DOCSLIB.ORG

REMEMBERING MICHAEL ATIYAH ICTS Address Survey No

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
REMEMBERING MICHAEL ATIYAH ICTS Address Survey No VOLUME V ISSUE 1 NEWS 2019 TATA INSTITUTE OF FUNDAMENTAL RESEARCH photograph by Peter Badge | www.abelprize.no Wall plaque at the ICTS Campus, Bangalore. REMEMBERING MICHAEL ATIYAH ICTS Address Survey No. 151, Shivakote village, Hesaraghatta Hobli, North Bengaluru, India 560089 Website www.icts.res.in 1 VOLUME V NEWS ISSUE 1 2019 TATA INSTITUTE OF FUNDAMENTAL RESEARCH L R: C. Musili, C.S. Seshadri, M.S. Narasimhan, M.S. Raghunathan and David Mumford at TIFR (1968) ⟶ www.cmi.ac.in/~ramadas/NS@50/ tifr archives Michael Atiyah lecturing at TIFR tifr archives Michael Atiyah during his visit to TIFR, Mumbai, in 1984. MICHAEL ATIYAH M.S. NARASIMHAN & C.S. SESHADRI M.S. NARASIMHAN mathematical aspects of gauge theory arising in of holomorphic vector bundles on the Riemann Sir Michael Atiyah was a great mathematician theoretical physics. surface which have been studied earlier by Seshadri who made path-breaking contributions to various and myself. This work of Atiyah and Bott was very His best known result is the celebrated Atiyah- fields in mathematics and played a crucial role in influential and popularised these moduli spaces Singer index theorem, which is a vast generalisation promoting the resurgence of contact between among physicists. of famous results like the Riemann-Roch theorem mathematics and theoretical physics in the latter and which is intimately related to K-Theory. A He had an effervescent and inspiring personality. half of the twentieth century. major related work is the Atiyah-Patodi-Singer He valued collaboration in mathematics and loved He was the recipient of many honours, including index theorem, which deals with elliptic operators intense mathematical discussions. He seemed to the Fields Medal and the Abel Prize. His work on manifolds with boundary and non-local derive many of his insights and ideas in the process combined creatively Analysis, Differential boundary conditions. (A version of the Hirzebruch of talking. Geometry and Topology. His major contributions signature theorem for manifolds with boundary) I maintained a close and fruitful contact with him have been to Topological K-Theory (which proved Atiyah and Bott made an extensive study of over the years and we had several conversations to be a remarkably powerful tool in Topology), the Yang-Mills equation on compact Riemann on moduli problems and some aspects of the index Index Theorem for linear elliptic operators and surfaces and its relationship to the moduli spaces theorem. Moreover, many of my best students did www.icts.res.in 2 | ICTS NEWS | VOLUME V | ISSUE 1 2019 3 their post-doctoral study with Atiyah, the best C.S. SESHADRI earliest to perceive the importance of NS and let his “… The fact that I had physics friends is an known example being Vijay K. Patodi. Patodi’s I had heard of Michael Atiyah as a brilliant student student Newstead to calculate the cohomology of indication that the frontiers between disciplines stay in Princeton, working with Atiyah, resulted of Hodge when I was a graduate student at the certain “moduli spaces” figuring in NS. Theory needs were breaking down, and this has become the in his celebrated joint work with Atiyah, Bott and Tata Institute of Fundamental Research (TIFR). main feature of our times. The old rigid disciplines Newstead got a position in Liverpool and a framework Singer. (Atiyah and I later edited the Collected of the past are giving way to a much more fluid Then I came across his paper on the classification of Atiyah arranged for a visit to Liverpool by Papers of Patodi. In the foreword Atiyah wrote in which to scene, which is why the ICTS is the right body for vector bundles on Elliptic Curves. This attracted me M. S. Narasimhan and me. Atiyah again perceived that the work of Atiyah-Patodi-Singer ‘was the future. The future belongs to the young and the since Grothendieck had classified vector bundles on the importance of “moduli spaces of parabolic a great collaboration, exploiting the different develop and, as a science that is now emerging will affect the lives of the projective line and I noticed in my thesis that vector bundles” – a work I had done with Vikram talents of the participants.’) everyone on the planet. The ICTS has a noble task, ’s result is a consequence of an early Mehta (i.e. the geometry and physics of knots, Grothendieck mathematician, that of providing the right atmosphere to inspire I will mention a few recollections of my work of G.D. Birkoff. by Atiyah). I met Atiyah several times during the next generation of scientists …” ■ mathematical interactions with him. The original 1975–76, at the Institute for Advanced Study, I believe that In 1957, I proved that a finitely generated proof of the Atiyah-Bott fixed point theorem Princeton, TIFR Colloquium on vector bundles, Read the full text of the speeches here: projective module over the ring of polynomials mathematics used a result of Takeshi Kotake and myself on 1982, sometime in Bonn when he expressed his https://bit.ly/Atiyah_ScienceAtICTS_Speech in two variables is free (the first non-trivial case fractional powers of linear elliptic operators. great regard for Patodi. provides https://bit.ly/Atiyah_ScienceAtICTS_youtube of Serre’s problem). I attended the International Atiyah asked me to give a talk on this result. On https://bit.ly/Atiyah_FoundationStoneRemarks Congress held at Edinburgh in 1958 and remember It is a great privilege for having known Atiyah and another occasion, during an Arbeitstagung in that unifying conversing with during a cruise trip at this to have had his interest in my work. ■ Bonn, I told him about the canonical filtration Atiyah Congress. He told me that a British mathematician Michael Atiyah delivers the ceremonial speech for framework … which Harder and I had introduced on vector M. S. Narasimhan and C. S. Seshadri are eminent the ICTS Inauguration Event: Science at ICTS (whose name I forget) also solved Serre’s problem bundles on curves and suggested that it could mathematicians, well known for the famous for two variables but found that I had already was a nice problem that taught me that at times be of use in the work on Yang-Mills equation on Narasimhan-Seshadri theorem.They were among will contain mathematically oriented scientists. Spenta R. Wadia is a theoretical physicist and the done this earlier. Atiyah was the star attraction in a path integral it is necessary to sum over both Riemann surfaces he was undertaking at that the main pillars of the famed School of Mathematics So the exact allocation is not important, so long as Founding Director, Infosys Homi Bhabha Chair at the International Conference on Differential stable and unstable classical solutions to obtain the flexibility and quality are maintained. time with Bott. of TIFR, where they spent most of their professional Professor and Professor Emeritus at ICTS–TIFR, Analysis in 1964 at TIFR. A few months later, the correct answer. I did not publish this result as my life. Narasimhan later spent some years building Bengaluru His passing is a great loss for the mathematical Narasimhan-Seshadri theorem (NS) was proved. Finally, I totally agree that your initial appointments mathematics at ICTP and Seshadri founded the PhD mentor Bunji Sakita had emphasized that one community. M.S. Narasimhan gave a talk at this conference should be made cautiously and not in haste.” Chennai School of Mathematics. should write up only very substantive papers! In on an ingredient of NS. Atiyah was among the retrospect I think it would have been been valuable He embraced the idea and purpose of the ICTS, which to publish this simple result. he put forth in his remarks for the Foundation Stone BETWEEN THE Ceremony of ICTS on 28 December 2009, Michael also played an important role in the SCIENCE “Science has the noble aim of trying to understand establishment of the ICTS. The committee he the natural world in human terms: to make sense chaired that reviewed Mathematics at the Tata REMEMBERING MICHAEL FRANCIS ATIYAH of what we see. This brief phrase encapsulates both Institute of Fundamental Research (TIFR) in 2005 ABHIRUP GHOSH’ s submission towards theory and experiment. What we see, in the broad made a strong recommendation for an interactive the Augmenting Writing Skills for Articulating sense, covers experiment and making sense is the program for physicists and mathematicians. This Research (AWSAR) Award 2018, organised by (1929-2019) task of theory. As the great French mathematician reinforced the recommendation of the Theoretical the Department of Science and Technology Henri Poincare said, science is no more a collection of SPENTA R. WADIA Physics review committee in 2006 to establish ICTS (DST), Government of India, has been facts than a house is a collection of bricks: it requires in 2007. selected among the top 100 entries.. theory to hold it together. Michael Francis Atiyah, one of the most significant 3-dimensional topology. String theory embodies He was deeply interested in bringing the math and He strongly supported the idea of creating the ICTS Theory needs a framework in which to develop and, ADHIP AGARWALA’s PhD thesis titled mathematicians of the second half of the 20th this essential unity in a natural way. physics communities together to talk and share. and was a member of its International Advisory as a mathematician, I believe that mathematics ‘Excursions in ill condensed quantum matter: century, passed away on 11th January 2019. His His enthusiasm was infectious. He attributed his Board from inception till his passing away. His active During my tenure at the Enrico Fermi Institute, provides that unifying framework … from amorphous topological insulators profound contributions to topology, geometry interest in physics to his collaborator and friend involvement and advice has been encouraging and Michael visited the University of Chicago for a to fractional spins’ has been selected by and mathematical physics earned him the highest Isa Singer, who also visited UC during Michael’s invaluable.
Load more

Recommended publications
  • Why U.S. Immigration Barriers Matter for the Global Advancement of Science
    DISCUSSION PAPER SERIES IZA DP No. 14016 Why U.S. Immigration Barriers Matter for the Global Advancement of Science Ruchir Agarwal Ina Ganguli Patrick Gaulé Geoff Smith JANUARY 2021 DISCUSSION PAPER SERIES IZA DP No. 14016 Why U.S. Immigration Barriers Matter for the Global Advancement of Science Ruchir Agarwal Patrick Gaulé International Monetary Fund University of Bath and IZA Ina Ganguli Geoff Smith University of Massachusetts Amherst University of Bath JANUARY 2021 Any opinions expressed in this paper are those of the author(s) and not those of IZA. Research published in this series may include views on policy, but IZA takes no institutional policy positions. The IZA research network is committed to the IZA Guiding Principles of Research Integrity. The IZA Institute of Labor Economics is an independent economic research institute that conducts research in labor economics and offers evidence-based policy advice on labor market issues. Supported by the Deutsche Post Foundation, IZA runs the world’s largest network of economists, whose research aims to provide answers to the global labor market challenges of our time. Our key objective is to build bridges between academic research, policymakers and society. IZA Discussion Papers often represent preliminary work and are circulated to encourage discussion. Citation of such a paper should account for its provisional character. A revised version may be available directly from the author. ISSN: 2365-9793 IZA – Institute of Labor Economics Schaumburg-Lippe-Straße 5–9 Phone: +49-228-3894-0 53113 Bonn, Germany Email: [email protected] www.iza.org IZA DP No. 14016 JANUARY 2021 ABSTRACT Why U.S.
    [Show full text]
  • The Work of Grigory Perelman
    The work of Grigory Perelman John Lott Grigory Perelman has been awarded the Fields Medal for his contributions to geom- etry and his revolutionary insights into the analytical and geometric structure of the Ricci flow. Perelman was born in 1966 and received his doctorate from St. Petersburg State University. He quickly became renowned for his work in Riemannian geometry and Alexandrov geometry, the latter being a form of Riemannian geometry for metric spaces. Some of Perelman’s results in Alexandrov geometry are summarized in his 1994 ICM talk [20]. We state one of his results in Riemannian geometry. In a short and striking article, Perelman proved the so-called Soul Conjecture. Soul Conjecture (conjectured by Cheeger–Gromoll [2] in 1972, proved by Perelman [19] in 1994). Let M be a complete connected noncompact Riemannian manifold with nonnegative sectional curvatures. If there is a point where all of the sectional curvatures are positive then M is diffeomorphic to Euclidean space. In the 1990s, Perelman shifted the focus of his research to the Ricci flow and its applications to the geometrization of three-dimensional manifolds. In three preprints [21], [22], [23] posted on the arXiv in 2002–2003, Perelman presented proofs of the Poincaré conjecture and the geometrization conjecture. The Poincaré conjecture dates back to 1904 [24]. The version stated by Poincaré is equivalent to the following. Poincaré conjecture. A simply-connected closed (= compact boundaryless) smooth 3-dimensional manifold is diffeomorphic to the 3-sphere. Thurston’s geometrization conjecture is a far-reaching generalization of the Poin- caré conjecture. It says that any closed orientable 3-dimensional manifold can be canonically cut along 2-spheres and 2-tori into “geometric pieces” [27].
    [Show full text]
  • Dishant M Pancholi
    Dishant M Pancholi Contact Chennai Mathematical Institute Phone: +91 44 67 48 09 59 Information H1 SIPCOT IT Park, Siruseri Fax: +91 44 27 47 02 25 Kelambakkam 603103 E-mail: [email protected] India Research • Contact and symplectic topology Interests Employment Assitant Professor June 2011 - Present Chennai Mathematical Institute, Kelambakkam, India Post doctoral Fellow (January 2008 - January 2010) International Centre for Theoratical physics,Trieste, Italy. Visiting Fellow, (October 2006 - December 2007) TIFR Centre, Bangalore, India. Education School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India. Ph.D, August 1999 - September 2006. Thesis title: Knots, mapping class groups and Kirby calculus. Department of Mathematics, M.S. University of Baroda, Vadodara, India. M.Sc (Mathematics), 1996 - 1998. M.S. University of Baroda, Vadodara, India B.Sc, 1993 - 1996. Awards and • Shree Ranchodlal Chotalal Shah gold medal for scoring highest marks in mathematics Scholarships at M S University of Baroda, Vadodara in M.Sc. (1998) • Research Scholarship at TIFR, Mumbai. (1999) • Best Thesis award TIFR, Mumbai (2007) Research • ( With Siddhartha Gadgil) Publications Homeomorphism and homology of non-orientable surfaces Proceedings of Indian Acad. of Sciences, 115 (2005), no. 3, 251{257. • ( With Siddhartha Gadgil) Non-orientable Thom-Pontryagin construction and Seifert surfaces. Journal of RMS 23 (2008), no. 2, 143{149. • ( With John Etnyre) On generalizing Lutz twists. Journal of the London Mathematical Society (84) 2011, Issue 3, p670{688 Preprints • (With Indranil Biswas and Mahan Mj) Homotopical height. arXiv:1302.0607v1 [math.GT] • (With Roger Casals and Francisco Presas) Almost contact 5{folds are contact. arXiv:1203.2166v3 [math.SG] (2012)(submitted) • (With Roger Casals and Francisco Presas) Contact blow-up.
    [Show full text]
  • Debashish Goswami Jyotishman Bhowmick Quantum Isometry Groups Infosys Science Foundation Series
    Infosys Science Foundation Series in Mathematical Sciences Debashish Goswami Jyotishman Bhowmick Quantum Isometry Groups Infosys Science Foundation Series Infosys Science Foundation Series in Mathematical Sciences Series editors Gopal Prasad, University of Michigan, USA Irene Fonseca, Mellon College of Science, USA Editorial Board Chandrasekhar Khare, University of California, USA Mahan Mj, Tata Institute of Fundamental Research, Mumbai, India Manindra Agrawal, Indian Institute of Technology Kanpur, India S.R.S. Varadhan, Courant Institute of Mathematical Sciences, USA Weinan E, Princeton University, USA The Infosys Science Foundation Series in Mathematical Sciences is a sub-series of The Infosys Science Foundation Series. This sub-series focuses on high quality content in the domain of mathematical sciences and various disciplines of mathematics, statistics, bio-mathematics, financial mathematics, applied mathematics, operations research, applies statistics and computer science. All content published in the sub-series are written, edited, or vetted by the laureates or jury members of the Infosys Prize. With the Series, Springer and the Infosys Science Foundation hope to provide readers with monographs, handbooks, professional books and textbooks of the highest academic quality on current topics in relevant disciplines. Literature in this sub-series will appeal to a wide audience of researchers, students, educators, and professionals across mathematics, applied mathematics, statistics and computer science disciplines. More information
    [Show full text]
  • Learning from Fields Medal Winners
    TechTrends DOI 10.1007/s11528-015-0011-6 COLUMN: RETHINKING TECHNOLOGY & CREATIVITY IN THE 21ST CENTURY Creativity in Mathematics and Beyond – Learning from Fields Medal Winners Rohit Mehta1 & Punya Mishra1 & Danah Henriksen2 & the Deep-Play Research Group # Association for Educational Communications & Technology 2016 For mathematicians, mathematics—like music, poetry, advanced mathematics through their work in areas as special- or painting—is a creative art. All these arts involve— ized and diverse as dynamical systems theory, the geometry of and indeed require—a certain creative fire. They all numbers, stochastic partial differential equations, and the dy- strive to express truths that cannot be expressed in ordi- namical geometry of Reimann surfaces. nary everyday language. And they all strive towards An award such as the Fields Medal is a recognition of beauty — Manjul Bhargava (2014) sustained creative effort of the highest caliber in a challenging Your personal life, your professional life, and your cre- domain, making the recipients worthy of study for scholars ative life are all intertwined — Skylar Grey interested in creativity. In doing so, we continue a long tradi- tion in creativity research – going all the way back to Galton in Every 4 years, the International Mathematical Union rec- the 19th century – of studying highly accomplished individ- ognizes two to four individuals under the age of 40 for their uals to better understand the nature of the creative process. We achievements in mathematics. These awards, known as the must point out that the focus of creativity research has shifted Fields Medal, have often been described as the “mathemati- over time, moving from an early dominant focus on genius, cian’s Nobel Prize” and serve both a form of peer-recognition towards giftedness in the middle of the 20th century, to a more of highly influential and creative mathematical work, as well contemporary emphasis on originality of thought and work as an encouragement of future achievement.
    [Show full text]
  • ANNEXURE 1 School of Mathematics Sciences —A Profile As in March 2010
    University Yearbook (Academic Audit Report 2005-10) ANNEXURE 1 School of Mathematics Sciences —A Profile as in March 2010 Contents 1. Introduction 1 1.1. Science Education: A synthesis of two paradigms 2 2. Faculty 3 2.1. Mathematics 4 2.2. Physics 12 2.3. Computer Science 18 3. International Links 18 3.1. Collaborations Set Up 18 3.2. Visits by our faculty abroad for research purposes 19 3.3. Visits by faculty abroad to Belur for research purposes 19 4. Service at the National Level 20 4.1. Links at the level of faculty 20 5. Student Placement 21 5.1. 2006-8 MSc Mathematics Batch 21 5.2. 2008-10 MSc Mathematics Batch 22 6. Research 23 1. Introduction Swami Vivekananda envisioned a University at Belur Math, West Bengal, the headquarters of the Ramakrishna Mission. In his conversation with Jamshedji Tata, it was indicated that the spirit of austerity, service and renun- ciation, traditionally associated with monasticism could be united with the quest for Truth and Universality that Science embodies. An inevitable fruit of such a synthesis would be a vigorous pursuit of science as an aspect of Karma Yoga. Truth would be sought by research and the knowledge obtained disseminated through teach- ing, making knowledge universally available. The VivekanandaSchool of Mathematical Sciences at the fledgling Ramakrishna Mission Vivekananda University, headquartered at Belur, has been set up keeping these twin ide- als in mind. We shall first indicate that at the National level, a realization of these objectives would fulfill a deeply felt National need. 1.1.
    [Show full text]
  • Tata Institute of Fundamental Research Prof
    Annual Report 1988-89 Tata Institute of Fundamental Research Prof. M. G. K. Menon inaugurating the Pelletron Accelerator Facility at TIFR on December 30, 1988. Dr. S. S. Kapoor, Project Director, Pelletron Accelerator Facility, explaining salient features of \ Ion source to Prof. M. G. K. Menon, Dr. M. R. Srinivasan, and others. Annual Report 1988-89 Contents Council of Management 3 School of Physics 19 Homi Bhabha Centre for Science Education 80 Theoretical Physics l'j Honorary Fellows 3 Theoretical A strophysics 24 Astronomy 2') Basic Dental Research Unit 83 Gravitation 37 A wards and Distinctions 4 Cosmic Ray and Space Physics 38 Experimental High Energy Physics 41 Publications, Colloquia, Lectures, Seminars etc. 85 Introduction 5 Nuclear and Atomic Physics 43 Condensed Matter Physics 52 Chemical Physics 58 Obituaries 118 Faculty 9 Hydrology M Physics of Semi-Conductors and Solid State Electronics 64 Group Committees 10 Molecular Biology o5 Computer Science 71 Administration. Engineering Energy Research 7b and Auxiliary Services 12 Facilities 77 School of Mathematics 13 Library 79 Tata Institute of Fundamental Research Homi Bhabha Road. Colaba. Bombav 400005. India. Edited by J.D. hloor Published by Registrar. Tata Institute of Fundamental Research Homi Bhabha Road, Colaba. Bombay 400 005 Printed bv S.C. Nad'kar at TATA PRESS Limited. Bombay 400 025 Photo Credits Front Cover: Bharat Upadhyay Inside: Bharat Upadhyay & R.A. A chary a Design and Layout by M.M. Vajifdar and J.D. hloor Council of Management Honorary Fellows Shri J.R.D. Tata (Chairman) Prof. H. Alfven Chairman. Tata Sons Limited Prof. S. Chandrasekhar Prof.
    [Show full text]
  • Alexander Grothendieck Heng Li
    Alexander Grothendieck Heng Li Alexander Grothendieck's life Alexander Grothendieck was a German born French mathematician. Grothendieck was born on March 28, 1928 in Berlin. His parents were Johanna Grothendieck and Alexander Schapiro. When he was five years old, Hitler became the Chancellor of the German Reich, and called to boycott all Jewish businesses, and ordered civil servants who were not of the Aryan race to retire. Grothendieck's father was a Russian Jew and at that time, he used the name Tanaroff to hide his Jewish Identity. Even with a Russian name it is still too dangerous for Jewish people to stay in Berlin. In May 1933, his father, Alexander Schapiro, left to Paris because the rise of Nazism. In December of the same year, his mother left Grothendieck with a foster family Heydorns in Hamburg, and joined Schapiro in Paris. While Alexander Grothendieck was with his foster family, Grothendieck's parents went to Spain and partici- pated in the Spanish Civil War. After the Spanish Civil War, Johanna(Hanka) Grothendieck and Alexander Schapiro returned to France. In Hamburg, Alexander Grothendieck attended to elementary schools and stud- ied at the Gymnasium (a secondary school). The Heydorns family are a part of the resistance against Hitler; they consider that it was too dangerous for young Grothendieck to stay in Germany, so in 1939 the Heydorns sent him to France to join his parents. However, because of the outbreak of World War II all German in France were required by law to be sent to special internment camps. Grothendieck and his parents were arrested and sent to the camp, but fortunately young Alexander Grothendieck was allowed to continue his education at a village school that was a few miles away from the camp.
    [Show full text]
  • To, Prof. Ajit Kembhavi, President, ASI CC
    To, Prof. Ajit Kembhavi, President, ASI CC : Prof. Dipankar Banerjee, Secretary, ASI Subject : Formation of Working Group for Gender Equity Dear Sir, We, the undersigned, members of the Indian Astronomy & Astrophysics community (from a number of academic institutions), would like to request you for the formation of a working group for gender equity under the aegis of ASI. To this effect we hereby submit a formal proposal giving details of the rational behind such a working group and a brief outline of the role we would like the working group to play. We hope you would take cognizance of the fact that a significant fraction of our community feel that the formation of such a group is the need of the hour. For the Working Group to better serve the needs of the entire astronomy community, it would be advantageous if its constituted members reflect the diversity of the community in gender, age, affiliation, geographical region etc. In addition, in the interest of promoting equity in gender representation, we feel it is best chaired by a woman astronomer. Hoping to receive a positive response from you. Yours sincerely, The Proposers of the Working Group The Proposers : A core group of people (Preeti Kharb, Sushan Konar, Niruj Mohan, L. Resmi, Jasjeet Singh Bagla, Nissim Kanekar, Prajval Shastri, Dibyendu Nandi etc.) have been working towards sensitising the Indian Astronomical community about gender related issues and proposing the formation of a working group, with significant input from a number of others. However, a large section of the community have been supportive of this activity and would like to be a part of this initiative.
    [Show full text]
  • LIST of PUBLICATIONS (1) (With S. Ramanan), an Infinitesimal Study of the Moduli of Hitchin Pairs. Journal of the London Mathema
    LIST OF PUBLICATIONS INDRANIL BISWAS (1) (With S. Ramanan), An infinitesimal study of the moduli of Hitchin pairs. Journal of the London Mathematical Society, Vol. 49, No. 2, (1994), 219{231. (2) A remark on a deformation theory of Green and Lazarsfeld. Journal f¨urdie Reine und Angewandte Mathematik, Vol. 449 (1994), 103{124. (3) (With N. Raghavendra), Determinants of parabolic bundles on Riemann surfaces. Proceedings of the Indian Academy of Sciences (Math. Sci.), Vol. 103, No. 1, (1993), 41{71. (4) (With K. Guruprasad), Principal bundles on open surfaces and invariant functions on Lie groups. International Journal of Mathematics, Vol. 4, No. 4, (1993), 535{ 544. (5) Secondary invariants of Higgs bundles. International Journal of Mathematics, Vol. 6, No. 2, (1995), 193{204. (6) On the mapping class group action on the cohomology of the representation space of a surface. Proceedings of the American Mathematical Society, Vol. 124, No. 6, (1996), 1959{1965. (7) (With K. Guruprasad), On some geometric invariants associated to the space of flat connections on an open surface. Canadian Mathematical Bulletin, Vol. 39, No. 2, (1996), 169{177. (8) A remark on Hodge cycles on moduli spaces of rank 2 bundles over curves. Journal f¨urdie Reine und Angewandte Mathematik, Vol. 370 (1996), 143{152. (9) On Harder{Narasimhan filtration of the tangent bundle. Communications in Anal- ysis and Geometry, Vol. 3, No. 1, (1995), 1{10. (10) (With N. Raghavendra), Canonical generators of the cohomology of moduli of parabolic bundles on curves. Mathematische Annalen, Vol. 306, No. 1, (1996), 1{14. (11) (With M.
    [Show full text]
  • Dynamics and Chaos in the Unified Scalar Field Cosmology
    Dynamics and chaos in the unified scalar field Cosmology Georgios Lukes-Gerakopoulos,1,2 Spyros Basilakos,1 and George Contopoulos1 1Academy of Athens, Research Center for Astronomy, Soranou Efesiou 4, GR-11527, Athens, GREECE 2University of Athens,Department of Physics, Section of Astrophysics, Astronomy and Mechanics We study the dynamics of the closed scalar field FRW cosmological models in the framework of the so called Unified Dark Matter (UDM) scenario. Performing a theoretical as well as a numerical analysis we find that there is a strong indication of chaos in agreement with previous studies. We find that a positive value of the spatial curvature is essential for the appearance of chaoticity, though the Lyapunov number seems to be independent of the curvature value. Models that are close to flat (k 0+) exhibit a chaotic behavior after a long time while pure flat models do not exhibit any chaos.→ Moreover, we find that some of the semiflat models in the UDM scenario exhibit similar dynamical behavior with the Λ cosmology despite their chaoticity. Finally, we compare the measured evolution of the Hubble parameter derived from the differential ages of passively evolving galaxies with that expected in the semiflat unified scalar field cosmology. Based on a specific set of initial conditions we find that the UDM scalar field model matches well the observational data. PACS numbers: 98.80.-k, 11.10.Ef, 11.10.Lm Keywords: Scalar field; Cosmology; Chaotic scattering 1. INTRODUCTION question which has to be addressed: how can we esti- mate the amount of chaos? It has been shown that in chaotic scattering problems classical tools, like the Lya- There is by now convincing evidence that the avail- punov characteristic number, are unable to provide a able high quality cosmological data (Type Ia supernovae, clear answer whether an orbit is chaotic or not [22].
    [Show full text]
  • To View Abstract Book
    86th Annual Conference of e Indian Mathematical Society – An International Meet (Online Conference) December 17-20, 2020 Organized under the auspices of Department of Mathematics School of Advanced Sciences, VIT Vellore IMS 2020 - Committee IMS - Executive President Prof. B. Sury Indian Statistical Institute, Bangaluru General Secretary Prof. Satya Deo Harish-Chandra Research Institute, Allahabad. Editor, e Journal of the Prof. Sudhir R. Ghorpade Indian Mathematical Society Indian Institute of Technology - Bombay, Mumbai. Editor, e Mathematics Student Prof. M. M. Shikare Center for Advanced Study in Mathematics, Pune. Academic Secretary Prof. Peeyush Chandra Formerly, IIT Kanpur Administrative Secretary Prof. B. N. Waphare Center for Advanced Study in Mathematics, Pune. Treasurer Prof. S. K. Nimbhorkar Dr Baba Saheb Ambedkar Marathawada University, Aurangabad. Librarian Prof. M. Pitchaimani Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai. Organizing Committee Patron Dr. G. Viswanathan Chancellor, VIT Vellore Co-Patron Mr. G.V. Selvam Vice President, VIT Vellore Prof. Rambabu Kodali Vice Chancellor, VIT Vellore Prof. S. Narayanan Pro-Vice Chancellor, VIT Vellore Chair Person Prof. A. Mary Saral Dean, SAS, VIT Vellore Co-Chair Person Prof. K. Karthikeyan Head, Department of Mathematics, SAS, VIT Vellore Organising Secretary Prof. B. Rushi Kumar Associate Professor, Department of Mathematics, SAS, VIT Vellore Organising Committee Prof. Dinesh Kumar S Prof. Sri Rama Varaprasad Bhuvanagiri Prof. Sivaraj R Prof. Raja Das Prof. V Madhu Members (Faculty) Prof. Chandrasekaran V M Prof. Mubashir Unnissa M Prof. Murugusundaramoorthy G Prof. Easwaramoorthy D Prof. Selvakumar R Prof. Subramanyam Reddy A Prof. Natarajan G Prof. Yogalakshmi T Prof. Vijaya K Prof. Mokesh Rayalu G Prof.
    [Show full text]