Completions of Categories. Seminar 1966, Zuerich

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Category Theory at McGill

We provide some criteria for a semicocycle to be linearizable as well as several easily verifiable sufficient conditions. This is joint work with Fiana Jacobzon and Guy Katriel. It can be understood as the functional inequality that arises from attempting to understand convergence of the so-called heat flow to its equilibrium state. This approach can be generalized to the setting of Markov semigroups, with a non-positive generator that posseses a spectral gap. A natural question that one can consider is: What happens if the generator does not have a spectral gap?

Can we still deduce a rate of convergence from a functional setting? This talk is based on a joint work with Jonathan Ben-Artzi. Abstract: Consider a polygon-shaped billiard table on which a ball can roll along straight lines and reflect off of edges infinitely. In work joint with Moon Duchin, Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of a polygonal billiard table and the set of possible infinite edge itineraries of balls travelling on it.

The classical results of Atiyah and Atiyah, Singer provide the homotopy types of the space FB H of Fredholm bounded operators on a Hilbert space H and of its subspace FBsa H consisting of self-adjoint operators. However, in many applications e. The space B H of bounded operators is then replaced by the space R H of regular that is, closed and densely defined operators.

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Finally, an analog, for regular operators, of results of Atiyah and Singer was obtained in by Joachim. I will describe in the talk how this result of Joachim can be included into broader picture. I will also give a simple definition of the family index for unbounded operators. All terminology will be explained during the talk. Affine Sobolev inequality of G. We also establish an analogous result for nonexpansive semigroups. Abstract: I will describe the rich connections between homogeneous dynamics and Diophantine approximation on manifolds with an emphasis on some recent developments.

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Abstract: In this talk I will give a brief survey on my Ph. The talk, correspondingly to my thesis, is divided into two parts. I will present some results from a joint work with Adam Dor-On, in which we studied maximal representations of graph tensor algebra. The second part is devoted to operator-algebras arising from noncommutative nc varieties and is based on a joint work with Orr Shalit and Eli Shamovich. The limiting absorption principle will be discussed as well. This is a joint work with S.

Joachim Lambek's Documents -

Abstract : Given n uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them? It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting flow yield such a partition—with exactly equal areas, no matter how the points are distributed.

This has an application to almost optimal matching of n uniform blue points to n uniform red points on the sphere. I will also describe open problems regarding greedy and electrostatic matching Joint work with Nina Holden and Alex Zhai Another topic where local and global optimization sharply differ appears starts from the classical overhang problem: Given n blocks supported on a table, how far can they be arranged to extend beyond the edge of the table without falling off?

With Paterson, Thorup, Winkler and Zwick we showed ten years ago that an overhang of order cube root of n is the best possible; a crucial element in the proof involves an optimal control problem for diffusion on a line segment and I will describe generalizations of this problem to higher dimensions with Florescu and Racz. Free boundary minimal surfaces in the unit 3-ball have recently attracted a lot of attention, and many new examples have been constructed.

In a seminal series of papers, A. Fraser and R. Schoen have shown that the existence of such surfaces is related to a maximisation problem for the first non-zero Steklov eigenvalue, on abstract surfaces with boundary.

Zbigniew Król: Category Theory and Philosophy: Inspirations, Problems and Methods

More precisely, it is interesting to relate this number to the topology of the surface. For the latter, a celebrated result of F.

Urbano characterises the closed minimal surfaces of the 3-sphere with minimal index. The problem of extending the definition to higher dimensions remained open until recently Georgieva, , and Solomon-Tukachinsky, If time permits, we will describe equations, a version of the open WDVV equations, which the resulting invariants satisfy. We show how to decompose the moduli-space of shapes of polyhedra and how such a decomposition can be used to solve geometric realization problems.

Abstract: Caprace and De Medts discovered that Thompson's V can be written as a group of tree almost automorphisms, allowing to embed it densely into a totally disconnected, locally compact t. Matui discovered that it can be written as the topological full group of the groupoid associated to a one-sided shift. Combining these, we find countably many different t. Handwriting comparison and identification, e. However, even in the case of modern documents, the proposed computerized solutions are quite unsatisfactory. We propose a new method for learning high quality and diverse lists using structured prediction models.

Our method is based on perturbations: learning a noise function that is particularly suitable for generating such lists. We further develop a novel method max over marginals that can distill a new high quality tree from the perturbation-based list. In experiments with cross-lingual dependency parsing across 16 languages, we show that our method can lead to substantial gains in parsing accuracy over existing methods. These curves, appropriately parametrized, emerge as traveling waves for a bi-stable Hamiltonian system that can be viewed as a conservative model for phase transitions.

This talk will be devoted to probabilistic constructions appearing in statistics and geometry. I will introduce the classical notion of VC dimension and discuss how it arises naturally in several problems. One of the questions will be the so-called epsilon-approximation problem. Lagrangian Floer cohomology is notoriously hard to compute, and is typically only possible in special cases. For the beginning of the talk, I will assume very little background knowledge of symplectic geometry.

Stable commutator length scl is a well established invariant of group elements g write scl g and has both geometric and algebraic meaning. We will show that in fact one may take a generalisation of homomorphisms letter-quasimorphisms to obtain this bound, in particular for some non-residually free groups.

This is not a mathematics or a physics talk but it is a talk about mathematicians for mathematicians and physicists. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse.

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The purpose of this talk is two-fold. It can be taught in less than one hour to first-year math students! If time permits I will mention recent developments, joint with P. Mironescu and I. Shafrir, concerning the energy required to pass from a given configuration to another one. While originated in topological data analysis, persistence modules and their barcodes provide an efficient way to book-keep homological information contained in Morse and Floer theories.

I shall describe applications of persistence barcodes to symplectic topology and geometry of Laplace eigenfunctions. This is joint work with Romain Tessera. I will talk about the critical exponent associated to an invariant random subgroup of a rank one simple Lie group G. Most of our results hold true more generally for IRS in the isometry group of any Gromov hyperbolic metric space. This is a joint work with Ilya Gehktman.

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  • Abstract : Attached. Let P be a second-order, symmetric, and nonnegative elliptic operator with real coefficients defined on noncompact Riemannian manifold M, and let V be a real valued function which belongs to the class of small perturbation potentials with respect to the heat kernel of P in M. If time permits we shall show that the parabolic Martin boundary is stable under such perturbations. This is a joint work with Prof. Yehuda Pinchover.

    The evens and odds form a partition of the integers into arithmetic progressions. It is natural to try to describe in general how the integers can be partitioned into arithmetic progressions. For example, a classic result from the 's shows that if a set of arithmetic progressions partitions the integers, there must be two arithmetic progressions with the same difference.

    In my talk I will give some of the more interesting results on this subject, report some relatively new results and present two generalizations of partitioning the integers by arithmetic progressions, namely:. The main conjecture in thefirst topic is due to A.

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    • Fraenkel and describes all the partitionshaving distinct moduli. The main conjecture in the second topic, dueto M. Schonheim, claims that in every coset partition of a group there must be two cosets of the same index. Technion—Israel Institute of Technology. Center for Mathematical Sciences. Nonpositively Curved Groups on the Mediterranean. For more information:. In this talk we examine the regularity theory of the solutions to a few examples of nonlinear PDEs.

      Completions of Categories

      Our techniques relate a problem of interest to another one - for which a richer theory is available - by means of a geometric structure, e. Ideally, information is transported along such a path, giving access to finer properties of the original equation. Our examples include elliptic and parabolic fully nonlinear problems, the Isaacs equation, degenerate examples and a double divergence model. We close the talk with a discussion on open problems and further directions of work. Bass-Serre theory is a useful tool to study groups which acts on simplicial trees by isometries.

      In this talk I discuss group actions on quasi-trees. A quasi-tree is a geodesic metric space which is quasi-isometric to a simplicial tree. I discuss an axiomatic method to produce group actions on quasi-trees for a given group.