- Bordered Floer homology
- Other papers on Heegard Floer homology
- Conformal geometry and rational maps
- Geometric topology
- Finite type and quantum invariants
- Cluster algebras and representation theory
- Discrete geometry
- Other papers
- Programs
- Notes

“**Bordered Heegaard Floer homology: Invariance and
pairing**”, with R. Lipshitz and P. Ozsváth,
*Memoirs of the AMS*,
accepted for publication,
arXiv:0810.0687.
An extension of Heegaard Floer homology to 3-manifolds with
parametrized boundary. We associate a differential graded algebra *A*
to every connected
surface *F*, and a module over *A* to a manifold with
boundary *F*, in a
way that lets us reconstruct the invariant of the closed manifold.

“**Bimodules in bordered Heegaard Floer homology**”, with R. Lipshitz and P. Ozsváth,
*Geometry and Topology*,
**19** (2015), 525–724,
arXiv:1003.0598.
We extend bordered Floer homology to allow for 3-manifolds with two boundary components, algebraically giving bimodules. In particular, this allows for reparametrization of the surface.

“**A tour of bordered Floer theory**”, with R. Lipshitz and P. Ozsváth,
*Proceedings of the National Academy of Sciences*,
**108**(20):8085--8092, May 17, 2011
arXiv:1107.5621.
A summary of the major structures and results of bordered Floer theory.

“**Slicing planar grid diagrams: A gentle introduction to bordered Heegaard Floer homology**”, with R. Lipshitz and P. Ozsváth,
In *Proceedings of the Gökova Geometry-Topology Conference 2008*,
arXiv:0810.0695.
A description of some of the algebra underlying the decomposition of
planar grid diagrams. This provides a useful toy model for bordered
Heegaard Floer homology. This paper is meant to serve as an
introduction to the subject, and does not itself have immediate
topological applications.

“**Heegaard Floer homology as morphism spaces**”, with R. Lipshitz and P. Ozsváth,
*Quanum Topology*,
**2**(4):381--449, 2011
arXiv:1005.1248.
Another version of the pairing theorem for bordered Floer
homology, using homomorphisms rather than tensor products. This
allows a more direct comparison with Fukaya-category
constructions, and leads to many new dualities.

“**A faithful linear-categorical action of the mapping class group of a surface with boundary**”, with R. Lipshitz and P. Ozsváth,
*Journal of the European Mathematical Society*,
**15**(4):1279--1307, 2013,
arXiv:1012.1032.
We show that the categorical action of the based mapping class
group given by bordered Floer homology is faithful. This paper
is partly intended as an introduction for geometric topologists
interested in the theory.

“**Computing HF^ by factoring mapping classes**”, with
R. Lipshitz and P. Ozsváth,

“**Bordered Floer homology and the spectral sequence of a branched double cover I**”, with R. Lipshitz and P. Ozsváth,
*Journal of Topology*,
**7** (2014), 1155–1199,
arXiv:1011.0499.
We explicitly calculate the spectral sequence starting at
Khovanov homology of a link and converging to the Heegaard Floer
homology of its branched double cover. In this paper we compute
an explicit combinatorial spectral sequence; in the sequel we
show that this spectral sequence agrees with the previously
known one.

“**Notes on bordered Floer homology**”, with R. Lipshitz and P. Ozsváth,
In *Contact and Symplectic Topology*, 275–355,
Bolyai Mathematical Studies 26, 2014,
arXiv:1211.6791.
A series of notes for lectures given by Robert Lipshitz at the
CaST conference in Budapest in the summer of 2012.

“**Relative ℚ-gradings from bordered Floer theory**”, with R. Lipshitz and P. Ozsváth.
*Turkish Journal of Mathematic*, accepted for publication,
arXiv:1211.6990.
We show how to reconstruct the relative ℚ-grading in
Heegaard Floer homology from the noncommutative grading on
bordered Floer homology.

“**Bordered Floer homology and the spectral sequence of a
branched double cover II: The spectral sequences agree**”, with
R. Lipshitz and P. Ozsváth,
*Journal of
Topology*,
**9** (2016), 607–686,
arXiv:1404.2894.
We show that the spectral sequence we constructed earlier, from
the Khovanov homology of a knot to its Heegaard Floer homology,
agrees with the previously-defined one using holomorphic
curves.

“**On combinatorial link Floer homology**”, with C. Manolescu,
P. Ozsváth, and Z. Szabó,
*Geometry and Topology*,
**11** (2007), 2339–2412,
arXiv:math.GT/0610559.
We give an
elementary explanation of the basic properties of the combinatorial Floer
homology introduced by Manolescu-Ozsváth-Sarkar, including a
self-contained proof of its invariance. We also extend the theory
to work with signs, over ℤ rather than ℤ/2ℤ.

“**Legendrian knots, transverse knots and combinatorial Floer homology**”,
with P. Ozsváth and Z. Szabó,
*Geometry and Topology*,
**12** (2008), 941–980,
arXiv:math.GT/0611841.
We
use the explicit chain maps used in the proof of invariance of
link Floer homology
to construct invariants of
Legendrian and transverse knots, with values in the combinatorial Floer
homology of the underlying topological knot.

“**Transverse knots distinguished by knot Floer homology**”, with L. Ng and P. Ozsváth,
*Journal of Symplectic Geometry*,
**6**(2008):461–490,
arXiv:math.GT/0703446.
We use the invariant of transverse knots above to find several
new examples of distinct transverse knots with the same
classical invariants.

“**Grid diagrams and Heegaard Floer invariants**”,
with C. Manolescu and P. Ozsváth,
arXiv:0910.0078.
We use a link surgery spectral sequence due to Manolescu and
Ozsváth to give a combinatorial description of Heegaard Floer
homology for arbitrary three-manifolds.

“**Naturality and mapping class groups in Heegaard Floer homology**”,
with A. Juhász,
arXiv:1210.4996.
We show that all flavors of Heegaard Floer homology are natural,
in the sense that they assign a group to a pointed 3-manifold,
not just an isomorphism class of groups.

“**From rubber bands to rational maps: Research report**”,
*Research in the
Mathematical Sciences*,
**3**:15, 2016,
doi:10.1186/s40687-015-0039-4,
arXiv:1502.02561.
A new positive characterization of which branched
self-covers of the sphere are equivalent to rational maps. See
also the program **Harmonic** below.

“**Conformal surface embeddings and extremal length**”,
with J. Kahn and K. Pilgrim.
Preprint, 2015,
arXiv:1507.05294.
A characterization of when one Riemann surface conformally
embeds inside another in a given homotopy class of maps, using
extremal length. This is used in the characterization of which
branched self-covers are equivalent to rational maps.

“**Elastic Graphs**”.
Preprint, 2016,
arXiv:1607.00340.
When is one elastic graph looser than another, regardless of
which target space you consider? This paper answers one version
of this question, and also introduces a family of energies for
maps between graphs. This is used in the characterization of
which branched self-covers are equivalent to rational maps.

“**A positive characterization of rational maps**”.
Preprint, 2016,
arXiv:1612.04424.
When is a topological branched self-cover of the sphere
equivalent to a rational map on **CP**^{1}? William
Thurston gave one answer in 1982, giving a negative criterion
(an obstruction to a map being rational). We give a
complementary, positive criterion: the branched self-cover is
equivalent to a rational map if and only if there is an elastic
spine that gets "looser" under backwards iteration.

“**The algebra of knotted trivalent graphs and Turaev's
shadow world**”,
in *Geometry and
Topology Monographs*,
Volume 4:
Invariants of knots and 3-manifolds (Kyoto 2001), T Ohtsuki, T
Kohno, T Le, J Murakami, J Roberts and V Turaev (editors),
arXiv:math.GT/0311458.
The algebra of knotted trivalent graphs may be thought of as a
generalisation of many different ways of representing knots. This
paper introduces the algebra of knotted trivalent graphs and
explains the connection to Turaev's shadow world diagrams.

With Ian Agol, Appendix to “**The volume of hyperbolic
alternating link complements**” by Marc Lackenby,
*Proc. London Math. Soc. (3)*,
**88** (2004), no. 1, 204–224,
arXiv:math.GT/0012185.
In his paper, Marc Lackenby proves
that the volume of the complement of a hyperbolic alternating link
is bounded above and below by linear functions of the twist
number, the number of non-parallel crossings. In the appendix,
Ian Agol and I improve the upper bound and show that it is
asymptotically sharp by constructing an explicit chain-link fence
link.

“**A random tunnel number one 3-manifold does not fiber over
the circle**”,
with N. Dunfield,
*Geometry and Topology*,
**10** (2006), 2431–2499,
arXiv:math.GT/0510129.
A proof that, in a measured lamination model, a random 3-manifold does not fiber over the circle. One motivation is to give insight into the Virtual Fibration Conjecture. The source code is also available.

“**3-manifolds efficiently bound 4-manifolds**”,
with F. Costantino,
*Journal of Topology*,
**1** (2008), 703–745,
arXiv:math.GT/0506577.
A proof that an oriented 3-manifold of complexity *d*
bounds a 4-manifold of complexity *O(d ^{2})*,
where complexity measured suitably. In particular, this
implies that surgery diagrams are not too inefficient as a way
of representing 3-manifolds. The proof uses the technology of
shadow surfaces.

“**Grid diagrams, braids, and contact geometry**”, with L. Ng,
in *Proceedings of the Gökova Geometry-Topology Conference 2008*,
arXiv:0812.3665.
We use grid diagrams to present a unified picture of braids,
Legendrian knots, and transverse knots.

“**Non-peripheral ideal decompositions of alternating knots**”,
with S. Garoufalidis and I. Moffatt,
arXiv:1610.09901.
We use the small cancellation property of the Dehn presentation
of alternating knot groups to show that several reasonable ideal
decompositions of alternating knot complements are not
peripheral.

“**A shadow
calculus for 3-manifolds**”
(draft), with F. Costantino.
Two versions of a finite set of moves that relate all shadow
surfaces representing the same 3-manifold (with slightly
different assumptions). The corresponding question for
4-manifolds is related to the Andrews-Curtis conjecture.

"**On geometric intersection of
curves in surfaces**" (draft).
An exploration of how curves on surfaces intersect. Among other
results, I derive simple formulas for the transformation of
Dehn-Thurston coordinates when you change the pair of pants
decomposition. (R. Penner had earlier found more complicated
formulas for the same problem.)

“**Integral expressions for the Vassiliev knot invariants**”,
senior thesis, Harvard University, May 1995, math.QA/9901110. Advisor:
R. Bott. Building on earlier work of Bott and Taubes, I construct here
a universal Vassiliev invariant using the topology of configuration
spaces. This is an alternative to the earlier Kontsevich integral,
which is less topological.

“**Wheels, wheeling, and the Kontsevich integral of the unknot**”,
with D. Bar-Natan, S. Garoufalidis, and L. Rozansky, *Israel J. Math*.
**119** (2000), 217–237, q-alg/9703025.
Here we make the “Wheels” and “Wheeling” conjectures for finite type
invariants. They are motivated
by some facts on the level of Lie algebras; we prove the conjectures on this
level. The conjectures involve the Bernoulli numbers in an intimate way.

“**The Aarhus invariant of rational homology 3-spheres I: A
highly non-trivial flat connection on S^{3}**”, with
D. Bar-Natan, S. Garoufalidis, and L. Rozansky,

“

“

These three papers introduce the “Aarhus integral”, a method for turning a universal finite-type invariant for knots into a universal finite-type invariant for 3-manifolds, using a diagrammatic version of integration. The first paper is introduction and intuition. The second we give precise definitions and prove that we construct a universal finite type invariant. In the third paper we show that this construction is equivalent to an earlier construction by Le, Murakami, and Ohtsuki; our contribution is motivation, and, in particular, a way of performing calculus on diagrams.

“**On the existence of finite type link homotopy invariants**”, with
Blake Mellor, *J. Knot Theory Ramifications* **10** (2001),
no. 7, 1025–1039, math.GT/0010206. We
show that there are finite type link homotopy invariants for links with 9
or more components, but none for links with 5 or fewer components.

“**Wheeling: A diagrammatic analogue of the Duflo isomorphism**”,
Ph.D. thesis, U.C. Berkeley, math.QA/0006083. Parts
are joint work with Dror Bar-Natan and Thang Le. Here we prove our earlier
“Wheels” and “Wheeling” conjectures. The “Wheels” conjecture computes the
Kontsevich integral of the unknot, the first knot for which this invariant
is known to all degrees, and the “Wheeling” conjecture is a stronger, diagrammatic,
analogue of the Duflo isomorphism. Our proof boils down to properly
interpreting some trivial topological facts, analogous to “1+1=2” and
“n*0=0”.

“**Two applications of elementary knot theory to Lie algebras and
Vassiliev invariants**”, with D. Bar-Natan and T. Le,
*Geometry and Topology*,
**7** (2003), no. 1, 1–31,
math.QA/0204311. This
is the published version of the parts of my Ph.D. thesis with the proof of the
Wheels and Wheeling conjectures.

“**Perturbative 3-manifold invariants by cut-and-paste topology**”,
with G. Kuperberg, UC Davis Math 1999-36, math.GT/9912167. In this
paper we construct a universal finite-type invariant for rational homology
spheres using configuration spaces. The construction is not particularly
new; the new part is the proof that the invariant is universal (finite type
with the correct weight system). In particular, this construction gives
a new, elementary definition of the Casson invariant.

“**The F_{4} and E_{6} families
have only a finite number of points**”
(draft).
Computations giving evidence against Deligne's conjecture on the
existence of an exceptional series of Lie algebras. (Version of 2004-11-13)

“**Cluster algebras and triangulated surfaces. Part I: Cluster complexes**”,
with S. Fomin and M. Shapiro,
*Acta Mathematica*,
**201** (2008), 83–146,
arXiv:math.RA/0608367.
We
study the cluster algebras arising from Teichmüller theory of
bordered surfaces, first introduced by Gekhtman-Shapiro-Vainshtein
and Fock-Goncharov. We describe these cluster algebras explicitly
in terms of
tagged triangulations and show that they give a large family of
cluster algebras which are “mutationally finite”:
although they are not finite type, there are only a finite number
of combinatorial types of coefficients. We furthermore determine
their homotopy type and growth rate.

“**Cluster algebras and triangulated surfaces. Part II: Lambda lengths**”,
with S. Fomin,
*Memoirs of the AMS*,
accepted for publication,
arXiv:1210.5569.
We explain the geometry behind the construction of cluster
algebras from tagged triangulations.

“**Positive basis for surface skein algebras**”,
*Proceedings of the National Academy of Sciences*,
**111** (2014), 9725–9732,
arXiv:1310.1959.
We give a natural basis for a certain skein algebra of
a surface, corresponding to the twisted *SL*_{2}
representations of the fundamental group. This basis is strongly
positive, in the sense that the structure constants for
multiplication are positive integers.

“**The complex volume of SL(n,ℂ) representations of 3-manifolds**”,
with S. Garoufaldis and C. Zickert,

“**From dominoes to hexagons**”, math.CO/0405482.
A generalisation of domino tilings to topological tilings by
hexagons, with connections to planar algebras, Legendrian knots,
and cluster algebras.

“**Discrete one-forms on meshes and applications to 3D mesh
parametrization**”, with S. Gortler and
C. Gotsman, *Computer Aided Geometric
Design*, **23** (2006),
83–112, Harvard Computer Science TR-12-04.
We use a discrete version of 1-forms on surfaces to give an easy
proof of Tutte's theorem and generalize it to other contexts,
including mesh decompositions of the torus.

“**Characterizing generic global rigidity**”,
with S. Gortler and A. Healy,
*American Journal of Mathematics*,
**132** (2010), no. 4, 897–939,
arXiv:0710.0926.
We prove of a
conjecture by R. Connelly on which graphs are generically globally
rigid. That is, for which graphs does a random rigid straight-line
embedding in *d* dimensions have an alternate embedding with
the same edge lengths? They can be characterized by a local and
efficiently checkable criterion.

“**Sensor network localization using sensor perturbation**”,
with Y. Zhu and S. Gortler,
*Transactions on Sensor Networks*,
**7** (2011), no. 4, Article 36.

“**On affine rigidity**”,
with S. Gortler, C. Gotsman, and L. Liu,
*Journal of Computational Geometry*,
**4** (2013), no. 1, 160–181,
arXiv:1011.5553.

“**Generic global rigidity in complex and pseudo-Euclidean spaces**”,
with S. Gortler,
in *Rigidity and Symetry*,
Fields Institute Comunications 70, 131–154, 2014,
arXiv:1212.6685.

“**Characterizing the universal rigidity of generic frameworks**”,
with S. Gortler,
*Discrete and Computational Geometry*,
**51** (2014), no. 4, 1017–1036,
arXiv:1001.0172.

“**Measurement isomorphism of graphs**”,
with S. Gortler,
arXiv:1212.6551.

“**A bulk inflaton from large volume extra dimensions**”,
with B. Greene, D. Kabat, and J. Levin,
*Physics Letters B*,
**694** (2011), no. 4, 485–490,
arXiv:1212.6551.

"**Markup
optimisation by probabilistic parsing**", with Chung-chieh Shan. First-place winner in
the ACM International
Conference on Functional Programming programming
contest. We solved the problem of markup optimisation in a
stripped-down version of HTML by reinterpreting it as a parsing
problem and applying standard techniques for optimal parsing.

**Harmonic**, a Haskell program
for computing harmonic maps of graphs and *stretch factors*.
This gives a criterion for when a given branched self-cover of the
sphere is equivalent to a rational map. This program is in a very
preliminary state, and currently has no user interface.

**Discrete quadratic
differentials**, a model for a discrete version of quadratic
differentials and measured foliations on a surface. Unlike other
representations (e.g., with train tracks), this gives a uniform
system of coordinates for all measured foliations on a closed
surface simultaneously. It is also useful for approximating actual
harmonic measured foliations.

**Extremal Length on Curves and
Graphs**, concrete estimates relating a notion of “extremal
length” on ribbon graphs to standard extremal length on a thickening
of the graph.

Dylan Thurston