L2

\ \ The cluster category of Dynkin type $A_\infty$ is a ubiquitous object with interesting properties, some of which will be explained in this talk.

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\ \ Let us denote the category by $\mathcal{D}$. Then $\mathcal{D}$ is a 2-Calabi-Yau triangulated category which can be defined in a standard way as an orbit category, but it is also the compact derived category $D^c(C^{∗}(S^2;k))$ of the singular cochain algebra $C^*(S^2;k)$ of the 2-sphere $S^{2}$. There is also a “universal” definition: $\mathcal{D}$ is the algebraic triangulated category generated by a 2-spherical object. It was proved by Keller, Yang, and Zhou that there is a unique such category.

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\ \ Just like cluster categories of finite quivers, $\mathcal{D}$ has many cluster tilting subcategories, with the crucial difference that in $\mathcal{D}$, the cluster tilting subcategories have infinitely many indecomposable objects, so do not correspond to cluster tilting objects.

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\ \ The talk will show how the cluster tilting subcategories have a rich combinatorial

structure: They can be parametrised by “triangulations of the $\infty$-gon”. These are certain maximal collections of non-crossing arcs between non-neighbouring integers.

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\ \ This will be used to show how to obtain a subcategory of $\mathcal{D}$ which has all the properties of a cluster tilting subcategory, except that it is not functorially finite. There will also be remarks on how $\mathcal{D}$ generalises the situation from Dynkin type $A_n$ , and how triangulations of the $\infty$-gon are new and interesting combinatorial objects.

What do historians of mathematics do? What sort of questions do they ask? What kinds of sources do they use? This series of four informal lectures will demonstrate some of the research on history of mathematics currently being done in Oxford. The subjects range from the late Renaissance mathematician Thomas Harriot (who studied at Oriel in 1577) to the varied and rapidly developing mathematics of the seventeenth century (as seen through the eyes of Savilian Professor John Wallis, and others) to the emergence of a new kind of algebra in Paris around 1830 in the work of the twenty-year old Évariste Galois.

Each lecture will last about 40 minutes, leaving time for questions and discussion. No previous knowledge is required: the lectures are open to anyone from the department or elsewhere, from undergraduates upwards.

What do historians of mathematics do? What sort of questions do they ask? What kinds of sources do they use? This series of four informal lectures will demonstrate some of the research on history of mathematics currently being done in Oxford. The subjects range from the late Renaissance mathematician Thomas Harriot (who studied at Oriel in 1577) to the varied and rapidly developing mathematics of the seventeenth century (as seen through the eyes of Savilian Professor John Wallis, and others) to the emergence of a new kind of algebra in Paris around 1830 in the work of the twenty-year old Évariste Galois.

Each lecture will last about 40 minutes, leaving time for questions and discussion. No previous knowledge is required: the lectures are open to anyone from the department or elsewhere, from undergraduates upwards.

What do historians of mathematics do? What sort of questions do they ask? What kinds of sources do they use? This series of four informal lectures will demonstrate some of the research on history of mathematics currently being done in Oxford. The subjects range from the late Renaissance mathematician Thomas Harriot (who studied at Oriel in 1577) to the varied and rapidly developing mathematics of the seventeenth century (as seen through the eyes of Savilian Professor John Wallis, and others) to the emergence of a new kind of algebra in Paris around 1830 in the work of the twenty-year old Évariste Galois.

Each lecture will last about 40 minutes, leaving time for questions and discussion. No previous knowledge is required: the lectures are open to anyone from the department or elsewhere, from undergraduates upwards.

Starting from Morel and Voevodsky's stable homotopy theory of schemes, one defines, for each noetherian scheme of finite dimension $X$, the triangulated category $DM(X)$ of motives over $X$ (with rational coefficients). These categories satisfy all the the expected functorialities (Grothendieck's six operations), from

which one deduces that $DM$ also satisfies cohomological proper

descent. Together with Gabber's weak local uniformisation theorem,

this allows to prove other expected properties (e.g. finiteness

theorems, duality theorems), at least for motivic sheaves over

excellent schemes.