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Horizontal and vertical mutation fans.

arXiv:1805.04799


These are lecture notes from my talk at Workshop on Cluster Algebras and Related Topics, July 1013, 2017. Chern Institute of Mathematics, Tianjin, China.
We introduce diagrams for mcluster categories which we call “horizontal” and “vertical” mutation fans. These are analogous to the mutation fans (also known as “semiinvariant pictures” or “scattering diagrams”) for the standard (m = 1) cluster case which are dual to the poset of finitely generated torsion classes. The purpose of these diagrams is to visualize mutations and analogues of maximal green sequences in the mcluster category with special emphasis on the cvectors (the “brick” labels).


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Tame hereditary algebras have finitely many mmaximal green sequences.

arXiv:1706.09118


The paper proves the statement in the title. Also, in an attempt to extend this statement to the clustertilted case, we show, with the same argument, that there are finitely many mred sequences between any two silting objects. The proof is really short since we quote lemmas from the m=1 case.


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Maximal green sequences for clustertilted algebras of finite type. (Appendix written jointly with Gordana Todorov)

arXiv:1706.06503


This paper proves the equivalence between three descriptions of MGS: (1) The standard one given by FominZelevinsky mutation at positive cvectors, (2) Wall crossing sequences, (3) Maximal forward homorthogonal sequences. This last characterization is a very useful one. In the Appendix, Gordana and I give a conjecture describing the MGSs of maximal length using the fact that clustertilted algebras are relation extensions of tilted algebras. We give several examples of this conjecture.


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Signed exceptional sequences and the cluster morphism category. (with Gordana Todorov)

arXiv:1706.02041


We introduce signed exceptional sequences as factorizations of morphisms in the cluster morphism category. The objects of this category are wide subcategories of the module category of a hereditary algebra. A morphism [T] : A → B is an equivalence class of rigid objects T in the cluster category of A so that B is the right homext perpendicular category of the underlying object T ∈ A. Factorizations of a morphism [T] are given by totally orderings of the components of T. This is equivalent to a “signed exceptional sequences.” For an algebra of finite representation type, the geometric realization of the cluster morphism category is the EilenbergMacLane space with fundamental group equal to the “picture group” introduced by the authors in [IOTW4]. The paper was updated following completion of IOTW3:1503.07945.


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Picture groups of finite type and cohomology in type A_n. (with Kent Orr, Gordana Todorov, Jerzy Weyman)

arXiv:1609.02636


For every quiver of finite type we define a finitely presented group called a picture group. We construct a finite CW complex which is shown in the signed exceptional sequences paper to be a K(pi,1) for this picture group. The special case of type A_n with straight orientation also has an independent proof in the noncrossing partitions paper. (But both of those other papers are based on ideas from a earlier version of this paper.) We use this cell complex to compute the cohomology ring of picture groups of type A_n with any orientation. (The ring is independent of orientation as long as it is acyclic.)


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The No Gap Conjectute for tame hereditary algebras. (with Stephen Hermes)

arXiv:1601.04054


The "No Gap Conjecture" of BrustleDupontPerotin states that the set of lengths of maximal green sequences for hereditary algebras over an algebraically closed field has no gaps. This follows from a stronger conjecture that any two maximal green sequences can be "polygonally deformed" into each other. We prove this stronger conjecture for all tame hereditary algebras over any field.


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Modulated semiinvariants. (with Kent Orr, Gordana Todorov, Jerzy Weyman)

arXiv:1507.03051
(v2, Sept 1, 2015)


We prove the basic properties of determinantal semiinvariants for presentation spaces over any finite dimensional hereditary algebra over any field. These include the virtual generic decomposition theorem, stability theorem and the cvector theorem which says that the cvectors of a cluster tilting object are, up to sign, the determinantal weights of the determinantal semiinvariants defined on the cluster tilting objects. Applications of these theorems are given in several concurrently written papers. There is also an appendix which compares determinantal weights to the ``true weights'' of semiinvariants.


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Combinatorics of exceptional sequences of type A. (with Al Garver, Jacob Matherne, Jonah Ostroff)

arXiv:1506.08927


We introduce a class of objects called strand diagrams and use
this model to classify exceptional sequences of representations of a quiver
whose underlying graph is a type A_n Dynkin diagram. We also use variations
of this model to classify cmatrices of such quivers, to interpret exceptional
sequences as linear extensions of posets, and to give a simple bijection
between exceptional sequences and certain chains in the lattice of noncrossing
partitions.


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Semiinvariant pictures and two conjectures about maximal green sequences. (with Thomas Brustle, Stephen Hermes and Gordana Todorov)

J Algebra 473, March 2017, 80109.
(with only minor corrections from arXiv version).


We use semiinvariant pictures to prove two conjectures about maximal green sequences. First: if Q is any acyclic valued quiver with an arrow j→i of infinite type then any maximal green sequence for Q must mutate at i before mutating at j. (This statement is not true without the word "acyclic".) Second: for any quiver Q′ obtained by mutating an acyclic valued quiver Q of tame type, there are only finitely many maximal green sequences for Q′. Both statements follow from the Rotation Lemma for reddening sequences and this in turn follows from the Mutation Formula for the semiinvariant picture for Q.


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The category of noncrossing partitions.

arXiv:1411.0196


This category has noncrossing partitions of n+1 as objects and binary forests as morphisms. I use a theorem of Gromov to show that the classifying space of this category is locally CAT(0). It is not too hard to show that the fundamental group of this space is the picture group of type An with straight orientation. This paper has no representation theory and is independent of the other papers in the series except for the last section where I use cluster categories to compare my category with HuberyKrause. The comparison is difficult even when n=2.


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Periodic trees and semiinvariants. (with Gordana Todorov, Jerzy Weyman)

arXiv1407.0619 July 2, 2014


Periodic trees are combinatorial structures which are in bijection with cluster tilting objects in cluster categories of affine type A_{n−1}. The internal edges of the tree encode the cvectors corresponding to the cluster tilting object, as well as the weights of the virtual semiinvariants associated to the cluster tilting object. We also show a direct relationship between the position of the edges of the tree and whether the corresponding summands of the cluster tilting object are preprojective, (preinjective or shifted projective) or regular. This paper is the first of two parts.


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Continuous cluster categories I. (with Gordana Todorov)

Algebras and Representation Theory, vol 18, no 1 (2015), 65101.


In "Continuous Frobenius categories" we constructed topological triangulated categories C_{c} as stable categories of certain topological Frobenius categories F_{c}. In this paper we show that these categories have a cluster structure for certain values of c including c=pi. The continuous cluster categories are those C_{c} which have cluster structure. We study the basic structure of these cluster categories and we show that C_{c} is isomorphic to an orbit category D_{r}/F_{s} of the continuous derived category D_{r} if c=r pi/s. In C_{pi}, a cluster is equivalent to a discrete lamination of the hyperbolic plane. We give the representation theoretic interpretation of these clusters and laminations.


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Continuous cluster categories of type D. (with Gordana Todorov)

arXiv1309.7409 Sept 30, 2013


We construct continuous Frobenius categories of type D. These are idempotent completions of the action of Z/2 on the continuous Frobenius category of rotation by pi. The stable category is a cluster category of type D when the characteristic of the field is not 2. We also show that we get a cluster category if the field has characteristic 2 or if Z/2 is replaced by Z/p for any odd prime. These last two give the "same" result.


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Mixed cobinary trees. (with Jonah Ostroff)

arXiv1307.3587 July 12, 2013


We develop basic cluster theory from an elementary point of vies using a variation of binary trees which we call mixed cobinary trees. We show the number of (isomorphism classes) of such trees is given by the Catalan number C_{n} where n is the number of internal nodes. We also consider the corresponding quiver Q_{e} of type A_{n1}. As a special case of more general known results about the relation between cvectors, representations of quivers and their semiinvariants, we explain the bijection between mixed cobinary trees and the vertices of the generalized associahedron corresponding to the quiver Q_{e}. v3: statements and definitions are more precise. Several misprints corrected.


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Cluster categories coming from cyclic posets. (with Gordana Todorov)

Final version Communications in Algebra, vol 43 (2015), 43674402.


In this paper we show that any cyclic poset gives rise to a Frobenius category over any discrete valuation ring R. The continuous cluster categories of [11] are examples of this construction. If we twist the construction using an admissible automorphism of the cyclic poset, we generate other examples such as the mcluster category of type A_{∞} (m ≥ 3). Version 2: minor revisions. References added.


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The first Hochschild cohomology group of a clustertilted algebra revisited. (with Ibrahim Assem, Juan Carlos Bustamante and Ralf Schiffler)

ArXiv:1209.2146 Sept 11, 2012


This paper improves some of the results of Assem, Redondo and Schiffler on HH^{1} of cluster tilted algebras (from ArXiv:1204.1607). Another paper by IBIS is planned.


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Continuous Frobenius categories. (with Gordana Todorov)

ArXiv:1209.0038 Proceedings of the Abel Symposium 2011 (2013), 115143.


We use representations of the circle S^{1} to construct various Frobenius categories whose stable categories are the continuous cluster categories. This is (logically) the first paper in a series of papers. This version describes the topology of these categories.


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A proof of the strong no loop conjecture. (with Shiping Liu and Charles Paquette)

ArXiv:1103.5361
Advances in Mathematics 228 (2011) 27312742.



In terms of a quiver with relations, the strong no loop conjecture states that, if the quiver has a loop at a vertex, the simple module at the vertex has infinite projective dimension.


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Exceptional sequences, braid groups and clusters,
Groups, Algebras and Applications

(Proceedings of XVIII LATINAMERICAN ALGEBRA COLLOQUIUM Sao Pedro, August 7, 2009, Cesar P. Milies, Ed.) Contemporary Mathematics (2011) 537, 227240.


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Exceptional sequences and clusters.
(with Ralf Schiffler)

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Cluster Complexes via SemiInvariants.
(with Kent Orr, Gordana Todorov, Jerzy Weyman)

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On the finitisitc dimension conjecture for Artin algebras.
(with Gordana Todorov)

Fields Inst. Comm, vol 45, Amer. Math. Soc., Providence RI, 2005, pp 201204. 


Papers in Topology

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An equivariant version of Hatcher's G/O construction. (with Tom Goodwillie and Chris Ohrt)

ArXiv:1307.5554



This paper gives an explicit construction of the generators of the rational homotopy groups of the space of stable hcobordisms of the classifying space of a cyclic group of order n and calculate the higher torsion of these bundle verifying that the cohomology of these spaces is generated by equivariant higher torsion invariants. This plays an important role in Chris Ohrt's axiomatization of the higher equivariant torsion invariants. J Topology (2015) 8 (3): 675690.


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Exotic smooth structures on topological fibre bundles I. (with Sebastian Goette and Bruce Williams)

ArXiv:1203.2203
March 16, 2012



This paper used to be the appendix of the Exotic smooth structures paper. It is now a separate paper at the suggestion of the referee. Given a smooth bundle, an exotic smooth structure is a smooth structure different from the one already given. Such structures are classified, rationally and stably, by a homology class in the total space of the bundle. We call it the relative smooth structure class. Trans. Amer. Math. Soc. 366 (2014) 749790.


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Exotic smooth structures on topological fibre bundles II. (with Sebastian Goette)

ArXiv:1011.4653
March 09, 2012



We use a variation of Hatcher's construction to construct virtually all stable exotic smooth structures on compact smooth manifold bundles whose fibers have sufficiently large odd dimension and and show that the relative higher Reidemeister torsion is the Poincare dual of the image of the relative smooth structure class (constructed in Part B) in the homology of the base. Trans. Amer. Math. Soc. 366 (2014) 791832.


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Iterated integrals of superconnections.

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Twisting cochains and higher torsion.

arXiv:0212383v3
Journal of Homotopy and Related Structures. 6 (2011), no.2, 213238.



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Families of regular matroids.


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Part A of "Algebraic Ktheory of Ainfty ring spaces'' (1982)

download 3.2MB, Algebraic KTheory, Part II (Oberwolfach,
1980), Springer, Berlin, 1982, pp. 146194. I talked about this in the Algebra Seminar at University of Connecticut on March 21, 2018.



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Stability theorem for smooth pseudoisotopies

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The generalized Grassmann invariant
(This is my original paper about pictures.)

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Postnikov Invariants and Pseudoisotopy
