Diagrammatics for Coxeter groups and their braid groups

We give a monoidal presentation of Coxeter and braid 2-groups, in terms of decorated planar graphs. This presentation extends the Coxeter presentation. We deduce a simple criterion for a Coxeter group or braid group to act on a category.Comment: Many figures, best viewed in color. Minor updates. This version agrees with the published versio

The Hodge theory of Soergel bimodules

We prove Soergel's conjecture on the characters of indecomposable Soergel bimodules. We deduce that Kazhdan-Lusztig polynomials have positive coefficients for arbitrary Coxeter systems. Using results of Soergel one may deduce an algebraic proof of the Kazhdan-Lusztig conjecture.Comment: 44 pages. v2: many minor changes, final versio

The geometry of Markov traces

We give a geometric interpretation of the Jones-Ocneanu trace on the Hecke algebra, using the equivariant cohomology of sheaves on SL(n). This construction makes sense for all simple algebraic groups, so we obtain a generalization of the Jones-Ocneanu trace to Hecke algebras of other types. We give a geometric expansion of this trace in terms of the irreducible characters of the Hecke algebra, and conclude that it agrees with a trace defined independently by Gomi. Based on our proof, we also prove that certain simple perverse sheaves on a reductive algebraic group G are equivariantly formal for the conjugation action of a Borel B, or equivalently, that the Hochschild homology of any Soergel bimodule is free, as the authors had previously conjectured. This construction is closely tied to knot homology. This interpretation of the Jones-Ocneanu trace is a more elementary manifestation of the geometric construction of HOMFLYPT homology given by the authors in a previous paper.Comment: 14 pages. v2; typos and minor errors fixed, simplified argument in final section. DVI may not render correctly on all computers; PDF is prefere

Poverty, policy, and industrialization : lessons from the distant past

Pessimists say industrialization increased poverty; optimists say it did not. The authors argue that how much industrialization eradicates poverty depends on the form industrialization takes. It is not economic growth by itself, but the processes and policies associated with different growth regimes which make the poor poorer. The authors address two questions : 1) what happened to the proportionate share of the population living in poverty, and to the living standards of the poor, during nineteenth century industrial revolutions?; and 2) why did poverty statistics behave the way they did? Modern economic growth may erode traditional entitlements that serve as safety nets in preindustrial societies. It may be convenient to think otherwise, but typically the poor in preindustrial European and North American societies were not supported by the family and private institutions. Much of the responsibility for the poor lay with the state and other formal, statelike institutions that intervened in food markets. Where laissez-faire policies were adopted during the Industrial Revolution, as in America and England, many of the poor (especially the extremely poor) became more vulnerable to adverse conditions.Environmental Economics&Policies,Services&Transfers to Poor,Rural Poverty Reduction,Safety Nets and Transfers,Governance Indicators

Thick Soergel calculus in type A

Let R be the polynomial ring in n variables, acted on by the symmetric group S_n. Soergel constructed a full monoidal subcategory of R-bimodules which categorifies the Hecke algebra, whose objects are now known as Soergel bimodules. Soergel bimodules can be described as summands of Bott-Samelson bimodules (attached to sequences of simple reflections), or as summands of generalized Bott-Samelson bimodules (attached to sequences of parabolic subgroups). A diagrammatic presentation of the category of Bott-Samelson bimodules was given by the author and Khovanov in previous work. In this paper, we extend it to a presentation of the category of generalized Bott-Samelson bimodules. We also diagrammatically categorify the representations of the Hecke algebra which are induced from trivial representations of parabolic subgroups. The main tool is an explicit description of the idempotent which picks out a generalized Bott-Samelson bimodule as a summand inside a Bott-Samelson bimodule. This description uses a detailed analysis of the reduced expression graph of the longest element of S_n, and the semi-orientation on this graph given by the higher Bruhat order of Manin and Schechtman.Comment: Changed title. Expanded the exposition of the main proof. This paper relies extensively on color figure

Student data systems and GovTech apps will increase competition and performance measurement in higher education

Current debates in higher education policy have drawn attention to the significant impacts of marketisation, metrics, and performance management on the sector. Ben Williamson argues that a restructuring of the data infrastructure is shaping these HE trends. An examination of the HE data infrastructure reveals the political aspirations coded into its architecture, the actors involved in its production, and its practical effects. The UK HE data infrastructure is about to get a massive upgrade, increasing competition, measurement, and consumer ranking through data platforms, dashboards, and new GovTech applications