Gromov-Witten Invariants of Symmetric Products of Projective Space
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Heterotic String Compactification with a View Towards Cosmology
Heterotic String Compactification with a View Towards Cosmology by Jørgen Olsen Lye Thesis for the degree Master of Science (Master i fysikk) Department of Physics Faculty of Mathematics and Natural Sciences University of Oslo May 2014 Abstract The goal is to look at what constraints there are for the internal manifold in phe- nomenologically viable Heterotic string compactification. Basic string theory, cosmology, and string compactification is sketched. I go through the require- ments imposed on the internal manifold in Heterotic string compactification when assuming vanishing 3-form flux, no warping, and maximally symmetric 4-dimensional spacetime with unbroken N = 1 supersymmetry. I review the current state of affairs in Heterotic moduli stabilisation and discuss merging cosmology and particle physics in this setup. In particular I ask what additional requirements this leads to for the internal manifold. I conclude that realistic manifolds on which to compactify in this setup are severely constrained. An extensive mathematics appendix is provided in an attempt to make the thesis more self-contained. Acknowledgements I would like to start by thanking my supervier Øyvind Grøn for condoning my hubris and for giving me free rein to delve into string theory as I saw fit. It has lead to a period of intense study and immense pleasure. Next up is my brother Kjetil, who has always been a good friend and who has been constantly looking out for me. It is a source of comfort knowing that I can always turn to him for help. Mentioning friends in such an acknowledgement is nearly mandatory. At least they try to give me that impression. -
Vincent Bouchard – Associate Professor
Vincent Bouchard | Associate Professor Rhodes Scholar (Québec and Magdalen, 2001) Department of Mathematical and Statistical Sciences, University of Alberta 632 CAB, Edmonton, AB, Canada – T6G 2G1 T +1 (780) 492 0099 • B [email protected] Í www.ualberta.ca/ vbouchar Education University of Oxford, Magdalen College Oxford, UK D.Phil. Mathematics 2001–2005 Université de Montréal Montréal, QC B.Sc. Physics, Excellence, Palmarès de la doyenne, average 4.3/4.3 1998–2001 Cégep Saint-Jean-sur-Richelieu Saint-Jean-sur-Richelieu, QC D.E.C. Natural Sciences, Excellence 1996–1998 D.Phil. thesis Title: Toric Geometry and String Theory [arXiv:hep-th/0609123] Supervisor: Rouse Ball Professor Philip Candelas Positions Associate Professor Edmonton, AB Department of Mathematical and Statistical Sciences, University of Alberta 2013–Present Assistant Professor Edmonton, AB Department of Mathematical and Statistical Sciences, University of Alberta 2009–2013 Postdoctoral Fellow Cambridge, MA Physics Department, Harvard University 2007–2009 NSERC Postdoctoral Fellow Waterloo, ON Perimeter Institute for Theoretical Physics 2006–2007 Postdoctoral Fellow Berkeley, CA Mathematical Sciences Research Institute 2006 “New Topological Structures in Physics” program, January–May Postdoctoral Fellow Phildelphia, PA Department of Mathematics, University of Pennsylvania 2005 Joint math/physics visiting position, September–December Research Grants NSERC Discovery Grant: $225,000 2013–2018 NSERC Discovery Grant: $130,000 2010–2013 University of Alberta Research -
Torification and Factorization of Birational Maps 3
TORIFICATION AND FACTORIZATION OF BIRATIONAL MAPS DAN ABRAMOVICH, KALLE KARU, KENJI MATSUKI, AND JAROSLAW WLODARCZYK Abstract. Building on work of the fourth author in [69], we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blowings up and blowings down with smooth centers. Contents 0. Introduction 1 1. Preliminaries 5 2. Birational Cobordisms 10 3. Torification 16 4. A proof of the weak factorization theorem 22 5. Generalizations 24 6. Problems related to weak factorization 27 References 28 0. Introduction We work over an algebraically closed field K of characteristic 0. We denote the multiplicative group of K by K∗. 0.1. Statement of the main result. The purpose of this paper is to give a proof for the following weak factorization conjecture of birational maps. We note that another proof of this theorem was given by the fourth author in [70]. See section 0.12 for a brief comparison of the two approaches. Theorem 0.1.1 (Weak Factorization). Let φ : X1 99K X2 be a birational map between complete nonsingular algebraic varieties X1 and X2 over an algebraically closed field K of characteristic zero, and let U ⊂ X1 be an open set where φ is an isomorphism. Then φ can be factored into a sequence of blowings up and blowings arXiv:math/9904135v4 [math.AG] 31 May 2000 down with smooth irreducible centers disjoint from U, namely, there exists a sequence of birational maps between complete nonsingular algebraic varieties ϕ1 ϕ2 ϕi ϕi+1 ϕi+2 ϕl−1 ϕl X1 = V0 99K V1 99K · · · 99K Vi 99K Vi+1 99K · · · 99K Vl−1 99K Vl = X2 where 1. -
Calabi-Yau Geometry and Higher Genus Mirror Symmetry
Calabi-Yau Geometry and Higher Genus Mirror Symmetry A dissertation presented by Si Li to The Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Mathematics Harvard University Cambridge, Massachusetts May 2011 © 2011 { Si Li All rights reserved. iii Dissertation Advisor: Professor Shing-Tung Yau Si Li Calabi-Yau Geometry and Higher Genus Mirror Symmetry Abstract We study closed string mirror symmetry on compact Calabi-Yau manifolds at higher genus. String theory predicts the existence of two sets of geometric invariants, from the A-model and the B-model on Calabi-Yau manifolds, each indexed by a non-negative inte- ger called genus. The A-model has been mathematically established at all genera by the Gromov-Witten theory, but little is known in mathematics for B-model beyond genus zero. We develop a mathematical theory of higher genus B-model from perturbative quantiza- tion techniques of gauge theory. The relevant gauge theory is the Kodaira-Spencer gauge theory, which is originally discovered by Bershadsky-Cecotti-Ooguri-Vafa as the closed string field theory of B-twisted topological string on Calabi-Yau three-folds. We generalize this to Calabi-Yau manifolds of arbitrary dimensions including also gravitational descen- dants, which we call BCOV theory. We give the geometric description of the perturbative quantization of BCOV theory in terms of deformation-obstruction theory. The vanishing of the relevant obstruction classes will enable us to construct the higher genus B-model. We carry out this construction on the elliptic curve and establish the corresponding higher genus B-model. -
New York Journal of Mathematics Lang's Conjectures, Fibered
New York Journal of Mathematics New York J. Math. 2 (1996) 20–34. Lang’s Conjectures, Fibered Powers, and Uniformity Dan Abramovich and Jos´e Felipe Voloch Abstract. We prove that the fibered power conjecture of Caporaso et al. (Con- jecture H, [CHM], §6) together with Lang’s conjecture implies the uniformity of rational points on varieties of general type, as predicted in [CHM]; a few applications on the arithmetic and geometry of curves are stated. In an opposite direction, we give counterexamples to some analogous results in positive characteristic. We show that curves that change genus can have arbitrarily many rational points; and that curves over Fp(t) can have arbitrarily many Frobenius orbits of non-constant points. Contents 1. Introduction 21 1.1. A Few Conjectures of Lang 21 1.2. The Fibered Power Conjecture 22 1.3. Summary of Results on the Implication Side 22 1.4. Summary of Results: Examples in Positive Characteristic 24 1.5. Acknowledgments 25 2. Proof of Theorem 1.5 25 2.1. Preliminaries 25 2.2. Prolongable Points 25 2.3. Proof of Theorem 1.5 26 3. A Few Refinements and Applications in Arithmetic and Geometry 26 3.1. Proof of Theorem 1.6 26 3.2. Uniformity in Terms of the Degree of an Extension 27 3.3. The Geometric Case 28 4. Examples in Positive Characteristic 30 4.1. Curves that Change Genus 30 Received December 20, 1995. Mathematics Subject Classification. 14G; 11G. Key words and phrases. arithmetic geometry, Lang’s conjecture, rational points. Abramovich partially supported by NSF grant DMS-9503276. -
Gauss-Manin Connection in Disguise: Calabi-Yau Threefolds 1
Gauss-Manin connection in disguise: Calabi-Yau threefolds 1 Murad Alim, Hossein Movasati, Emanuel Scheidegger, Shing-Tung Yau Abstract We describe a Lie Algebra on the moduli space of Calabi-Yau threefolds enhanced with differential forms and its relation to the Bershadsky-Cecotti-Ooguri-Vafa holo- morphic anomaly equation. In particular, we describe algebraic topological string alg partition functions Fg , g ≥ 1, which encode the polynomial structure of holomorphic and non-holomorphic topological string partition functions. Our approach is based on Grothendieck’s algebraic de Rham cohomology and on the algebraic Gauss-Manin connection. In this way, we recover a result of Yamaguchi-Yau and Alim-L¨ange in an algebraic context. Our proofs use the fact that the special polynomial generators defined using the special geometry of deformation spaces of Calabi-Yau threefolds cor- respond to coordinates on such a moduli space. We discuss the mirror quintic as an example. Contents 1 Introduction 2 2 Holomorphic anomaly equations 7 2.1 Specialgeometry ................................. 7 2.2 Specialcoordinates.............................. 8 2.3 Holomorphic anomaly equations . ... 9 2.4 Polynomialstructure. .. .. .. .. .. .. .. .. 9 3 Algebraic structure of topological string theory 10 3.1 Specialpolynomialrings . 10 3.2 Different choices of Hodge filtrations . ..... 12 3.3 LieAlgebradescription . 13 3.4 Algebraic master anomaly equation . ..... 14 arXiv:1410.1889v1 [math.AG] 7 Oct 2014 4 Proofs 15 4.1 Generalizedperioddomain . 15 4.2 ProofofTheorem1................................ 16 4.3 ProofofTheorem2................................ 17 4.4 The Lie Algebra G ................................ 18 4.5 Two fundamental equalities of the special geometry . .......... 18 5 Mirror quintic case 19 6 Final remarks 21 1 Math. -
MEMORIA DEL CENTRO DE CIENCIAS DE BENASQUE Barcelona, 31 De Diciembre De 2005
MEMORIA DEL CENTRO DE CIENCIAS DE BENASQUE Barcelona, 31 de Diciembre de 2005 I – INTRODUCCIÓN En unos pocos años España se ha convertido en uno de los focos mundiales de atracción de actividades veraniegas, principalmente de carácter lúdico y turístico. Hasta este momento se han usado poco las características de nuestro país para convertirlo en un foco de actividades de tipo científico. Hace unas pocas décadas la representación española en el concierto científico internacional era, prácticamente, inexistente. Aún en 1977 sólo el 0.64% de los artículos científicos que aparecían en la base de datos del Science Citation Index (SCI) tenían algún autor español. En el período 1977-1993 los artículos científicos del SCI en los que al menos un autor era español ha aumentado a un ritmo superior al 9% anual y en 1993 un 2.0% de todos los artículos contenidos en el SCI tienen, al menos, un autor español. Se ha avanzado mucho en pocos años, pero este esfuerzo debe continuar. En España era habitual, en 1994, la celebración de conferencias científicas sobre distintas temáticas y con características diversas. Todas estas reuniones, más o menos multitudinarias, se caracterizan por su corta duración (alrededor de una semana) y gran número diario de conferencias, lo cual hace que quede poco tiempo para poder trabajar o discutir con otros científicos. Sin embargo ni en nuestro país ni en el resto de Europa existían, por aquel entonces, verdaderos centros donde, en un ambiente agradable, los científicos se pudieran dedicar a trabajar tranquilamente y a discutir con sus colegas. Un centro de este tipo lleva muchos años funcionando, con gran éxito, en la localidad norteamericana de Aspen, en Colorado. -
Albert Einstein
L B. A Department of Physics (MC 0435) Tel:(+1)2155100693 Virginia Te Fax:(+1)5402317511 850 Campus Drive email: [email protected] Blasburg, VA, 24061, USA. URL: hp://www.phys.vt.edu/ lara/ Employment - Assistant Professor of Physics and Affiliate Professor of Mathematics Virginia Tech (Blacksburg, VA, USA), - Postdoctoral Researer in eoretical Physics Harvard University (Boston, MA, USA), - Postdoctoral Researer in eoretical Physics University of Pennsylvania (Philadelphia, PA, USA), - Visiting Researer in eoretical/Mathematical Physics Institute for Advanced Study (Princeton, NJ, USA), - Stipendiary Lecturer in Applied Mathematics Pembroke College, University of Oxford (Oxford, UK) Stipendiary Lecturer in Applied Mathematics Somerville College, University of Oxford (Oxford, UK) V P/F Kavli Institute for eoretical Physics, UC Santa Barbara (). Isaac Newton Institute for Mathemat- ical Sciences, University of Cambridge (). Simons Center for Geometry and Physics (). Simons Center for Geometry and Physics (). Resear Interests: Physics – String eory (including M-theory, F-theory, and Heterotic string theory), String Phenomenol- ogy, High Energy Particle Physics, Gauge eory, Supersymmetry Mathematics – Algebraic Geometry and Topology (including computational algebraic geometry), Ge- ometric Invariants, Moduli Spaces, Special Holonomy/Special Structure Manifolds. Education - University of Oxford (Oxford, UK), PhD in Mathematical Physics. esis Advisors: Philip Candelas (Mathematics) and Andre Lukas (eoretical Physics) - Utah State University (Logan, UT, USA), Master of Science, Physics. Summa Cum Laude. - Utah State University (Logan, UT, USA), Bachelor of Science, Physics. Summa Cum Laude. (4.0 GPA. Ranked 1st in graduating class). Bachelor of Science, Mathematics. Summa Cum Laude. ! S F S Royal Society University Research Fellowship (UK) ( years and £600K of funding). (Declined to accept position at Virginia Tech). -
2020-2021 Annual Report
Institute for Computational and Experimental Research in Mathematics Annual Report May 1, 2020 – April 30, 2021 Brendan Hassett, Director, PI Mathew Borton, IT Director Ruth Crane, Assistant Director and Chief of Staff Juliet Duyster, Assistant Director Finance and Administration Sigal Gottlieb, Deputy Director Jeffrey Hoffstein, Consulting Associate Director Caroline Klivans, Deputy Director Benoit Pausader, co-PI Jill Pipher, Consulting Director Emerita, co-PI Kavita Ramanan, Associate Director, co-PI Bjorn Sandstede, Associate Director, co-PI Ulrica Wilson, Associate Director for Diversity and Outreach Table of Contents Mission ........................................................................................................................................... 5 Annual Report for 2020-2021 ........................................................................................................ 5 Core Programs and Events ............................................................................................................ 5 Participant Summaries by Program Type ..................................................................................... 7 ICERM Funded Participants ................................................................................................................. 7 ICERM Funded Speakers ...................................................................................................................... 9 All Speakers (ICERM funded and Non-ICERM funded) ................................................................ -
Eighteenth En.Pdf
ICMAT Newsletter #18 First semester 2019 CSIC - UAM - UC3M - UCM EDITORIAL A new year has begun, and another has come to an end, which makes us especially aware of the ICMAT as a “home for mathe- matics” that is open to the international community. This News- letter contains information about some of the most interesting results that have recently been obtained at the Institute. How- Image: ICMAT. ever, it is also worth pointing out that the influence of a centre like the ICMAT is not confined to a mere list of its publications and its members, but will always extend to cover all the results, projects and collaborations that originate in our facilities and go on to bear fruit later, very often far from where they started. The prime objective of the ICMAT is mathematical research, arising from the stimulus of that original thought which produces deci- sive advances in science, enriching our knowledge of nature and leading to applications that were frequently once unimaginable. Appointments for the next four years of new faculty members be- Antonio Córdoba. longing to the three universities of the Institute were completed in January, with the ratification of the Steering Committee for the proposal drawn up by the Joint Committee (consisting of the three universities and the CSIC), and which had already received approv- CONTENTS al from the Governing Board of the ICMAT. We welcome these new members and congratulate those renewing their membership of Editorial: Antonio Córdoba (ICMAT)........................1 the Institute. There is no doubt that their abilities, good work and Report: The Theory of Moduli Spaces: dedication will help the ICMAT continue to advance, with an excit- The Treasure Map....................................................3 ing project which this year includes the application for the third Severo Ochoa Centre of Excellence award. -
BRYCE SELIGMAN DEWITT January 8, 1923–September 23, 2004
NATIONAL ACADEMY OF SCIENCES B R Y C E S E L I G M A N D EWITT 1 9 2 3 — 2 0 0 4 A Biographical Memoir by S T E V E N W EIN B ERG Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. Biographical Memoir COPYRIGHT 2008 NATIONAL ACADEMY OF SCIENCES WASHINGTON, D.C. Larry Murphy BRYCE SELIGMAN DEWITT January 8, 1923–September 23, 2004 BY STEVEN WEIN BERG RYCE SELIGMAN DEWITT, PROFESSOR EMERITUS IN THE physics Bdepartment of the University of Texas at Austin, died on September 23, 2004. His career was marked by major contributions to classical and quantum field theories, in particular, to the theory of gravitation. DeWitt was born Carl Bryce Seligman on January 8, 1923, in Dinuba, California, the eldest of four boys. His paternal grandfather, Emil Seligman, left Germany around 1875 at the age of 17 and emigrated to California, where he and his brother established a general store in Traver. Emil mar- ried Anna Frey, a young woman who had emigrated from Switzerland at about the same time. They had 11 children, whom Anna raised in the Methodist church. In 1921 DeWitt’s father, who had become a country doctor, married the local high school teacher of Latin and mathematics. Her ancestors were French Huguenots and Scottish Presbyterians. DeWitt was raised in the Presbyterian Church, and the only Jewish elements in his early life were the matzos that his grandfather bought around the time of Passover. -
Arxiv:Math/9809023V1 [Math.AG] 5 Sep 1998 .W a That Say We 0
UNIFORMITY OF STABLY INTEGRAL POINTS ON PRINCIPALLY POLARIZED ABELIAN VARIETIES OF DIMENSION ≤ 2 DAN ABRAMOVICH AND KENJI MATSUKI Abstract. The purpose of this paper is to prove, assuming that the conjecture of Lang and Vojta holds true, that there is a uniform bound on the number of stably integral points in the complement of the theta divisor on a principally polarized abelian surface defined over a number field. Most of our argument works in arbitrary dimension and the restriction on the dimension ≤ 2 is used only at the last step, where we apply Pacelli’s stronger uniformity results for elliptic curves. Preliminary version, July 13, 2021. 0. Introduction 0.1. The conjecture of Lang and Vojta. Let X be a variety over a field of characteristic 0. We say that X is a variety of logarithmic general type, if there exists a desingularization X˜ → X, and a projective embedding X˜ ⊂ Y where D = Y X˜ is a divisor of normal crossings, such that the invertible sheaf ωY (D) is big. We note first that this peoperty is independent of the choices of X˜ and Y , and that it is a proper birational invariant, namely, if X′ → X is a proper birational morphism (or an inverse of such) then X is of logarithmic general type if and only if X′ is. Now let X be a variety of logarithmic general type defined over a number field K. Let S be a finite set of places in K and let OK,S ⊂ K be the ring of S-integers. Fix a model X of X over OK,S.