Summer Term 2015

Dale Rolfsen (University of British Columbia). Knots, groups and orderings. Friday May 8th, 1.30-2.30pm, Huxley 140.

Abstract: The mathematical study of knots has a long and fascinating history and is currently a very lively area of research and application. I will discuss one of the earliest and most powerful algebraic invariants of knots: the fundamental group of its complement, commonly called the knot’s group. Although it is routine to find generators and relations for its group from a picture of a given knot, the problem of whether two presentations actually define isomorphic groups can be quite intractable. On the positive side, and rather surprisingly, all knot groups are left-orderable. This means that the elements of the group can be given a strict total ordering which is invariant under left multiplication (or right multiplication, if one prefers, but not necessarily both simultaneously). However, the groups of some knots, such as the figure-eight knot, can be given a two-sided ordering. These facts give new algebraic information about knot groups and have consequences regarding surgery and similar application of knot theory to other branches of topology. I’ll conclude by discussing some open questions, such as a conjecture relating left-orderability with Heegaard-Floer homology of 3-dimensional manifolds.

Ciprian Manolescu (UCLA). The triangulation conjecture. Friday May 15th, 1.30-2.30pm, Huxley 140.

Abstract: The triangulation conjecture stated that any ndimensional topological manifold is homeomorphic to a simplicial complex. It is true in dimensions at most 3, but false in dimension 4 by the work of Casson and Freedman. In this talk I will explain the proof that the conjecture is also false in higher dimensions. This result is based on previous work of Galewski-Stern and Matumoto, who reduced the problem to a question in low dimensions (the existence of elements of order 2 and Rokhlin invariant one in the 3-dimensional homology cobordism group). The low-dimensional question can be answered in the negative using a variant of Floer homology, Pin(2)-equivariant Seiberg-Witten Floer homology.

Peter Topping (University of Warwick). Flowing a map to a minimal immersion: dealing with singularities. Friday May 22th, 1.30-2.30pm, Huxley 140.

Abstract: Given a map from a closed surface to an arbitrary closed Riemannian manifold, there is a gradient flow of the harmonic map energy that would like to flow it to be a branched minimal immersion. This is impossible in general, so the flow develops singularities. In recent work we described what sort of singularities can happen asymptotically at infinite time. In forthcoming work we plan to describe what can happen at finite time. When combined, the work gives a description of how the arbitrary map decomposes automatically into a collection of minimal immersions.
I’m planning to make this accessible to a general audience of geometers and/or analysts.
Joint work particularly with Melanie Rupflin.

Olivier Benoist (Université de Strasbourg). Complete families of smooth space curves. Friday May 29th, 1.30-2.30pm, Huxley 140.

Abstract: In this talk, we will study complete families of smooth space curves, that is complete subvarieties of the Hilbert scheme of smooth curves in P^3. On the one hand, we will construct non-trivial examples of such families. On the other hand, we obtain necessary conditions for a complete family of smooth polarized curves to induce a complete family of non-degenerate smooth space curves. Both results rely on the study of the strong semistability of certain vector bundles.

Lu Wang (Imperial College). A topological property of asymptotically conical self-shrinkers with small entropy. Friday June 5th, 1.30-2.30pm, Huxley 340.

Abstract: We show that given an asymptotically conical self-shrinker with entropy less than or equal to that of a cylinder, the link of the asymptotic cone must separate the unit sphere into exactly two connected components, both diffeomorphic to the self-shrinker. Combining this with recent work of Brendle, we conclude that the round sphere uniquely minimizes the entropy among all non-flat two-dimensional self-shrinkers. This confirms a conjecture of Colding-Ilmanen-Minicozzi-White in dimension two. This is a joint work with Jacob Bernstein.

Dmitry Kaledin (Steklov Mathematical Institute). Cyclic homology of a different kind. Friday June 12th, 1.30-2.30pm, Huxley 340.

Abstract: Periodic cyclic homology $HP_*(A)$ of an associative algebra $A$ is a non-commutative generalization of de Rham cohomology discovered by A. Connes and B. Tsygan 30 years ago (when $A$ is the algebra of functions on a smooth algebraic variety $X$, $HP_*(A)$ reduces to the de Rham cohomology of $X$). In the very definition of $HP_*(A)$, one needs to take the total complex of a certain bicomplex. There are two ways to do it. One of them gives $0$ in characteristic $0$, so it has been largely ingored.
However, about 10 years ago it has been suggested by Kontsevich that in positive characteristic, taking the “wrong” total complex is not stupid at all and gives an interesting new homology theory. I am going to give a brief review of the classic theory of $HP_*(A)$ and de Rham cohomology, and then show that Kontsevich’s suggestion is indeed true — there is an interesting new homology theory for algebras and DG algebras that behaves as nicely as $HP_*(A)$, but differs from it at least in some important examples.

Alessandro Carlotto (Imperial College). The large-scale structure of asymptotically flat spaces. Friday June 19th, 1.30-2.30pm, Huxley 140.

Abstract: Asymptotically flat spaces naturally arise in General Relativity as models for isolated gravitational systems, very much similarly to the way our solar system is classically thought of. From a geometric perspective, they are nothing but the most basic generalization of the Euclidean space and still some of the fundamental questions about their large-scale structure (most remarkably: the problem of existence and uniqueness of solutions for the isoperimetric problem, the problem of existence and uniqueness of constant mean curvature foliations, the problem of geometrically defining the center of mass etc…) have been elusive for decades. In this talk, I will present a gallery of recent results that shed new light on these classical questions and determine a rather complete landscape of surprising beauty and simplicity. These are achieved through a variety of methods, ranging from minimal surfaces theory to (inverse) mean-curvature flow to gluing methods for scalar curvature prescription problems.
The talk will be expository and fully accessible to the general mathematical audience.

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