A research blog about microlocal sheaves.

__New__

Microlocal category for Weinstein manifolds via h-principle

Mirror symmetry for very affine hypersurfaces

+ Benjamin Gammage

__Covariantly functorial Floer theory on Liouville sectors__

+ Sheel Ganatra, John Pardon

__Articles in preparation:__

P=W for graphs

*.*

+ Zsuszsana Dancso, Michael McBreen

Localizing the Fukaya category of a Weinstein manifold

+ Sheel Ganatra, John Pardon

Contact Fukaya Categories

*+ Tobias Ekholm, Lenhard Ng*

The wild character variety of an irregular singularity

and the HOMFLY homology of its asymptotic link

__Articles in repair:__

Generating families and constructible sheaves

Note: This paper has some serious mistakes, which are currently under repair in work with Sylvain Courte.

Symplectic structures from topological Fukaya categories

+ Alex Takeda

Note: In this paper we showed that the proper topological Fukaya category was right Calabi-Yau; it was pointed out to us by Chris Brav and Tobias Dyckerhoff that, in order to get symplectic structures, we actually needed to show that the smooth topological Fukaya category was left Calabi-Yau. Our methods extend to prove this statement as well; a rewrite should appear soon.

__Articles:__

__Sheafy symplectic geometry:__

Legendrian knots

Legendrian knots

Legendrian knots and constructible sheaves

+ David Treumann, Eric Zaslow

Inventiones Mathematicae (DOI: 10.1007/s00222-016-0681-5).

Augmentations are sheaves

+ Lenhard Ng, Daniel Rutherford, Steven Sivek, Eric Zaslow.

The cardinality of the augmentation category

+ Lenhard Ng, Daniel Rutherford, Steven Sivek,

to appear in Mathematics Research Letters.

Conormal torus and usual knots

Conormal torus and usual knots

The conormal torus is a complete knot invariant

A complete knot invariant from contact homology

+ Lenhard Ng, Tobias Ekholm,

Inventiones Mathematicae (DOI: 10.1007/s00222-017-0761-1).

*Cluster varieties*

Cluster varieties and Legendrian knots

+ David Treumann, Harold Williams, Eric Zaslow.

On the combinatorics of exact Lagrangian surfaces

+ David Treumann, Harold Williams.

__Algebraic geometry (I haven't done this in awhile):__*Character varieties.*

The weights of the tautological classes of the character varieties

IMRN (2016). (DOI: 10.1093/imrn/rnv363)

*Higher discriminants and applications.*

Higher discriminants and the topology of algebraic maps

+ Luca Migliorini,

to appear in Algebraic Geometry.

A support theorem for Hilbert schemes of planar curves

+ Luca Migliorini,

Journal of the European Mathematical Society 15.6 (2013), 2353--2367.

A support theorem for Hilbert schemes of planar curves, II

+ Luca Migliorini, Filippo Viviani

Equidistribution on the space of rank two vector bundles over the projective line

+ Jacob Tsimerman

to appear in Duke Mathematical Journal.

*Hilbert schemes of points on singular plane curves.*

*... and curve counting*

Hilbert schemes of points on a locally planar curve

and the Severi strata of its versal deformation

Compositio Mathematica 148.2 (2012), 531--547.

A short proof of the Goettsche conjecture

+ Martijn Kool and Richard Thomas

Geometry and Topology 15.1 (2011), 397--406.

On the Goettsche threshold

+ Steven Kleiman

“A celebration of Algebraic Geometry -- in honor of Joe Harris’s 60th birthday” (AMS 2013), 429--449.

Refined curve counting on complex surfaces

+ Lothar Goettsche

Geometry and Topology 18.4 (2014) 2245--2307.

The chi-y genera for relative Hilbert schemes

for linear systems on K3 and Abelian surfaces

+ Lothar Goettsche

Algebraic Geometry 4.2 (2015).

*... and knot invariants*

The Hilbert scheme of a plane curve singularity

and the HOMFLY polynomial of its link

+ Alexei Oblomkov

Duke Mathematical Journal, 161.7 (2012), 1277-1303.

The Hilbert scheme of a plane curve singularity

and the HOMFLY homology of its link

+ Alexei Oblomkov, Jacob Rasmussen, (with an appendix by Eugene Gorsky)

to appear in Geometry and Topology.

Large N duality, Lagrangian cycles, and algebraic knots

+ Duiliu-Emanuel Diaconescu, Cumrun Vafa

Communications in Mathematical Physics 319.3 (2013), 813--863.

Torus knots and the Rational DAHA

+ Eugene Gorsky, Alexei Oblomkov, Jacob Rasmussen

Duke Mathematical Journal 163.14 (2014), 2709-2794.

In a different lifetime, I worked on quantum circuits. I don't have an undergraduate or master's degree, but I think of the first five of these papers as my undergraduate thesis, and the sixth and seventh as my master's thesis.

*Quantum circuits.*

Synthesis of reversible logic circuits

+ Aditya Prasad, Igor Markov, John Hayes.

IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 22.6 (2003), 710-722.

Data structures and algorithms for simplifying reversible circuits

+ Aditya Prasad, Igor Markov, John Hayes, and Ketan Patel,

ACM Journal on Emerging Technologies in Computing Systems (JETC) 2.4, 277-293

Minimal universal two-qubit controlled-not based circuits

+ Igor Markov and Steven Bullock

Physical Review A 69.6 (2004), 062321.

Recognizing small-circuit structure in two-qubit operators

+ Igor Markov and Steven Bullock

Physical Review A 70.1 (2004), 012310.

Quantum circuits for incompletely specified two-qubit operators

+ Igor Markov

Quantum Information and Computation 5.1 (2004), 48-56.

Synthesis of quantum logic circuits

+ Igor Markov and Steven Bullock

IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25.6 (2006), 1000-1010.

On the CNOT-cost of Toffoli gates

+ Igor Markov

Quantum Information and Computation 9.5-6 (2009), 461--486.