We discuss recent results and open questions on the broad theme of (Nielsen) realization problems. Beyond realizing subgroups of mapping class groups, there are many other natural instances where one can ask if a surjection from a group of diffeomorphisms of a manifold to another group admits a section over particular subgroups. This survey includes many open problems, and some short proofs of new results that are illustrative of key techniques; drawing attention to parallels between problems arising in different areas.
{"title":"Realization problems for diffeomorphism\u0000 groups","authors":"Kathryn Mann, Bena Tshishiku","doi":"10.1090/pspum/102/11","DOIUrl":"https://doi.org/10.1090/pspum/102/11","url":null,"abstract":"We discuss recent results and open questions on the broad theme of (Nielsen) realization problems. Beyond realizing subgroups of mapping class groups, there are many other natural instances where one can ask if a surjection from a group of diffeomorphisms of a manifold to another group admits a section over particular subgroups. This survey includes many open problems, and some short proofs of new results that are illustrative of key techniques; drawing attention to parallels between problems arising in different areas.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"124 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131432576","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 discuss how to apply work of L. Rudolph to braid conjugacy class invariants to obtain potentially effective obstructions to a slice knot being ribbon. We then apply these ideas to a family of braid conjugacy class invariants coming from Khovanov-Lee theory and explain why we do not obtain effective ribbon obstructions in this case.
{"title":"On braided, banded surfaces and ribbon\u0000 obstructions","authors":"J. E. Grigsby, Elisenda Grigsby","doi":"10.1090/pspum/102/07","DOIUrl":"https://doi.org/10.1090/pspum/102/07","url":null,"abstract":"We discuss how to apply work of L. Rudolph to braid conjugacy class invariants to obtain potentially effective obstructions to a slice knot being ribbon. We then apply these ideas to a family of braid conjugacy class invariants coming from Khovanov-Lee theory and explain why we do not obtain effective ribbon obstructions in this case.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"90 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126241348","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 establish a quadratic identity for the Yamada polynomial of ribbon cubic graphs in 3-space, extending the Tutte golden identity for planar cubic graphs. An application is given to the structure of the flow polynomial of cubic graphs at zero. The golden identity for the flow polynomial is conjectured to characterize planarity of cubic graphs, and we prove this conjecture for a certain infinite family of non-planar graphs. Further, we establish exponential growth of the number of chromatic polynomials of planar triangulations, answering a question of D. Treumann and E. Zaslow. The structure underlying these results is the chromatic algebra, and more generally the SO(3) topological quantum field theory.
{"title":"Structure of the flow and Yamada polynomials\u0000 of cubic graphs","authors":"I. Agol, Vyacheslav Krushkal","doi":"10.1090/pspum/102/01","DOIUrl":"https://doi.org/10.1090/pspum/102/01","url":null,"abstract":"We establish a quadratic identity for the Yamada polynomial of ribbon cubic graphs in 3-space, extending the Tutte golden identity for planar cubic graphs. An application is given to the structure of the flow polynomial of cubic graphs at zero. The golden identity for the flow polynomial is conjectured to characterize planarity of cubic graphs, and we prove this conjecture for a certain infinite family of non-planar graphs. \u0000Further, we establish exponential growth of the number of chromatic polynomials of planar triangulations, answering a question of D. Treumann and E. Zaslow. The structure underlying these results is the chromatic algebra, and more generally the SO(3) topological quantum field theory.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127139264","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 show that there exists a transverse link in the standard contact structures on the 3-sphere such that all contact 3-manifolds are contact branched covers over this transverse link.
{"title":"Transverse universal links","authors":"Roger Casals, John B. Etnyre","doi":"10.1090/pspum/102/04","DOIUrl":"https://doi.org/10.1090/pspum/102/04","url":null,"abstract":"We show that there exists a transverse link in the standard contact structures on the 3-sphere such that all contact 3-manifolds are contact branched covers over this transverse link.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128806535","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 survey the Dirichlet problem for the complex Homogeneous Monge-Amp`ere Equation, both in the case of domains in $mathbb C^n$ and the case of compact K"ahler manifolds parametrized by a Riemann surface with boundary. We then give a self-contained account of previous work of the authors that connects this with the Hele-Shaw flow, and give several concrete examples illustrating various phenomena that solutions to this problem can display.
{"title":"The Dirichlet problem for the complex\u0000 homogeneous Monge-Ampère equation","authors":"J. Ross, D. Nystrom","doi":"10.1090/PSPUM/099/01744","DOIUrl":"https://doi.org/10.1090/PSPUM/099/01744","url":null,"abstract":"We survey the Dirichlet problem for the complex Homogeneous Monge-Amp`ere Equation, both in the case of domains in $mathbb C^n$ and the case of compact K\"ahler manifolds parametrized by a Riemann surface with boundary. We then give a self-contained account of previous work of the authors that connects this with the Hele-Shaw flow, and give several concrete examples illustrating various phenomena that solutions to this problem can display.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121097130","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}
Pawel Ciosmak, L. Hadasz, Masahide Manabe, P. Sułkowski
We show that quantum curves arise in infinite families and have the structure of singular vectors of a relevant symmetry algebra. We analyze in detail the case of the hermitian one-matrix model with the underlying Virasoro algebra, and the super-eigenvalue model with the underlying super-Virasoro algebra. In the Virasoro case we relate singular vector structure of quantum curves to the topological recursion, and in the super-Virasoro case we introduce the notion of super-quantum curves. We also discuss the double quantum structure of the quantum curves and analyze specific examples of Gaussian and multi-Penner models.
{"title":"Singular vector structure of quantum\u0000 curves","authors":"Pawel Ciosmak, L. Hadasz, Masahide Manabe, P. Sułkowski","doi":"10.1090/pspum/100/01766","DOIUrl":"https://doi.org/10.1090/pspum/100/01766","url":null,"abstract":"We show that quantum curves arise in infinite families and have the structure of singular vectors of a relevant symmetry algebra. We analyze in detail the case of the hermitian one-matrix model with the underlying Virasoro algebra, and the super-eigenvalue model with the underlying super-Virasoro algebra. In the Virasoro case we relate singular vector structure of quantum curves to the topological recursion, and in the super-Virasoro case we introduce the notion of super-quantum curves. We also discuss the double quantum structure of the quantum curves and analyze specific examples of Gaussian and multi-Penner models.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129584719","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}
Gopakumar, Ooguri and Vafa famously proposed the existence of a correspondence between a topological gauge theory on one hand ($U(N)$ Chern-Simons theory on the three-sphere) and a topological string theory on the other (the topological A-model on the resolved conifold). On the physics side, this duality provides a concrete instance of the large $N$ gauge/string correspondence where exact computations can be performed in detail; mathematically, it puts forward a triangle of striking relations between quantum invariants (Reshetikhin-Turaev-Witten) of knots and 3-manifolds, curve-counting invariants (Gromov-Witten/Donaldson-Thomas) of local Calabi-Yau 3-folds, and the Eynard-Orantin recursion for a specific class of spectral curves. I quickly survey recent results on the most general frame of validity of this correspondence and discuss some of its implications.
{"title":"On the Gopakumar–Ooguri–Vafa correspondence\u0000 for Clifford–Klein 3-manifolds","authors":"A. Brini","doi":"10.1090/pspum/100/01759","DOIUrl":"https://doi.org/10.1090/pspum/100/01759","url":null,"abstract":"Gopakumar, Ooguri and Vafa famously proposed the existence of a correspondence between a topological gauge theory on one hand ($U(N)$ Chern-Simons theory on the three-sphere) and a topological string theory on the other (the topological A-model on the resolved conifold). On the physics side, this duality provides a concrete instance of the large $N$ gauge/string correspondence where exact computations can be performed in detail; mathematically, it puts forward a triangle of striking relations between quantum invariants (Reshetikhin-Turaev-Witten) of knots and 3-manifolds, curve-counting invariants (Gromov-Witten/Donaldson-Thomas) of local Calabi-Yau 3-folds, and the Eynard-Orantin recursion for a specific class of spectral curves. I quickly survey recent results on the most general frame of validity of this correspondence and discuss some of its implications.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131407692","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 show that any virtual or welded period of a classical knot $K$ can be realized as a classical period. A direct consequence is that a classical knot admits only finitely many virtual or welded periods.
{"title":"Virtual and welded periods of classical\u0000 knots","authors":"H. Boden, Andrew J. Nicas","doi":"10.1090/pspum/102/03","DOIUrl":"https://doi.org/10.1090/pspum/102/03","url":null,"abstract":"We show that any virtual or welded period of a classical knot $K$ can be realized as a classical period. A direct consequence is that a classical knot admits only finitely many virtual or welded periods.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126030174","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 survey the theory of K"ahler-Einstein metrics, with particular focus on the circle of ideas surrounding the Yau-Tian-Donaldson conjecture for Fano manifolds.
{"title":"Kähler-Einstein metrics","authors":"Gábor Székelyhidi","doi":"10.1090/PSPUM/099/01745","DOIUrl":"https://doi.org/10.1090/PSPUM/099/01745","url":null,"abstract":"We survey the theory of K\"ahler-Einstein metrics, with particular focus on the circle of ideas surrounding the Yau-Tian-Donaldson conjecture for Fano manifolds.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123299399","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}
Nahm's equations are viewed in a more general context where they appear as a vector field on a moduli space of co-Higgs bundles on the projective line. Zeros of this vector field correspond to torsion-free sheaves on a singular spectral curve which we translate in terms of a smooth curve in three-dimensional projective space. We also show how generalizations of Nahm's equations are required when the spectral curve is non-reduced and deduce the existence of non-classical conserved quantities in this situation.
{"title":"Remarks on Nahm’s equations","authors":"N. Hitchin","doi":"10.1090/pspum/099/01738","DOIUrl":"https://doi.org/10.1090/pspum/099/01738","url":null,"abstract":"Nahm's equations are viewed in a more general context where they appear as a vector field on a moduli space of co-Higgs bundles on the projective line. Zeros of this vector field correspond to torsion-free sheaves on a singular spectral curve which we translate in terms of a smooth curve in three-dimensional projective space. We also show how generalizations of Nahm's equations are required when the spectral curve is non-reduced and deduce the existence of non-classical conserved quantities in this situation.","PeriodicalId":384712,"journal":{"name":"Proceedings of Symposia in Pure\n Mathematics","volume":"237 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133932182","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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