Notices of the International Congress of Chinese Mathematicians最新文献

{"title":"Enumerative and Algebraic Combinatorics in the 1960’s and 1970’s","authors":"R. Stanley","doi":"10.4310/iccm.2021.v9.n2.a2","DOIUrl":"https://doi.org/10.4310/iccm.2021.v9.n2.a2","url":null,"abstract":"The period 1960–1979 was an exciting time for enumerative and algebraic combinatorics (EAC). During this period EAC was transformed into an independent subject which is even stronger and more active today. I will not attempt a comprehensive analysis of the development of EAC but rather focus on persons and topics that were relevant to my own career. Thus the discussion will be partly autobiographical. There were certainly deep and important results in EAC before 1960. Work related to tree enumeration (including the Matrix-Tree theorem), partitions of integers (in particular, the Rogers-Ramanujan identities), the Redfield-Pólya theory of enumeration under group action, and especially the representation theory of the symmetric group, GL(n,C) and some related groups, featuring work by Georg Frobenius (1849–1917), Alfred Young (1873–1940), and Issai Schur (1875–1941), are some highlights. Much of this work was not concerned with combinatorics per se; rather, combinatorics was the natural context for its development. For readers interested in the development of EAC, as well as combinatorics in general, prior to 1960, see Biggs [14], Knuth [77, §7.2.1.7], Stein [147], and Wilson and Watkins [153]. Before 1960 there are just a handful of mathematicians who did a substantial amount of enumerative combinatorics. The most important and influential of these is Percy Alexander MacMahon (1854–1929). He was a highly original pioneer, whose work was not properly appreciated during his lifetime except for his contributions to invariant theory and integer partitions. Much of the work in EAC in the","PeriodicalId":415664,"journal":{"name":"Notices of the International Congress of Chinese Mathematicians","volume":"809 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133216289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Roman Jackiw and Chern–Simons theories","authors":"R. Pisarski","doi":"10.4310/iccm.2021.v9.n1.a3","DOIUrl":"https://doi.org/10.4310/iccm.2021.v9.n1.a3","url":null,"abstract":"Recently, the Center of Mathematical Sciences and Applications at Harvard initiated an excellent series of talks on mathematics. In the inaugural talk on March 13, 2020, S.-T. Yau spoke about S. S. Chern as a great geometer of the 20th century [1]. Particularly in the discussion section after the talk, Prof. Yau emphasized the essential role which Roman Jackiw played in bringing Chern-Simons theories into physics. In this note I wish to share some recollections from those times, and especially my interactions with Roman. I do this because I most strongly agree with Prof. Yau’s assessment of Roman’s contribution. In this as other areas, Roman’s work exhibits both the sheer joy of computation, while continually pushing to understand the greater significance of what is truly new and exciting. As a graduate student of David Gross, with Larry Yaffe we computed the fluctuations to one loop order about a single instanton at nonzero temperature [2]. I then went off to Yale University as a postdoc [3], where I worked with Tom Appelquist on gauge theories in three dimensions. Our motivation was to understand the behavior of gauge theories at high temperature, which for the static mode reduces to a gauge theory in three dimensions. We computed the gluon self energy to one and (in part) to two loop order. The gluon self energy isn’t gauge invariant, so the greater significance of our analysis wasn’t clear. But we also computed in an Abelian theory for a large number of scalars, which is a nice soluble model, and from Tom I learned the most useful craft of power counting diagrams. We concentrated on the gluons, since quarks are fermions, and decouple in the static limit. Sometime in the fall of 1980, I went to MIT to give a talk on this work. It was a joint seminar with Harvard, and I remember it well to this day. Talks were then on transparencies, and the evening before I thought I would be clever, and added a comment that while topological charge in four dimensions is quantized classically, perhaps it isn’t quantum mechanically. Sidney Coleman was sitting near the front, in a purple crushed velvet suit which to me looked very much like that of Superfly. When he saw that slide, however, Coleman pounced, and did not let up until I surrendered abjectly, admitting to the idiocy of my suggestion. During the talk and for the entire afternoon after, Roman grilled me about details of the calculation, how gauge theories in three dimensions work, what about gauge invariance, everything. It was very intense and quite exhilarating. Roman and S. Templeton then wrote a paper on theories in three dimensions [4], which because Roman is a superb calculator, appeared as a preprint a few weeks before ours. When their paper came out, I remember looking at it, and thinking, ah, very good, they were spending their time on something irrelevant, two component fermions in three dimensions. Now if you forget about nonzero temperature and just do dimensional reduction, the natural th","PeriodicalId":415664,"journal":{"name":"Notices of the International Congress of Chinese Mathematicians","volume":" 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120832841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Compact Dupin hypersurfaces","authors":"Thomas E. Cecil","doi":"10.4310/iccm.2021.v9.n1.a4","DOIUrl":"https://doi.org/10.4310/iccm.2021.v9.n1.a4","url":null,"abstract":"A hypersurface M in R is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if the number of distinct principal curvatures is constant on M , i.e., each continuous principal curvature function has constant multiplicity on M . These conditions are preserved by stereographic projection, so this theory is essentially the same for hypersurfaces in R or S. The theory of compact proper Dupin hypersurfaces in S is closely related to the theory of isoparametric hypersurfaces in S, and many important results in this field concern relations between these two classes of hypersurfaces. This problem was formulated in 1985 in a conjecture of Cecil and Ryan [17, p. 184], which states that every compact, connected proper Dupin hypersurface M ⊂ S is equivalent to an isoparametric hypersurface in S by a Lie sphere transformation. This paper gives a survey of progress on this conjecture and related developments. 1 Dupin hypersurfaces This paper is a survey of the main results in the theory of compact, proper Dupin hypersurfaces in Euclidean space R, which began with the study of the cyclides of Dupin in R in a book by Dupin [25] in 1822. This theory is closely related to the well-known theory of isoparametric hypersurfaces in S, i.e., hypersurfaces with constant principal curvatures in S, introduced by E. Cartan [2]–[5] and developed by many mathematicians (see [7], [23], [65], [18, pp. 85–184] for surveys).","PeriodicalId":415664,"journal":{"name":"Notices of the International Congress of Chinese Mathematicians","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116011701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hilbert schemes, Donaldson–Thomas theory, Vafa–Witten and Seiberg–Witten theory","authors":"A. Sheshmani","doi":"10.4310/ICCM.2019.V7.N2.A3","DOIUrl":"https://doi.org/10.4310/ICCM.2019.V7.N2.A3","url":null,"abstract":"This article provides a summary of arXiv:1701.08899 and arXiv:1701.08902 where the authors studied the enumerative geometry of nested Hilbert schemes of points and curves on algebraic surfaces and their connections to threefold theories, and in particular relevant Donaldson-Thomas, Vafa-Witten and Seiberg-Witten theories.","PeriodicalId":415664,"journal":{"name":"Notices of the International Congress of Chinese Mathematicians","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131340626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Liouville properties","authors":"T. Colding, W. Minicozzi","doi":"10.4310/iccm.2019.v7.n1.a10","DOIUrl":"https://doi.org/10.4310/iccm.2019.v7.n1.a10","url":null,"abstract":"The classical Liouville theorem states that a bounded harmonic function on all of $RR^n$ must be constant. In the early 1970s, S.T. Yau vastly generalized this, showing that it holds for manifolds with nonnegative Ricci curvature. Moreover, he conjectured a stronger Liouville property that has generated many significant developments. We will first discuss this conjecture and some of the ideas that went into its proof. We will also discuss two recent areas where this circle of ideas has played a major role. One is Kleiner's new proof of Gromov's classification of groups of polynomial growth and the developments this generated. Another is to understanding singularities of mean curvature flow in high codimension. We will see that some of the ideas discussed in this survey naturally lead to a new approach to studying and classifying singularities of mean curvature flow in higher codimension. This is a subject that has been notoriously difficult and where much less is known than for hypersurfaces.","PeriodicalId":415664,"journal":{"name":"Notices of the International Congress of Chinese Mathematicians","volume":"80 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132535930","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Positive Structures in Lie Theory","authors":"G. Lusztig","doi":"10.4310/iccm.2020.v8.n1.a4","DOIUrl":"https://doi.org/10.4310/iccm.2020.v8.n1.a4","url":null,"abstract":"0.1. In late 19th century and early 20th century, a new branch of mathematics was born: Lie theory or the study of Lie groups and Lie algebras (Lie, Killing, E.Cartan, H.Weyl). It has become a central part of mathematics with applications everywhere. More recent developments in Lie theory are as follows. -Analogues of simple Lie groups over any field (including finite fields where they explain most of the finite simple groups): Chevalley 1955; -infinite dimensional versions of the simple Lie algebras/simple Lie groups: Kac and Moody 1967, Moody and Teo 1972; -theory of quantum groups: Drinfeld and Jimbo 1985.","PeriodicalId":415664,"journal":{"name":"Notices of the International Congress of Chinese Mathematicians","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133921600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"History of Leningrad mathematics in the first half of the 20th century","authors":"A. Nazarov, G. Sinkevich","doi":"10.4310/ICCM.2019.V7.N2.A5","DOIUrl":"https://doi.org/10.4310/ICCM.2019.V7.N2.A5","url":null,"abstract":"The first half of the 20th century in the history of Russian mathematics is striking with a combination of dramaticism, sometimes a tragedy, and outstanding achievements. \u0000The paper is devoted to St. Petersburg-Leningrad Mathematical School. It is based on a chapter in the multi-author monograph \"Mathematical Petersburg. History, science, sights\" (SPb: Educational projects, 2018, 336 pp. in Russian).","PeriodicalId":415664,"journal":{"name":"Notices of the International Congress of Chinese Mathematicians","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125454940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A brief chronicle of the Levi (Hartog’s inverse) problem, coherence and open problem","authors":"J. Noguchi","doi":"10.4310/ICCM.2019.V7.N2.A2","DOIUrl":"https://doi.org/10.4310/ICCM.2019.V7.N2.A2","url":null,"abstract":"Here we chronologically summarize briefly the developments of the Levi (Hartogs' Inverse) Problem together with the notion of coherence and its solution, shedding light on some records which have not been discussed in the past references. In particular, we will discuss K. Oka's unpublished papers 1943 which solved the Levi (Hartogs' Inverse) Problem for unramified Riemann domains of arbitrary dimension $n geq 2$, usually referred as it was solved by Oka IX in 1953, H.J. Bremermann and F. Norguet in 1954 for univalent domains, independently. \u0000At the end we emphasize an open problem in a ramified case.","PeriodicalId":415664,"journal":{"name":"Notices of the International Congress of Chinese Mathematicians","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124989874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hypergeometric heritage of W. N. Bailey","authors":"W. Zudilin","doi":"10.4310/ICCM.2019.v7.n2.a4","DOIUrl":"https://doi.org/10.4310/ICCM.2019.v7.n2.a4","url":null,"abstract":"We review some of W.N. Bailey's work on hypergeometric functions that found solid applications in number theory. The text is complemented by Bailey's letters to Freeman Dyson from the 1940s.","PeriodicalId":415664,"journal":{"name":"Notices of the International Congress of Chinese Mathematicians","volume":"102 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121613761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hodge bundles on smooth compactifications of Siegel varieties and applications","authors":"S. Yau, Yi Zhang","doi":"10.4310/ICCM.2019.V7.N2.A1","DOIUrl":"https://doi.org/10.4310/ICCM.2019.V7.N2.A1","url":null,"abstract":"We study Hodge bundles on Siegel varieties and their various extensions to smooth toroidal compactifications. Precisely, we construct a canonical Hodge bundle on an arbitrary Siegel variety so that the holomorphic tangent bundle can be embedded into the Hodge bundle, and we observe that the Bergman metric on the Siegel variety is compatible with the induced Hodge metric. Therefore we obtain the asymptotic estimate of the Bergman metric explicitly. Depending on these properties and the uniformitarian of K\"ahler-Einstein manifold, we study extensions of the tangent bundle over any smooth toroidal compactification. We also apply this result, together with Siegel cusp modular forms, to study general type for Siegel varieties.","PeriodicalId":415664,"journal":{"name":"Notices of the International Congress of Chinese Mathematicians","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121543869","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}