@article{439,
abstract = {We count points over a finite field on wild character varieties,of Riemann surfaces for singularities with regular semisimple leading term. The new feature in our counting formulas is the appearance of characters of Yokonuma–Hecke algebras. Our result leads to the conjecture that the mixed Hodge polynomials of these character varieties agree with previously conjectured perverse Hodge polynomials of certain twisted parabolic Higgs moduli spaces, indicating the
possibility of a P = W conjecture for a suitable wild Hitchin system.},
author = {Hausel, Tamas and Mereb, Martin and Wong, Michael},
journal = {Journal of the European Mathematical Society},
publisher = {European Mathematical Society},
title = {{Arithmetic and representation theory of wild character varieties}},
year = {2019},
}
@article{441,
author = {Kalinin, Nikita and Shkolnikov, Mikhail},
journal = {European Journal of Mathematics},
publisher = {Springer Nature},
title = {{Tropical formulae for summation over a part of SL(2,Z)}},
doi = {10.1007/s40879-018-0218-0},
year = {2018},
}
@article{303,
abstract = {The theory of tropical series, that we develop here, firstly appeared in the study of the growth of pluriharmonic functions. Motivated by waves in sandpile models we introduce a dynamic on the set of tropical series, and it is experimentally observed that this dynamic obeys a power law. So, this paper serves as a compilation of results we need for other articles and also introduces several objects interesting by themselves.},
author = {Kalinin, Nikita and Shkolnikov, Mikhail},
journal = {Discrete and Continuous Dynamical Systems- Series A},
number = {6},
pages = {2827 -- 2849},
publisher = {AIMS},
title = {{Introduction to tropical series and wave dynamic on them}},
doi = {10.3934/dcds.2018120},
volume = {38},
year = {2018},
}
@article{322,
abstract = {We construct quantizations of multiplicative hypertoric varieties using an algebra of q-difference operators on affine space, where q is a root of unity in C. The quantization defines a matrix bundle (i.e. Azumaya algebra) over the multiplicative hypertoric variety and admits an explicit finite étale splitting. The global sections of this Azumaya algebra is a hypertoric quantum group, and we prove a localization theorem. We introduce a general framework of Frobenius quantum moment maps and their Hamiltonian reductions; our results shed light on an instance of this framework.},
author = {Ganev, Iordan V},
journal = {Journal of Algebra},
pages = {92 -- 128},
publisher = {World Scientific Publishing},
title = {{Quantizations of multiplicative hypertoric varieties at a root of unity}},
doi = {10.1016/j.jalgebra.2018.03.015},
volume = {506},
year = {2018},
}
@article{5,
abstract = {In this paper, we introduce a quantum version of the wonderful compactification of a group as a certain noncommutative projective scheme. Our approach stems from the fact that the wonderful compactification encodes the asymptotics of matrix coefficients, and from its realization as a GIT quotient of the Vinberg semigroup. In order to define the wonderful compactification for a quantum group, we adopt a generalized formalism of Proj categories in the spirit of Artin and Zhang. Key to our construction is a quantum version of the Vinberg semigroup, which we define as a q-deformation of a certain Rees algebra, compatible with a standard Poisson structure. Furthermore, we discuss quantum analogues of the stratification of the wonderful compactification by orbits for a certain group action, and provide explicit computations in the case of SL2.},
author = {Ganev, Iordan V},
journal = {Journal of the London Mathematical Society},
publisher = {Wiley},
title = {{The wonderful compactification for quantum groups}},
doi = {10.1112/jlms.12193},
year = {2018},
}
@article{5999,
abstract = {We introduce for each quiver Q and each algebraic oriented cohomology theory A, the cohomological Hall algebra (CoHA) of Q, as the A-homology of the moduli of representations of the preprojective algebra of Q. This generalizes the K-theoretic Hall algebra of commuting varieties defined by Schiffmann-Vasserot. When A is the Morava K-theory, we show evidence that this algebra is a candidate for Lusztig's reformulated conjecture on modular representations of algebraic groups.
We construct an action of the preprojective CoHA on the A-homology of Nakajima quiver varieties. We compare this with the action of the Borel subalgebra of Yangian when A is the intersection theory. We also give a shuffle algebra description of this CoHA in terms of the underlying formal group law of A. As applications, we obtain a shuffle description of the Yangian. },
author = {Yang, Yaping and Zhao, Gufang},
issn = {0024-6115},
journal = {Proceedings of the London Mathematical Society},
number = {5},
pages = {1029--1074},
publisher = {Oxford University Press (OUP)},
title = {{The cohomological Hall algebra of a preprojective algebra}},
doi = {10.1112/plms.12111},
volume = {116},
year = {2018},
}
@unpublished{196,
abstract = {The abelian sandpile serves as a model to study self-organized criticality, a phenomenon occurring in biological, physical and social processes. The identity of the abelian group is a fractal composed of self-similar patches, and its limit is subject of extensive collaborative research. Here, we analyze the evolution of the sandpile identity under harmonic fields of different orders. We show that this evolution corresponds to periodic cycles through the abelian group characterized by the smooth transformation and apparent conservation of the patches constituting the identity. The dynamics induced by second and third order harmonics resemble smooth stretchings, respectively translations, of the identity, while the ones induced by fourth order harmonics resemble magnifications and rotations. Starting with order three, the dynamics pass through extended regions of seemingly random configurations which spontaneously reassemble into accentuated patterns. We show that the space of harmonic functions projects to the extended analogue of the sandpile group, thus providing a set of universal coordinates identifying configurations between different domains. Since the original sandpile group is a subgroup of the extended one, this directly implies that it admits a natural renormalization. Furthermore, we show that the harmonic fields can be induced by simple Markov processes, and that the corresponding stochastic dynamics show remarkable robustness over hundreds of periods. Finally, we encode information into seemingly random configurations, and decode this information with an algorithm requiring minimal prior knowledge. Our results suggest that harmonic fields might split the sandpile group into sub-sets showing different critical coefficients, and that it might be possible to extend the fractal structure of the identity beyond the boundaries of its domain. },
author = {Lang, Moritz and Shkolnikov, Mikhail},
booktitle = {ArXiv},
pages = {25},
publisher = {ArXiv},
title = {{Harmonic dynamics of the Abelian sandpile}},
year = {2018},
}
@article{64,
abstract = {Tropical geometry, an established field in pure mathematics, is a place where string theory, mirror symmetry, computational algebra, auction theory, and so forth meet and influence one another. In this paper, we report on our discovery of a tropical model with self-organized criticality (SOC) behavior. Our model is continuous, in contrast to all known models of SOC, and is a certain scaling limit of the sandpile model, the first and archetypical model of SOC. We describe how our model is related to pattern formation and proportional growth phenomena and discuss the dichotomy between continuous and discrete models in several contexts. Our aim in this context is to present an idealized tropical toy model (cf. Turing reaction-diffusion model), requiring further investigation.},
author = {Kalinin, Nikita and Guzmán Sáenz, Aldo and Prieto, Y and Shkolnikov, Mikhail and Kalinina, V and Lupercio, Ernesto},
issn = {00278424},
journal = {PNAS: Proceedings of the National Academy of Sciences of the United States of America},
number = {35},
pages = {E8135 -- E8142},
publisher = {National Academy of Sciences},
title = {{Self-organized criticality and pattern emergence through the lens of tropical geometry}},
doi = {10.1073/pnas.1805847115},
volume = {115},
year = {2018},
}
@inbook{6525,
abstract = {This chapter finds an agreement of equivariant indices of semi-classical homomorphisms between pairwise mirror branes in the GL2 Higgs moduli space on a Riemann surface. On one side of the agreement, components of the Lagrangian brane of U(1,1) Higgs bundles, whose mirror was proposed by Hitchin to be certain even exterior powers of the hyperholomorphic Dirac bundle on the SL2 Higgs moduli space, are present. The agreement arises from a mysterious functional equation. This gives strong computational evidence for Hitchin’s proposal.},
author = {Hausel, Tamás and Mellit, Anton and Pei, Du},
booktitle = {Geometry and Physics: Volume I},
isbn = {9780198802013},
pages = {189--218},
publisher = {Oxford University Press},
title = {{Mirror symmetry with branes by equivariant verlinde formulas}},
doi = {10.1093/oso/9780198802013.003.0009},
year = {2018},
}
@article{687,
abstract = {Pursuing the similarity between the Kontsevich-Soibelman construction of the cohomological Hall algebra (CoHA) of BPS states and Lusztig's construction of canonical bases for quantum enveloping algebras, and the similarity between the integrality conjecture for motivic Donaldson-Thomas invariants and the PBW theorem for quantum enveloping algebras, we build a coproduct on the CoHA associated to a quiver with potential. We also prove a cohomological dimensional reduction theorem, further linking a special class of CoHAs with Yangians, and explaining how to connect the study of character varieties with the study of CoHAs.},
author = {Davison, Ben},
issn = {00335606},
journal = {Quarterly Journal of Mathematics},
number = {2},
pages = {635 -- 703},
publisher = {Oxford University Press},
title = {{The critical CoHA of a quiver with potential}},
doi = {10.1093/qmath/haw053},
volume = {68},
year = {2017},
}