## Publications

6 results found

Arbesfeld N, Enriquez B, 2015, ON A LOWER CENTRAL SERIES FILTRATION OF THE GROTHENDIECK-TEICHMULLER LIE ALGEBRA grt(1), *MOSCOW MATHEMATICAL JOURNAL*, Vol: 15, Pages: 205-256, ISSN: 1609-3321

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- Citations: 1

Arbesfeld N, 2013, Partial permutations avoiding pairs of patterns, *DISCRETE MATHEMATICS*, Vol: 313, Pages: 2614-2625, ISSN: 0012-365X

Arbesfeld N, Schiffmann O, A presentation of the deformed W_{1+\infty} algebra, *Proceedings of the conference Infinite Analysis 11—Frontier of Integrability held at the University of Tokyo, Tokyo, July 25–29, 2011, and the conference "Symmetries, Integrable Systems and Representations'' held at the Université Claude Bernard Lyon 1, Lyon, December 13–16, 2011.*

We provide a generators and relation description of the deformedW_{1+\infty}-algebra introduced in previous joint work of E. Vasserot and thesecond author. This gives a presentation of the (spherical) cohomological Hallalgebra of the one-loop quiver, or alternatively of the spherical degeneratedouble affine Hecke algebra of GL(\infty).

Arbesfeld N, Jordan D, 2010, New results on the lower central series quotients of a free associative algebra, *JOURNAL OF ALGEBRA*, Vol: 323, Pages: 1813-1825, ISSN: 0021-8693

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- Citations: 8

Arbesfeld N, Johnson D, Lim W, et al., The virtual K-theory of Quot schemes of surfaces

We study virtual invariants of Quot schemes parametrizing quotients ofdimension at most 1 of the trivial sheaf of rank N on nonsingular projectivesurfaces. We conjecture that the generating series of virtual K-theoreticinvariants are given by rational functions. We prove rationality for severalgeometries including punctual quotients for all smooth projective surfaces anddimension 1 quotients for surfaces X with p_g>0. We also show that thegenerating series of virtual cobordism classes can be irrational. Given a K-theory class on X of rank r, we associate natural series of virtualSegre and Verlinde numbers. We show that the Segre and Verlinde series match inthe following three cases: Quot schemes of dimension 0 quotients, Hilbertschemes of points and curves over surfaces with p_g>0, Quot schemes of minimalelliptic surfaces for quotients supported on fiber classes. Moreover, forpunctual quotients of the trivial sheaf of rank N, we prove a new symmetry ofthe Segre/Verlinde series exchanging r and N. The Segre/Verlinde statementshave analogues for punctual Quot schemes over curves.

Arbesfeld N, K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface

Integrals of characteristic classes of tautological sheaves on the Hilbertscheme of points on a surface frequently arise in enumerative problems. We usethe K-theoretic Donaldson-Thomas theory of certain toric Calabi-Yau threefoldsto study K-theoretic variants of such expressions. We study limits of theK-theoretic Donaldson-Thomas partition function of a toric Calabi-Yau threefoldunder certain one-parameter subgroups called slopes, and formulate a conditionunder which two such limits coincide. We then explicitly compute the limits ofcomponents of the partition function under so-called preferred slopes,obtaining explicit combinatorial expressions related to the refined topologicalvertex of Iqbal, Kos\c{c}az and Vafa. Applying these results to specificCalabi-Yau threefolds, we deduce dualities satisfied by a generating functionbuilt from tautological bundles on the Hilbert scheme of points on$\mathbb{C}^2$. We then use this duality to study holomorphic Eulercharacteristics of exterior and symmetric powers of tautological bundles on theHilbert scheme of points on a general surface.

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