Curriculum Vitae 2021
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Arxiv:2011.02915V1 [Math.LO] 3 Nov 2020 the Otniy Ucin Pnst Tctr)Ue Nti Ae R T Are Paper (C 1.1
REVERSE MATHEMATICS OF THE UNCOUNTABILITY OF R: BAIRE CLASSES, METRIC SPACES, AND UNORDERED SUMS SAM SANDERS Abstract. Dag Normann and the author have recently initiated the study of the logical and computational properties of the uncountability of R formalised as the statement NIN (resp. NBI) that there is no injection (resp. bijection) from [0, 1] to N. On one hand, these principles are hard to prove relative to the usual scale based on comprehension and discontinuous functionals. On the other hand, these principles are among the weakest principles on a new com- plimentary scale based on (classically valid) continuity axioms from Brouwer’s intuitionistic mathematics. We continue the study of NIN and NBI relative to the latter scale, connecting these principles with theorems about Baire classes, metric spaces, and unordered sums. The importance of the first two topics re- quires no explanation, while the final topic’s main theorem, i.e. that when they exist, unordered sums are (countable) series, has the rather unique property of implying NIN formulated with the Cauchy criterion, and (only) NBI when formulated with limits. This study is undertaken within Ulrich Kohlenbach’s framework of higher-order Reverse Mathematics. 1. Introduction The uncountability of R deals with arbitrary mappings with domain R, and is therefore best studied in a language that has such objects as first-class citizens. Obviousness, much more than beauty, is however in the eye of the beholder. Lest we be misunderstood, we formulate a blanket caveat: all notions (computation, continuity, function, open set, et cetera) used in this paper are to be interpreted via their higher-order definitions, also listed below, unless explicitly stated otherwise. -
Fundamental Algebraic Geometry
http://dx.doi.org/10.1090/surv/123 hematical Surveys and onographs olume 123 Fundamental Algebraic Geometry Grothendieck's FGA Explained Barbara Fantechi Lothar Gottsche Luc lllusie Steven L. Kleiman Nitin Nitsure AngeloVistoli American Mathematical Society U^VDED^ EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 14-01, 14C20, 13D10, 14D15, 14K30, 18F10, 18D30. For additional information and updates on this book, visit www.ams.org/bookpages/surv-123 Library of Congress Cataloging-in-Publication Data Fundamental algebraic geometry : Grothendieck's FGA explained / Barbara Fantechi p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 123) Includes bibliographical references and index. ISBN 0-8218-3541-6 (pbk. : acid-free paper) ISBN 0-8218-4245-5 (soft cover : acid-free paper) 1. Geometry, Algebraic. 2. Grothendieck groups. 3. Grothendieck categories. I Barbara, 1966- II. Mathematical surveys and monographs ; no. 123. QA564.F86 2005 516.3'5—dc22 2005053614 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. -
Category Theory and Lambda Calculus
Category Theory and Lambda Calculus Mario Román García Trabajo Fin de Grado Doble Grado en Ingeniería Informática y Matemáticas Tutores Pedro A. García-Sánchez Manuel Bullejos Lorenzo Facultad de Ciencias E.T.S. Ingenierías Informática y de Telecomunicación Granada, a 18 de junio de 2018 Contents 1 Lambda calculus 13 1.1 Untyped λ-calculus . 13 1.1.1 Untyped λ-calculus . 14 1.1.2 Free and bound variables, substitution . 14 1.1.3 Alpha equivalence . 16 1.1.4 Beta reduction . 16 1.1.5 Eta reduction . 17 1.1.6 Confluence . 17 1.1.7 The Church-Rosser theorem . 18 1.1.8 Normalization . 20 1.1.9 Standardization and evaluation strategies . 21 1.1.10 SKI combinators . 22 1.1.11 Turing completeness . 24 1.2 Simply typed λ-calculus . 24 1.2.1 Simple types . 25 1.2.2 Typing rules for simply typed λ-calculus . 25 1.2.3 Curry-style types . 26 1.2.4 Unification and type inference . 27 1.2.5 Subject reduction and normalization . 29 1.3 The Curry-Howard correspondence . 31 1.3.1 Extending the simply typed λ-calculus . 31 1.3.2 Natural deduction . 32 1.3.3 Propositions as types . 34 1.4 Other type systems . 35 1.4.1 λ-cube..................................... 35 2 Mikrokosmos 38 2.1 Implementation of λ-expressions . 38 2.1.1 The Haskell programming language . 38 2.1.2 De Bruijn indexes . 40 2.1.3 Substitution . 41 2.1.4 De Bruijn-terms and λ-terms . 42 2.1.5 Evaluation . -
Curriculum Vitae - Rogier Bos
Curriculum Vitae - Rogier Bos Personal Full name: Rogier David Bos. Date and place of birth: August 7, 1978, Ter Aar, The Netherlands. Email: rogier bos [at] hotmail.com. Education 2010 M. Sc. Mathematics Education at University of Utrecht, 2007 Ph. D. degree in Mathematics at the Radboud University Nijmegen, thesis: Groupoids in geometric quantization, advisor: Prof. dr. N. P. Landsman. Manuscript committee: Prof. dr. Gert Heckman, Prof. dr. Ieke Moerdijk (Universiteit Utrecht), Prof. dr. Jean Renault (Universit´ed'Orl´eans), Prof. dr. Alan Weinstein (University of California), Prof. dr. Ping Xu (Penn State University). 2001 M. Sc. (Doctoraal) in Mathematics at the University of Utrecht, thesis: Operads in deformation quantization, advisor: Prof. dr. I. Moerdijk; minors in philosophy and physics. 1997 Propedeuse in Mathematics at the University of Utrecht. 1997 Propedeuse in Physics at the University of Utrecht. 1996 Gymnasium at Erasmus College Zoetermeer. General Working experience 2010-now Co-author dutch mathematics teaching series \Wagen- ingse Methode". 2010-now Lecturer at Master Mathematics Education at Hogeschool Utrecht. 2009-now Mathematics and general science teacher at the Chris- telijk Gymnasium Utrecht. 2007-2008 Post-doctoral researcher at the Center for Mathematical Analysis, Geometry, and Dynamical Systems, Instituto Superior T´ecnicoin Lisbon (Portugal). 2002-2007 Ph.D.-student mathematical physics at the University of Amsterdam (Radboud University Nijmegen from 2004); funded by FOM; promotor: Prof. dr. N. P. Landsman. 2002 march-august Mathematics teacher at the highschool Minkema College in Woerden. 1 1999-2001 Teacher of mathematics exercise classes at the Univer- sity of Utrecht. University Teaching Experience 2007 Mathematics deficiency repair course for highschool students (in preparation of studying mathematics or physics), Radboud Univer- sity Nijmegen. -
An Outline of Algebraic Set Theory
An Outline of Algebraic Set Theory Steve Awodey Dedicated to Saunders Mac Lane, 1909–2005 Abstract This survey article is intended to introduce the reader to the field of Algebraic Set Theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, admitting adjustment in several respects to model different theories including classical, intuitionistic, bounded, and predicative ones. Under this scheme some familiar set theoretic properties are related to algebraic ones, like freeness, while others result from logical constraints, like definability. The overall theory is complete in two important respects: conventional elementary set theory axiomatizes the class of algebraic models, and the axioms provided for the abstract algebraic framework itself are also complete with respect to a range of natural models consisting of “ideals” of sets, suitably defined. Some previous results involving realizability, forcing, and sheaf models are subsumed, and the prospects for further such unification seem bright. 1 Contents 1 Introduction 3 2 The category of classes 10 2.1 Smallmaps ............................ 12 2.2 Powerclasses............................ 14 2.3 UniversesandInfinity . 15 2.4 Classcategories .......................... 16 2.5 Thetoposofsets ......................... 17 3 Algebraic models of set theory 18 3.1 ThesettheoryBIST ....................... 18 3.2 Algebraic soundness of BIST . 20 3.3 Algebraic completeness of BIST . 21 4 Classes as ideals of sets 23 4.1 Smallmapsandideals . .. .. 24 4.2 Powerclasses and universes . 26 4.3 Conservativity........................... 29 5 Ideal models 29 5.1 Freealgebras ........................... 29 5.2 Collection ............................. 30 5.3 Idealcompleteness . .. .. 32 6 Variations 33 References 36 2 1 Introduction Algebraic set theory (AST) is a new approach to the construction of models of set theory, invented by Andr´eJoyal and Ieke Moerdijk and first presented in [16]. -
E.W. Beth Als Logicus
E.W. Beth als logicus ILLC Dissertation Series 2000-04 For further information about ILLC-publications, please contact Institute for Logic, Language and Computation Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam phone: +31-20-525 6051 fax: +31-20-525 5206 e-mail: [email protected] homepage: http://www.illc.uva.nl/ E.W. Beth als logicus Verbeterde electronische versie (2001) van: Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magni¯cus prof.dr. J.J.M. Franse ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Aula der Universiteit op dinsdag 26 september 2000, te 10.00 uur door Paul van Ulsen geboren te Diemen Promotores: prof.dr. A.S. Troelstra prof.dr. J.F.A.K. van Benthem Faculteit Natuurwetenschappen, Wiskunde en Informatica Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam Copyright c 2000 by P. van Ulsen Printed and bound by Print Partners Ipskamp. ISBN: 90{5776{052{5 Ter nagedachtenis aan mijn vader v Inhoudsopgave Dankwoord xi 1 Inleiding 1 2 Levensloop 11 2.1 Beginperiode . 12 2.1.1 Leerjaren . 12 2.1.2 Rijpingsproces . 14 2.2 Universitaire carriere . 21 2.2.1 Benoeming . 21 2.2.2 Geleerde genootschappen . 23 2.2.3 Redacteurschappen . 31 2.2.4 Beth naar Berkeley . 32 2.3 Beth op het hoogtepunt van zijn werk . 33 2.3.1 Instituut voor Grondslagenonderzoek . 33 2.3.2 Oprichting van de Centrale Interfaculteit . 37 2.3.3 Logici en historici aan Beths leiband . -
Requirements and Theories of Meaning
INTERNATIONAL CONFERENCE ON ENGINEERING DESIGN, ICED07 28-31 AUGUST 2007, CITE DES SCIENCE ET DE L’INDUSTRIE, PARIS, FRANCE REQUIREMENTS AND THEORIES OF MEANING Ichiro Nagasaka Kobe University ABSTRACT In this paper, the fundamental characteristics of the concept of meaning in the traditional theory of design are discussed and the similarities between the traditional theory of design and the truth-conditional theory of meaning are pointed out. The difficulties brought by the interpretation of the model are explained in the context of theory of design and the argument that the sequence-of-use based proof-conditional theory of meaning could resolve the difficulties by changing the notion of the requirement is presented. Finally, based on the ideas of the proof-conditional theory of meaning we show the perspective of the formulation of the sequence-of-use based theory and discuss how the theory deals with the sequence of use in a constructive way. Keywords: Requirements, theories of meaning, constructive theory of design 1 INTRODUCTION As we all know, the requirement or the need has an essential role to play in the process of design. Most of the theories of design proposed since the early 1960s share the view on the requirement. For example: • Design begins with the recognition of a need and the conception of an idea to satisfy this need [1, p.iii]. • The main task of engineers is to apply their scientific and engineering knowledge to the solution of technical problems, and then to optimize those solutions within the requirements and constraints [...] [2, p.1]. • It is widely recognized that any serious attempt to make the environment work, must begin with a statement of user needs. -
On The´Etale Fundamental Group of Schemes Over the Natural Numbers
On the Etale´ Fundamental Group of Schemes over the Natural Numbers Robert Hendrik Scott Culling February 2019 A thesis submitted for the degree of Doctor of Philosophy of the Australian National University It is necessary that we should demonstrate geometrically, the truth of the same problems which we have explained in numbers. | Muhammad ibn Musa al-Khwarizmi For my parents. Declaration The work in this thesis is my own except where otherwise stated. Robert Hendrik Scott Culling Acknowledgements I am grateful beyond words to my thesis advisor James Borger. I will be forever thankful for the time he gave, patience he showed, and the wealth of ideas he shared with me. Thank you for introducing me to arithmetic geometry and suggesting such an interesting thesis topic | this has been a wonderful journey which I would not have been able to go on without your support. While studying at the ANU I benefited from many conversations about al- gebraic geometry and algebraic number theory (together with many other inter- esting conversations) with Arnab Saha. These were some of the most enjoyable times in my four years as a PhD student, thank you Arnab. Anand Deopurkar helped me clarify my own ideas on more than one occasion, thank you for your time and help to give me new perspective on this work. Finally, being a stu- dent at ANU would not have been nearly as fun without the chance to discuss mathematics with Eloise Hamilton. Thank you Eloise for helping me on so many occasions. To my partner, Beth Vanderhaven, and parents, Cherry and Graeme Culling, thank you for your unwavering support for my indulgence of my mathematical curiosity. -
PHIL 149 - Special Topics in Philosophy of Logic and Mathematics: Nonclassical Logic Professor Wesley Holliday Tuth 11-12:30 UC Berkeley, Fall 2018 Dwinelle 88
PHIL 149 - Special Topics in Philosophy of Logic and Mathematics: Nonclassical Logic Professor Wesley Holliday TuTh 11-12:30 UC Berkeley, Fall 2018 Dwinelle 88 Syllabus Description A logical and philosophical exploration of alternatives to the classical logic students learn in PHIL 12A. The focus will be on intuitionistic logic, as a challenger to classical logic for reasoning in mathematics, and quantum logic, as a challenger to classical logic for reasoning about the physical world. Prerequisites PHIL 12A or equivalent. Readings All readings will be available on the bCourses site. Requirements { Weekly problem sets, due on Tuesdays in class (45% of grade) { Term paper of around 7-8 pages due December 7 on bCourses (35% of grade) { Final exam on December 12, 8-11am with location TBA (20% of grade) Class, section, and Piazza participation will be taken into account for borderline grades. For graduate students in philosophy, this course satisfies the formal philosophy course requirement. Graduate students should see Professor Holliday about assignments. Sections All enrolled students must attend a weekly discussion section with our GSI, Yifeng Ding, a PhD candidate in the Group in Logic and the Methodology of Science. Contact Prof. Holliday | [email protected] | OHs: TuTh 1-2, 246 Moses Yifeng Ding | [email protected] | OHs: Tu 2-4, 937 Evans Schedule Aug. 23 (Th) Course Overview Reading: none. 1/5 Part I: Review of Classical Logic Aug. 28 (Tu) Review of Classical Natural Deduction Videos: Prof. Holliday’s videos on natural deduction. Aug. 30 (Th) Nonconstructive Proofs Reading: Chapter 1, Section 2 of Constructivism in Mathematics by Anne Sjerp Troelstra and Dirk van Dalen. -
Two Lectures on Constructive Type Theory
Two Lectures on Constructive Type Theory Robert L. Constable July 20, 2015 Abstract Main Goal: One goal of these two lectures is to explain how important ideas and problems from computer science and mathematics can be expressed well in constructive type theory and how proof assistants for type theory help us solve them. Another goal is to note examples of abstract mathematical ideas currently not expressed well enough in type theory. The two lec- tures will address the following three specific questions related to this goal. Three Questions: One, what are the most important foundational ideas in computer science and mathematics that are expressed well in constructive type theory, and what concepts are more difficult to express? Two, how can proof assistants for type theory have a large impact on research and education, specifically in computer science, mathematics, and beyond? Three, what key ideas from type theory are students missing if they know only one of the modern type theories? The lectures are intended to complement the hands-on Nuprl tutorials by Dr. Mark Bickford that will introduce new topics as well as address these questions. The lectures refer to recent educational material posted on the PRL project web page, www.nuprl.org, especially the on-line article Logical Investigations, July 2014 on the front page of the web cite. Background: Decades of experience using proof assistants for constructive type theories, such as Agda, Coq, and Nuprl, have brought type theory ever deeper into computer science research and education. New type theories implemented by proof assistants are being developed. More- over, mathematicians are showing interest in enriching these theories to make them a more suitable foundation for mathematics. -
An Alpine Bouquet of Algebraic Topology
708 An Alpine Bouquet of Algebraic Topology Alpine Algebraic and Applied Topology Conference August 15–21, 2016 Saas-Almagell, Switzerland Christian Ausoni Kathryn Hess Brenda Johnson Ieke Moerdijk Jérôme Scherer Editors 708 An Alpine Bouquet of Algebraic Topology Alpine Algebraic and Applied Topology Conference August 15–21, 2016 Saas-Almagell, Switzerland Christian Ausoni Kathryn Hess Brenda Johnson Ieke Moerdijk Jérôme Scherer Editors EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Catherine Yan 2010 Mathematics Subject Classification. Primary 11S40, 18Axx, 18Gxx, 19Dxx, 55Nxx, 55Pxx, 55Sxx, 55Uxx, 57Nxx. Library of Congress Cataloging-in-Publication Data Names: Conference on Alpine Algebraic and Applied Topology (2016: Saas-Almagell, Switzerland) | Ausoni, Christian, 1968– editor. Title: An alpine bouquet of algebraic topology: Conference on Alpine Algebraic and Applied Topology, August 15–21, 2016, Saas-Almagell, Switzerland / Christian Ausoni [and four oth- ers], editors. Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Contem- porary mathematics; volume 708 | Includes bibliographical references. Identifiers: LCCN 2017055549 | ISBN 9781470429119 (alk. paper) Subjects: LCSH: Algebraic topology–Congresses. | AMS: Number theory – Algebraic number theory: local and p-adic fields – Zeta functions and L-functions. msc | Category theory; homological algebra – General theory of categories and functors – General theory of categories and functors. msc | Category theory; homological algebra – Homological algebra – Homological algebra. msc | K-theory – Higher algebraic K-theory – Higher algebraic K-theory. msc | Algebraic topology – Homology and cohomology theories – Homology and cohomology theories. msc | Algebraic topology – Homotopy theory – Homotopy theory. msc | Algebraic topology – Operations and obstructions – Operations and obstructions. msc | Algebraic topology – Applied homological algebra and category theory – Applied homological algebra and category theory. -
Classifying Toposes and Foliations Annales De L’Institut Fourier, Tome 41, No 1 (1991), P
ANNALES DE L’INSTITUT FOURIER IEKE MOERDIJK Classifying toposes and foliations Annales de l’institut Fourier, tome 41, no 1 (1991), p. 189-209 <http://www.numdam.org/item?id=AIF_1991__41_1_189_0> © Annales de l’institut Fourier, 1991, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Fourier, Grenoble 41, 1 (1991), 189-209 CLASSIFYING TOPOSES AND FOLIATIONS by leke MOERDIJK This paper makes no special claim to originality. Its sole purpose is to point out that in some circumstances, classifying toposes are more convenient to work with than classifying spaces. Let G be a topological groupoid. I will focus on the case where G is etale, in the sense that the domain and codomain maps do and di : GI =4 GQ are etale (that is, are local homeomorphisms). A prime example is Haefliger's [14] groupoid ^q of germs of local diffeomorphisms of R9, which enters into the construction of the classifying space B^q for foliations of codimension q. But the holonomy groupoid [15] of any foliation is an example of an etale topological groupoid, as is any 5-atlas [9]. For such a groupoid G, one can construct its classifying space BG by taking the geometric realization of the nerve (?• of G (Segal [29], [30], Haefliger [14]).