Young Mathematicians Workshop on Cluster Algebras CMC Thematic Program on Cluster Algebras in Mathematics and Physics Date: Dec. 12, 2014 / Place: Rm.1423, KIAS, Seoul, Korea |
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Name
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Alexander Garver
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Title
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A combinatorial model for exceptional sequences in type A
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Abstract
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Exceptional sequences are certain ordered sequences of quiver representations. We use noncrossing edge-labeled trees in a disk with boundary vertices (expanding on T. Araya's work) to classify exceptional sequences of representations of Q, the linearly ordered quiver with n vertices. We also show how to use variations of this model to classify c-matrices of Q, to interpret exceptional sequences as linear extensions, and to give an elementary bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. In the case of c-matrices, we also give an interpretation of c-matrix mutation in terms of our noncrossing trees with directed edges. This is part of ongoing joint work with Jacob Matherne.
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Hyun Kyu Kim
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Title
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Central extensions of mapping class groups from quantum Teichmuller theory
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Quantum Teichmuller theory yields projective representations of the mapping class groups of surfaces, and hence central extensions of mapping class groups. In the universal case, one obtains central extensions of the Thompson group T, and we show that these central extensions coincide with the ones constructed by a completely different topological method. I will also briefly review the formulation of quantum Teichmuller theory, in relation to quantum cluster varieties. Arxiv:1211.4300
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Name
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Myungho Kim
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Title
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Cluster monomials and upper global basis
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Abstract
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The quantum coordinate ring $A_q(n)$ of the unipotent subgroup $N$ of a simple algebraic group of type A,D,E has the structure of a quantum cluster algebra (Geiss-Leclerc-Schroer). In a joint work with S.-J. Kang, M. Kashiwara and S.-j. Oh, we showed that the quantum cluster monomials belong to the upper global basis of $A_q(n)$. In this talk, I will explain basic materials around this problem and will give a sketch of the proof.
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Name
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John Lawson
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Title
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Minimal mutation-infinite quivers
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Abstract
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Following work by A. Seven classifying all minimal infinite-type diagrams and the classification of all mutation-finite quivers by Felikson, Shapiro and Tumarkin, we classify all minimal mutation-infinite quivers. Orientations of simply-laced Coxeter diagrams from hyperbolic Coxeter simplices give a family of minimal mutation-infinite quivers, but are not the only such quivers. These quivers admit a number of moves through which they can be classified, so we will discuss this classification, its links to hyperbolic Coxeter simplices and those exceptional classes not related to simplices.
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Name
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Matthew Mills
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Title
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A dyck path formula for certain elements of the upper cluster algebra
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Abstract
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We develop an elementary formula for certain non-trivial elements of upper
cluster algebras in terms of Dyck paths. These elements have positive coefficients. Using this formula, we show that each non-acyclic skew-symmetric cluster algebras of rank 3 is properly contained in its upper cluster algebra. Joint work with Kyungyong Lee and Li Li.
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Jonathan Wilson
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Abstract
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It is well known that cluster algebras arise from surfaces. Given a Riemann surface with boundary we may allocate marked points to each boundary component in order to triangulate our surface. Additionally, endowing our surface with a hyperbolic structure we have that each arc in a triangulation can be represented by a unique geodesic. Moreover putting horocycles around each marked point we have a way of measuring the lengths of these arcs. By uniformly adjusting these lengths (to what we call the lambda length of the arc), we see that the lambda length of a flipped arc is transformed under the Ptolemy relation.
I will discuss Dupont and Palesi's expanded version of this construction to include non-orientable surfaces, and will then concentrate on quasi-cluster algebras of finite type and the structure of their exchange graph.
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