Proceedings of the International Congress of Mathematicians (ICM 2018)最新文献

{"title":"QUANTITATIVE ESTIMATES FOR ADVECTIVE EQUATION WITH DEGENERATE ANELASTIC CONSTRAINT","authors":"D. Bresch, P. Jabin","doi":"10.1142/9789813272880_0134","DOIUrl":"https://doi.org/10.1142/9789813272880_0134","url":null,"abstract":"In these proceedings we are interested in quantitative estimates for advective equations with an anelastic constraint in presence of vacuum. More precisely, we derive a quantitative stability estimate and obtain the existence of renormalized solutions. Our main objective is to show the flexibility of the method introduced recently by the authors for the compressible Navier-Stokes’ system. This method seems to be well adapted in general to provide regularity estimates on the density of compressible transport equations with possible vacuum state and low regularity of the transport velocity field; the advective equation with degenerate anelastic constraint considered here is another good example of that. As a final application we obtain the existence of global renormalized solution to the so-called lake equation with possibly vanishing topography.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122067598","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":"INVERSE PROBLEMS FOR LINEAR AND NON-LINEAR HYPERBOLIC EQUATIONS","authors":"M. Lassas","doi":"10.1142/9789813272880_0199","DOIUrl":"https://doi.org/10.1142/9789813272880_0199","url":null,"abstract":"We consider inverse problems for hyperbolic equations and systems and the solutions of these problems based on the focusing of waves. Several inverse problems for linear equations can be solved using control theory. When the coefficients of the modelling equation are unknown, the construction of the point sources requires solving blind control problems. For non-linear equations we consider a new artificial point source method that applies the non-linear interaction of waves to create microlocal points sources inside the unknown medium. The novel feature of this method is that it utilizes the non-linearity as a tool in imaging, instead of considering it as a difficult perturbation of the system. To demonstrate the method, we consider the non-linear wave equation and the coupled Einstein and scalar field equations.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128698281","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":"LIMIT SHAPES AND THEIR ANALYTIC PARAMETERIZATIONS","authors":"R. Kenyon","doi":"10.1142/9789813272880_0175","DOIUrl":"https://doi.org/10.1142/9789813272880_0175","url":null,"abstract":"A “limit shape” is a form of the law of large numbers, and happens when a large random system, typically consisting of many interacting particles, can be described, after an appropriate normalization, by a certain nonrandom object. Limit shapes occur in, for example, random integer partitions or in random interface models such as the dimer model. Typically limit shapes can be described by some variational formula based on a large deviations estimate. We discuss limit shapes for certain 2-dimensional interface models, and explain how their underlying analytic structure is related to a (conjectural in some cases) conformal invariance property for the models. 1 Limit shapes: integer partitions We illustrate the notion of limit shape with a fundamental example. Given a uniform random integer partition of n for n large, a theorem of Vershik and Kerov [1981] asserts that, when both axes are scaled by p n, the graph of (that is, the Young diagram associated to ) converges with probability tending to 1 to a nonrandom curve, given by the equation e cx + e cy = 1, with c = p 2/6, see Figure 1. This is an example (in fact, one of the first examples) of a limit shape theorem: in the limit of large system size, the typical random object will, when appropriately scaled, concentrate on a fixed nonrandom shape. One way to make a more precise formulation of this statement is say that for each n, the random partition of n defines a certain probability measure n (on the space of nonincreasing functions f : [0; 1) ! [0; 1) of integral 1) and as n ! 1 this sequence of measures converges in probability1 to a point mass on the Vershik-Kerov curve. MSC2010: 82B20. 1 The topology of convergence for the sequence of random functions can be taken to be uniform convergence on compact subsets of (0; 1).","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121953417","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":"BACK MATTER","authors":"B. Sirakov, Paulo Ney de Souza, M. Viana","doi":"10.1142/9789813272880_bmatter04","DOIUrl":"https://doi.org/10.1142/9789813272880_bmatter04","url":null,"abstract":"","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128188073","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":"THE SUBCONVEXITY PROBLEM FOR L-FUNCTIONS","authors":"R. Munshi","doi":"10.1142/9789813272880_0058","DOIUrl":"https://doi.org/10.1142/9789813272880_0058","url":null,"abstract":"Estimating the size of automorphic L-functions on the critical line is a central problem in analytic number theory. An easy consequence of the standard analytic properties of theL-function is the convexity bound, whereas the generalised Riemann Hypothesis predicts a much sharper bound. Breaking the convexity barrier is a hard problem. The moment method has been used to surpass convexity in the case of Lfunctions of degree one and two. In this talk I will discuss a different method, which has been quite successful to settle certain longstanding open problems in the case of degree three. At the 1994 International Congress at Zürich, J. B. Friedlander [1995] briefly described the essence of the amplified moment method which he was developing in a series of joint works with Duke and Iwaniec, with the aim of obtaining non-trivial bounds for Lfunctions. Since then the amplification technique has proved to be very effective in a number of scenarios involvingGL(2)L-functions (see J. Friedlander and Iwaniec [1992], Duke, J. B. Friedlander, and Iwaniec [1993, 1994, 1995, 2001, 2002], Kowalski, Michel, and VanderKam [2002], Michel [2004], Harcos and Michel [2006], and Blomer and Harcos [2008]). But there are major hurdles in extending the method far beyond. In the last decade the automorphic period approach has been developed in great detail and generality (over number fields), by Michel, Venkatesh and others (see Bernstein and Reznikov [2010], Michel and Venkatesh [2010], Wu [2014]). This puts the moment method in a proper perspective and gives a satisfactory explanation to the ‘mysterious identities between families of L-functions’ that already occurs in the study of the moments of the Rankin-Selberg L-functions Harcos and Michel [2006], Michel [2004]. This has been the topic of Michel’s address at the 2006 International Congress at Madrid Michel and Venkatesh [2006]. Here I will briefly describe a new approach to tackle subconvexity, which has not only settled some of the longstanding open problems in the field, but has also matched in strength the existing benchmarks. As there are several excellent accounts MSC2010: primary 11F66; secondary 11M41.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132033510","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":"TOWARDS A THEORY OF DEFINABLE SETS","authors":"Stephen Jackson","doi":"10.1142/9789813272880_0043","DOIUrl":"https://doi.org/10.1142/9789813272880_0043","url":null,"abstract":"The subject of descriptive set theory is traditionally concerned with the theory of definable subsets of Polish spaces. By introducing large cardinal/determinacy axioms, a theory of definable subsets of Polish spaces and their associated ordinals has been developed over the last several decades which extends far up in the definability hierarchy. Recently, much interest has been focused on trying to extend the theory of definable objects to more general types of sets, not necessarily subsets of a Polish space or an ordinal. A large class of these objects are represented by equivalence relations on Polish spaces. Even for some of the simpler of these relations, an interesting combinatorial theory is emerging. We consider both problems of extending further the theory of definable subsets of Polish spaces, and that of determining the structure of these new types of definable sets.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130443088","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":"ON GROTHENDIECK–SERRE CONJECTURE CONCERNING PRINCIPAL BUNDLES","authors":"I. Panin","doi":"10.1142/9789813272880_0051","DOIUrl":"https://doi.org/10.1142/9789813272880_0051","url":null,"abstract":"Let R be a regular local ring. Let G be a reductive group scheme over R. A wellknown conjecture due to Grothendieck and Serre assertes that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R. In other words, if K is the fraction field of R, then the map of non-abelian cohomology pointed sets Hét(R;G) ! H 1 ét(K;G); induced by the inclusion of R into K, has a trivial kernel. The conjecture is solved in positive for all regular local rings contaning a field. More precisely, if the ring R contains an infinite field, then this conjecture is proved in a joint paper due to R. Fedorov and I. Panin published in 2015 in Publications l’IHES. If the ring R contains a finite field, then this conjecture is proved in 2015 in a preprint due to I. Paninwhich can be found on preprint server Linear Algebraic Groups and Related Structures. A more structured exposition can be found in Panin’s preprint of the year 2017 on arXiv.org. This and other results concerning the conjecture are discussed in the present paper. We illustrate the exposition by many interesting examples. We begin with couple results for complex algebraic varieties and develop the exposition step by step to its full generality.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"168 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127555312","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":"FROM BLACKBOARD TO BEDSIDE - GAUβ PRIZE LECTURE","authors":"D. Donoho","doi":"10.1142/9789813272880_0010","DOIUrl":"https://doi.org/10.1142/9789813272880_0010","url":null,"abstract":"","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130904419","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":"BACK MATTER","authors":"B. Sirakov, Paulo Ney de Souza, M. Viana","doi":"10.1142/9789813272880_bmatter01","DOIUrl":"https://doi.org/10.1142/9789813272880_bmatter01","url":null,"abstract":"","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128877256","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":"ASYMPTOTIC ENUMERATION OF GRAPHS WITH GIVEN DEGREE SEQUENCE","authors":"N. Wormald","doi":"10.1142/9789813272880_0180","DOIUrl":"https://doi.org/10.1142/9789813272880_0180","url":null,"abstract":"We survey results on counting graphs with given degree sequence, focusing on asymptotic results, and mentioning some of the applications of these results. The main recent development is the proof of a conjecture that facilitates access to the degree sequence of a random graph via a model incorporating independent binomial random variables. The basic method used in the proof was to examine the changes in the counting function when the degrees are perturbed. We compare with several previous uses of this type of method.","PeriodicalId":318252,"journal":{"name":"Proceedings of the International Congress of Mathematicians (ICM 2018)","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123999206","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}