Index Theorems and Loop Space Geometry

We investigate the evaluation of the Dirac index using symplectic geometry in the loop space of the corresponding supersymmetric quantum mechanical model. In particular, we find that if we impose a simple first class constraint, we can evaluate the Callias index of an odd dimensional Dirac operator directly from the quantum mechanical model which yields the Atiyah-Singer index of an even dimensional Dirac operator in one more dimension. The effective action obtained by BRST quantization of this constrained system can be interpreted in terms of loop space symplectic geometry, and the corresponding path integral for the index can be evaluated exactly using the recently developed localization techniques.Comment: 15 pages, report CERN-TH-6471 and HU-TFT-92-1

Derivation of Index Theorems by Localization of Path Integrals

We review the derivation of the Atiyah-Singer and Callias index theorems using the recently developed localization method to calculate exactly the relevant supersymmetric path integrals. (Talk given at the III International Conference on Mathematical Physics, String Theory and Quantum Gravity, Alushta, Ukraine, June 13-24, 1993)Comment: 11 pages in LaTeX, HU-TFT-93-3

Symmetries and Observables for BF-theories in Superspace

The supersymmetric version of a topological quantum field theory describing flat connections, the super BF-theory, is studied in the superspace formalism. A set of observables related to topological invariants is derived from the curvature of the superspace. Analogously to the non-supersymmetric versions, the theory exhibits a vector-like supersymmetry. The role of the vector supersymmetry and an additional new symmetry of the action in the construction of observables is explained.Comment: 11 pages, LaTe

Rollen lause polynomeille

Rollen lauseen nimi tulee tyypillisesti lauseen julkaisijan nimestä. Michel Rolle määritteli lauseen ensimmäisenä vuonna 1691. Yleensä Rollen lauseeseen törmää matematiikassa analyysin puolella. Rollen lause onkin yksi erikoistapaus differentiaalilaskennan väliarvolauseesta. Sitä myös hyödynnetään differentiaalilaskennan väliarvolauseen todistuksessa. Tutkielmassa lähestyn Rollen lausetta algebrallisesti, mikä on harvinaisempi lähestymistapa. Algebrallinen lähestymistapa edellyttää algebran perusteiden, muun muassa kompleksilukujen ja polynomien hallintaa, sillä esitän ja todistan Rollen lauseen reaalikertoimisilla polynomeilla. Kompleksilukuja tarvitaan toisen asteen polynomin juuria ja niiden lukumääriä tutkittaessa. Tutkielmassa on käytetty pääasiassa Maurice Mignotten kirjoittamaa kirjaa: Mathematics for Computer Algebra. Jotta lukijan on helpompi hahmottaa algebrallisesti polynomeilla esitetyn Rollen lauseen eron analyysissä esiintyvästä lauseesta, esitän Rollen lauseen kahdella tavalla. Ensimmäinen muoto on analyysissä esiintyvä muoto, ja toinen on se, johon palataan tutkielmani loppupuolella. Tutkielman ensisijainen tavoite on todistaa Rollen lause algebrallisesti reaalikertoimisilla polynomeilla

European Social Work Research Association SIG to Study Decisions, Assessment, and Risk

This is an Accepted Manuscript of an article published by Taylor & Francis Group in Journal of Evidence-Informed Social Work on 13/12/2017, available online: http://www.tandfonline.com/doi/full/10.1080/23761407.2017.1394244 Purpose: The increasing interest in professional judgement and decision making is often separate from the discourse about “risk,” and the time-honored focus on assessment. Method: The need to develop research in and across these topics was recognized in the founding of a Decisions, Assessment, and Risk Special Interest Group (DARSIG) by the European Social Work Research Association in 2014. Results: The Group's interests include cognitive judgements; decision processes with clients, families, other professionals and courts; assessment tools and processes; the assessment, communication, and management of risk; and legal, ethical, and emotional aspects of these. This article outlines the founding and scope of DARSIG; gives an overview of decision making, assessment, and risk for practice; illustrates connections between these; and highlights future research directions. Discussion: Professional knowledge about decision making, assessment, and risk complements knowledge about effectiveness of interventions. Conclusion: DARSIG promises to be a useful mechanism for the purpose