@article{8539, abstract = {Cohomological and K-theoretic stable bases originated from the study of quantum cohomology and quantum K-theory. Restriction formula for cohomological stable bases played an important role in computing the quantum connection of cotangent bundle of partial flag varieties. In this paper we study the K-theoretic stable bases of cotangent bundles of flag varieties. We describe these bases in terms of the action of the affine Hecke algebra and the twisted group algebra of KostantKumar. Using this algebraic description and the method of root polynomials, we give a restriction formula of the stable bases. We apply it to obtain the restriction formula for partial flag varieties. We also build a relation between the stable basis and the Casselman basis in the principal series representations of the Langlands dual group. As an application, we give a closed formula for the transition matrix between Casselman basis and the characteristic functions.}, author = {Su, C. and Zhao, Gufang and Zhong, C.}, issn = {0012-9593}, journal = {Annales Scientifiques de l'Ecole Normale Superieure}, number = {3}, pages = {663--671}, publisher = {Société Mathématique de France}, title = {{On the K-theory stable bases of the springer resolution}}, doi = {10.24033/asens.2431}, volume = {53}, year = {2020}, } @article{7004, abstract = {We define an action of the (double of) Cohomological Hall algebra of Kontsevich and Soibelman on the cohomology of the moduli space of spiked instantons of Nekrasov. We identify this action with the one of the affine Yangian of gl(1). Based on that we derive the vertex algebra at the corner Wr1,r2,r3 of Gaiotto and Rapčák. We conjecture that our approach works for a big class of Calabi–Yau categories, including those associated with toric Calabi–Yau 3-folds.}, author = {Rapcak, Miroslav and Soibelman, Yan and Yang, Yaping and Zhao, Gufang}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, pages = {1803--1873}, publisher = {Springer Nature}, title = {{Cohomological Hall algebras, vertex algebras and instantons}}, doi = {10.1007/s00220-019-03575-5}, volume = {376}, year = {2020}, } @article{7683, abstract = {For any free oriented Borel–Moore homology theory A, we construct an associative product on the A-theory of the stack of Higgs torsion sheaves over a projective curve C. We show that the resulting algebra AHa0C admits a natural shuffle presentation, and prove it is faithful when A is replaced with usual Borel–Moore homology groups. We also introduce moduli spaces of stable triples, heavily inspired by Nakajima quiver varieties, whose A-theory admits an AHa0C-action. These triples can be interpreted as certain sheaves on PC(ωC⊕OC). In particular, we obtain an action of AHa0C on the cohomology of Hilbert schemes of points on T∗C.}, author = {Minets, Sasha}, issn = {14209020}, journal = {Selecta Mathematica, New Series}, number = {2}, publisher = {Springer Nature}, title = {{Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces}}, doi = {10.1007/s00029-020-00553-x}, volume = {26}, year = {2020}, } @article{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}, issn = {1091-6490}, journal = {Proceedings of the National Academy of Sciences}, number = {8}, pages = {2821--2830}, publisher = {National Academy of Sciences}, title = {{Harmonic dynamics of the Abelian sandpile}}, doi = {10.1073/pnas.1812015116}, volume = {116}, year = {2019}, } @article{6986, abstract = {Li-Nadler proposed a conjecture about traces of Hecke categories, which implies the semistable part of the Betti geometric Langlands conjecture of Ben-Zvi-Nadler in genus 1. We prove a Weyl group analogue of this conjecture. Our theorem holds in the natural generality of reflection groups in Euclidean or hyperbolic space. As a corollary, we give an expression of the centralizer of a finite order element in a reflection group using homotopy theory. }, author = {Li, Penghui}, issn = {1088-6826}, journal = {Proceedings of the American Mathematical Society}, number = {11}, pages = {4597--4604}, publisher = {AMS}, title = {{A colimit of traces of reflection groups}}, doi = {10.1090/proc/14586}, volume = {147}, year = {2019}, } @article{7436, abstract = {For an ordinary K3 surface over an algebraically closed field of positive characteristic we show that every automorphism lifts to characteristic zero. Moreover, we show that the Fourier-Mukai partners of an ordinary K3 surface are in one-to-one correspondence with the Fourier-Mukai partners of the geometric generic fiber of its canonical lift. We also prove that the explicit counting formula for Fourier-Mukai partners of the K3 surfaces with Picard rank two and with discriminant equal to minus of a prime number, in terms of the class number of the prime, holds over a field of positive characteristic as well. We show that the image of the derived autoequivalence group of a K3 surface of finite height in the group of isometries of its crystalline cohomology has index at least two. Moreover, we provide a conditional upper bound on the kernel of this natural cohomological descent map. Further, we give an extended remark in the appendix on the possibility of an F-crystal structure on the crystalline cohomology of a K3 surface over an algebraically closed field of positive characteristic and show that the naive F-crystal structure fails in being compatible with inner product. }, author = {Srivastava, Tanya K}, issn = {1431-0643}, journal = {Documenta Mathematica}, pages = {1135--1177}, publisher = {EMS Press}, title = {{On derived equivalences of k3 surfaces in positive characteristic}}, doi = {10.25537/dm.2019v24.1135-1177}, volume = {24}, year = {2019}, } @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}, number = {3}, pages = {778--806}, publisher = {Wiley}, title = {{The wonderful compactification for quantum groups}}, doi = {10.1112/jlms.12193}, volume = {99}, year = {2019}, } @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}, issn = {1435-9855}, journal = {Journal of the European Mathematical Society}, number = {10}, pages = {2995--3052}, publisher = {European Mathematical Society}, title = {{Arithmetic and representation theory of wild character varieties}}, doi = {10.4171/JEMS/896}, volume = {21}, year = {2019}, } @article{441, author = {Kalinin, Nikita and Shkolnikov, Mikhail}, issn = {2199-6768}, journal = {European Journal of Mathematics}, number = {3}, pages = {909–928}, publisher = {Springer Nature}, title = {{Tropical formulae for summation over a part of SL(2,Z)}}, doi = {10.1007/s40879-018-0218-0}, volume = {5}, year = {2019}, } @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{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}, title = {{The cohomological Hall algebra of a preprojective algebra}}, doi = {10.1112/plms.12111}, volume = {116}, year = {2018}, } @inbook{61, abstract = {We prove that there is no strongly regular graph (SRG) with parameters (460; 153; 32; 60). The proof is based on a recent lower bound on the number of 4-cliques in a SRG and some applications of Euclidean representation of SRGs. }, author = {Bondarenko, Andriy and Mellit, Anton and Prymak, Andriy and Radchenko, Danylo and Viazovska, Maryna}, booktitle = {Contemporary Computational Mathematics}, pages = {131 -- 134}, publisher = {Springer}, title = {{There is no strongly regular graph with parameters (460; 153; 32; 60)}}, doi = {10.1007/978-3-319-72456-0_7}, 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}, }