The cluster category of Dynkin type $A_\infty$
Abstract
\ \ 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.
17:00
Editing the manuscripts of Évariste Galois (1811–1832)
Abstract
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.
17:00
The extensive correspondence of John Wallis (1616–1703)
Abstract
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.
17:00
The life, work, and reputation of Thomas Harriot (1560–1621)
Abstract
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.
(HoRSe seminar) Motivic sheaves over excellent schemes
Abstract
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.
Bilinear Forms and Differential Forms under Field Extensions
Abstract
An important problem in algebra is the study of algebraic objects
defined over fields and how they behave under field extensions,
for example the Brauer group of a field, Galois cohomology groups
over fields, Milnor K-theory of a field, or the Witt ring of bilinear
forms over
a field. Of particular interest is the determination
of the kernel of the restriction map when passing to a field extension.
We will give an overview over some known results concerning the
kernel of the restriction map from the Witt ring of a field to the
Witt ring of an extension field. Over fields of characteristic
not two, general results are rather sparse. In characteristic two,
we have a much more complete picture. In this talk, I will
explain the full solution to this problem for extensions that are
given by function fields of hypersurfaces over fields of
characteristic two. An important tool is the study of the
behaviour of differential forms over fields of positive
characteristic under field extensions. The result for
Witt rings in characteristic two then follows by applying earlier
results by Kato, Aravire-Baeza, and Laghribi. This is joint
work with Andrew Dolphin.