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8/13/2019 Research Stat1

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Research Statement

Anton Khoroshkin

1 Summary and Goals

In my research I develop and apply new homological methods and theories to various questionsof mathematical physics, representation theory, and algebraic topology.

Section 2 below contains the description of my (joint) results that have already been pub-lished. Section 3 is devoted to the future directions and projects. Among the most importantfuture projects/problems I should mention the following:

Describe a natural homological theory of BV-algebras with coefficients in a right moduleapplicable to the perturbative Chern-Simons theory, topological quantum field theory and

realization of nonbased mapping spaces; Implement homological computations for operads over finite fields using the theory of

Grobner bases;

Develop combinatorial and asymptotical methods in the theory of operads;

Develop the theory of Grobner bases for various operad-like algebraic structures such asdioperads, coloured operads, algebras over PBW operads, n-operads;

2 Research overview, important results

My research is primarily concentrated on homological computations and methods that mightbe applied in different areas of mathematics such as representation theory, algebraic topologyand mathematical physics.

2.1 Lie algebra cohomology and Koszul property for algebras

Homological computations for infinite dimensional Lie algebras was initiated in the 70s bymeans of what is nowadays known as the formal geometry [23]. Similar cohomology theorywhich turns out to be extremely useful in mathematical physics is known as BRST-cohomology.The Koszul property is an important homological tool for quadratic algebras discovered byPriddy [39] and was extensively used in different areas of algebra and geometry [38].

Using these two general methods mentioned in the title, I was able to make two particularcomputations in my PhD thesis [7]. Firstly, I was able to carry out a new computation ofthe syzygies of the Plucker embedding of the Grassmanian of two-dimensional planes whichis much simpler than the one previously known. Secondly, I was able to compute the Liealgebra cohomology of the Lie algebra of formal vector fields on the plane with coefficients inthe symmetric powers of its coadjoint representation. The conjectural answer for the secondquestion was well-known as one of the main conjectures of formal geometry due to B. L. Feigin,D. B. Fuchs and I. M. Gelfand ([21, 22, 13]).

In order to do these computations, I developed several new general methods that mightbe applicable in many other situations. For example, in a joint work with A. Gorodentsev,A. Rudakov [6] which was motivated by our attempts to understand the crucial result of N. Berkovits

on superstring theory [11, 12] we present a very general unified method of syzygy computations.

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2.2 Operads and Koszul property

Later on I concentrated on the study of the notion of Koszul duality for various algebraicstructures since it was known as a helpful method in different mathematical contexts, see [28].The notion of an operadwas first invented by topologists. (Operads were known in pure algebraunder the name of a variety of algebras and was introduced for the purposes of describingidentities for various types of algebras, see [18].) Namely, if one takes an algebra of some type, i.e.a set with some operations on it, and removes the underlying set, what remains is the collection

of all operations one can define. This collection with the prescribed rules of compositions is whatone calls anoperad. The machinery provided by operads while studying various types of algebrasbecame extremely useful in different areas of mathematics. For example, the rational homotopytheory due to Quillen and Sullivan [40] as well as Tamarkins approach to the deformationquantization [45, 33] were based on this machinery.

My first application of the operad theory was the proof of several conjectures by B. Feiginand An. Kirillov about the dimensions of the double Orlik-Solomon algebras (certain quadraticalgebras which generalize the cohomology ring of a hyperplane arrangement of type An, see[31]). The final results are published in joint with M. Berstein and V. Dotsenko articles [3, 1].Namely, we found an operadic structure on the union of the double Orlik-Solomon algebrasand shown the isomorphism of this operad with the operad of Bi-Hamiltonian algebras. While

making manipulations with the Hilbert series ofSn-characters we found an interesting nontrivialexample of an application of the Koszul duality for operads in the description of the symmetricgroup action on the spaces ofn-ary operations. The method we use in [3] is quite general andmay be applied in other similar situations.

2.3 Grobner bases for operads

It is now well-known that applications of the operad theory in general (and, in particular, toverifications of the Koszul property) are really difficult in particular computations. There wasno known arithmetic of operations similar to the arithmetic of integers or polynomials (by anarithmetic we mean the usual notion of divisibility). The good analogue of multiplication forthe operadic data is the composition of operations. But the action of the symmetric groups onthe entries of operations do not allow to give a natural definition of divisibility. I found a solutionto this problem using the notion ofShuffle operadsintroduced in a joint with V. Dotsenko paper[2]. The key idea was to forget about a certain part of the action of the symmetric group. Inspite of being a very simple idea, it allowed us to introduce a theory of monomials, theirdivisibility and compatible orderings of monomials for operads. Summarizing these notions, wecame up with the notion of Grobner bases for operads (also presented in [2]). Grobner bases is aremarkable technical tool initiated in the commutative algebra setting by Buchberger [14] whichallows one to solve systems of equations with many unknowns (see [44] for a brief introduction).The theory of Grobner bases for operads made it possible to provide a unified proof of theexisting computational results in the field as well as to prove some new results. It is clear thatthere are many topics that can be successfully approached by these new methods.

2.4 Resolutions and homology computations via Grobner bases.

In particular, I want to mention some homological applications of the theory of Gr obner bases.With each given Grobner basis of an operad we associate a free resolution of an operad whichis much smaller than the standard one (joint work with V. Dotsenko [4]). This presents aneffective method of homology computation of an operad. A general construction we gave isbased on a analog of the inclusion-exclusion principle. Among other examples we found adirect homology computation for some non-quadratic operads. We were able to compute thehomology of the so-called BV-operad [25] (also known as the homology of the framed little discsoperad). The answer we obtained explains the mysterious connection between the BV operad

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and the operad of compactified moduli spaces of stable curves of genus zero [26] (noticed byBarranikovKontsevich [9]).

Though a very recent development, shuffle operads have been covered in a forthcomingtextbook on algebraic operads by J.-L. Loday and B. Vallette [34]. In joint with V. Dotsenko[5] and B. Shapiro [8] papers we apply shuffle operads to the so-called pattern avoidance inpermutations which is a very active area of algebraic combinatorics. (As we were informed byS. Kitaev, the material of [5, 8] will be included in his forthcoming textbook on this subject.)

3 Ongoing projects

3.1 BV algebras and mapping spaces

A structure of Batalin-Vilkovisky algebra [25] naturally appears in different subjects of algebraictopology and mathematical physics. For example, the homology of the space of nonbased loopsMaps(S1, X) admits a BV structure ifX is a compact manifold (due to Chas-Sullivan stringtopology [16]). Another construction naturally appears in the perturbative Chern-Simons theory[15] and in BF theories [37]. BV-algebras are extensively used in TCFT [17], TQFT [42] andin Calabi-Yau categories [20].

We want to develop a natural homology theory of BV algebras such that the resultinginvariants of the classes of BV-algebras will give new approaches to simplicial perturbativeChern-Simons theory, quantum cohomology and many other theories. Namely, since the operadfE2of framed little discs is formal in dimension 2 ([43]) we may identify homotopy BV-algebrasand fE2-algebras. The union of configuration spaces fConf(M) of points (with frames) on agiven surfaceMadmits a natural structure of a topological right module over the framed littlediscs operad. Therefore one may consider the derived tensor product over the operad fE2 ofthe right module of chains on fConf(M) and any BV-algebra considered as a left fE2-module.Let us mention two possible application of this definition.

3.1.1 Application to mapping spaces. The mentioned below application should beimportant for the purposes of algebraic topology. Mays recognition principle explains that any

topological algebra over the operad of little discs Ed is homotopy equivalent to the space ofbased loops of dimension d. I.e. there is a natural action ofEd on Maps((S

d, pt), (X,pt)). Theorthogonal groupSO(d 1) acts on (Sd, pt) by rotations around the earth axis and thereforethere is a natural structure of topologicalfEd-algebra on the space Maps((S

d, pt), (X,pt)). Thedescribed above tensor product of modules over an operad is well defined for any d-dimensionalmanifold Md and we come up with a nice approximation of the space of nonbased maps:

Maps(Md, X) is homotopy equivalent to fConf(Md) fEdMaps((Sd, pt), (X,pt)).

3.1.2 Application to the TQFT. fE1-algebras coincide with E1-algerbas that are justhomotopy associative algebras. The derived tensor product of configurations on a circle andassociative algebra A coincides with its Hochshcild homology HH(A) = TorAAop(A,A).

Therefore our homology theory (fibered product over operad fEd) should be also consideredas a natural generalization to higher dimension of the Hochschild homology: LetA be a fEd-algebra. The natural embedding of a (d 1)-dimensional disc into a d-dimensional disc viaequator defines a morphism of operads: fEd1 fEd and therefore one has a fEd1-algebrastructure onA. LetHSd1(A) :=fConf(Sd1)fEd1Abe the corresponding fibered product.The gluing of cylinders (Sd1 [0, 1/2]) (Sd1 [1/2, 1]) (Sd1 [0, 1]) defines an algebrastructure on HSd1(A). Moreover there are left and right action of the algebraHSd1(A) onAitself. The following isomorphism exists for every f Ed-algebra A:

HSd(A) :=fConf(Sd)fEdA=T orHS

d1(A)

(A,A)

If, in addition, the algebra A has an invariant scalar product one may identify T or and Extin the above relation. Therefore the inner product on Ext-groups defines an algebra structure

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on HSd(A). The action of the orthogonal group SO(d+ 1) (via rotations of a sphere Sd)and multiplication (from Ext) generate a fEd+1-algebra structure on HS

d(A) that may begeneralized even to a (d+ 1)-dimensional TQFT by the similar construction with configurationspaces. We are going to clarify this rough definition of a functor from fEd-algebras withinvariant pairing to (d + 1)-TQFTs. Ford = 1 this coincides with a well known equivalence ofcategories of Frobenius algebras and (1 + 1)-TQFTs ([32]).

Our computation of the homology of BV-operad made in [4] and methods discovered bythe theory of Grobner bases should provide us with direct definitions and computations of the

prescribed homology. (This is a joint project with N. Markarian.)

3.2 Operads in positive characteristic and over a ring

The importance to understand the operad theory while working over a field of positive character-istic can be illustrated by a well-known topological result formulated, for example, in Mandellspaper [35]. The statement is (roughly) as follows. There is a unique (up to homotopy) actionof the cofibrant resolution (over Z) of an operad of commutative algebras on the space of sin-gular cochains of a simply-connected topological space. Moreover, two (nilpotent, finite type)spaces are weakly homotopically equivalent if and only if their singular cochain complexes arequasi-isomorphic as modules over the corresponding operad (the so-called E-operad).

Very little is known about operads over Z except for their definitions and general state-ments that can hardly be applied in concrete computations. For example, any realization ofthe E-operad is too big for any direct computations. The main problem is the modularityof representations of the symmetric groups over finite fields when the number of permutingelements is large compared to the characteristic of the base field. On the other hand, the notionof shuffle operads and the corresponding theory of Grobner bases are well defined even in thiscase. Therefore, if the coefficients of the leading monomials of the corresponding Grobner basisare invertible elements of the base ring then one can obtain the free resolutions as it was donein [4]. These resolutions will be free operads but will not be cofibrant objects in the corre-sponding category. In spite of non-cofibrance, the obtained free resolutions still contain a lot ofinformation about the homotopy theory of algebras over the initial operad. For example, the

natural homology theories and deformation complexes known for commutative algebras withdivided powers and for restricted Lie algebras [29] may be covered by this approach. The paperof E.Hoffbeck [30] should be considered as a possible attempt in this direction.

3.3 Hilbert series of operads with finite Grobner bases

The Hilbert series of an associative algebra with a finite Gr obner basis is a rational function[47]. This statement is extremely useful in the asymptotical and combinatorial approach to thetheory of associative algebras [48].

A natural question about appropriate class of generating functions can be posed within theoperad theory. In a joint project with D. Piontkovski, we plan to show that the Hilbert series

of a non-symmetric operad with a finite Grobner basis should be an algebraic function. On theother hand, exponential Hilbert series of symmetric operad with a finite shuffle Grobner basis(under some mild assumptions) should be a solution of a differential equation with algebraiccoefficients. Moreover we are going to present an effective algorithm of computing the Hilbertseries for an operad with a given Grobner basis as well as to give some bounds on the order ofoccurring differential equation. As a corollary, this should allow us to present a list of possibleHilbert series of PBW operads with a small number of generators.

The special case of shuffle operads with finite Grobner (even monomial) bases is directly re-lated to the theory of (consecutive) pattern avoidance. Homological theory for operads adoptedto this special case was applied by the applicant jointly with B. Shapiro. For a given collection ofpatterns (with mild assumptions satisfied in a huge amount of cases) we find a linear differential

equation with polynomial coefficients such that the inverse function to the exponential gener-

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ating function of the number of permutations avoiding all patterns from the given collection isa solution of this equation, see [8]. In a current joint with B. Shapiro project we plan to studyseveral asymptotical problems in the theory of pattern avoidance where one might be able toachieve a substantial progress using our homological approach.

3.4 Grobner bases for various algebraic structures.

Another area to be approached in this project is to develop a constructive arithmetic theory

of operad-like structures. In particular, the adopted theory of Grobner bases will be extremelyuseful for various associative structures built from labeled graphs. For example, the case ofdioperads [24], which describes tree-type operations with several inputs and several outputs willbe beneficial for differential geometry. The case ofcoloured operads [49], where the inputs andthe outputs could be of several different types (colours), has applications to the operad theoryitself and will be useful while obtaining formulas connecting the simplicial and differential worlds(at present only the case of simplexes of dimension one is known to the specialists [36]). Highercategory theory and the case ofn-operads[10] should also be attacked by similar methods.

Besides developing the general methods we are going to consider particular examples wherethe method of Grobner bases should be useful. Several non-quadratic operads naturally ap-pear in projective geometry and in quantum mechanics. Namely, Alternative algebras, Jordan

algebras and special Jordan algebras [27, 46]. (This is a joint project with V.Dotsenko.)

References

Cited publications and preprints by A. Khoroshkin

[1] M. Bershtein, V. Dotsenko, A. Khoroshkin. Quadratic algebras related to the bihamiltonianoperad.IMRN 2007, no. 24, Art. ID rnm122, 30 pp.

[2] V. Dotsenko, A. Khoroshkin. Grobner bases for operads Duke Math. J. 153 (2010), no. 2,363396.

[3] V. Dotsenko, A. Khoroshkin. Character formulas for the operad of two compatible bracketsand for the bi-Hamiltonian operad Funct. Anal. Appl. 41 (2007), no. 1, 117

[4] V. Dotsenko, A. Khoroshkin. Free resolutions via Grobner bases Preprintmath.arxiv:0912.4895, 24 pp.

[5] V. Dotsenko, A. Khoroshkin. Anick-type resolutions and consecutive pattern avoidancePreprint math.arXiv:1002.2761, 16 pp.

[6] A. Gorodentsev, A. Khoroshkin, A. Rudakov. On syzygies of highest weight orbits. MoscowSeminar on Mathematical Physics. II, 79120, Amer. Math. Soc. Transl. Ser. 2, 221, Amer.

Math. Soc., Providence, RI, 2007

[7] A. Khoroshkin. Formal geometry and algebraic invariants of geometric structures. PhDthesis Moscow State University, (russian), 2007.

[8] A. Khoroshkin; B. Shapiro. Using homological duality in consecutive pattern avoidance.Preprint math.arXiv:1009.5308, 12 pp.

Cited publications by other authors

[9] S. Barannikov, M. Kontsevich. Frobenius manifolds and formality of Lie algebras of polyvec-

tor fields.Internat. Math. Res. Notices 1998, no. 4, 201215.

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[10] M. Batanin. The Eckmann-Hilton argument and higher operads. Adv. Math. 217 (2008),no. 1, 334-385.

[11] N. Berkovits.Super-Poincare Covariant Quantization of the Superstring.JHEP 0004 (2000)018, see also arXiv:hep-th/0001035.

[12] N. Berkovits. ICTP Lectures on Covariant Quantization of the Superstring. Preprint

arXiv:hep-th/0209059 .

[13] J. Bernstein, B. Rosenfeld. Homogeneous spaces of infinite-dimensional Lie algebras andthe characteristic classes of foliations. Russ. Math.Surv. 28 (1973), no. 4(172), 103138.

[14] B. Buchberger.An Algorithm for Finding the Basis Elements of the Residue Class Ring ofa Zero Dimensional Polynomial Ideal. Ph.D. dissertation, University of Innsbruck. 1965.English translation by Michael Abramson in Journal of Symbolic Computation 41 (2006):471511.

[15] A. Cattaneo, P. Mnev. Remarks on Chern-Simons invariants. Comm. Math. Phys. 293(2010), no. 3, 803836.

[16] M. Chas, D. Sullivan.String Topology. Preprint arXiv:math/9911159v1

[17] K. Costello.Topological conformal field theories and Calabi-Yau categories.Adv. Math. 210(2007), no. 1, 165214

[18] V. Drensky. Free algebras and PI-algebras. Graduate course in algebra. Singapore, 2000.271 pp.

[19] S. Elizalde, M. Noy. Consecutive patterns in permutations. Formal power series and al-gebraic combinatorics (Scottsdale, AZ, 2001). Adv. in Appl. Math. 30 (2003), no. 1-2,110125.

[20] C-H. Eu, T. Schedler. Calabi-Yau Frobenius algebras.J. Algebra 321 (2009), no. 3, 774815

[21] B. Feigin., Characteristic classes of flags of foliations. Funct. Anal. Appl. 9 (1975), no. 4,4956.

[22] B. Feigin, D. Fuchs, I. Gelfand. Cohomology of the Lie algebra of formal vector fields withcoefficients in its dual space and variations of characteristic classes of foliations. Funct.Anal. Appl. 8 (1974), no. 2, 1329.

[23] D. B. Fuks. Cohomology of infinite-dimensional Lie algebras. Contemporary Soviet Math-ematics. Consultants Bureau, New York, 1986. 339 pp.

[24] W. L.Gan. Koszul duality for dioperads. Math. Res. Lett. 10 (2003), no. 1, 109124

[25] E. Getzler. Batalin-Vilkovisky algebras and two-dimensional topological field theories.Comm. Math. Phys. 159 (1994), no. 2, 265285.

[26] E. Getzler. Operads and moduli spaces of genus0 Riemann surfaces. The moduli space ofcurves (Texel Island, 1994), 199230, Progr. Math., 129, Boston, MA, 1995.

[27] C. Glennie. some identities valid in special Jordan algebras but not valid in all Jordanalgebras. Pacific J. Math. 16 1966 4759

[28] V. Ginzburg, M. Kapranov. Koszul duality for operads, Duke Math. J., 76:1 (1994), 203272.

[29] G. Hochschild. Cohomology of restricted Lie algebras. Amer. J. Math. 76, (1954). 555580

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[30] E. Hoffbeck,-homology of algebras over an operad. Algebr. Geom. Topol. 10 (2010), no.3, 17811806

[31] An. Kirillov. On some quadratic algebras. L. D. Faddeevs Seminar on MathematicalPhysics, 91113, Amer. Math. Soc. Transl. Ser. 2, 201, 2000

[32] J. Kock. Frobenius algebras and 2D topological quantum field theories. London Math. Soc.St. Texts, 59. Cambridge University Press, Cambridge, 2004. 240 pp

[33] M. Kontsevich. Operads and motives in deformation quantization. Lett. Math. Phys. 48(1999), no. 1, 3572

[34] J.-L. Loday, B. Vallette. Algebraic Operads. Latest draft of this book available athttp://www-irma.u-strasbg.fr/~loday/PAPERS/JLLBV.pdf

[35] M. Mandell. Cochains and homotopy type. Publ. Math. Inst. HautesEtudes Sci. No. 103(2006), 213246.

[36] S. Merkulov. Operad of formal homogeneous spaces and Bernoulli numbers. Algebra Num-ber Theory 2 (2008), no. 4, 407433,

[37] P. Mnev. Notes on simplicial BF theory. Mosc. Math. J. 9 (2009), no. 2, 371410

[38] A. Polishchuk, L. Positselski. Quadratic algebras. University Lecture Series, 37. AmericanMathematical Society, Providence, RI, 2005. 159 pp.

[39] S. Priddy. Koszul resolutions. Trans. Amer. Math. Soc. 152 1970 3960.

[40] D. Quillen. Rational homotopy theory. Ann. of Math. (2) 90, 1969, 205295.

[41] D. Quillen.Homotopical algebra. Lecture Notes in Mathematics, No. 43, 1967, 156 pp.

[42] G. Segal.Notes of Lectures on field theories, http://www.cgtp.duke.edu/ITP99/segal/

[43] P. SeveraFormality of the chain operad of framed little disks. Lett. Math. Phys. 93 (2010),No. 1, 2935

[44] B. Sturmfels.What is . . . a Grobner Basis?, Notices of the American Mathematical Soci-ety 52 (10): 11991200, http://math.berkeley.edu/ bernd/what-is.pdf, a brief introduction.

[45] D. Tamarkin. Another proof of M. Kontsevich formality theorem. PreprintarXiv:math/9803025

[46] A. Thedy.A naturals-identity of Jordan algebras. Comm. Algebra 15 (1987), no. 10, 20812098.

[47] V. Ufnarovski. Criterion for the growth of graphs and algebras given by words. Mat. Za-metki 31 (1982), no. 3, 465472, 476

[48] V. Ufnarovski. Combinatorial and asymptotic methods in algebra. Algebra, VI, 1196, En-cyclopaedia Math. Sci., 57, Springer, Berlin, 1995,

[49] P. Van der Laan. Coloured Koszul duality and strongly homotopy operads. PreprintarXiv:math/0312147v2

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