
Papers in Topology

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A Legendrian Turaev torsion via generating families[formerly "Turaev torsion of ribbon Legendrians"] (with Daniel ÁlvarezGavela)

Journal de l'École polytechnique  Mathematics, Volume 8 (2021), pp. 57119.



We introduce a Legendrian invariant built out of the Turaev torsion of generating families. This invariant is defined for a certain class of Legendrian submanifolds of 1jet spaces, which we call of Euler type. We use our invariant to study mesh Legendrians: a family of 2dimensional Euler type Legendrian links whose linking pattern is determined by a bicolored trivalent ribbon graph. The Turaev torsion of mesh Legendrians is related to a certain monodromy of handle slides, which we compute in terms of the combinatorics of the graph. As an application, we exhibit pairs of Legendrian links in the 1jet space of any orientable closed surface which are formally equivalent, cannot be distinguished by any natural Legendrian invariant, yet are not Legendrian isotopic. These examples appeared in a different guise in the work of the second author with J. Klein on pictures for K_{3} and the higher Reidemeister torsion of circle bundles.


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

J Topology (2015) 8 (3): 675690.



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.


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

ArXiv:1203.2203
Trans. Amer. Math. Soc. 366 (2014) 749790.



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.


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

ArXiv:1011.4653
Trans. Amer. Math. Soc. 366 (2014) 791832.



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.


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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.

Papers in Algebra

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Exceptional sequences and rooted labeled forests. (with Emre Sen)

arXiv:2108.11351


We give a representationtheoretic bijection between rooted labeled forests with n vertices and complete exceptional sequences for the quiver of type An with straight orientation, i.e., for each labeled rooted forest we construct a unique exceptional sequence. This is equivalent to GouldenYong. But the point is: the ascending and descending vertices in the forest correspond to relatively projective and relatively injective objects in the exceptional sequence. We conclude that every object in an exceptional sequence is either relatively projective or relatively injective or both. We construct a natural action of the braid group on rooted labeled forests and show that it agrees with the known action on complete exceptional sequences. We also give a new proof of the known bijection between complete exceptional sequences and parking functions. Then, we compare this bijection, using Prüfer codes with our bijection with rooted labeled forests.


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Infinitesimal semiinvariant pictures and coamagamation. (with Eric Hanson, Moses Kim and Gordana Todorov)

arXiv:2012.06572


The purpose of this paper is to study the local structure of the semiinvariant picture of a tame hereditary algebra near the null root. Using a construction that we call coamalgamation, we show that this local structure is completely described by the semiinvariant pictures of a collection of Nakayama algebras. We then show that this local structure is (piecewise linearly) invariant under cluster tilting.


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Picture groups and maximal green sequences. (with Gordana Todorov)

Electronic Research Archive 0 (2021), The special issue of algebraic representation theory
;


We show that picture groups are directly related to maximal green sequences for valued Dynkin quivers of finite type. Namely, there is a bijection between maximal green sequences and positive expressions (words in the generators without inverses) for the Coxeter element of the picture group. We actually prove the theorem for the more general set up of finite “vertically and laterally ordered” sets of positive real Schur roots for any hereditary algebra (not necessarily of finite type).
Furthermore, we show that every picture for such a set of positive roots is a linear combination of “atoms” and we give a precise description of atoms as special semiinvariant pictures. We also give a history of pictures going back to Renée Peiffer.


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Universal quivers. (joint with Sergey Fomin and Kyungyong Lee.)

arXiv:2003.01244


We show that for any positive integer n, there exists a quiver Q with O(n^{2}) vertices and O(n^{2}) edges such that any quiver on n vertices is a full subquiver of a quiver mutation equivalent to Q. We generalize this statement to skewsymmetrizable matrices, and obtain other related results.


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A counterexample to the phidimension conjecture. (joint with Eric Hanson)

arXiv:1911.00614


The counterexample is given by three triangles: amalgamate two and tensor with a third. We construct a sequence of pairs of modules X_{k},Y_{k} so that their 3kth syzygies are not isomorphic but their higher syzygies are.


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Resolution quiver and cyclic homology criteria for Nakayama algebras. (joint with Eric Hanson)

Journal of Algebra, Volume 553, 1 July 2020, Pages 138153


By D.Shen, a Nakayama algebra has finite global dimension if and only if its "resolution quiver" R(A) (defined by Ringel) is connected with weight 1. KI and D. Zacharia proved many year ago that a monomial relation algebra has finite global dimension if and only if the cyclic homology of its radical is zero. For Nakayama algbras, this is equivalent to saying that the "relation complex" L(A) has Euler characteristic 1. In this paper we go directly from L(A) to the R(A) by showing that the Euler characteristic of L(A) is equal to the number of components of R(A) with weight one.


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Continuous quivers of type A (I) The generalized barcode theorem. (with Job Rock and Gordana Todorov)

arXiv:1909.10499


The quotient of the continuous cluster category by the full subcategory given by a discrete cluster is isomorphic to the category of infinite dimensional modules over the endomorphism ring of the cluster which have finitely many summands. This is being generalized in a series of papers by Job Rock.


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Continuous cluster categories II: continuous clustertilted categories. (with Gordana Todorov)

arXiv:1909.05340


The quotient of the continuous cluster category by the full subcategory given by a discrete cluster is isomorphic to the category of infinite dimensional modules over the endomorphism ring of the cluster which have finitely many summands. This is being generalized in a series of papers by Job Rock.


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Continuously triangulating the continuous cluster category. (with Mathew Garcia)

Topology and its Applications
Volume 285, 1 November 2020, 107411 (free download until Dec 24, 2020.)


This paper, part of Matt's PhD thesis, classifies all the ``equivalence covers'' of the continuous cluster category. They must be even coverings of the Moebius band. For 2fold coverings, we found three! The connected covering was given in "Continuous cluster categories I" by KI and G. Todorov and the disconnected covering was given (by a general formula for singularity categories) by Orlov. The third one was very unexpected.


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Frieze varieties are invariant under Coxeter mutation. (with Ralf Schiffler)

arXiv:1906.00106
Contemp. Math. 761 (2021) 103116;


Frieze varieties were introduced by Lee, Li, Mills, Seceleanu and the second author. We define generalized frieze varieties and show that the components are cyclically permuted by the Coxeter mutation (mutate at all the vertices in admissible order). So, they have the same dimension. We also give a method of finding the defining equations for the frieze variety. A sequel is planned with three authors.


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Pairwise compatibility of 2simple minded collections of gentle algebras. (joint with Eric Hanson)

Journal of Pure and Applied Algebra 225 (2021), no. 6


This paper shows that the picture space of a gentle algebra is locally CAT0 if and only if the gentle algebra is Nakayama.


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Cyclic posets and triangulation clusters. (with Gordana Todorov)

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Maximal green sequences of quivers with multiple edges. (joint with Ying Zhou)

arXiv:arXiv:1902.07375


If an acyclic quiver has a multiple edge, removal of all the arrows of this multiple edge will increase the number of maximal green sequences!


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tauCluster Morphism Categories and Picture Groups. (joint with Eric Hanson)

arXiv:1809.08989


Abstract: taucluster morphism categories were introduced in [arXiv:1802.03812] as a generalization of cluster morphism categories to tautilting finite algebras. In this paper, we show that the classifying space of such a category is a cube complex, generalizing a result of [arXiv:1411.0196] and [arXiv:1706.02041]. We further show that the fundamental group of this space is isomorphic to a generalized version of the picture group of the algebra, as defined in [arXiv:1609.02636]. We end this paper by showing that if our algebra is Nakayama, then this space is locally CAT(0), and hence a K(pi,1). We do this by constructing a combinatorial interpretation of the 2simple minded collections of Nakayama algebras.


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Weighted quivers. (joint with Moses Kim)

arXiv:1803.03582


Abstract: A "weight" on a quiver Q with values in a group G is a function which assigns an element of G for each arrow in Q. This paper shows that the essential steps in the mutation of quivers with potential [DWZ] goes through with weights provided that the weights on each cycle in the potential have trivial product. This gives another proof of the sign coherence of cvectors. We also classify all weights on tame quivers.


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Stability conditions for affine type A.
(joint with P.J. Apruzzese)

Algebras and Representation Theory, (Nov 2019)
(Third paper on the linearity problem)


Abstract: We construct maximal green sequences of maximal length for any affine quiver of type A. We determine which sets of modules (equivalently cvectors) can occur in such sequences and, among these, which are given by a linear stability condition (also called a central charge). There is always at least one such maximal set which is linear. The proofs use representation theory and three kinds of diagrams shown in Figure 1. Background material is reviewed with details presented in two separate papers [arxiv:1706.06986] and [arXiv:1706.06503].


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

Algebraic Combinatorics, (Oct 2019)
(Second of three papers on Reineke's linearity question)


Abstract: We show that, for any clustertilted algebra of finite representation type over an algebraically closed field, the following three definitions of a maximal green sequence are equivalent: (1) the usual definition in terms of FominZelevinsky mutation of the extended exchange matrix, (2) a forward homorthogonal sequence of Schurian modules, (3) the sequence of wall crossings of a generic green path. Together with [arXiv:1706.06986], this completes the foundational work needed to support the author’s work with P.J. Apruzzese [arXiv:1804.09100], namely, to determine all lengths of all maximal green sequences for all quivers whose underlying graph is an oriented or unoriented cycle and to determine which are “linear”.
In an Appendix, written jointly with G. Todorov, we give a conjectural description of maximal green sequences of maximum length for any clustertilted algebra of finite representation type.


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Linearity of stability conditions.

Communications in Algebra, Vol 48 (2020), No 4.16711696. (First of three papers on Reineke's linearity question)
Get free copy of final published paper here (while supplies last).


Abstract: We study different concepts of stability for modules over a finite dimensional algebra: linear stability, given by a "central charge", and nonlinear stability given by the wallcrossing sequence of a "green path". Two other concepts, finite HarderNarasimhan stratification of the module category and maximal forward homorthogonal sequences of Schurian modules, which are always equivalent to each other, are shown to be equivalent to nonlinear stability and to a maximal green sequence, defined using FominZelevinsky quiver mutation, in the case the algebra is hereditary.
This is the first of a series of three papers whose purpose is to determine all maximal green sequences of maximal length for quivers of affine type Ã and determine which are linear. The complete answer will be given in the final paper with PJ.


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mnoncrossing trees.

Journal of Algebra and Its Applications, Vol. 17, No. 10 (2018) 1850187 (20 pages)


We examine mcluster theory from an elementary point of view using a generalization
of m + 1ary trees which we call mnoncrossing trees. We show that these trees are in
bijection with mclusters in the mcluster category of a quiver of type A. Similar trees
are in bijection with complete exceptional sequences. Most of this paper is expository,
explaining definitions and known results about these topics in representation theory.
One application of these trees is that the mutation formula for mclusters is derived
from the more elementary mutation of trees. The main new result is that the natural
map of an mnoncrossing tree into the plane is an embedding. We also explain the
relationship between mnoncrossing trees and finite Harder–Narasimhan systems in the
derived category of the module category of type A.


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Horizontal and vertical mutation fans.

Sci. China Math. (2018). https://doi.org/10.1007/s1142501793167


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. (with Ying Zhou)

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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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)

Journal of Pure and Applied Algebra 223 (2019) 1040–1053


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)

The Electronic Journal of Combinatorics, Vol 26, Issue 1 (2019), p1.20.


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)

Journal of Algebra and Its Applications: Vol. 17, No. 09, 1850170 (2018)


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)

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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. 



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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
