The Experts below are selected from a list of 270 Experts worldwide ranked by ideXlab platform
Yisong Yang - One of the best experts on this subject based on the ideXlab platform.
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Existence of Energy Minimizers as Stable Knotted Solitons in the Faddeev Model
Communications in Mathematical Physics, 2004Co-Authors: Fanghua Lin, Yisong YangAbstract:In this paper, we study the existence of knot-like solitons realized as the energy-minimizing configurations in the Faddeev quantum field theory model. Topologically, these solitons are characterized by an Hopf invariant, Q , which is an integral class in the Homotopy Group π_3( S ^ 2 )= . We prove in the full space situation that there exists an infinite subset of such that for any m ∈ , the Faddeev energy, E , has a minimizer among the topological class Q = m . Besides, we show that there always exists a least-positive-energy Faddeev soliton of non-zero Hopf invariant. In the bounded domain situation, we show that the existence of an energy minimizer holds for = . As a by-product, we obtain an important technical result which says that E and Q satisfy the sublinear inequality E ≤ C | Q |^3/4, where C >0 is a universal constant. Such a fact explains why knotted (clustered soliton) configurations are preferred over widely separated unknotted (multisoliton) configurations when | Q | is sufficiently large.
Brent Doran - One of the best experts on this subject based on the ideXlab platform.
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A^1-Homotopy Groups, excision, and solvable quotients
arXiv: Algebraic Geometry, 2009Co-Authors: Aravind Asok, Brent DoranAbstract:We study some properties of A^1-Homotopy Groups: geometric interpretations of connectivity, excision results, and a re-interpretation of quotients by free actions of connected solvable Groups in terms of covering spaces in the sense of A^1-Homotopy theory. These concepts and results are well-suited to the study of certain quotients via geometric invariant theory. As a case study in the geometry of solvable Group quotients, we investigate A^1-Homotopy Groups of smooth toric varieties. We give simple combinatorial conditions (in terms of fans) guaranteeing vanishing of low degree A^1-Homotopy Groups of smooth (proper) toric varieties. Finally, in certain cases, we can actually compute the "next" non-vanishing A^1-Homotopy Group (beyond \pi_1^{A^1}) of a smooth toric variety. From this point of view, A^1-Homotopy theory, even with its exquisite sensitivity to algebro-geometric structure, is almost "as tractable" (in low degrees) as ordinary Homotopy for large classes of interesting varieties.
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A1-Homotopy Groups, excision, and solvable quotients
Advances in Mathematics, 2009Co-Authors: Aravind Asok, Brent DoranAbstract:Abstract We study some properties of A 1 -Homotopy Groups: geometric interpretations of connectivity, excision results, and a re-interpretation of quotients by free actions of connected solvable Groups in terms of covering spaces in the sense of A 1 -Homotopy theory. These concepts and results are well suited to the study of certain quotients via geometric invariant theory. As a case study in the geometry of solvable Group quotients, we investigate A 1 -Homotopy Groups of smooth toric varieties. We give simple combinatorial conditions (in terms of fans) guaranteeing vanishing of low degree A 1 -Homotopy Groups of smooth (proper) toric varieties. Finally, in certain cases, we can actually compute the “next” non-vanishing A 1 -Homotopy Group (beyond π 1 A 1 ) of a smooth toric variety. From this point of view, A 1 -Homotopy theory, even with its exquisite sensitivity to algebro-geometric structure, is almost “as tractable” (in low degrees) as ordinary Homotopy for large classes of interesting varieties.
Paul G. Mezey - One of the best experts on this subject based on the ideXlab platform.
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The topology of catchment regions of potential energy hypersurfaces
Theoretical Chemistry Accounts, 1999Co-Authors: Paul G. MezeyAbstract:A simple proof is presented for a fundamental topological property of catchment regions of potential energy hypersurfaces: each catchment region C(λ,i), representing a chemical species and its conformational range on the potential energy hypersurface, is simply k-connected for each dimension k=1,2,…3N−6−λ, where λ is the index of the catchment region. The consequences of this property on the structure of the fundamental Group of reaction mechanisms (the one-dimensional Homotopy Group of reaction paths) is discussed.
Roman Mikhailov - One of the best experts on this subject based on the ideXlab platform.
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Group and Lie algebra filtrations and Homotopy Groups of spheres.
arXiv: Group Theory, 2021Co-Authors: Laurent Bartholdi, Roman MikhailovAbstract:We establish a bridge between Homotopy Groups of spheres and commutator calculus in Groups, and solve in this manner the "dimension problem" by providing a converse to Sjogren's theorem: every abelian Group of bounded exponent can be embedded in the dimension quotient of a Group. This is proven by embedding for arbitrary $s,d$ the torsion of the Homotopy Group $\pi_s(S^d)$ into a dimension quotient, via a result of Wu. In particular, this invalidates some long-standing results in the literature, since for every prime $p$, there is some $p$-torsion in $\pi_{2p}(S^2)$ by a result of Serre. We explain in this manner Rips's famous counterexample to the dimension conjecture in terms of the Homotopy Group $\pi_4(S^2)=\mathbb Z/2\mathbb Z$. We finally obtain analogous results in the context of Lie rings: for every prime $p$ there exists a Lie ring with $p$-torsion in some dimension quotient.
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On the third Homotopy Group of Orr's space
Algebraic & Geometric Topology, 2018Co-Authors: Emmanuel Dror Farjoun, Roman MikhailovAbstract:K. Orr defined a Milnor-type invariant of links that lies in the third Homotopy Group of a certain space $K_\omega.$ The problem of non-triviality of this third Homotopy Group has been open. We show that it is an infinitely generated Group. The question of realization of its elements as links remains open.
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on the third Homotopy Group of orr s space
Algebraic & Geometric Topology, 2018Co-Authors: Emmanuel Dror Farjoun, Roman MikhailovAbstract:K Orr defined a Milnor-type invariant of links that lies in the third Homotopy Group of a certain space Kω. The problem of nontriviality of this third Homotopy Group has been open. We show that it is an infinitely generated Group. The question of realization of its elements as links remains open.
Hanieh Mirebrahimi - One of the best experts on this subject based on the ideXlab platform.
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Some results in quasitopological Homotopy Groups
arXiv: Algebraic Topology, 2017Co-Authors: T. Nasri, Hanieh Mirebrahimi, H. TorabiAbstract:In this paper we show that the nth quasitopological Homotopy Group of a topological space is isomorphic to (n-1)th quasitopological Homotopy Group of its loop space and by this fact we obtain some results about quasitopological Homotopy Groups. Finally, using the long exact sequence of a based pair and a fibration in qTop introduced by Brazas in 2013, we obtain some results in this field.
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On quasitopological Homotopy Groups of Inverse Limit Spaces
arXiv: Algebraic Topology, 2012Co-Authors: T. Nasri, Behrooz Mashayekhy, Hanieh MirebrahimiAbstract:The paper is devoted to study the behavior of quasitopological Homotopy Groups on inverse limit spaces. More precisely, we present some conditions under which the quasitopological Homotopy Group of an inverse limit space and especially a product space is a topological Group. Finally, we give some conditions for countability of Homotopy Groups.