What's Next?Pub Date : 2020-07-07DOI: 10.2307/j.ctvthhdvv.9
Benson Farb, J. Franks
{"title":"Groups of Homeomorphisms of One-Manifolds, I:","authors":"Benson Farb, J. Franks","doi":"10.2307/j.ctvthhdvv.9","DOIUrl":"https://doi.org/10.2307/j.ctvthhdvv.9","url":null,"abstract":"","PeriodicalId":404905,"journal":{"name":"What's Next?","volume":"26 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114037253","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}
What's Next?Pub Date : 2019-12-31DOI: 10.2307/j.ctvthhdvv.13
Yi Liu, V. Marković
{"title":"Construction of Subsurfaces via Good Pants","authors":"Yi Liu, V. Marković","doi":"10.2307/j.ctvthhdvv.13","DOIUrl":"https://doi.org/10.2307/j.ctvthhdvv.13","url":null,"abstract":"","PeriodicalId":404905,"journal":{"name":"What's Next?","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129155890","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}
What's Next?Pub Date : 2019-12-31DOI: 10.2307/j.ctvthhdvv.10
M. Gromov
{"title":"Morse Spectra, Homology Measures, Spaces of Cycles and Parametric Packing Problems","authors":"M. Gromov","doi":"10.2307/j.ctvthhdvv.10","DOIUrl":"https://doi.org/10.2307/j.ctvthhdvv.10","url":null,"abstract":"An ”ensemble” Ψ = Ψ(X) of (finitely or infinitely many) particles in a space X, e.g. in the Euclidean 3-space, is customary characterised by the set function U ↦ entU(Ψ) = ent(Ψ∣U), U ⊂ X, that assigns the entropies of the U-reductions Ψ∣U of Ψ, to all bounded open subsets U ⊂ X. In the physicists’ parlance, this entropy is ”the logarithm of the number of the states of E that are effectively observable from U”, This ”definition”, in the context of mathematical statistical mechanics, is translated to the language of the measure/probability theory.1 But what happens if ”effectively observable number of states” is replaced by ”the number of effective/persistent degrees of freedom of ensembles of moving particles”? We suggest in this paper several mathematical counterparts to the idea of ” persistent degrees of freedom ” and formulate specific questions, many of which are inspired by Larry Guth’s results and ideas on the Hermann Weyl kind of asymptotics of the Morse (co)homology spectra of the volume energy function on the spaces of cycles in balls.2 And often we present variable aspects of the same idea in different sections of this paper. Hardly anything that can be called ”new theorem” can be found in our paper but we reshuffle many known results and expose them from a particular angle. This article is meant as an introductory chapter to something yet to be written with much of what we present here extracted from my yet unfinished manuscript Number of Questions. 1 Overview of Concepts and Examples. We introduce below the idea of ”parametric packing” and of related concepts which are expanded in detail in the rest of the paper. A. Let X be a topological space, e.g. a manifold, and I is a countable index set that may be finite, especially if X is compact. A collection of I-tuples of non-empty open (sometimes closed) subsets Ui ⊂ X, i ∈ I, is called a packing or an I-packing of X if these subsets do not intersect. Denote by Ψ(X; I) the space of these packings with some natural topology, where, observe there are several candidates for such a topology if X is noncompact. 1See: Lanford’s Entropy and equilibrium states in classical statistical mechanics, Lecture Notes in Physics, Volume 20, pp. 1-113, 1973 and Ruelle’s Thermodynamic formalism : the mathematical structures of classical equilibrium statistical mechanics, 2nd Edition, Cambridge Mathematical Library 2004, where the emphasis is laid upon (discrete) lattice systems. Also a categorical rendition of Boltzmann-Shannon entropy is suggested in ”In a Search for a Structure, Part 1: On Entropy”, www.ihes.fr/∼gromov/PDF/structre-serch-entropy-july5-2012.pdf 2Minimax problems related to cup powers and Steenrod squares, Geometric and Functional Analysis, 18 (6), 1917-1987 (2009).","PeriodicalId":404905,"journal":{"name":"What's Next?","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126109800","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}
What's Next?Pub Date : 2019-06-12DOI: 10.1515/9780691185897-013
W. Thurston, Hyungryul Baik, Yan Gao, J. Hubbard, Tan Lei, Kathryn A. Lindsey, D. Thurston
{"title":"Degree-d-invariant Laminations","authors":"W. Thurston, Hyungryul Baik, Yan Gao, J. Hubbard, Tan Lei, Kathryn A. Lindsey, D. Thurston","doi":"10.1515/9780691185897-013","DOIUrl":"https://doi.org/10.1515/9780691185897-013","url":null,"abstract":"Degree-$d$-invariant laminations of the disk model the dynamical action of a degree-$d$ polynomial; such a lamination defines an equivalence relation on $S^1$ that corresponds to dynamical rays of an associated polynomial landing at the same multi-accessible points in the Julia set. Primitive majors are certain subsets of degree-$d$-invariant laminations consisting of critical leaves and gaps. The space $textrm{PM}(d)$ of primitive degree-$d$ majors is a spine for the set of monic degree-$d$ polynomials with distinct roots and serves as a parameterization of a subset of the boundary of the connectedness locus for degree-$d$ polynomials. The core entropy of a postcritically finite polynomial is the topological entropy of the action of the polynomial on the associated Hubbard tree. Core entropy may be computed directly, bypassing the Hubbard tree, using a combinatorial analogue of the Hubbard tree within the context of degree-$d$-invariant laminations.","PeriodicalId":404905,"journal":{"name":"What's Next?","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126529007","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}
What's Next?Pub Date : 2017-12-31DOI: 10.12987/9780300175080-007
Anatole Kaletsky
{"title":"5. The World Bets On Eu Rope, But The United States Will Probably Win","authors":"Anatole Kaletsky","doi":"10.12987/9780300175080-007","DOIUrl":"https://doi.org/10.12987/9780300175080-007","url":null,"abstract":"","PeriodicalId":404905,"journal":{"name":"What's Next?","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131973231","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}
What's Next?Pub Date : 2017-12-31DOI: 10.12987/9780300175080-011
K. Jefferis
{"title":"9. Prospects For Sub- Saharan Africa In 2010–2011","authors":"K. Jefferis","doi":"10.12987/9780300175080-011","DOIUrl":"https://doi.org/10.12987/9780300175080-011","url":null,"abstract":"","PeriodicalId":404905,"journal":{"name":"What's Next?","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129156658","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}
What's Next?Pub Date : 2017-12-31DOI: 10.12987/9780300175080-017
Narimon Safavi
{"title":"15. In The Aftermath Of Iran’S Latest Revolution","authors":"Narimon Safavi","doi":"10.12987/9780300175080-017","DOIUrl":"https://doi.org/10.12987/9780300175080-017","url":null,"abstract":"","PeriodicalId":404905,"journal":{"name":"What's Next?","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121180740","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}