We give a general overview of the influence of William Thurston on the French mathematical school and we show how some of the major problems he solved are rooted in the French mathematical tradition. At the same time, we survey some of Thurston's major results and their impact. The final version of this paper will appear in the Surveys of the European Mathematical Society.
{"title":"W. P. Thurston and French mathematics","authors":"F. Laudenbach, A. Papadopoulos","doi":"10.4171/emss/32","DOIUrl":"https://doi.org/10.4171/emss/32","url":null,"abstract":"We give a general overview of the influence of William Thurston on the French mathematical school and we show how some of the major problems he solved are rooted in the French mathematical tradition. At the same time, we survey some of Thurston's major results and their impact. The final version of this paper will appear in the Surveys of the European Mathematical Society.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":2.3,"publicationDate":"2019-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/emss/32","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48200159","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}
Motivated by Emmanuel Kowalski's exponential sums over definable sets in finite fields, we generalize theorems of Ax and Chatzidakis-Van den Dries-Macintyre to pseudo-finite fields with an additive character, in a continuous logic setting.
受Emmanuel Kowalski在有限域中可定义集上的指数和的启发，我们在连续逻辑环境中将Ax和Chatzidakis Van den Dries Macintyre的定理推广到具有加性的伪有限域。
{"title":"Ax’s theorem with an additive character","authors":"E. Hrushovski","doi":"10.4171/emss/47","DOIUrl":"https://doi.org/10.4171/emss/47","url":null,"abstract":"Motivated by Emmanuel Kowalski's exponential sums over definable sets in finite fields, we generalize theorems of Ax and Chatzidakis-Van den Dries-Macintyre to pseudo-finite fields with an additive character, in a continuous logic setting.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":2.3,"publicationDate":"2019-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47808906","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}
We describe the Special Kahler structure on the base of the so-called Hitchin system in terms of the geometry of the space of spectral curves. It yields a simple formula for the Kahler potential. This extends to the case of a singular spectral curve and we show that this defines the Special Kahler structure on certain natural integrable subsystems. Examples include the extreme case where the metric is flat.
{"title":"Integrable systems and Special Kähler metrics","authors":"N. Hitchin","doi":"10.4171/emss/46","DOIUrl":"https://doi.org/10.4171/emss/46","url":null,"abstract":"We describe the Special Kahler structure on the base of the so-called Hitchin system in terms of the geometry of the space of spectral curves. It yields a simple formula for the Kahler potential. This extends to the case of a singular spectral curve and we show that this defines the Special Kahler structure on certain natural integrable subsystems. Examples include the extreme case where the metric is flat.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":2.3,"publicationDate":"2019-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47813737","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}
We review the theory of non-commutative deformations of sheaves and describe a versal deformation by using an A-infinity algebra and the change of differentials of an injective resolution. We give some explicit non-trivial examples.
{"title":"On non-commutative formal deformations of coherent sheaves on an algebraic variety","authors":"Y. Kawamata","doi":"10.4171/emss/49","DOIUrl":"https://doi.org/10.4171/emss/49","url":null,"abstract":"We review the theory of non-commutative deformations of sheaves and describe a versal deformation by using an A-infinity algebra and the change of differentials of an injective resolution. We give some explicit non-trivial examples.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":2.3,"publicationDate":"2019-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46971597","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}
This is an expository article detailing results concerning large arcs in finite projective spaces, which attempts to cover the most relevant results on arcs, simplifying and unifying proofs of known old and more recent theorems. The article is mostly self-contained and includes a proof of the most general form of Segre's lemma of tangents and a short proof of the MDS conjecture over prime fields based on this lemma.
{"title":"Arcs in finite projective spaces","authors":"Simeon Ball, M. Lavrauw","doi":"10.4171/emss/33","DOIUrl":"https://doi.org/10.4171/emss/33","url":null,"abstract":"This is an expository article detailing results concerning large arcs in finite projective spaces, which attempts to cover the most relevant results on arcs, simplifying and unifying proofs of known old and more recent theorems. The article is mostly self-contained and includes a proof of the most general form of Segre's lemma of tangents and a short proof of the MDS conjecture over prime fields based on this lemma.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":"1 1","pages":""},"PeriodicalIF":2.3,"publicationDate":"2019-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/emss/33","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44449589","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}
In this review article we discuss a number of recent results concerning wild weak solutions of the incompressible Euler and Navier-Stokes equations. These results build on the groundbreaking works of De Lellis and Szekelyhidi Jr., who extended Nash's fundamental ideas on $C^1$ flexible isometric embeddings, into the realm of fluid dynamics. These techniques, which go under the umbrella name convex integration, have fundamental analogies the phenomenological theories of hydrodynamic turbulence. Mathematical problems arising in turbulence (such as the Onsager conjecture) have not only sparked new interest in convex integration, but certain experimentally observed features of turbulent flows (such as intermittency) have also informed new convex integration constructions. First, we give an elementary construction of nonconservative $C^{0+}_{x,t}$ weak solutions of the Euler equations, first proven by De Lellis-Szekelyhidi Jr.. Second, we present Isett's recent resolution of the flexible side of the Onsager conjecture. Here, we in fact follow the joint work of De Lellis-Szekelyhidi Jr. and the authors of this paper, in which weak solutions of the Euler equations in the regularity class $C^{frac 13-}_{x,t}$ are constructed, attaining any energy profile. Third, we give a concise proof of the authors' recent result, which proves the existence of infinitely many weak solutions of the Navier-Stokes in the regularity class $C^0_t L^{2+}_x cap C^0_t W^{1,1+}_x$. We conclude the article by mentioning a number of open problems at the intersection of convex integration and hydrodynamic turbulence.
{"title":"Convex integration and phenomenologies in turbulence","authors":"T. Buckmaster, V. Vicol","doi":"10.4171/emss/34","DOIUrl":"https://doi.org/10.4171/emss/34","url":null,"abstract":"In this review article we discuss a number of recent results concerning wild weak solutions of the incompressible Euler and Navier-Stokes equations. These results build on the groundbreaking works of De Lellis and Szekelyhidi Jr., who extended Nash's fundamental ideas on $C^1$ flexible isometric embeddings, into the realm of fluid dynamics. These techniques, which go under the umbrella name convex integration, have fundamental analogies the phenomenological theories of hydrodynamic turbulence. Mathematical problems arising in turbulence (such as the Onsager conjecture) have not only sparked new interest in convex integration, but certain experimentally observed features of turbulent flows (such as intermittency) have also informed new convex integration constructions. \u0000First, we give an elementary construction of nonconservative $C^{0+}_{x,t}$ weak solutions of the Euler equations, first proven by De Lellis-Szekelyhidi Jr.. Second, we present Isett's recent resolution of the flexible side of the Onsager conjecture. Here, we in fact follow the joint work of De Lellis-Szekelyhidi Jr. and the authors of this paper, in which weak solutions of the Euler equations in the regularity class $C^{frac 13-}_{x,t}$ are constructed, attaining any energy profile. Third, we give a concise proof of the authors' recent result, which proves the existence of infinitely many weak solutions of the Navier-Stokes in the regularity class $C^0_t L^{2+}_x cap C^0_t W^{1,1+}_x$. We conclude the article by mentioning a number of open problems at the intersection of convex integration and hydrodynamic turbulence.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":2.3,"publicationDate":"2019-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/emss/34","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47043551","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}
Donaldon constructed a hyperk"ahler moduli space $mathcal{M}$ associated to a closed oriented surface $Sigma$ with $textrm{genus}(Sigma) geq 2$. This embeds naturally into the cotangent bundle $T^*mathcal{T}(Sigma)$ of Teichm"uller space or can be identified with the almost-Fuchsian moduli space associated to $Sigma$. The later is the moduli space of quasi-Fuchsian threefolds which contain a unique incompressible minimal surface with principal curvatures in $(-1,1)$. Donaldson outlined various remarkable properties of this moduli space for which we provide complete proofs in this paper: On the cotangent-bundle of Teichm"uller space, the hyperk"ahler structure on $mathcal{M}$ can be viewed as the Feix--Kaledin hyperk"ahler extension of the Weil--Petersson metric. The almost-Fuchsian moduli space embeds into the $textrm{SL}(2,mathbb{C})$-representation variety of $Sigma$ and the hyperk"ahler structure on $mathcal{M}$ extends the Goldman holomorphic symplectic structure. Here the natural complex structure corresponds to the second complex structure in the first picture. Moreover, the area of the minimal surface in an almost-Fuchsian manifold provides a K"ahler potential for the hyperk"ahler metric. The various identifications are obtained using the work of Uhlenbeck on germs of hyperbolic $3$-manifolds, an explicit map from $mathcal{M}$ to $mathcal{T}(Sigma)times bar{mathcal{T}(Sigma)}$ found by Hodge, the simultaneous uniformization theorem of Bers, and the theory of Higgs bundles introduced by Hitchin.
{"title":"The hyperkähler metric on the almost-Fuchsian moduli space","authors":"Samuel Trautwein","doi":"10.4171/emss/30","DOIUrl":"https://doi.org/10.4171/emss/30","url":null,"abstract":"Donaldon constructed a hyperk\"ahler moduli space $mathcal{M}$ associated to a closed oriented surface $Sigma$ with $textrm{genus}(Sigma) geq 2$. This embeds naturally into the cotangent bundle $T^*mathcal{T}(Sigma)$ of Teichm\"uller space or can be identified with the almost-Fuchsian moduli space associated to $Sigma$. The later is the moduli space of quasi-Fuchsian threefolds which contain a unique incompressible minimal surface with principal curvatures in $(-1,1)$. Donaldson outlined various remarkable properties of this moduli space for which we provide complete proofs in this paper: On the cotangent-bundle of Teichm\"uller space, the hyperk\"ahler structure on $mathcal{M}$ can be viewed as the Feix--Kaledin hyperk\"ahler extension of the Weil--Petersson metric. The almost-Fuchsian moduli space embeds into the $textrm{SL}(2,mathbb{C})$-representation variety of $Sigma$ and the hyperk\"ahler structure on $mathcal{M}$ extends the Goldman holomorphic symplectic structure. Here the natural complex structure corresponds to the second complex structure in the first picture. Moreover, the area of the minimal surface in an almost-Fuchsian manifold provides a K\"ahler potential for the hyperk\"ahler metric. The various identifications are obtained using the work of Uhlenbeck on germs of hyperbolic $3$-manifolds, an explicit map from $mathcal{M}$ to $mathcal{T}(Sigma)times bar{mathcal{T}(Sigma)}$ found by Hodge, the simultaneous uniformization theorem of Bers, and the theory of Higgs bundles introduced by Hitchin.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":2.3,"publicationDate":"2018-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/emss/30","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46337118","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}
A new framework for the optimal control of probability density functions (PDF) of stochastic processes is reviewed. This framework is based on Fokker-Planck (FP) partial differential equations that govern the time evolution of the PDF of stochastic systems and on control objectives that may require to follow a given PDF trajectory or to minimize an expectation functional. Corresponding to different stochastic processes, different FP equations are obtained. In particular, FP equations of parabolic, fractional parabolic, integro parabolic, and hyperbolic type are discussed. The corresponding optimization problems are deterministic and can be formulated in an open-loop framework and within a closed-loop model predictive control strategy. The connection between the Dynamic Programming scheme given by the Hamilton-Jacobi-Bellman equation and the FP control framework is discussed. Under appropriate assumptions, it is shown that the two strategies are equivalent. Some applications of the FP control framework to different models are discussed and its extension in a mean-field framework is elucidated. This is a preprint of the paper Mario Annunziato and Alfio Borzì A Fokker–Planck control framework for stochastic systems EMS Surveys In Mathematical Sciences, 5 (2018), 65 98. (DOI: 10.4171/EMSS/27)
{"title":"A Fokker–Planck control framework for stochastic systems","authors":"M. Annunziato, A. Borzì","doi":"10.4171/EMSS/27","DOIUrl":"https://doi.org/10.4171/EMSS/27","url":null,"abstract":"A new framework for the optimal control of probability density functions (PDF) of stochastic processes is reviewed. This framework is based on Fokker-Planck (FP) partial differential equations that govern the time evolution of the PDF of stochastic systems and on control objectives that may require to follow a given PDF trajectory or to minimize an expectation functional. Corresponding to different stochastic processes, different FP equations are obtained. In particular, FP equations of parabolic, fractional parabolic, integro parabolic, and hyperbolic type are discussed. The corresponding optimization problems are deterministic and can be formulated in an open-loop framework and within a closed-loop model predictive control strategy. The connection between the Dynamic Programming scheme given by the Hamilton-Jacobi-Bellman equation and the FP control framework is discussed. Under appropriate assumptions, it is shown that the two strategies are equivalent. Some applications of the FP control framework to different models are discussed and its extension in a mean-field framework is elucidated. This is a preprint of the paper Mario Annunziato and Alfio Borzì A Fokker–Planck control framework for stochastic systems EMS Surveys In Mathematical Sciences, 5 (2018), 65 98. (DOI: 10.4171/EMSS/27)","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":"1 1","pages":""},"PeriodicalIF":2.3,"publicationDate":"2018-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/EMSS/27","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46310046","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}
Kazhdan-Lusztig-Stanley polynomials are a combinatorial generalization of Kazhdan-Lusztig polynomials of for Coxeter groups that include g-polynomials of polytopes and Kazhdan-Lusztig polynomials of matroids. In the cases of Weyl groups, rational polytopes, and realizable matroids, one can count points over finite fields on flag varieties, toric varieties, or reciprocal planes to obtain cohomological interpretations of these polynomials. We survey these results and unite them under a single geometric framework.
{"title":"The algebraic geometry of Kazhdan–Lusztig–Stanley polynomials","authors":"N. Proudfoot","doi":"10.4171/EMSS/28","DOIUrl":"https://doi.org/10.4171/EMSS/28","url":null,"abstract":"Kazhdan-Lusztig-Stanley polynomials are a combinatorial generalization of Kazhdan-Lusztig polynomials of for Coxeter groups that include g-polynomials of polytopes and Kazhdan-Lusztig polynomials of matroids. In the cases of Weyl groups, rational polytopes, and realizable matroids, one can count points over finite fields on flag varieties, toric varieties, or reciprocal planes to obtain cohomological interpretations of these polynomials. We survey these results and unite them under a single geometric framework.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":2.3,"publicationDate":"2017-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/EMSS/28","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48380508","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}
The goal of this note is to give an introduction to locally conformally symplectic and Kahler geometry. In particular, Sections 1 and 3 aim to provide the reader with enough mathematical background to appreciate this kind of geometry. The reference book for locally conformally Kahler geometry is "Locally conformal Kahler Geometry" by Sorin Dragomir and Liviu Ornea. Many progresses in this field, however, were accomplished after the publication of this book, hence are not contained there. On the other hand, there is no book on locally conformally symplectic geometry and many recent advances lie scattered in the literature. Sections 2 and 4 would like to demonstrate how these geometries can be used to give precise mathematical formulations to ideas deeply rooted in classical and modern Physics.
{"title":"Locally conformally symplectic and Kähler geometry","authors":"Giovanni Bazzoni","doi":"10.4171/EMSS/29","DOIUrl":"https://doi.org/10.4171/EMSS/29","url":null,"abstract":"The goal of this note is to give an introduction to locally conformally symplectic and Kahler geometry. In particular, Sections 1 and 3 aim to provide the reader with enough mathematical background to appreciate this kind of geometry. The reference book for locally conformally Kahler geometry is \"Locally conformal Kahler Geometry\" by Sorin Dragomir and Liviu Ornea. Many progresses in this field, however, were accomplished after the publication of this book, hence are not contained there. On the other hand, there is no book on locally conformally symplectic geometry and many recent advances lie scattered in the literature. Sections 2 and 4 would like to demonstrate how these geometries can be used to give precise mathematical formulations to ideas deeply rooted in classical and modern Physics.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":"1 1","pages":""},"PeriodicalIF":2.3,"publicationDate":"2017-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/EMSS/29","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45957807","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}
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