{"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":null,"pages":null},"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}
M. Fradelizi, M. Madiman, Arnaud Marsiglietti, A. Zvavitch
{"title":"The convexification effect of Minkowski summation","authors":"M. Fradelizi, M. Madiman, Arnaud Marsiglietti, A. Zvavitch","doi":"10.4171/EMSS/26","DOIUrl":"https://doi.org/10.4171/EMSS/26","url":null,"abstract":"Let us define for a compact set $A subset mathbb{R}^n$ the sequence $$ A(k) = left{frac{a_1+cdots +a_k}{k}: a_1, ldots, a_kin Aright}=frac{1}{k}Big(underset{k {rm times}}{underbrace{A + cdots + A}}Big). $$ It was independently proved by Shapley, Folkman and Starr (1969) and by Emerson and Greenleaf (1969) that $A(k)$ approaches the convex hull of $A$ in the Hausdorff distance induced by the Euclidean norm as $k$ goes to $infty$. We explore in this survey how exactly $A(k)$ approaches the convex hull of $A$, and more generally, how a Minkowski sum of possibly different compact sets approaches convexity, as measured by various indices of non-convexity. The non-convexity indices considered include the Hausdorff distance induced by any norm on $mathbb{R}^n$, the volume deficit (the difference of volumes), a non-convexity index introduced by Schneider (1975), and the effective standard deviation or inner radius. After first clarifying the interrelationships between these various indices of non-convexity, which were previously either unknown or scattered in the literature, we show that the volume deficit of $A(k)$ does not monotonically decrease to 0 in dimension 12 or above, thus falsifying a conjecture of Bobkov et al. (2011), even though their conjecture is proved to be true in dimension 1 and for certain sets $A$ with special structure. On the other hand, Schneider's index possesses a strong monotonicity property along the sequence $A(k)$, and both the Hausdorff distance and effective standard deviation are eventually monotone (once $k$ exceeds $n$). Along the way, we obtain new inequalities for the volume of the Minkowski sum of compact sets, falsify a conjecture of Dyn and Farkhi (2004), demonstrate applications of our results to combinatorial discrepancy theory, and suggest some questions worthy of further investigation.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2017-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/EMSS/26","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49230040","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":"Exploring the unknown: The work of Louis Nirenberg on partial differential equations","authors":"Tristan Riviere","doi":"10.1090/noti1328","DOIUrl":"https://doi.org/10.1090/noti1328","url":null,"abstract":"They were for a long time restricted only to the study of natural phenomena or questions pertaining to geometry, before becoming over the course of time, and especially in the last century, a eld in itself. The second half of the XXth century was the golden age\" of the exploration of partial dierential equations from a theoretical perspective. The mathematical work of Louis Nirenberg since the early 1950s has to a large extent contributed to the growth of this fundamental area of human knowledge. The name Nirenberg is associated with many of the milestones in the study of PDEs. The award of the Abel Prize to Louis Nirenberg marks a special occasion for us to revisit the development of the eld of PDEs and the work of one of the main actors of its exploration.","PeriodicalId":43833,"journal":{"name":"EMS Surveys in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2015-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60563949","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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