Vivek Shende

I'm an algebraic geometer, but much of my work involves bridges to other subjects: from the enumerative theory of curve singularities to quantum and categorified knot invariants to the representation theory of Cherednik algebras; from the geometry of special divisors in hyperelliptic curves through general results on how the cohomology of algebraic varieties can vary in families to questions of equidistribution. I also think about counting nodal curves in surfaces, and about higher discriminants of maps between algebraic varieties. Most recently, also about sheaftheoretic categorical invariants of Legendrian knots, and how these are related to the items in the first sentence.
Here's the technical part of my NSF proposal "Curves, Knots, Hilbert schemes, and the Hitchin system." It contains a summary of some of my current research and a programme which proposes to use the relations between singular curves, legendrian knot invariants, and wild character varieties to approach the conjecture (of Oblomkov, Rasmussen, and myself) relating the cohomology of Hilbert schemes of points on a singular plane curve to the KhovanovRozansky homology of its link. I've posted this largely because this programme does not currently appear elsewhere in print.
I am now (spring 2014) teaching a section of math 113, intro to algebra. Here's the syllabus. Soon there will be a bcourses website.
In fall 2013 I taught representation theory, here are some of the notes from the lectures.
Legendrian knots and constructible sheaves,
with David Treumann and Eric Zaslow. 

Equidistribution on the space of rank two vector bundles over the projective line,
with Jacob Tsimerman. 

The chiy genera of relative Hilbert schemes for linear systems on Abelian and K3 surfaces,
with Lothar Göttsche. 

Higher discriminants and the topology of algebraic maps,
with Luca Migliorini. 

Refined curve counting on complex surfaces,
with Lothar Göttsche. to appear in Geometry and Topology. 

Torus knots and the rational DAHA,
with Alexei Oblomkov, Jacob Rasmussen, and Eugene Gorsky. to appear in Duke Mathematical Journal.  
On the Göttsche Threshold,
with Steven Kleiman. in A celebration of algebraic geometry  in honor of Joe Harris's 60th birthday (AMS 2013), 429449. 

The Hilbert scheme of a plane curve singularity and the
HOMFLY homology of its link,
with Alexei Oblomkov and Jacob Rasmussen; and with an appendix by Eugene Gorsky. 

Large N duality, lagrangian cycles, and algebraic knots,
with Duiliu Emanuel Diaconescu and Cumrun Vafa. Communications in Mathematical Physics 319.3 (2013), 813863. 

A support theorem for Hilbert schemes of planar curves,
with Luca Migliorini. Journal of the European Mathematical Society 15.6 (2013), 23532367. 

A short proof of the Göttsche conjecture,
with Martijn Kool and Richard Thomas. Geometry and Topology 15 (2011), 397406. 

Hilbert schemes of points on a locally planar curve and the
Severi strata of its versal deformation. Compositio Mathematica 148.2 (2012), 531547. 

The Hilbert scheme of a plane curve singularity and the HOMFLY
polynomial of its link, with Alexei Oblomkov. Duke Mathematical Journal 161.7 (2012), 12771303. 