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DTSTART;TZID=America/New_York:20221026T090000
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DTSTAMP:20260410T175811
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SUMMARY:Kähler bands—Chern insulators\, holomorphicity and induced quantum geometry
DESCRIPTION:Topological Quantum Matter Seminar \n\n\nSpeaker: Bruno Mera\, Tohoku University\n\nTitle: Kähler bands—Chern insulators\, holomorphicity and induced quantum geometry\n\nAbstract: The notion of topological phases has dramatically changed our understanding of insulators. There is much to learn about a band insulator beyond the assertion that it has a gap separating the valence bands from the conduction bands. In the particular case of two dimensions\, the occupied bands may have a nontrivial topological twist determining what is called a Chern insulator. This topological twist is not just a mathematical observation\, it has observable consequences—the transverse Hall conductivity is quantized and proportional to the 1st Chern number of the vector bundle of occupied states over the Brillouin zone. Finer properties of band insulators refer not just to the topology\, but also to their geometry. Of particular interest is the momentum-space quantum metric and the Berry curvature. The latter is the curvature of a connection on the vector bundle of occupied states. The study of the geometry of band insulators can also be used to probe whether the material may host stable fractional topological phases. In particular\, for a Chern band to have an algebra of projected density operators which is isomorphic to the W∞ algebra found by Girvin\, MacDonald and Platzman—the GMP algebra—in the context of the fractional quantum Hall effect\, certain geometric constraints\, associated with the holomorphic character of the Bloch wave functions\, are naturally found and they enforce a compatibility relation between the quantum metric and the Berry curvature of the band. The Brillouin zone is then endowed with a Kähler structure which\, in this case\, is also translation-invariant (flat). Motivated by the above\, we will provide an overview of the geometry of Chern insulators from the perspective of Kähler geometry\, introducing the notion of a Kähler band which shares properties with the well-known ideal case of the lowest Landau level. Furthermore\, we will provide a prescription\, borrowing ideas from geometric quantization\, to generate a flat Kähler band in some appropriate asymptotic limit. Such flat Kähler bands are potential candidates to host and realize fractional Chern insulating phases. Using geometric quantization arguments\, we then provide a natural generalization of the theory to all even dimensions.\n\n\nReferences:\n[1] Tomoki Ozawa and Bruno Mera. Relations between topology and the quantum metric for Chern insulators. Phys. Rev. B\, 104:045103\, Jul 2021.\n[2] Bruno Mera and Tomoki Ozawa. Kähler geometry and Chern insulators: Relations between topology and the quantum metric. Phys. Rev. B\, 104:045104\, Jul 2021.\n[3] Bruno Mera and Tomoki Ozawa. Engineering geometrically flat Chern bands with Fubini-Study  Kähler structure. Phys. Rev. B\, 104:115160\, Sep 2021.
URL:https://cmsa.fas.harvard.edu/event/9062/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Topological Quantum Matter Seminar
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Topological-Seminar-10.26.22.png
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BEGIN:VEVENT
DTSTART;TZID=America/New_York:20221026T123000
DTEND;TZID=America/New_York:20221026T133000
DTSTAMP:20260410T175811
CREATED:20230817T174027Z
LAST-MODIFIED:20240121T174027Z
UID:10001269-1666787400-1666791000@cmsa.fas.harvard.edu
SUMMARY:Clique listing algorithms
DESCRIPTION:Speaker: Virginia Vassilevska Williams (MIT) \nTitle: Clique listing algorithms \nAbstract: A k-clique in a graph G is a subgraph of G on k vertices in which every pair of vertices is linked by an edge. Cliques are a natural notion of social network cohesiveness with a long history. \nA fundamental question\, with many applications\, is “How fast can one list all k-cliques in a given graph?”. \nEven just detecting whether an n-vertex graph contains a k-Clique has long been known to be NP-complete when k can depend on n (and hence no efficient algorithm is likely to exist for it). If k is a small constant\, such as 3 or 4 (independent of n)\, even the brute-force algorithm runs in polynomial time\, O(n^k)\, and can list all k-cliques in the graph; though O(n^k) time is far from practical. As the number of k-cliques in an n-vertex graph can be Omega(n^k)\, the brute-force algorithm is in some sense optimal\, but only if there are Omega(n^k) k-cliques. In this talk we will show how to list k-cliques faster when the input graph has few k-cliques\, with running times depending on the number of vertices n\, the number of edges m\, the number of k-cliques T and more. We will focus on the case when k=3\, but we will note some extensions. \n(Based on joint work with Andreas Bjorklund\, Rasmus Pagh\, Uri Zwick\, Mina Dalirrooyfard\, Surya Mathialagan and Yinzhan Xu)
URL:https://cmsa.fas.harvard.edu/event/collquium_102722/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Colloquium-10.26.22.png
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BEGIN:VEVENT
DTSTART;TZID=America/New_York:20221026T140000
DTEND;TZID=America/New_York:20221026T150000
DTSTAMP:20260410T175811
CREATED:20230808T185319Z
LAST-MODIFIED:20240115T103149Z
UID:10001214-1666792800-1666796400@cmsa.fas.harvard.edu
SUMMARY:From Engine to Auto
DESCRIPTION:New Technologies in Mathematics Seminar \nSpeakers: João Araújo\, Mathematics Department\, Universidade Nova de Lisboa and Michael Kinyon\, Department of Mathematics\, University of Denver \n\nTitle: From Engine to Auto \n\n\nAbstract: Bill McCune produced the program EQP that deals with first order logic formulas and in 1996 managed to solve Robbins’ Conjecture. This very powerful tool reduces to triviality any result that can be obtained by encoding the assumptions and the goals. The next step was to turn the program into a genuine assistant for the working mathematician: find ways to help the prover with proofs; reduce the lengths of the automatic proofs to better crack them;  solve problems in higher order logic; devise tools that autonomously prove results of a given type\, etc.\n\nIn this talk we are going to show some of the tools and strategies we have been producing. There will be real illustrations of theorems obtained for groups\, loops\, semigroups\, logic algebras\, lattices and generalizations\, quandles\, and many more.
URL:https://cmsa.fas.harvard.edu/event/nt-102622/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:New Technologies in Mathematics Seminar
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-NTM-Seminar-10.26.2022.png
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