The Materiality & Ontology of Digital Subjectivity
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Describing the Universal Cover of a Compact Limit ∗
Describing the Universal Cover of a Compact Limit ∗ John Ennis Guofang Wei Abstract If X is the Gromov-Hausdorff limit of a sequence of Riemannian manifolds n Mi with a uniform lower bound on Ricci curvature, Sormani and Wei have shown that the universal cover X˜ of X exists [13, 14]. For the case where X is compact, we provide a description of X˜ in terms of the universal covers M˜ i of the manifolds. More specifically we show that if X¯ is the pointed Gromov- Hausdorff limit of the universal covers M˜ i then there is a subgroup H of Iso(X¯) such that X˜ = X/H.¯ 1 Introduction In 1981 Gromov proved that any finitely generated group has polynomial growth if and only if it is almost nilpotent [7]. In his proof, Gromov introduced the Gromov- Hausdorff distance between metric spaces [7, 8, 9]. This distance has proven to be especially useful in the study of n-dimensional manifolds with Ricci curvature uniformly bounded below since any sequence of such manifolds has a convergent subsequence [10]. Hence we can follow an approach familiar to analysts, and consider the closure of the class of all such manifolds. The limit spaces of this class have path metrics, and one can study these limit spaces from a geometric or topological perspective. Much is known about the limit spaces of n-dimensional Riemannian manifolds with a uniform lower bound on sectional curvature. These limit spaces are Alexandrov spaces with the same curvature bound [1], and at all points have metric tangent cones which are metric cones. -
Group Knowledge and Mathematical Collaboration: a Philosophical Examination of the Classification of Finite Simple Groups
Group Knowledge and Mathematical Collaboration: A Philosophical Examination of the Classification of Finite Simple Groups Joshua Habgood-Coote and Fenner Stanley Tanswell Forthcoming in Episteme, please refer to final version. Abstract In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but have received very little attention. In this paper, we will consider the philosophical tensions that Steingart uncovers, and use them to argue that the best account of the epistemic status of the Classification Theorem will be essentially and ineliminably social. This forms part of the broader argument that in order to understand mathematical proofs, we must appreciate their social aspects. 1 Introduction Popular conceptions of mathematics are gripped by the myth of the ‘lone genius’. This myth is inspired by famous figures like Andrew Wiles, Grigori Perelman, and Srinivasa Ramanujan whose individual efforts are taken to be both representative and exemplary. In this paper, we push back against this individualistic view of how mathematics is and should be practiced, examining the significance of social and group level features of mathematical practices. In particular, we will discuss the epistemology of mathematics, and argue that in order to understand mathematics and its epistemology, we need to pay attention to collaboration, group knowledge, and other social factors. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
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. -
On the Sormani-Wenger Intrinsic Flat Convergence of Alexandrov Spaces 3
ON THE SORMANI-WENGER INTRINSIC FLAT CONVERGENCE OF ALEXANDROV SPACES NAN LI AND RAQUEL PERALES Abstract. We study sequences of integral current spaces (X j, d j, T j) such that the integral current structure T j has weight 1 and no boundary and, all (X j, d j) are closed Alexandrov spaces with curvature uniformly bounded from below and diameter uniformly bounded from above. We prove that for such sequences either their limits collapse or the Gromov- Hausdorff and Sormani-Wenger Intrinsic Flat limits agree. The latter is done showing that the lower n dimensional density of the mass measure at any regular point of the Gromov- Hausdorff limit space is positive by passing to a filling volume estimate. In an appendix we show that the filling volume of the standard n dimensional integral current space coming from an n dimensional sphere of radius r > 0 in Euclidean space equals rn times the filling volume of the n dimensional integral current space coming from the n dimensional sphere of radius 1. Introduction Burago, Gromov and Perelman proved that sequences of Alexandrov spaces with cur- vature uniformly bounded from below, diameter and dimension uniformly bounded from above, have subsequences which converge in the Gromov-Hausdorff sense to an Alexan- drov space with the same curvature and diameter bounds. The properties of Alexandrov spaces and their Gromov-Hausdorff limit spaces have been amply studied by Alexander- Bishop [1], Alexander-Kapovitch-Petrunin [2], Burago-Gromov-Perelman [5], Burago- Burago-Ivanov [4], Li-Rong [15], Otsu-Shioya [20], and many others. The Gromov Hausdorff distance between two metric spaces, Xi, is defined as, Z (0.1) dGH (X , X ) = inf d (ϕ (X ) , ϕ (X )) , 1 2 { H 1 1 2 2 } Z ff where dH denotes the Hausdor distance and the infimum is taken over all complete metric spaces Z and all distance preserving maps ϕi : Xi Z, [7]. -
Large Scale Geometry by Nowak and Yu
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 52, Number 1, January 2015, Pages 141–149 S 0273-0979(2014)01460-0 Article electronically published on August 20, 2014 Large scale geometry, by P. Nowak and G. Yu, EMS Textbooks in Mathematics, European Mathematical Society, Zu¨rich, 2012, xiv+189 pp., ISBN 978-3-03719- 112-5, AC38.00, $41.80 Everyone knows that many phenomena are best studied at one scale or another, and that deep problems often involve the interactions of several scales. From the broad conception of our understanding of physics, based on the atomic hypothesis to the germ theory of disease to the success of calculus with its move to the infin- itesimal, the value of the study of phenomena at small scales and the techniques of integrating such information to the macroscopic level are now the most basic intuition in the scientific world view—with attendant backlashes against “blind reductionism”. In the subject of global geometry, the important general direction of “under- standing manifolds under some condition of curvature” is of this sort. Curvature is the infinitesimal measure of how the space di®ers from Euclidean space, and one tries to obtain global conclusions from uniform assumptions on this type of local hypothesis. There is also another extreme: going from the cosmic, the large scale, back to the local. In retrospect, at least, one can see this trend in, say, Liouville’s theorem that bounded analytic functions are constant: boundedness is a large scale condition; surely, if one looks from a very large scale, a bounded function should not be viewed as di®erent than a constant, and Liouville’s theorem tells us that under a condition of analyticity, the large scale information tells all. -
An Intrinsic Parallel Transport in Wasserstein Space
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 145, Number 12, December 2017, Pages 5329–5340 http://dx.doi.org/10.1090/proc/13655 Article electronically published on July 10, 2017 AN INTRINSIC PARALLEL TRANSPORT IN WASSERSTEIN SPACE JOHN LOTT (Communicated by Guofang Wei) Abstract. If M is a smooth compact connected Riemannian manifold, let P (M) denote the Wasserstein space of probability measures on M. We describe a geometric construction of parallel transport of some tangent cones along geodesics in P (M). We show that when everything is smooth, the geometric parallel transport agrees with earlier formal calculations. 1. Introduction Let M be a smooth compact connected Riemannian manifold without boundary. The space P (M) of probability measures of M carries a natural metric, the Wasser- stein metric, and acquires the structure of a length space. There is a close relation between minimizing geodesics in P (M) and optimal transport between measures. For more information on this relation, we refer to Villani’s book [13]. Otto discovered a formal Riemannian structure on P (M), underlying the Wasser- stein metric [10]. One can do formal geometric calculations for this Riemannian structure [6]. It is an interesting problem to make these formal considerations into rigorous results in metric geometry. If M has nonnegative sectional curvature, then P (M) is a compact length space with nonnegative curvature in the sense of Alexandrov [8, Theorem A.8], [12, Propo- sition 2.10]. Hence one can define the tangent cone TμP (M)ofP (M)atameasure μ ∈ P (M). If μ is absolutely continuous with respect to the volume form dvolM , then TμP (M) is a Hilbert space [8, Proposition A.33]. -
Nominations for President
Nominations for President combinatorics which was then starting to become fashion- Nomination of able; and some number theory, which has never gone out George Andrews of fashion although the parts of it George dealt with were not fashionable then. George spent a year on leave at MIT. Richard Askey At Rota’s suggestion, he edited the “Collected Papers” of P. A. MacMahon. He also wrote a book, “The Theory of Who is George Andrews? According to Freeman Dyson, Partitions”, for the Encyclopedia of Mathematics and Its George Andrews is the chief gardener in Ramanujan’s Gar- Applications series which Rota edited. den. This is true, but is only part of who George Andrews George came to the University of Wisconsin-Madison for is. He is a number theorist with an honorary doctorate in a year, 1976-77. In the spring he went to Europe for two physics. He is a long-time user of computers in his own meetings in France. Since he was not teaching and airfare research, who has written about the harm technology can abroad was less expensive if you spent 21 to 45 days, he do in mathematics education. He is primarily a problem also went to Cambridge to look for old work in the Wren solver, yet one paper with Rodney Baxter and Peter For- Library of Trinity College. This changed his life. rester has had a major impact in mathematical physics What George found was a bit over 100 pages of math- with currently 490 citations in the Web of Science. ematical claims in the distinctive handwriting of Srinivasa George Andrews received his Ph.D. -
Applications at the International Congress by Marty Golubitsky
From SIAM News, Volume 39, Number 10, December 2006 Applications at the International Congress By Marty Golubitsky Grigori Perelman’s decision to decline the Fields Medal, coupled with the speculations surrounding this decision, propelled the 2006 Fields Medals to international prominence. Stories about the medals and the award ceremony at the International Congress of Mathematicians in Madrid this summer appeared in many influential news outlets (The New York Times, BBC, ABC, . .) and even in popular magazines (The New Yorker). In Madrid, the topologist John Morgan gave an excellent account of the history of the Poincaré conjecture and the ideas of Richard Hamilton and Perelman that led to the proof that the three-dimensional conjecture is correct. As Morgan pointed out, proofs of the Poincaré con- jecture and its direct generalizations have led to four Fields Medals: to Stephen Smale (1966), William Thurston (1982), Michael Freedman (1986), and now Grigori Perelman. The 2006 ICM was held in the Palacio Municipal de Congressos, a modern convention center on the outskirts of Madrid, which easily accommodated the 3600 or so participants. The interior of the convention center has a number of intriguing views—my favorite, shown below, is from the top of the three-floor-long descending escalator. Alfio Quarteroni’s plenary lecture on cardiovascular mathematics was among the many ses- The opening ceremony included a welcome sions of interest to applied mathematicians. from Juan Carlos, King of Spain, as well as the official announcement of the prize recipients—not only the four Fields Medals but also the Nevanlinna Prize and the (newly established) Gauss Prize. -
Prize Is Awarded Every Three Years at the Joint Mathematics Meetings
AMERICAN MATHEMATICAL SOCIETY LEVI L. CONANT PRIZE This prize was established in 2000 in honor of Levi L. Conant to recognize the best expository paper published in either the Notices of the AMS or the Bulletin of the AMS in the preceding fi ve years. Levi L. Conant (1857–1916) was a math- ematician who taught at Dakota School of Mines for three years and at Worcester Polytechnic Institute for twenty-fi ve years. His will included a bequest to the AMS effective upon his wife’s death, which occurred sixty years after his own demise. Citation Persi Diaconis The Levi L. Conant Prize for 2012 is awarded to Persi Diaconis for his article, “The Markov chain Monte Carlo revolution” (Bulletin Amer. Math. Soc. 46 (2009), no. 2, 179–205). This wonderful article is a lively and engaging overview of modern methods in probability and statistics, and their applications. It opens with a fascinating real- life example: a prison psychologist turns up at Stanford University with encoded messages written by prisoners, and Marc Coram uses the Metropolis algorithm to decrypt them. From there, the article gets even more compelling! After a highly accessible description of Markov chains from fi rst principles, Diaconis colorfully illustrates many of the applications and venues of these ideas. Along the way, he points to some very interesting mathematics and some fascinating open questions, especially about the running time in concrete situ- ations of the Metropolis algorithm, which is a specifi c Monte Carlo method for constructing Markov chains. The article also highlights the use of spectral methods to deduce estimates for the length of the chain needed to achieve mixing. -
AROUND PERELMAN's PROOF of the POINCARÉ CONJECTURE S. Finashin After 3 Years of Thorough Inspection, Perelman's Proof of Th
AROUND PERELMAN'S PROOF OF THE POINCARE¶ CONJECTURE S. Finashin Abstract. Certain life principles of Perelman may look unusual, as it often happens with outstanding people. But his rejection of the Fields medal seems natural in the context of various activities followed his breakthrough in mathematics. Cet animal est tr`esm¶echant, quand on l'attaque, il se d¶efend. Folklore After 3 years of thorough inspection, Perelman's proof of the Poincar¶eConjecture was ¯nally recognized by the consensus of experts. Three groups of researchers not only veri¯ed its details, but also published their own expositions, clarifying the proofs in several hundreds of pages, on the level \available to graduate students". The solution of the Poincar¶eConjecture was the central event discussed in the International Congress of Mathematicians in Madrid, in August 2006 (ICM-2006), which gave to it a flavor of a historical congress. Additional attention to the personality of Perelman was attracted by his rejec- tion of the Fields medal. Some other of his decisions are also non-conventional and tradition-breaking, for example Perelman's refusal to submit his works to journals. Many people ¯nd these decisions strange, but I ¯nd them logical and would rather criticize certain bad traditions in the modern organization of science. One such tradition is a kind of tolerance to unfair credit for mathematical results. Another big problem is the journals, which are often transforming into certain clubs closed for aliens, or into pro¯table businesses, enjoying the free work of many mathemati- cians. Sometimes the role of the \organizers of science" in mathematics also raises questions. -
In Defense of Punk Science
In defense of punk science T. P. Leão* This work is not related to my aliation or attributions nor does it represent the opinions or policies of the institution to which I am currently aliated. *email address: [email protected] April 15, 2021 v0.0 (Draft) Abstract We discuss the current state of commodication of science and marginalization of individuals and groups outside of the main institutions and research groups. We analyze how the num- ber of publications and not the quality of the knowledge produced has become the de facto currency to buy resources, prestige and power in academia, further oppressing and excluding individuals and smaller groups from building the scientic consensus. Due to the unfortunate and misguided policies of countries, states and institutions privileging numbers over quality, many groups operate true pyramid schemes where those at the bottom, undergraduate, gradu- ate students, lab technicians and post-docs, bear the brunt of the manual and intellectual labor while the upper strata of the pyramid concentrates publications, resources and power. In this system, those at the lower strata are often seen as disposable. We defend that an open-source model of presentation of scientic results with minimal voluntary curation from democratically elected representatives from the scientic community is an alternative to corporations. Public funds used to pay for publishing and access to publications in major corporations could be em- ployed to maintain open-access databases. We defend that the distribution of public funding for research should not follow a free market capitalism approach, based on the number of publica- tions as currency, as this leads to the accumulation of power and resources and oppression and exclusion of individuals and smaller groups, especially in developing countries.