Comlab

Nature and the world of human technology are full of
networks. People like to draw diagrams of networks: flow charts,
electrical circuit diagrams, signal flow diagrams, Bayesian networks,
Feynman diagrams and the like. Mathematically-minded people know that
in principle these diagrams fit into a common framework: category
theory. But we are still far from a unified theory of networks.

 Nature and the world of human technology are full of
networks. People like to draw diagrams of networks: flow charts,
electrical circuit diagrams, signal flow diagrams, Bayesian networks,
Feynman diagrams and the like. Mathematically-minded people know that
in principle these diagrams fit into a common framework: category
theory. But we are still far from a unified theory of networks.

Nature and the world of human technology are full of
networks. People like to draw diagrams of networks: flow charts,
electrical circuit diagrams, signal flow diagrams, Bayesian networks,
Feynman diagrams and the like. Mathematically-minded people know that
in principle these diagrams fit into a common framework: category
theory. But we are still far from a unified theory of networks.

Nature and the world of human technology are full of
networks. People like to draw diagrams of networks: flow charts,
electrical circuit diagrams, signal flow diagrams, Bayesian networks,
Feynman diagrams and the like. Mathematically-minded people know that
in principle these diagrams fit into a common framework: category
theory. But we are still far from a unified theory of networks.

In recent years, surprising connections between type theory and homotopy theory have been discovered. In this talk I will recall the notions of intensional type theories and identity types. I will describe "infinity groupoids", formal algebraic models of topological spaces, and explain how identity types carry the structure of an infinity groupoid. I will finish by discussing categorical semantics of intensional type theories.

The talk will take place in Lecture Theatre B, at the Department of Computer Science.

Algebraic theories, locally presentable categories and their application to type theories. The seminar will take place in Lecture Theatre A of the Department of Computer Science.

The seminar will take place in Lecture Theatre A, Department of Computer Science.

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Contextuality and non-locality are features of quantum mechanics which stand in sharp contrast to the realistic picture underlying classical physics. We shall describe a unified geometric perspective on these notions in terms of *obstructions to the existence of global sections*. This allows general results and structural notions to be uncovered, with quantum mechanics appearing as a special case. The natural language to use here is that of sheaves and presheaves; and cohomological obstructions can be defined which witness contextuality in a number of salient examples.

This is joint work with Adam Brandenburger
 http://iopscience.iop.org/1367-2630/13/11/113036/
 http://arxiv.org/abs/1102.0264

and Shane Mansfield and Rui Soares Barbosa
 http://arxiv.org/abs/1111.3620

Introductory talk on topological quantum field theories (TQFTs) and the cobordism hypothesis, focusing on the conceptual issues involved.

The lecture will take place this Friday at 11am in Lecture Theatre A of the Department of Computer Science

We will demonstrate the following. Category theory, usually conceived as some very abstract form of metamathematics, is present everywhere around us. Explicitly, we show how it provides a kindergarten version of quantum theory, an how it will help Google to understand sentences rather than words.

Some references are:

-[light] BC (2010) "Quantum picturalism". Contemporary Physics 51, 59-83. arXiv:0908.1787 
-[a bit heavier] BC and Ross Duncan (2011) "Interacting quantum observables: categorical algebra and diagrammatics". New Journal of Physics 13, 043016. arXiv:0906.4725
-[light] New Scientist (8 December 2010) "Quantum links let computers understand language". www.cs.ox.ac.uk/people/bob.coecke/NewScientist.pdf
-[a bit heavier] BC, Mehrnoosh Sadrzadeh and Stephen Clark (2011) "Mathematical foundations for a compositional distributional model of meaning". Linguistic Analysis - Lambek Festschrift. arXiv:1003.439

In contrast to spectra, pseudospectra of large Toeplitz matrices

behave as nicely as one could ever expect. We demonstrate some

basic phenomena of the asymptotic distribution of the spectra

and pseudospectra of Toeplitz matrices and show how by employing

a few simple $C^*$-algebra arguments one can prove rigorous

convergence results for the pseudospectra. The talk is a survey

of the development since a 1992 paper by Reichel and Trefethen

and is not addressed to specialists, but rather to a general

mathematically interested audience.