• A
• A
• A
• ABC
• ABC
• ABC
• А
• А
• А
• А
• А
Regular version of the site
Of all publications in the section: 9
Sort:
by name
by year
Article
Brav C. I., Ben-Bassat O., Bussi V. et al. Geometry and Topology. 2015. Vol. 19. No. 3. P. 1287-1359.
Article
Esterov A. I., Lang L. Geometry and Topology. 2021.

Let C_d be the space of non-singular, univariate polynomials of degree d. The Viète map V sends a polynomial to its unordered set of roots. It is a classical fact that the induced map V_∗ at the level of fundamental groups realises an isomorphism between π_1(C_d) and the Artin braid group B_d. For fewnomials, or equivalently for the intersection C of C_d with a collection of coordinate hyperplanes, the image of the map V_∗:π_1(C)→B_d is not known in general. In the present paper, we show that the map V_∗ is surjective provided that the support of the corresponding polynomials spans Z as an affine lattice. If the support spans a strict sublattice of index b, we show that the image of V_∗ is the expected wreath product of Z/bZ with B_d/b. From these results, we derive an application to the computation of the braid monodromy for collections of univariate polynomials depending on a common set of parameters.

Article
Gorsky E. Geometry and Topology. 2018. Vol. 22. P. 645-691.

We conjecture an expression for the dimensions of the Khovanov–Rozansky HOMFLY homology groups of the link of a plane curve singularity in terms of the weight polynomials of Hilbert schemes of points scheme-theoretically supported on the singularity. The conjecture specializes to our previous conjecture (2012) relating the HOMFLY polynomial to the Euler numbers of the same spaces upon setting t=−1. By generalizing results of Piontkowski on the structure of compactified Jacobians to the case of Hilbert schemes of points, we give an explicit prediction of the HOMFLY homology of a (k,n) torus knot as a certain sum over diagrams.

The Hilbert scheme series corresponding to the summand of the HOMFLY homology with minimal “a” grading can be recovered from the perverse filtration on the cohomology of the compactified Jacobian. In the case of (k,n) torus knots, this space furnishes the unique finite-dimensional simple representation of the rational spherical Cherednik algebra with central character k∕n. Up to a conjectural identification of the perverse filtration with a previously introduced filtration, the work of Haiman and Gordon and Stafford gives formulas for the Hilbert scheme series when k=mn+1.

Article
Guere J., Rossi P., Buryak A. Geometry and Topology. 2019. Vol. 23. No. 7. P. 3537-3600.

We present a family of conjectural relations in the tautological ring of the moduli spaces of stable curves which implies the strong double ramification/Dubrovin–Zhang equivalence conjecture introduced by the authors with Dubrovin. Our tautological relations have the form of an equality between two different families of tautological classes, only one of which involves the double ramification cycle. We prove that both families behave the same way upon pullback and pushforward with respect to forgetting a marked point. We also prove that our conjectural relations are true in genus $0$ and $1$ and also when first pushed forward from $\overline{\mathcal{M}}_{g,n+m}$ to $\overline{\mathcal{M}}_{g,n}$ and then restricted to $\mathcal{M}_{g,n}$ for any $g,n,m\ge 0$. Finally we show that, for semisimple CohFTs, the DR/DZ equivalence only depends on a subset of our relations, finite in each genus, which we prove for $g\le 2$. As an application we find a new formula for the class $\lambda_g$ as a linear combination of dual trees intersected with kappa- and psi-classes, and we check it for $g\le 3$.

Article
Verbitsky M. Geometry and Topology. 2011. Vol. 15. P. 2111-2133.
Article
Kondo S., Siegel C., Wolfson J. Geometry and Topology. 2017. No. 21. P. 903-922.

For each k≥5k≥5, we construct a modular operad ¯¯¯¯Ekℰ¯k of “kk–log-canonically embedded” curves. We also construct, for k≥2k≥2, a stable cyclic operad ¯¯¯¯Ekcℰ¯ck of such curves, and, for k≥1k≥1, a cyclic operad ¯¯¯¯Ek0,cℰ¯0,ck of “kk–log-canonically embedded” rational curves.

Article
Cheltsov I., Shramov K. Geometry and Topology. 2011. No. 15. P. 1843-1882.