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Convex Analysisの二,三の進展について

Part One: One-Dimensional Dynamics Examples of Dynamical Systems Preliminaries from Calculus Elementary Definitions Hyperbolicity An example: the quadratic family An Example: the Quadratic Family Symbolic Dynamics Topological Conjugacy Chaos Structural Stability Sarlovskiis Theorem The Schwarzian Derivative Bifurcation Theory Another View of Period Three Maps of the Circle Morse-Smale Diffeomorphisms Homoclinic Points and Bifurcations The Period-Doubling Route to Chaos The Kneeding Theory Geneaology of Periodic Units Part Two: Higher Dimensional Dynamics Preliminaries from Linear Algebra and Advanced Calculus The Dynamics of Linear Maps: Two and Three Dimensions The Horseshoe Map Hyperbolic Toral Automorphisms Hyperbolicm Toral Automorphisms Attractors The Stable and Unstable Manifold Theorem Global Results and Hyperbolic Sets The Hopf Bifurcation The Hnon Map Part Three: Complex Analytic Dynamics Preliminaries from Complex Analysis Quadratic Maps Revisited Normal Families and Exceptional Points Periodic Points The Julia Set The Geometry of Julia Sets Neutral Periodic Points The Mandelbrot Set An Example: the Exponential Function

An introduction to chaotic dynamical systems (2nd edition), by Robert L. Devaney. Pp 336. £34·95. 1989. ISBN 0-201-13046-7 (Addison-Wesley)

An algebra is said to be \emph{$\tau$-tilting finite} provided it has only a finite number of $\tau$-rigid objects up to isomorphism We associate a category to each such algebra The objects are the wide subcategories of its category of finite dimensional modules, and the morphisms are indexed by support $\tau$-tilting pairs

A category of wide subcategories

τ-exceptional sequences

Advances in Representation Theory of Algebras

τ-cluster morphism categories, introduced by Buan and Marsh, are a generalization of cluster morphism categories (defined by Igusa and Todorov). We show the classifying space of such a category is ...

τ-Cluster morphism categories and picture groups

Pairwise compatibility for 2-simple minded collections

$\tau$-cluster morphism categories were introduced by Buan and Marsh as a generalization of cluster morphism categories (as defined by Igusa and Todorov) to $\tau$-tilting finite algebras. In this paper, we show that the classifying space of such a category is a cube complex, generalizing results of Igusa and Todorov and Igusa. 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 by Igusa, Todorov, and Weyman. We end this paper by showing that if the algebra is Nakayama, then this space is locally $\mathsf{CAT}(0)$, and hence a $K(\pi,1)$. We do this by constructing a combinatorial interpretation of the 2-simple minded collections for Nakayama algebras. A key step in the proof is to show that, for Nakayama algebras, 2-simple minded collections are characterized by pairwise compatibility conditions, a fact not true in general.

{\tau}-Cluster Morphism Categories and Picture Groups

We prove that pointwise finite-dimensional S^1 persistence modules over an arbitrary field decompose uniquely, up to isomorphism, into the direct sum of a bar code and finitely-many Jordan cells These persistence modules have also been called angle-valued or circular persistence modules We allow either a cyclic order or partial order on S^1 and do not have additional finiteness requirements on the modules We also show that a pointwise finite-dimensional S^1 persistence module is indecomposable if and only if it is a bar or Jordan cell (a string or a band module, respectively, in representation theory) Along the way we classify the isomorphism classes of such indecomposable modules

Decomposition of Pointwise Finite-Dimensional S^1 Persistence Modules

Let {\Lambda} be a finite-dimensional associative algebra over a field. A semibrick pair is a collection of bricks in mod-{\Lambda} for which certain Hom- and Ext-sets vanish. We say Lambda has the pairwise 2-simple minded completability property if every set of bricks which is not contained in a 2-term simple minded collection has a subset of size 2 which is likewise not contained in a 2-term simple minded collection. We prove that a preprojective algebra of Dynkin type has this property if and only if it is of type A_1, A_2, or A_3. We then reduce the 2-simple minded completability property to a condition on semibrick pairs of size 3 and prove that all {\tau}-tilting finite algebras with 3 simple modules have this property. We conclude by giving a combinatorial proof that for preprojective algebras of type A, any semibrick pair of "maximal size" is a 2-term simple minded collection.

Eric J. Hanson

Papers

τ-Cluster morphism categories and picture groups

Pairwise compatibility for 2-simple minded collections

{\tau}-Cluster Morphism Categories and Picture Groups

Decomposition of Pointwise Finite-Dimensional S^1 Persistence Modules

Pairwise Compatibility for 2-Simple Minded Collections II: Preprojective Algebras and Semibrick Pairs of Full Rank