Matthew Hogancamp: Research Statement

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1 Matthew Hogancamp: Research Statement Introduction I am interested in low dimensional topology, representation theory, and categorification, especially the categorification of structures related to quantum topology. Quantum topology concerns invariants of links and 3-manifolds derived from the representation theory of quantum groups, and categorification is the realization of such invariants as Euler characteristics of homological invariants. The most famous example of this is Khovanov homology [Kho00] which categorifies the Jones polynomial. My current research can be roughly separated into three interconnected projects: Categorification of quantum invariants. In [EHb], Ben Elias and I construct complexes of Soergel bimodules which categorify the Young symmetrizers. These can be used to construct arbitrarily colored triply-graded link homology and colored sl N link homology. This link homology should be isomorphic to the constructions of Cautis [Cau12] and Webster [Web10], but the construction of mine and Elias has an advantage in that it also produces finite versions of colored sl N -projectors. These generalize the finite versions of categorified Jones- Wenzl projectors that I constructed in [Hog14]. I expect that the resulting finite-dimensional colored homology is functorial under link cobordisms. Functoriality gives link homology its 4-dimensional flavor, which is the primary reason for the importance of these invariants to topology. I am also investigating, in a joint project with Louis-Hadrien Robert, a colored sl N homology which more closely follows the approach of Khovanov [Kho05] to colored sl 2 link homology. Homology of torus links and the conjectures of Gorsky, Oblomkov, Rasmussen, Shende. In this project I plan to study a number recent conjectures which relate certain structures in link homology with certain Hilbert schemes and rational Cherednik algebras. I recently had success in studying stable limits of the triply graded Khovanov-Rozansky homology of torus links in [Hog15, AH15] (the latter is joint with Michael Abel). These prove stable versions of conjectures in [GORS14]. We show that the stable limits can be computed from certain categorified Young symmetrizers. These, in turn, are related to the flag Hilbert scheme by a beautiful recent conjecture of Gorsky-Rasmussen, and our results prove this conjecture in two special cases. The theory of categorical diagonalization (below) endows the categorified Young symmetrizers with the structure of finitely generated dg-modules over a polynomial ring, which may ultimately be responsible for the relation with algebraic geometry. Diagonalization of functors and categorical representation theory. In a joint project with Ben Elias, we introduce a notion of diagonalizability of functors and give a categorical analogue of the usual minimal polynomial condition for diagonalizability in linear algebra. Elias and I show that the Rouquier complex associated to the full twist braid acts diagonalizably on the category of Soergel bimodules, and that eigenprojections yield categorified Young symmetrizers. We plan to use our results to study the categorical representation theory of Hecke algebras, for instance by categorifying the Okounkov-Vershik approach to the representation theory of the symmetric group. We also plan to further develop our theory and apply it to other contexts, for instance the categorified sl 2 Casimir operator of Beliakova, Khovanov, and Lauda [BKL12]. 1

2 2 Categorification of Quantum Invariants Previous and current work. The Jones-Wenzl projector p n is a certain idempotent element of the Temperley-Lieb algebra, and is an essential ingredient in constructing the sl 2 Reshetikhin-Turaev invariant. This idempotent is categorified by the complex P n, constructed in [CK12, Roz10] in the setting of Khovanov homology. Cabling link diagrams and inserting P n gives rise to a categorification of the n-colored Jones polynomial. In the Cooper-Krushkal theory, the invariant of the n-colored unknot is graded infinite dimensional, being isomorphic to the ring of endomorphisms of the infinite complex P n. This infiniteness obstructs the theory s being functorial under 4-dimensional link cobordisms. In [Hog14] I propose a solution to this problem, by showing that the categorified projector is a dg-module over a polynomial ring Z[u 1,..., u n ], and then considering the complex K n obtained from P n by killing the variables u 2,..., u n. Here, killing variables means forming the Koszul complex associated to the sequence u 2,..., u n acting on P n. In [Hog14] I prove: Theorem 1 (H, [Hog14]). The complex K n is bounded and categorifies a multiple of the Jones- Wenzl projector. This complex is a homotopy quasi-idempotent, in the sense that K n K n is homotopy equivalent to a finite direct sum of K n with shifts. Further, replacing P n by K n yields a categorification of the sl 2 Reshetikhin-Turaev invariant by bounded complexes. These constructions generalize to all sl N, by recent work of mine and Ben Elias. The behavior of the sl N Reshetikhin-Turaev invariant for large N is governed by the Hecke algebra. In [EHb], Elias and I categorify a standard collection of primitive idempotents in the Hecke algebra, known as Young symmetrizers (see Theorem 7 below). From these one can derive categorifications of the Jones-Wenzl projectors and their sl N generalizations, previously accomplished in [CK12, Roz14, Ros12] for N = 2, 3 and [Cau12] for all N. The categorified Young symmetrizers give rise to a colored, triply graded link homology which categorifies the λ-colored HOMFLY-PT link invariant, where λ is an arbitrary Young diagram. Our link homology has infinite rank unless all colors are one-column partitions. The categorified Young symmetrizers are dg modules over a polynomial ring, and killing certain of these polynomial generators gives bounded complexes. Applying a functor to, say sl N foams [QR14], yields an arbitrarily colored, finite sl N homology. This homology is the first of its kind. A different, earlier approach to colored sl 2 homology was introduced by Khovanov in [Kho05]. An approach to arbitrarily colored sl N homology in a similar vein will be proposed by myself and Louis-Hadrien Robert in [HR]. At the moment we can construct an explicit resolution of the irreducible sl N representation V λ by representations of the form Λ k 1 (C N ) Λ kr (C n ). Translating this resolution into the language of sl N foams will define our colored homology theory in terms of existing constructions [QR14, Wu09] of Λ k (C N )-colored link homology. Future work. (1) Show that the finite, colored sl N projectors are dg Frobenius algebra objects. Then, prove that the resulting colored sl N -link homology is functorial under link cobordisms up to sign. The expected homology of the unknot can be extracted in some special cases from the conjectures in [GORS14, GOR13, GR]. As a nontrivial check of those conjectures, it should be verified that killing some polynomial generators yields Frobenius algebras. (2) Show that the construction of colored sl N homology with Louis-Hadrien Robert produces a topological invariant. Investigate the relationship of this homology theory with the colored sl N homology constructed above via categorified projectors.

3 (3) Investigate finite categorifications of spin networks. The finite complex K n yields categorified spin networks; it will be very interesting to study the resulting evaluations of tetrahedral and theta networks. The categorified 6j-symbols should have an interpretation in this setting as well. Homology of Torus Lniks and Hilbert Schemes Current and previous work. Computing Khovanov-Rozansky homology of torus links is a difficult and important problem. The importance of this problem is enhanced by [ORS12, GORS14] which relate link homology of algebraic links to Hilbert schemes and rational Cherednik algebras. Through work of Cherednik [Che13], these conjectures can be thought of as a concrete manifestation of a deep connection between the known, mathematically rigorous constructions of link homology with physical approaches to link homology. An encouraging step in this direction was recently made by myself and Ben Elias: Theorem 2 (Elias-H, [EHa]). The triply graded Khovanov-Rozansky homology of the (n, n) torus link is supported in even homological degrees. We are in fact able to compute this homology exactly. I have also had success in studying stable limits of the triply graded homology of torus links. As observed by Rozansky in the case of Khovanov homology [Roz14, Roz10], in the triply graded setting there are two stable limits, and each can be computed from a categorified idempotent. The two limits were studied by myself [Hog15] and in joint work with Michael Abel [AH15]. The limits can be computed from the categorified Young symmetrizers P n and P 1 n associated to one-row partitions and one-column partitions, respectively. Theorem 3 (H, [Hog15]). The triply graded homology of the (n, m) torus link stabilizes as m, with stable limit the Hochschild homology HHH(P n ). Finally, HHH(P n ) is an explicit super-polynomial ring. Theorem 4 (Abel-H, [AH15]). There is a stable limit H n,m of the triply graded homology of the (n, nk + m) torus link, as k. This homology is such that H n,0 = HHH(P1 n) is an explicit quotient of a super-polynomial ring, and H n,m is an explicit quotient of H n,0 which depends only on gcd(n, m). Each of the previous results proves a stable version of conjectures in [ORS12, GORS14] for torus knots. The homology HHH(P T ) of categorified Young symmetrizers is related to an object called the flag Hilbert scheme, via a beautiful recent conjecture of Gorsky-Rasmussen [GR]. The above results prove this conjecture in the special cases of one-row and one-column partitions. Future work. (1) Compare the computation of the triply graded homology of (n, n) torus links with that predicted by conjectures in [GORS14]. (2) Extend the methods of mine and Elias from (n, n) torus links to other cases. Of particular interest is the case of the (n, n + 1)-torus knots, which by [GORS14] are conjecturally related to an interesting combinatorial object called the space of diagonal harmonics [Hai94]. (3) Compute HHH(P T ). There are a number of fascinating conjectures regarding these homologies. First, the Gorsky-Rasmussen conjecture [GR] can be thought of as given an explicit 3

4 4 set of relations on the polynomial action on HHH(P T ) implied by Remark 6. Secondly work of Gorsky-Rasmussen and Gukov-Stosic [GS12] suggests that there is a mirror symmetry which relates HHH(P T ) with HHH(P T ), where T is the transpose tableau. I plan to investigate these conjectures, starting with the case where the shape of T is a hook partition. Diagonalization of Functors and Categorical Representation Theory of Hecke Algebras Current and previous work. In [EHb], Ben Elias and I introduce the notion of diagonalizability to category theory. To state our main theorem, let A be an additive category with direct sum, tensor product, and neutral object 1, for instance a category of bimodules over a ring R, with 1 = R. Let K (A) be the homotopy category of chain complexes over A which are bounded above. Fix a chain complex F K (A), which we may also regard as a functor F ( ). Let k denote a shift in homological degree; note that M k = 1 k M. In the categorical setting, one has not only eigenvalues and eigenvectors, but eigenmaps. We call a map α : 1 k F an eigenmap if there is a nonzero object M such that α Id M and Id M α are homotopy equivalences. In this case M is called an α-eigenobject. In particular, an α-eigenobject satisfies F M M k. An elementary property of mapping cones implies that M is an α-eigenobject if and only if M Cone(α) and Cone(α) M are contractible. The theorem below gives a notion of what it means for F to have enough eigenmaps, and states that in this case F satisfies a categorical analogue of diagonalizability: Theorem 5 (Elias-H, [EHb]). Fix F K (A) be as above. Let λ i = 1 a i be shifted copies of 1 (1 i r), and assume that a 1 < < a r. Suppose α i : λ i F are chain maps such that (1) r Cone(λ i i=1 Then there exist complexes P i K (A) such that: α i F ) 0 the P i are mutually orthogonal idempotents: P i P j P i if j = i, and zero otherwise. the P i are complete: the identity object 1 can be recovered as 1 Tot(P 1 P 2 P r ). This is not the total complex of a bicomplex, but rather the total complex of a Postnikov system in the sense of triangulated categories. In particular it is a filtered complex with subquotients P i. P i is an α i -eigenobject. We also prove that object M is an α i -eigenobject if and only if P i M M, so that P i projects onto the α i -eigencategory. The decomposition of K (A) into its eigencategories for F is an example of a semi-orthogonal decomposition in the sense of triangulated categories. Remark 6. The presence of eigenmaps imparts some additional structure on the eigenprojections which is not visible on the level of linear algebra. Tensoring the eigenmap α j : λ j F with the projection P i yields a map λ j P i λ i P i if i j, which we will regard as an endomorphism U i j of P i, of degree λ j λ 1 i, and which is homotopic to the identity if i = j. Thus, the eigenprojections are equipped with an action of a polynomial ring Z[U1 i,..., Û i i,..., U r] i on P i, where the variable Ui i is omitted.

5 Functors which are diagonalizable seem to be important, with examples coming from group cohomology and the tube-cutting bimodule in bordered Floer homology [LT12]. By recent work of myself and Ben Elias, an application of the above theory to the setting of Soergel bimodules yields categorifications of the q-young symmetrizers, which are certain idempotent elements of the Hecke algebra. Recall that associated to each braid β one has the Rouquier complex F (β) of Soergel bimodules, and these satisfy F (β) F (β ) F (ββ ). This categorifies a well known action of the braid group on the Hecke algebra. Theorem 7 (Elias-H, [EHb]). The Rouquier complex FT n associated to the full-twist is diagonalizable in the sense of Theorem 5. By tensoring together the eigenprojections for FT 2, FT 3,..., FT n in all possible ways, one obtains a family of complexes P T, indexed by standard tableaux T of size n, which categorify the q-young symmetrizers. These complexes form a complete set of projections in the sense of Theorem 5. They project onto simultaneous eigencategories for the action of FT 2,..., FT n. Applying a functor to sl 2 foams recovers the categorified Temperley-Lieb idempotents constructed by myself and Ben Cooper [CH12]. Future work. (1) Further develop the theory of categorical diagonalization, and find examples in nature. For instance, diagonalize the categorified Casimir operator [BKL12], with relevance to categorified sl 2 -representations. (2) Categorify the Okounkov-Vershik approach to the representation theory of S n. In this approach, one analyzes the spectrum of the Jucys-Murphy operators y 2,..., y n Q[S n ]. In the categorified situation, the role of the Jucys-Murphy operators is played by the Rouquier complexes FT k FT 1 k 1 or simply FT k, the categorical spectral theory of which is given in Theorem 7. Early results in this direction are encouraging. (3) Extend the previous results to Coxeter groups other than the symmetric group. The case of dihedral groups will appear in forthcoming work with Ben Elias. Our methods may be particularly well-suited to tackle Coxeter groups of type B n as well. (4) Diagonalize the full-twist in characteristic p. In characteristic p, one expects the categorified full twist to have more eigenvalues than its decategorification, which is very intriguing. Perhaps this will give a new measure of the badness of Soergel bimodules in characteristic p. This is an important area of research, and recently led to Williamson s disproving the Lusztig conjecture [Wil13]. (5) Study the center of the Soergel category. It is well known that the center of the Hecke algebra (in type A n ) consists of the symmetric functions of the Jucys-Murphy operators. It should be possible to categorify these elements. From the projections P T it is possible to extract a collection of central idempotents P λ, indexed by partitions of n, and also finite versions K λ of these. We conjecture that the K λ are central. It will be very important to better understand these K λ s, perhaps shedding light on the structure of the center of the Soergel category. (6) Investigate Morita equivalence of idempotents. That is to say, look for complexes E T,U of Soergel bimodules, associated to pairs of standard tableaux of the same shape, such that E T,U E U,T P T and E U,T E T,U P U. This would provide a categorical analaogue of the fact that the q-young symmetrizers p T and p U define isomorphic representations of the Hecke algebra if they have the same shape. 5

6 6 References [AH15] M. Abel and M. Hogancamp, Stable homology of torus links via categorified Young symmetrizers II: one-column partitions, arxiv: , October , 3 [BKL12] Anna Beliakova, Mikhail Khovanov, and Aaron D. Lauda, A categorification of the Casimir of quantum sl(2), Adv. Math. 230 (2012), no. 3, MR , 5 [Cau12] S. Cautis, Clasp technology to knot homology via the affine Grassmannian, ArXiv e-prints (2012). 1, [CH12] 2 B. Cooper and M. Hogancamp, An Exceptional Collection For Khovanov Homology, ArXiv e-prints (2012). 5 [Che13] Ivan Cherednik, Jones polynomials of torus knots via DAHA, Int. Math. Res. Not. IMRN (2013), no. 23, MR [CK12] B. Cooper and V. Krushkal, Categorification of the Jones-Wenzl projectors, Quantum Topol. 3 (2012), no. 2, MR [EHa] B. Elias and M. Hogancamp, Homology of the (n, n) torus links (in preparation). 3 [EHb], A new approach to the categorical representation theory of Hecke algebras I (in preparation). 1, 2, 4, 5 [GOR13] E. Gorsky, A. Oblomkov, and J. Rasmussen, On stable Khovanov homology of torus knots, Exp. Math. 22 (2013), no. 3, MR [GORS14] E. Gorsky, A. Oblomkov, J. Rasmussen, and V. Shende, Torus knots and the rational DAHA, Duke Math. J. 163 (2014), no. 14, MR , 2, 3 [GR] E. Gorsky and J. Rasmussen, On colored projectors in Khovanov-Rozansky homology (in preparation). 2, 3 [GS12] Sergei Gukov and Marko Stošić, Homological algebra of knots and BPS states, Proceedings of the Freedman Fest, Geom. Topol. Monogr., vol. 18, Geom. Topol. Publ., Coventry, 2012, pp MR [Hai94] Mark D. Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3 (1994), no. 1, MR (95a:20014) 3 [Hog14] M. Hogancamp, A polynomial action on colored sl(2) link homology, ArXiv e-prints (2014). 1, 2 [Hog15] [HR], Stable homology of torus links via categorified Young symmetrizers I: one-row partitions, ArXiv: , May , 3 M. Hogancamp and L.-H. Robert, Arbitrarily colored sl n link homology by foams and resolutions (in preparation). 2 [Kho00] M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, MR (2002j:57025) 1 [Kho05], Categorifications of the colored Jones polynomial, J. Knot Theory Ramifications 14 (2005), no. 1, MR (2006a:57016) 1, 2 [LT12] R. Lipshitz and D. Treumann, Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers, arxiv: , March [ORS12] A. Oblomkov, J. Rasmussen, and V. Shende, The Hilbert scheme of a plane curve singularity and the [QR14] [Ros12] [Roz10] HOMFLY homology of its link, ArXiv: , January H. Queffelec and D. E. V. Rose, The sl n foam 2-category: a combinatorial formulation of Khovanov- Rozansky homology via categorical skew Howe duality, ArXiv e-prints (2014). 2 D. E. V. Rose, Categorification of Quantum sl(3) Projectors and the sl(3) Reshetikhin-Turaev Invariant of Framed Tangles, ProQuest LLC, Ann Arbor, MI, 2012, Thesis (Ph.D.) Duke University. MR L. Rozansky, A categorification of the stable SU(2) Witten-Reshetikhin-Turaev invariant of links in S 2 S 1, ArXiv: , November , 3 [Roz14] L. Rozansky, An infinite torus braid yields a categorified Jones-Wenzl projector, Fund. Math. 225 (2014), , 3 [Web10] B. Webster, Knot invariants and higher representation theory II: the categorification of quantum knot invariants, ArXiv e-prints (2010). 1 [Wil13] G. Williamson, Schubert calculus and torsion explosion, ArXiv e-prints (2013). 5 [Wu09] H. Wu, A colored sl(n)-homology for links in Sˆ3, ArXiv e-prints (2009). 2