Higher-rank Groups Research Articles

Character rigidity for lattices and commensurators

abstract: We prove an operator algebraic superrigidity statement for homomorphisms of irreducible lattices, and also their commensurators, from certain higher-rank groups into unitary groups of finite factors. This extends the authors' previous work regarding non-free measure-preserving actions, and also answers a question of Connes for such groups.

3-Manifolds and VOA Characters

By studying the properties of q-series Z^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widehat{Z}$$\\end{document}-invariants, we develop a dictionary between 3-manifolds and vertex algebras. In particular, we generalize previously known entries in this dictionary to Lie groups of higher rank, to 3-manifolds with toral boundaries, and to BPS partition functions with line operators. This provides a new physical realization of logarithmic vertex algebras in the framework of the 3d-3d correspondence and opens new avenues for their future study. For example, we illustrate how invoking a knot-quiver correspondence for Z^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widehat{Z}$$\\end{document}-invariants leads to many infinite families of new fermionic formulae for VOA characters.

A new geographical record of Abelisauridae (Theropoda, Dinosauria) for the Bauru Group (Upper Cretaceous)

Theropod teeth allow high-rank group assignments based on both phylogenetic positioning and morphometry. To the Bauru Group, abelisaurid distribution is known mostly from isolated teeth from relatively few localities. Based on teeth we extend their distribution to Ituiutaba, NNE of the basin. Teeth specimens from the neighboring Prata county, historically attributed to Carcharodontosauridae and Abelisauridae were also analyzed. For taxon assignment, we used methods such as phylogenetic, discriminant analysis, and direct comparisons. In our phylogenetic analysis, all teeth were recovered within abelisaurids whereas in the discriminant analyses, they nested close or within abelisaurids, neovenatorids, and dromaeosaurids. Therefore, our analyses support that teeth from Ituiutaba/Prata can be assigned to Abelisauridae, evidencing a continuous spatial distribution along the Bauru Group and those sites overlaying the Serra Geral formation in the southeastern state of Mato Grosso. Further studies at Ituiutaba, including information on the associated fauna, may subsidize the understanding of abelisaurid ecological requirements and the paleoenvironment of the Adamantina Formation.

Empathic forecasting of the big-fish-little-pond effect

ABSTRACT The big-fish-little-pond effect (BFLPE) is the tendency for students to evaluate themselves more favourably when they have high rank in a low rank school than low rank in a high rank school. Research has documented the BFLPE on experienced emotions. We conducted three studies that examined forecasts of how the BFLPE influences other people’s emotions (i.e. empathic forecasts). In Study 1, participants received performance feedback about themselves or another person and reported their own affect or anticipated the other person’s affect. Results extended the BFLPE to empathic forecasting. Moreover, forecasters anticipated that the BFLPE had a stronger influence on negative emotion than it actually did. Study 2 tested whether neglect of group rank underlies the BFLPE in empathic forecasting. Empathic forecasts were strongly influenced by another person’s rank in their group, but only weakly influenced by group rank. Study 3 tested whether extremity of group ranks exacerbates the BFLPE in empathic forecasting. Empathic forecasts were especially favourable (unfavourable) when a target had very high (low) rank in a very low (high) rank group. These data support the BFLPE in empathic forecasting, but also illustrate ways in which it is both similar to and different from actual experience of the effect.

Enhancing Diagnosis of Rotating Elements in Roll-to-Roll Manufacturing Systems through Feature Selection Approach Considering Overlapping Data Density and Distance Analysis.

Roll-to-roll manufacturing systems have been widely adopted for their cost-effectiveness, eco-friendliness, and mass-production capabilities, utilizing thin and flexible substrates. However, in these systems, defects in the rotating components such as the rollers and bearings can result in severe defects in the functional layers. Therefore, the development of an intelligent diagnostic model is crucial for effectively identifying these rotating component defects. In this study, a quantitative feature-selection method, feature partial density, to develop high-efficiency diagnostic models was proposed. The feature combinations extracted from the measured signals were evaluated based on the partial density, which is the density of the remaining data excluding the highest class in overlapping regions and the Mahalanobis distance by class to assess the classification performance of the models. The validity of the proposed algorithm was verified through the construction of ranked model groups and comparison with existing feature-selection methods. The high-ranking group selected by the algorithm outperformed the other groups in terms of training time, accuracy, and positive predictive value. Moreover, the top feature combination demonstrated superior performance across all indicators compared to existing methods.

𝐴𝑢𝑡-invariant quasimorphisms on groups

For a large class of groups, we exhibit an infinite-dimensional space of homogeneous quasimorphisms that are invariant under the action of the automorphism group. This class includes non-elementary hyperbolic groups, infinitely-ended finitely generated groups, some relatively hyperbolic groups, and a class of graph products of groups that includes all right-angled Artin and Coxeter groups that are not virtually abelian. This was known for F 2 F_2 by a result of Brandenbursky and Marcinkowski [Comment. Math. Helv. 94 (2019), pp. 661–687], but is new even for free groups of higher rank, settling a question of Miklós Abért. The case of graph products of finitely generated abelian groups settles a question of Michał Marcinkowski. As a consequence, we deduce that a variety of A u t Aut -invariant norms on such groups are unbounded.

Conjugation-invariant norms on SL2(R) for rings of S-algebraic integers with infinitely many units

This paper studies conjugation-invariant norms on for R a ring of S-algebraic integers with infinitely many units in a number field. We will show that barring some differences in the proofs the properties of these norms strongly resemble related properties of conjugation-invariant norms on arithmetic Chevalley groups of higher rank. Specifically, we prove two properties: First, we show that the diameter of each word norm on defined by finitely many conjugacy classes has an upper bound linear in the number of conjugacy-classes. Second, we show that all non-discrete conjugation-invariant norms on have a profinite norm-completion. Both of these properties of conjugation-invariant norms are already known for higher rank arithmetic Chevalley groups.

Formulae of ı-divided powers in [formula omitted], III

The ı-divided powers (depending on a parity) form the ıcanonical basis for the split rank 1 ıquantum group and they are a basic ingredient for ıquantum groups of higher rank. We obtain closed formulae for the structure constants for multiplication of the ı-divided powers. Closed formulae for the comultiplication of the ı-divided powers are also obtained. These structure constants are integral and positive.

Fourier non-uniqueness sets from totally real number fields

Let K be a totally real number field of degree n \geq 2 . The inverse different of K gives rise to a lattice in \mathbb{R}^n . We prove that the space of Schwartz Fourier eigenfunctions on \mathbb{R}^n which vanish on the “component-wise square root” of this lattice, is infinite dimensional. The Fourier non-uniqueness set thus obtained is a discrete subset of the union of all spheres \sqrt{m}S^{n-1} for integers m \geq 0 and, as m \rightarrow \infty , there are \sim c_{K} m^{n-1} many points on the m -th sphere for some explicit constant c_{K} , proportional to the square root of the discriminant of K . This contrasts a recent Fourier uniqueness result by Stoller (2021) Using a different construction involving the codifferent of K , we prove an analogue for discrete subsets of ellipsoids. In special cases, these sets also lie on spheres with more densely spaced radii, but with fewer points on each. We also study a related question about existence of Fourier interpolation formulas with nodes “ \sqrt{\Lambda} ” for general lattices \Lambda \subset \mathbb{R}^n . Using results about lattices in Lie groups of higher rank we prove that if n \geq 2 and a certain group \Gamma_{\Lambda} \leq \operatorname{PSL}_2(\mathbb{R})^n is discrete, then such interpolation formulas cannot exist. Motivated by these more general considerations, we revisit the case of one radial variable and prove, for all n \geq 5 and all real \lambda > 2 , Fourier interpolation results for sequences of spheres \sqrt{2 m/ \lambda}S^{n-1} , where m ranges over any fixed cofinite set of non-negative integers. The proof relies on a series of Poincaré type for Hecke groups of infinite covolume and is similar to the one in Stoller (2021).

Big principal series, -adic families and -invariants

In earlier work, the first named author generalized the construction of Darmon-style$\mathcal {L}$-invariants to cuspidal automorphic representations of semisimple groups of higher rank, which are cohomological with respect to the trivial coefficient system and Steinberg at a fixed prime. In this paper, assuming that the Archimedean component of the group has discrete series we show that these automorphic$\mathcal {L}$-invariants can be computed in terms of derivatives of Hecke eigenvalues in$p$-adic families. Our proof is novel even in the case of modular forms, which was established by Bertolini, Darmon and Iovita. The main new technical ingredient is the Koszul resolution of locally analytic principal series representations by Kohlhaase and Schraen. As an application of our results we settle a conjecture of Spieß: we show that automorphic$\mathcal {L}$-invariants of Hilbert modular forms of parallel weight$2$are independent of the sign character used to define them. Moreover, we show that they are invariant under Jacquet–Langlands transfer and, in fact, equal to the Fontaine–Mazur$\mathcal {L}$-invariant of the associated Galois representation. Under mild assumptions, we also prove the equality of automorphic and Fontaine–Mazur$\mathcal {L}$-invariants for representations of definite unitary groups of arbitrary rank. Finally, we study the case of Bianchi modular forms to show how our methods, given precise results on eigenvarieties, can also work in the absence of discrete series representations.

An ethnobotanical study of medicinal plants used to treat skin diseases by the native communities of Murree, Pakistan

The objective of this study was to document traditional medicinal plants (including their applications and associated procedures) used to treat skin diseases by native communities of Murree Hills (Pakistan). Data was collected from 5 different sampling sites and 138 informants were interviewed. Respondents were queried in their local language for the type of herbal treatment, famous to that area. It was observed that elderly people and health practitioners of the study area possessed more knowledge about herbal remedies in contrast to young aged people. Results reported that total 56 plants belonging to 36 families and 54 genera were in use by local communities for treatment of multiple skin diseases. It was detected that herbs (41%) constituted the highest proportion of medicinal plants followed by shrubs. Leaves were most reported plant part (43 %) followed by fruits (20%), whereas infusion (30%) was a widely used mode of preparation of traditional remedies and direct application on skin was the dominant route of administration. Relative frequency citation (RFC) was recorded highest for Mentha royleana (0.913) and least for Ajuga bracteosa (0.369). Cluster analysis grouped these medicinal species into high, medium and low-ranking groups. High ranking group comprised of 18 medicinal plants, which reflected the popularity and trust of local communities of these plants. Therefore, further pharmacological studies are highly recommended. This study is first proper documentation based on the valuable information attained by the local tribes and it can be utilized by scientific community for further validation and characterization.

Babai\u2019s conjecture for high-rank classical groups with random generators

Let G = {text {SCl}}_n(q) be a quasisimple classical group with n large, and let x_1, ldots , x_k in G be random, where k ge q^C. We show that the diameter of the resulting Cayley graph is bounded by q^2 n^{O(1)} with probability 1 - o(1). In the particular case G = {text {SL}}_n(p) with p a prime of bounded size, we show that the same holds for k = 3.

Automorphic Forms and Arithmetic

The workshop was at the interface of automorphic forms and analytic number theory. The aim was to disseminate, discuss and develop important new methods and results in the analytic theory of automorphic forms, in particular on higher rank groups, as well as their arithmetic applications. This includes, for instance, the study of various aspects of L -functions such as moments, reciprocity laws, and probabilistic aspects, counting problems with automorphic forms, applications of results of algebraic geometry to automorphic forms, as well as analytic aspects of automorphic forms over function fields.

Infinite co-minimal pairs involving lacunary sequences and generalisations to higher dimensions

The study of minimal complements in a group or a semigroup was initiated by Nathanson. The notion of minimal complements and being a minimal complement leads to the notion of co-minimal pairs which was considered in a prior work of the authors. In this article, we study which type of subsets in the integers and free abelian groups of higher rank can be a part of a co-minimal pair. We show that a majority of lacunary sequences have this property. From the conditions established, one can show that any infinite subset of any finitely generated abelian group has uncountably many subsets which is a part of a co-minimal pair. Further, the uncountable collection of sets can be chosen so that they satisfy certain algebraic properties.

The sparsity of character tables of high rank groups of Lie type

In the high rank limit, the fraction of non-zero character table entries of finite simple groups of Lie type goes to zero.

Density of the quasi $r$-rank Artin problem

For a given finitely generated multiplicative subgroup of the rationals which possibly contain negative numbers, we derive, subject to GRH, formulas for the densities of primes for which the index of the reduction group has a given value. We completely classify the cases of rank one, torsion groups for which the density vanishes and the the set of primes for which the index of the reduction group has a given value, is finite. For higher rank groups we propose some partial results. Finally, we present some computations comparing the approximated density computed with primes up to $10^{10}$ and those predicted by the Riemann Hypothesis.

The aspirational income hypothesis: On the limits of the relative income hypothesis

Abstract According to the “relative income hypothesis,” decision makers derive positive utility from identifying with a group that performs more poorly than they do. We hypothesize that decision makers simultaneously derive negative utility from identifying with such a group. The reason is that decision makers aspire toward status, and thus prefer to identify with a group that is more successful than they are. We call such proposed reason the “aspirational income hypothesis.” If the aspirational income effect dominates the relative income effect, decision makers would prefer to join groups with higher rank than their own. We report data from an experiment that supports the aspirational income hypothesis. It shows that the hypothesis holds (at a weaker intensity) even when it is costly to join a higher ranking group.

Pilot Study on the Reliability of the Coach's Eye: Identifying Talent Throughout a 4-Day Cadet Judo Camp.

A typical assumption found in talent identification literature is that different coaches, given the same athletes and circumstances, will identify the same subset of athletes as “talented”. However, while coaches play a major role during talent identification in practical sport settings, there is limited empirical research exploring the processes which underpin this. The purpose of this study was to explore the reliability of “the coach's eye” during the assessment of talent in a group of athletes. Specifically, this project compared inter-coach agreement between nine judo coaches (ages 35.8 ± 10.6 years) with varying levels of experience (12.9 ± 8.9 years) in the evaluation of 24 talented cadet judo athletes (13–15 years) at seven timepoints throughout a 4-day development training camp. Without discussion of their scores with other coaches, coaches provided a single score representing each athlete's “potential for future performance” on an 11-point Likert scale at each timepoint. Scores from each coach were converted into rankings from 1 to 24 to create a normalized scale to facilitate comparison of athletes. Based on their rankings at each timepoint, athletes were placed into one of three evenly distributed groups (high, medium, and low rank). Inter-coach agreement at each timepoint was determined by the number of coaches who ranked each athlete in the same group, categorized at three levels: 50, 75 or 100% agreement. Overall results showed that at completion of the camp, coaches reached 100% agreement on only two athletes, both of whom were in the high rank group. When inter-coach agreement was set at 50%, 15 athletes (62.5%) were placed into like groups. The first timepoint at which coaches were able to differentiate between the majority of athletes was Timepoint 3 (end of day 2). The findings suggest that, in isolation, coaches do not agree on the talent or potential of athletes. This indicates that the “coach's eye” is subjective and variable, and, given the same context, there is poor inter-coach agreement in the identification of talented athletes. In turn, these findings may have significant implications for both future talent identification research and athlete selection processes by sport organizations.

The Aspirational Income Hypothesis: On the Limits of the Relative Income Hypothesis

According to the “relative income hypothesis,” decision makers derive positive utility from identifying with a group that performs more poorly than they do. We hypothesize that decision makers simultaneously derive negative utility from identifying with such a group. The reason is that decision makers aspire toward status, and thus prefer to identify with a group that is more successful than they are. We call such proposed reason the “aspirational income hypothesis.” If the aspirational income effect dominates the relative income effect, decision makers would prefer to join groups with higher rank than their own. We report data from an experiment that supports the aspirational income hypothesis. It shows that the hypothesis holds (at a weaker intensity) even when it is costly to join a higher ranking group.

Integrability and cycles of deformed [formula omitted] gauge theory

To analyse pure N=2SU(2) gauge theory in the Nekrasov-Shatashvili (NS) limit (or deformed Seiberg-Witten (SW)), we use the Ordinary Differential Equation/Integrable Model (ODE/IM) correspondence, and in particular its (broken) discrete symmetry in its extended version with two singular irregular points. Actually, this symmetry appears to be ‘manifestation’ of the spontaneously broken Z2 R-symmetry of the original gauge problem and the two deformed SW one-cycle periods are simply connected to the Baxter's T and Q functions, respectively, of the Liouville conformal field theory at the self-dual point. The liaison is realised via a second order differential operator which is essentially the ‘quantum’ version of the square of the SW differential. Moreover, the constraints imposed by the broken Z2 R-symmetry acting on the moduli space (Bilal-Ferrari equations) seem to have their quantum counterpart in the TQ and the T periodicity relations, and integrability yields also a useful Thermodynamic Bethe Ansatz (TBA) for the periods (Y(θ,±u) or their square roots, Q(θ,±u)). A latere, two efficient asymptotic expansion techniques are presented. Clearly, the whole construction is extendable to gauge theories with matter and/or higher rank groups.