Geometry & Topology Monograph 18:  Proceedings of the Freedman Fest

 URL: http://www.msp.warwick.ac.uk/gtm/2012/18-01/

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) 1-7
   The bosonic birthday paradox
     by Alex Arkhipov and Greg Kuperberg
   URL: http://www.msp.warwick.ac.uk/gtm/2012/18-01/p001.xhtml
   DOI: 10.2140/gtm.2012.18.1

(2) Geometry & Topology Monographs 18 (2012) 9-34
   Broken Lefschetz fibrations and smooth structures on 4-manifolds
     by R Inanc Baykur
   URL: http://www.msp.warwick.ac.uk/gtm/2012/18-01/p002.xhtml
   DOI: 10.2140/gtm.2012.18.9

(3) Geometry & Topology Monographs 18 (2012) 35-60
   Universal quadratic forms and Whitney tower intersection invariants
     by James Conant, Rob Schneiderman and Peter Teichner
   URL: http://www.msp.warwick.ac.uk/gtm/2012/18-01/p003.xhtml
   DOI: 10.2140/gtm.2012.18.35

(4) Geometry & Topology Monographs 18 (2012) 61-67
   On Wigner's theorem
     by Daniel S Freed
   URL: http://www.msp.warwick.ac.uk/gtm/2012/18-01/p004.xhtml
   DOI: 10.2140/gtm.2012.18.61

(5) Geometry & Topology Monographs 18 (2012) 69-79
   Kernel(J) warns of false vacua
     by Michael Freedman
   URL: http://www.msp.warwick.ac.uk/gtm/2012/18-01/p005.xhtml
   DOI: 10.2140/gtm.2012.18.69

(6) Geometry & Topology Monographs 18 (2012) 81-101
   Surgery on nullhomologous tori
     by Ronald Fintushel and Ronald Stern
   URL: http://www.msp.warwick.ac.uk/gtm/2012/18-01/p006.xhtml
   DOI: 10.2140/gtm.2012.18.81

(7) Geometry & Topology Monographs 18 (2012) 103-114
   Reconstructing 4-manifolds from Morse 2-functions
     by David Gay and Robion Kirby
   URL: http://www.msp.warwick.ac.uk/gtm/2012/18-01/p007.xhtml
   DOI: 10.2140/gtm.2012.18.103

(8) Geometry & Topology Monographs 18 (2012) 115-160
   Matrix product operators and central elements: classical   
   description of a quantum state
     by Matthew B Hastings
   URL: http://www.msp.warwick.ac.uk/gtm/2012/18-01/p008.xhtml
   DOI: 10.2140/gtm.2012.18.115

(9) Geometry & Topology Monographs 18 (2012) 161-190
   Cohomotopy sets of 4-manifolds
     by Robion Kirby, Paul Melvin and Peter Teichner
   URL: http://www.msp.warwick.ac.uk/gtm/2012/18-01/p009.xhtml
   DOI: 10.2140/gtm.2012.18.161

(10) Geometry & Topology Monographs 18 (2012) 191-197
   Solutions to generalized Yang-Baxter equations via ribbon fusion
categories
     by Alexei Kitaev and Zhenghan Wang
   URL: http://www.msp.warwick.ac.uk/gtm/2012/18-01/p010.xhtml
   DOI: 10.2140/gtm.2012.18.191

(11) Geometry & Topology Monographs 18 (2012) 199-234
   Link groups of 4-manifolds
     by Vyacheslav Krushkal
   URL: http://www.msp.warwick.ac.uk/gtm/2012/18-01/p011.xhtml
   DOI: 10.2140/gtm.2012.18.199

(12) Geometry & Topology Monographs 18 (2012) 235-251
   The principal fibration sequence and the second cohomotopy set
     by Laurence R Taylor
   URL: http://www.msp.warwick.ac.uk/gtm/2012/18-01/p012.xhtml
   DOI: 10.2140/gtm.2012.18.235

(13) Geometry & Topology Monographs 18 (2012) 253-289
   An introduction to categorifying quantum knot invariants
     by Ben Webster
   URL: http://www.msp.warwick.ac.uk/gtm/2012/18-01/p013.xhtml
   DOI: 10.2140/gtm.2012.18.253

(14) Geometry & Topology Monographs 18 (2012) 291-308
   Khovanov homology and gauge theory
     by Edward Witten
   URL: http://www.msp.warwick.ac.uk/gtm/2012/18-01/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 4-manifolds
     by R Inanc Baykur

The broken genera are orientation preserving diffeomorphism invariants
of closed oriented 4-manifolds, defined via broken Lefschetz
fibrations. We study the properties of the broken genera invariants,
and calculate them for various 4-manifolds, 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 non-commutative 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 4-ball 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 4-ball.

(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 Fubini-Study 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 \sigma-models 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 \sigma-model 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^2-bar we
illustrate how surgery on a single embedded nullhomologous torus can
be utilized to change the symplectic structure, the Seiberg-Witten
invariant, and hence the smooth structure on a 4-manifold.

(7) Reconstructing 4-manifolds from Morse 2-functions
     by David Gay and Robion Kirby

Given a Morse 2-function 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 two-dimensional 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 one-dimensional
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 Levin-Wen models, as
either product operators in the Abelian case or matrix product
operators with low bond dimension in the non-Abelian case; this matrix
product operator representation may have practical application in
engineering the complicated multi-spin interactions in the Levin--Wen
models.

(9) Cohomotopy sets of 4-manifolds
     by Robion Kirby, Paul Melvin and Peter Teichner

Elementary geometric arguments are used to compute the group of
homotopy classes of maps from a 4-manifold X to the 3-sphere, and to
enumerate the homotopy classes of maps from X to the 2-sphere.  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
4-manifolds and provide a framework for the study of Morse 2-functions
on 4-manifolds, a subject that has garnered considerable recent
attention.

(10) Solutions to generalized Yang-Baxter 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+m-2 copies of V for
some fixed vector space V such that each braid generator sigma_i,
i=1,...,n-1, acts on m consecutive tensor factors from i through
i+m-1. The braid relation for m=2 is essentially the Yang-Baxter
equation, and the cases for m>2 are called generalized Yang-Baxter
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
Jones-Kauffman 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 4-manifolds
     by Vyacheslav Krushkal

The notion of a Bing cell is introduced, and it is used to define
invariants, link groups, of 4-manifolds.  Bing cells combine some
features of both surfaces and 4-dimensional handlebodies, and the link
group lambda(M) measures certain aspects of the handle structure of a
4-manifold 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 4-space.  Applications to the A-B 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 well-known 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 well-known
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 4--complex.  This is
relevant to questions involving broken Lefschetz fibrations on
4--manifolds.  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 five-dimensional 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.