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Publications started of the Proceedings of the Freedman Fest
Posted:
Oct 14, 2012 7:03 PM


Geometry & Topology Publications announces the start of Publication of:
Geometry & Topology Monograph 18: Proceedings of the Freedman Fest
URL: http://www.msp.warwick.ac.uk/gtm/2012/1801/
Fourteen papers have been published so far with further papers to be added later. Details and abstracts of these fourteen follow.
(1) Geometry & Topology Monographs 18 (2012) 17 The bosonic birthday paradox by Alex Arkhipov and Greg Kuperberg URL: http://www.msp.warwick.ac.uk/gtm/2012/1801/p001.xhtml DOI: 10.2140/gtm.2012.18.1
(2) Geometry & Topology Monographs 18 (2012) 934 Broken Lefschetz fibrations and smooth structures on 4manifolds by R Inanc Baykur URL: http://www.msp.warwick.ac.uk/gtm/2012/1801/p002.xhtml DOI: 10.2140/gtm.2012.18.9
(3) Geometry & Topology Monographs 18 (2012) 3560 Universal quadratic forms and Whitney tower intersection invariants by James Conant, Rob Schneiderman and Peter Teichner URL: http://www.msp.warwick.ac.uk/gtm/2012/1801/p003.xhtml DOI: 10.2140/gtm.2012.18.35
(4) Geometry & Topology Monographs 18 (2012) 6167 On Wigner's theorem by Daniel S Freed URL: http://www.msp.warwick.ac.uk/gtm/2012/1801/p004.xhtml DOI: 10.2140/gtm.2012.18.61
(5) Geometry & Topology Monographs 18 (2012) 6979 Kernel(J) warns of false vacua by Michael Freedman URL: http://www.msp.warwick.ac.uk/gtm/2012/1801/p005.xhtml DOI: 10.2140/gtm.2012.18.69
(6) Geometry & Topology Monographs 18 (2012) 81101 Surgery on nullhomologous tori by Ronald Fintushel and Ronald Stern URL: http://www.msp.warwick.ac.uk/gtm/2012/1801/p006.xhtml DOI: 10.2140/gtm.2012.18.81
(7) Geometry & Topology Monographs 18 (2012) 103114 Reconstructing 4manifolds from Morse 2functions by David Gay and Robion Kirby URL: http://www.msp.warwick.ac.uk/gtm/2012/1801/p007.xhtml DOI: 10.2140/gtm.2012.18.103
(8) Geometry & Topology Monographs 18 (2012) 115160 Matrix product operators and central elements: classical description of a quantum state by Matthew B Hastings URL: http://www.msp.warwick.ac.uk/gtm/2012/1801/p008.xhtml DOI: 10.2140/gtm.2012.18.115
(9) Geometry & Topology Monographs 18 (2012) 161190 Cohomotopy sets of 4manifolds by Robion Kirby, Paul Melvin and Peter Teichner URL: http://www.msp.warwick.ac.uk/gtm/2012/1801/p009.xhtml DOI: 10.2140/gtm.2012.18.161
(10) Geometry & Topology Monographs 18 (2012) 191197 Solutions to generalized YangBaxter equations via ribbon fusion categories by Alexei Kitaev and Zhenghan Wang URL: http://www.msp.warwick.ac.uk/gtm/2012/1801/p010.xhtml DOI: 10.2140/gtm.2012.18.191
(11) Geometry & Topology Monographs 18 (2012) 199234 Link groups of 4manifolds by Vyacheslav Krushkal URL: http://www.msp.warwick.ac.uk/gtm/2012/1801/p011.xhtml DOI: 10.2140/gtm.2012.18.199
(12) Geometry & Topology Monographs 18 (2012) 235251 The principal fibration sequence and the second cohomotopy set by Laurence R Taylor URL: http://www.msp.warwick.ac.uk/gtm/2012/1801/p012.xhtml DOI: 10.2140/gtm.2012.18.235
(13) Geometry & Topology Monographs 18 (2012) 253289 An introduction to categorifying quantum knot invariants by Ben Webster URL: http://www.msp.warwick.ac.uk/gtm/2012/1801/p013.xhtml DOI: 10.2140/gtm.2012.18.253
(14) Geometry & Topology Monographs 18 (2012) 291308 Khovanov homology and gauge theory by Edward Witten URL: http://www.msp.warwick.ac.uk/gtm/2012/1801/p014.xhtml DOI: 10.2140/gtm.2012.18.291
Abstracts =========
(1) The bosonic birthday paradox by Alex Arkhipov and Greg Kuperberg
We motivate and prove a version of the birthday paradox for k identical bosons in n possible modes. If the bosons are in the uniform mixed state, also called the maximally mixed quantum state, then we need k~\sqrt{n} bosons to expect two in the same state, which is smaller by a factor of \sqrt{2} than in the case of distinguishable objects (boltzmannons). While the core result is elementary, we generalize the hypothesis and strengthen the conclusion in several ways. One side result is that boltzmannons with a randomly chosen multinomial distribution have the same birthday statistics as bosons. This last result is interesting as a quantum proof of a classical probability theorem; we also give a classical proof.
(2) Broken Lefschetz fibrations and smooth structures on 4manifolds by R Inanc Baykur
The broken genera are orientation preserving diffeomorphism invariants of closed oriented 4manifolds, defined via broken Lefschetz fibrations. We study the properties of the broken genera invariants, and calculate them for various 4manifolds, while showing that the invariants are sensitive to exotic smooth structures.
(3) Universal quadratic forms and Whitney tower intersection invariants by James Conant, Rob Schneiderman and Peter Teichner
A general algebraic theory of quadratic forms is developed and then specialized from the noncommutative to the commutative to, finally, the symmetric settings. In each of these contexts we construct universal quadratic forms. We then show that the intersection invariant for twisted Whitney towers in the 4ball is such a universal symmetric refinement of the framed intersection invariant. As a corollary, we obtain a short exact sequence, Theorem 11, that has been essential in a sequence of papers by the authors on the classification of Whitney towers in the 4ball.
(4) On Wigner's theorem by Daniel S Freed
Wigner's theorem asserts that any symmetry of a quantum system is unitary or antiunitary. In this short note we give two proofs based on the geometry of the FubiniStudy metric.
(5) Kernel(J) warns of false vacua by Michael Freedman
JHC Whitehead defined a map J_r: pi_r(SO)> pi_r^s from the homotopy of the special orthogonal group to the stable homotopy of spheres. Within a toy model we show how the known computation for kernel(J) leads to nonlinear \sigmamodels with spherical source (space) and spherical target which admit false vacua separated from the true vacuum by an energy barrier. In this construction, the dimension of space must be at least 8 and the dimension of the \sigmamodel target at least 5.
(6) Surgery on nullhomologous tori by Ronald Fintushel and Ronald Stern
By studying the example of smooth structures on CP^2 # 3CP^2bar we illustrate how surgery on a single embedded nullhomologous torus can be utilized to change the symplectic structure, the SeibergWitten invariant, and hence the smooth structure on a 4manifold.
(7) Reconstructing 4manifolds from Morse 2functions by David Gay and Robion Kirby
Given a Morse 2function f: X^4 > S^2, we give minimal conditions on the fold curves and fibers so that X^4 and f can be reconstructed from a certain combinatorial diagram attached to S^2. Additional remarks are made in other dimensions.
(8) Matrix product operators and central elements: classical description of a quantum state by Matthew B Hastings
We study planar twodimensional quantum systems on a lattice whose Hamiltonian is a sum of local commuting projectors of bounded range. We consider whether or not such a system has a zero energy ground state. To do this, we consider the problem as a onedimensional problem, grouping all sites along a column into ``supersites''; using C^*algebraic methods (Bravyi and Vyalyi, Quantum Inf. and Comp. 5, 187 (2005)), we can solve this problem if we can characterize the central elements of the interaction algebra on these supersite. Unfortunately, these central elements may be very complex, making brute force impractical. Instead, we show a characterization of these elements in terms of matrix product operators with bounded bond dimension. This bound can be interpreted as a bound on the number of particle types in lattice theories with bounded Hilbert space dimension on each site. Topological order in this approach is related to the existence of certain central elements which cannot be ``broken'' into smaller pieces without creating an end excitation. Using this bound on bond dimension, we prove that several special cases of this problem are in NP, and we give part of a proof that the general case is in NP. Further, we characterize central elements that appear in certain specific models, including toric code and LevinWen models, as either product operators in the Abelian case or matrix product operators with low bond dimension in the nonAbelian case; this matrix product operator representation may have practical application in engineering the complicated multispin interactions in the LevinWen models.
(9) Cohomotopy sets of 4manifolds by Robion Kirby, Paul Melvin and Peter Teichner
Elementary geometric arguments are used to compute the group of homotopy classes of maps from a 4manifold X to the 3sphere, and to enumerate the homotopy classes of maps from X to the 2sphere. The former completes a project initiated by Steenrod in the 1940's, and the latter provides geometric arguments for and extensions of recent homotopy theoretic results of Larry Taylor. These two results complete the computation of all the cohomotopy sets of closed oriented 4manifolds and provide a framework for the study of Morse 2functions on 4manifolds, a subject that has garnered considerable recent attention.
(10) Solutions to generalized YangBaxter equations via ribbon fusion categories by Alexei Kitaev and Zhenghan Wang
Inspired by quantum information theory, we look for representations of the braid groups B_n on the tensor product of n+m2 copies of V for some fixed vector space V such that each braid generator sigma_i, i=1,...,n1, acts on m consecutive tensor factors from i through i+m1. The braid relation for m=2 is essentially the YangBaxter equation, and the cases for m>2 are called generalized YangBaxter equations. We observe that certain objects in ribbon fusion categories naturally give rise to such representations for the case m=3. Examples are given from the Ising theory (or the closely related SU(2)_2), SO(N)_2 for N odd, and SU(3)_3. The solution from the JonesKauffman theory at a 6th root of unity, which is closely related to SO(3)_2 or SU(2)_4, is explicitly described in the end.
(11) Link groups of 4manifolds by Vyacheslav Krushkal
The notion of a Bing cell is introduced, and it is used to define invariants, link groups, of 4manifolds. Bing cells combine some features of both surfaces and 4dimensional handlebodies, and the link group lambda(M) measures certain aspects of the handle structure of a 4manifold M. This group is a quotient of the fundamental group, and examples of manifolds are given with pi_1(M) not = lambda(M). The main construction of the paper is a generalization of the Milnor group, which is used to formulate an obstruction to embeddability of Bing cells into 4space. Applications to the AB slice problem and to the structure of topological arbiters are discussed.
(12) The principal fibration sequence and the second cohomotopy set by Laurence R Taylor
Let p: E>B be a principal fibration with classifying map w: B>C. It is wellknown that the group [X,\Omega(C)] acts on [X,E] with orbit space the image of p_#, where p_#: [X,E]>[X,B]. The isotropy subgroup of the map of X to the base point of E is also wellknown to be the image of [X,\Omega(B)]. The isotropy subgroups for other maps e: X>E can definitely change as e does. The set of homotopy classes of lifts of f: X>B to the free loop space on B is a group. If f has a lift to E, the set p_#^{1}(f) is identified with the cokernel of a natural homomorphism from this group of lifts to [X,\Omega(C)]. As an example, [X,S^2] is enumerated for X a 4complex. This is relevant to questions involving broken Lefschetz fibrations on 4manifolds. Kirby, Melvin and Teichner (in these proceedings) have a different approach to this enumeration.
(13) An introduction to categorifying quantum knot invariants by Ben Webster
We construct knot invariants categorifying the quantum knot invariants for all representations of quantum groups, based on categorical representation theory. This paper gives a condensed description of the construction from the author's earlier papers on the subject, without proofs and certain constructions used only indirectly in the description of these invariants.
(14) Khovanov homology and gauge theory by Edward Witten
In these notes, I will sketch a new approach to Khovanov homology of knots and links based on counting the solutions of certain elliptic partial differential equations in four and five dimensions. The equations are formulated on four and fivedimensional manifolds with boundary, with a rather subtle boundary condition that encodes the knots and links. The construction is formally analogous to Floer and Donaldson theory in three and four dimensions. It was discovered using quantum field theory arguments but can be described and understood purely in terms of classical gauge theory.
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