further consequences include that finitely generated torsion groups cannot act without a global fixed point on geodesically complete CAT(0) spaces of bounded geometry that are either visibility spaces or have compact Tits boundary. The methods involve scalings of actions by ultralimits and random walks.

math.GR↗

arXiv2025

A discrete approach to Dirichlet L-functions, their special values and zeros

We develop a discrete spectral framework for Dirichlet $L$-functions that reveals a combinatorial structure underlying their special values and connects this to their zeros. Our approach approximates the classical Dirichlet series by finite spectral sums $L_n(s,\chi)$ associated with cyclic graphs $\mathbb{Z}/n\mathbb{Z}$ and studies their asymptotics as $n\rightarrow \infty$. Combining a refined Euler Maclaurin expansion with a structural polynomiality property, we show that at integer arguments the asymptotic expansions terminate and yield exact identities. This asymptotic to exact principle produces new infinite families of relations among special values of Dirichlet $L$-functions and recovers, by a different mechanism, formulas previously obtained by Xie, Zhao and Zhao. An interesting feature of our method is that $\zeta(2n)$ and the corresponding special values for all Dirichlet $L$-functions thereby admit a finite combinatorial interpretation in terms of rooted spanning forests on any fixed cyclic graph. Concerning zeros, the same framework leads to some remarks about real zeros and a reformulation of the Generalized Riemann Hypothesis in the case of odd primitive characters in terms of an asymptotic functional equation relating $\xi_n(1-s,\overline{\chi})$ to $\xi_n(s,\chi)$ of the completed discrete functions. This establishes the remaining case of the one dimensional picture obtained in earlier works.

math.NT↗

arXiv2025

The uniqueness of inverse scattering problems, reciprocity principles, and nonradiating sources related to low-signature structures

This paper is about perfectly electrically conducting structures designed to produce negligible scattered power when exposed to a time-harmonic plane electromagnetic wave. The structures feature cavities capable of concealing objects. Theoretical investigations of the properties of the structures combined with accurate numerical computations lead to three key findings: the first concerns the uniqueness of the solution to an inverse scattering problem, the second establishes a reciprocity relation for the far-field scattering amplitude, and the third reveals the existence of non-radiating sources that generate substantial electromagnetic fields near the source region. The results have applications in low-observable technology.

physics.comp-ph↗

arXiv2025

On the heat kernel of a Cayley graph of $\operatorname{PSL}_2\mathbb{Z}$

In this paper, we obtain an explicit formula for the heat kernel on the Cayley graph of the modular group $PSL_2(Z)$, given by the presentation $\langle a,b\mid a^2=1, b^3=1\rangle$. Our approach extends a method of Chung--Yau by observing that the Cayley graph strongly and regularly covers a weighted infinite line. We solve the spectral problem on this line to obtain an integral expression for its heat kernel, and then lift this to the Cayley graph using spectral transfer principles for strongly regular coverings. The explicit formula allows us to determine the Laplace spectrum, containing eigenvalues and continuous parts. As a by-product, we suggest a conjecture on the lower bound for the spectral gap of Cayley graphs of $\operatorname{PSL}_2\mathbb{F}_p$ with our generators, inspired by the analogy with Selberg's $1/4$-conjecture. Numerical evidence to this conjecture is provided for small primes.

math.GR↗

arXiv2025

Periodic approximation of topological Lyapunov exponents and the joint spectral radius for cocycles of mapping classes of surfaces

We study cocycles taking values in the mapping class group of closed surfaces and investigate their leading topological Lyapunov exponent. Under a natural closing property, we show that the top topological Lyapunov exponent can be approximated by periodic orbits. We also extend the notion of the joint spectral radius to this setting, interpreting it via the exponential growth of curves under iterated mapping classes. Our approach connects ideas from ergodic theory, Teichmüller geometry, and spectral theory, and suggests a broader framework for similar results.

math.DS↗

arXiv2025

How iteration order influences convergence and stability in deep learning

Despite exceptional achievements, training neural networks remains computationally expensive and is often plagued by instabilities that can degrade convergence. While learning rate schedules can help mitigate these issues, finding optimal schedules is time-consuming and resource-intensive. This work explores theoretical issues concerning training stability in the constant-learning-rate (i.e., without schedule) and small-batch-size regime. Surprisingly, we show that the composition order of gradient updates affects stability and convergence in gradient-based optimizers. We illustrate this new line of thinking using backward-SGD, which produces parameter iterates at each step by reverting the usual forward composition order of batch gradients. Our theoretical analysis shows that in contractive regions (e.g., around minima) backward-SGD converges to a point while the standard forward-SGD generally only converges to a distribution. This leads to improved stability and convergence which we demonstrate experimentally. While full backward-SGD is computationally intensive in practice, it highlights that the extra freedom of modifying the usual iteration composition by reusing creatively previous batches at each optimization step may have important beneficial effects in improving training. Our experiments provide a proof of concept supporting this phenomenon. To our knowledge, this represents a new and unexplored avenue in deep learning optimization.

cs.LG↗

arXiv2024

The discrete analogue of the Gaussian

This paper illustrates the utility of the heat kernel on $\mathbb{Z}$ as the discrete analogue of the Gaussian density function. It is the two-variable function $K_{\mathbb{Z}}(t,x)=e^{-2t}I_{x}(2t)$ involving a Bessel function and variables $x\in\mathbb{Z}$ and real $t\geq 0$. Like its classic counterpart it appears in many mathematical and physical contexts and has a wealth of applications. Some of these will be reviewed here, concerning Bessel integrals, trigonometric sums, hypergeometric functions and asymptotics of discrete models appearing in statistical and quantum physics. Moreover, we prove a new local limit theorem for sums of integer-valued random variables, obtain novel special values of the spectral zeta function of Bethe lattices, and provide a discussion on how $e^{-2t}I_{x}(2t)$ could be useful in differential privacy.

math-ph↗

arXiv2024

Constructing heat kernels on infinite graphs

Let $G$ be an infinite, edge- and vertex-weighted graph with certain reasonable restrictions. We construct the heat kernel of the associated Laplacian using an adaptation of the parametrix approach due to Minakshisundaram-Pleijel in the setting of Riemannian geometry. This is partly motivated by the wish to relate the heat kernels of a graph and a subgraph, or of a domain and a discretization of it. As an application, assuming that the graph is locally finite, we express the heat kernel $H_G(x,y;t)$ as a Taylor series with the lead term being $a(x,y)t^r$, where $r$ is the combinatorial distance between $x$ and $y$ and $a(x,y)$ depends (explicitly) upon edge and vertex weights. In the case $G$ is the regular $(q+1)$-tree with $q\geq 1$, our construction reproves different explicit formulas due to Chung-Yau and to Chinta-Jorgenson-Karlsson. Assuming uniform boundedness of the combinatorial vertex degree, we show that a dilated Gaussian depending on any distance metric on $G$, which is uniformly bounded from below can be taken as a parametrix in our construction. Our work extends in part the recent articles [LNY21, CJKS23] in that the graphs are infinite and weighted.

math.AP↗

arXiv2024

Torsion groups of subexponential growth cannot act on finite-dimensional CAT(0)-spaces without a fixed point

We show that finitely generated groups which are Liouville and without infinite finite-dimensional linear representations must have a global fixed point whenever they act by isometry on a finite-dimensional complete CAT(0)-space. This provides a partial answer to an old question in geometric group theory and proves partly a conjecture formulated by Norin, Osajda, and Przytycki. It applies in particular to Grigorchuk's groups of intermediate growth and other branch groups as well as to simple groups with the Liouville property such as those found by Matte Bon and by Nekrashevych. The method of proof uses ultralimits, equivariant harmonic maps, subharmonic functions, horofunctions and random walks.

math.GR↗

arXiv2024

Design of perfectly conducting objects that are invisible to an incident plane wave

This work concerns the design of perfectly conducting objects that are invisible to an incident transverse magnetic plane wave. The object in question is a finite planar waveguide with a finite periodic array of barriers. By optimizing this array, the amplitude of the scattered field is reduced to less than $10^{-9}$ times the amplitude of the incident plane wave everywhere outside the waveguide. To accurately evaluate such minute amplitudes, we employ a recently developed boundary integral equation technique, adapted for objects whose boundaries have endpoints, corners, and branch points.

physics.comp-ph↗

arXiv2024

Backward dynamics of non-expanding maps in Gromov hyperbolic metric spaces

We study the interplay between the backward dynamics of a non-expanding self-map $f$ of a proper geodesic Gromov hyperbolic metric space $X$ and the boundary regular fixed points of $f$ in the Gromov boundary. To do so, we introduce the notion of stable dilation at a boundary regular fixed point of the Gromov boundary, whose value is related to the dynamical behaviour of the fixed point. This theory applies in particular to holomorphic self-maps of bounded domains $Ω\subset\subset \mathbb{C}^q$, where $Ω$ is either strongly pseudoconvex, convex finite type, or pseudoconvex finite type with $q=2$, and solves several open problems from the literature. We extend results of holomorphic self-maps of the disc $\mathbb{D}\subset \mathbb{C}$ obtained by Bracci and Poggi-Corradini. In particular, with our geometric approach we are able to answer a question, open even for the unit ball $\mathbb{B}^q\subset \mathbb{C}^q$, namely that for holomorphic parabolic self-maps any escaping backward orbit with bounded step always converges to a point in the boundary.

math.CV↗

arXiv2023

The parametrix construction of the heat kernel on a graph

In this paper we develop the parametrix approach for constructing the heat kernel on a graph $G$. In particular, we highlight two specific cases. First, we consider the case when $G$ is embedded in a Eulidean domain or manifold $Ω$, and we use a heat kernel associated to $Ω$ to obtain a formula for the heat kernel on $G$. Second, we consider when $G$ is a subgraph of a larger graph $\widetilde{G}$, and we obtain a formula for the heat kernel on $G$ from the heat kernel on $\widetilde{G}$ restricted to $G$.

math.AP↗

arXiv2023

The stars at infinity in several complex variables

This text reviews certain notions in metric geometry that may have further applications to problems in complex geometry and holomorphic dynamics in several variables. The discussion contains a few unrecorded results and formulates a number of questions related to the asymptotic geometry and boundary estimates of bounded complex domains, boundary extensions of biholomorphisms, the dynamics of holomorphic self-maps, Teichmüller theory, and the existence of constant scalar curvature metrics on compact Kähler manifolds.

math.CV↗

arXiv2023

The resolvent kernel on the discrete circle and twisted cosecant sums

Let $X_m$ denote the discrete circle with $m$ vertices. For $x,y\in X_{m}$ and complex $s$, let $G_{X_m,χ_β}(x,y;s)$ be the resolvent kernel associated to the combinatorial Laplacian which acts on the space of functions on $X_{m}$ that are twisted by a character $χ_β$. We will compute $G_{X_m,χ_β}(x,y;s)$ in two different ways. First, using the spectral expansion of the Laplacian, we show that $G_{X_m,χ_β}(x,y;s)$ is a generating function for certain trigonometric sums involving powers of the cosecant function; by choosing $β$ or $s$ appropriately, the sums in question involve powers of the secant function. Second, by viewing $X_{m}$ as a quotient space of $\mathbb{Z}$, we prove that $G_{X_m,χ_β}(x,y;s)$ is a rational function which is given in terms of Chebyshev polynomials. From the existence and uniqueness of $G_{X_m,χ_β}(x,y;s)$, these two evaluations are equal. From the resulting identity, we obtain a means by which one can obtain explicit evaluations of cosecant and secant sums. The identities we prove depend on a number of parameters, and when we specialize the values of these parameters we obtain several previously known formulas. Going further, we derive a recursion formula for special values of the $L$-functions associated to the cycle graph $X_{m}$, thus answering a question from arXiv:2212.13687v1.

math.CO↗

arXiv2023

A metric fixed point theorem and some of its applications

A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. It is new even for isometries of Banach spaces as well as for non-locally compact CAT(0)-spaces and injective spaces. Examples of actions on non-proper CAT(0)-spaces come from the study of diffeomorphism groups, birational transformations, and compact Kähler manifolds. A special case of the fixed point theorem provides a novel mean ergodic theorem that in the Hilbert space case implies von Neumann's theorem. The theorem accommodates classically fixed-point-free isometric maps such as those of Kakutani, Edelstein, Alspach and Prus. Moreover, from the main theorem together with some geometric arguments of independent interest, one can deduce that every bounded invertible operator of a Hilbert space admits a nontrivial invariant metric functional on the space of positive operators. This is a result in the direction of the invariant subspace problem although its full meaning is dependent on a future determination of such metric functionals.

math.FA↗

arXiv2022

Generalized Lyapunov exponents and aspects of the theory of deep learning

We discuss certain recent metric space methods and some of the possibilities these methods provide, with special focus on various generalizations of Lyapunov exponents originally appearing in the theory of dynamical systems and differential equations. These generalizations appear for example in topology, group theory, probability theory, operator theory and deep learning.

math.DS↗

arXiv2022

An efficient full wave solver for eddy currents

An integral equation reformulation of the Maxwell transmission problem is presented. The reformulation uses techniques such as tuning of free parameters and augmentation of close-to-rank-deficient operators. It is designed for the eddy current regime and works both for surfaces of genus $0$ and $1$. Well-conditioned systems and field representations are obtained despite the Maxwell transmission problem being ill-conditioned for genus $1$ surfaces due to the presence of Neumann eigenfields. Furthermore, it is shown that these eigenfields, for ordinary conductors in the eddy current regime, are different from the classical Neumann eigenfields for superconductors. Numerical examples, based on the reformulation, give an unprecedented $13$-digit accuracy both for transmitted and scattered fields.

math.AP↗

arXiv2022

Volumes of spheres and special values of zeta functions of $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$

The volume of the unit sphere in every dimension is given a new interpretation as a product of special values of the zeta function of $\mathbb{Z}$, akin to volume formulas of Minkowski and Siegel in the theory of arithmetic groups. A product formula is found for this zeta function that specializes to Catalan numbers. Moreover, certain closed-form expressions for various other zeta values are deduced, in particular leading to an alternative perspective on Euler's values of the Riemann zeta function.

math.NT↗