Geometry seminar

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Date / Time

Speaker

Title

Location

30 September 2020
16:15-17:15

Dr. Patrick Orson
ETH Zurich, Switzerland

Event Details

Title	Topologically embedding spheres in knot traces
Speaker, Affiliation	Dr. Patrick Orson, ETH Zurich, Switzerland
Date, Time	30 September 2020, 16:15-17:15
Location	HG G 19.1
Abstract	The problem of when a given 2-dimensional homology class in a 4-manifold can be represented by an embedded 2-sphere has been studied by many authors. In this talk we will consider the question for knot traces; these are particularly simple 4-manifolds, described by taking the closed 4-ball and glueing a 2-handle to it along a framed knot in the boundary 3-sphere. The resulting manifold is homotopy equivalent to the 2-sphere, and we consider the embedding problem for a generator of the second homology group. I will describe recent joint work in which we completely characterise when this homology class can be represented by an embedded 2-sphere whose complement has abelian fundamental group. The answer is in terms of classical and easily computable invariants of the knot in the 3-sphere. Given time I will describe some knot-theoretic corollaries, relating to the study of slice knots, and a uniqueness result for our sphere embeddings.

Topologically embedding spheres in knot tracesread_more

7 October 2020
16:15-17:15

Merlin Incerti-Medici
ETH Zurich, Switzerland

Event Details

Title	Circumcenter extensions on Hadamard manifolds
Speaker, Affiliation	Merlin Incerti-Medici, ETH Zurich, Switzerland
Date, Time	7 October 2020, 16:15-17:15
Location	HG G 19.1
Abstract	Given a geodesically complete CAT(-1) space, we can associate to it a boundary at infinity which can be equipped with a cross ratio. It is well known that the boundary, together with the cross ratio contains a lot of information about the geometry of the interior space. One way to make this relationship explicit uses a construction called the circumcenter extension. It turns out that the circumcenter extension can be defined for a large class of Hadamard manifolds that have flat regions and still provide interesting results. In this talk, we will survey known results, explain the construction of the circumcenter extension, and show that the boundary roughly determines the interior space for a large class of manifolds.

Circumcenter extensions on Hadamard manifoldsread_more

14 October 2020
16:15-17:15

Dr. Benjamin Brück
ETH Zurich, Switzerland

Event Details

Title	Between Tits buildings and free factor complexes
Speaker, Affiliation	Dr. Benjamin Brück, ETH Zurich, Switzerland
Date, Time	14 October 2020, 16:15-17:15
Location	HG G 19.1
Abstract	Much of the modern treatment of automorphism groups of free groups ismotivated by analogies with arithmetic groups. I will present a new family of simplicial complexes interpolating between two well-studied objects associated to these classes of groups: the free factor complex and the Tits building of GL_n(Q). Each of the new complexes is associated to the automorphism group Aut(A_\Gamma) of a right-angled Artin group and has the homotopy type of a wedge of spheres. The dimension of these spheres forms a new invariant associated to Aut(A_\Gamma). These complexes can also be seen as an Aut(A_\Gamma)-analogue of the curve complex.

Between Tits buildings and free factor complexesread_more (CANCELLED)

21 October 2020
16:15-17:15

Dr. Claire Burrin
ETH Zurich, Switzerland

Event Details

Title	A sparse equidistribution problem for expanding horocycles on the modular surface
Speaker, Affiliation	Dr. Claire Burrin, ETH Zurich, Switzerland
Date, Time	21 October 2020, 16:15-17:15
Location	HG G 19.1
Abstract	The orbits of the horocycle flow on surfaces are classified: each orbit is either dense or a closed horocycle around a cusp. In particular, expanding closed horocycles are themselves asymptotically dense, and in fact become equidistributed on the surface. The precise rate of equidistribution is of interest; on the modular surface, Zagier observed that a particular rate is equivalent to the Riemann hypothesis being true. In this talk, we explore the asymptotic behavior of evenly spaced points along an expanding closed horocycle on the modular surface. In this problem, the number of sparse points is made to depend on the expansion rate, and the difficulty is that these points are no more invariant under the horocycle flow. This is based on joint work with Uri Shapira and Shucheng Yu.

A sparse equidistribution problem for expanding horocycles on the modular surfaceread_more

28 October 2020
16:15-17:15

Dr. Benjamin Brück
ETH Zurich, Switzerland

Event Details

Title	Between Tits buildings and free factor complexes
Speaker, Affiliation	Dr. Benjamin Brück, ETH Zurich, Switzerland
Date, Time	28 October 2020, 16:15-17:15
Location	HG G 19.1
Abstract	Much of the modern treatment of automorphism groups of free groups ismotivated by analogies with arithmetic groups. I will present a new family of simplicial complexes interpolating between two well-studied objects associated to these classes of groups: the free factor complex and the Tits building of GL_n(Q). Each of the new complexes is associated to the automorphism group Aut(A_\Gamma) of a right-angled Artin group and has the homotopy type of a wedge of spheres. The dimension of these spheres forms a new invariant associated to Aut(A_\Gamma). These complexes can also be seen as an Aut(A_\Gamma)-analogue of the curve complex.

Between Tits buildings and free factor complexesread_more

Organizers: Marc Burger, Manfred Einsiedler, Peter Feller, Alessandra Iozzi, Urs Lang, Viktor Schröder

Archive: SS 24 AS 23 SS 23 AS 22 SS 22 AS 21 AS 20 SS 20 AS 19 SS 19 AS 18 SS 18 AS 17 SS 17 AS 16 SS 16 AS 15 SS 15 AS 14 SS 14 AS 13 SS 13 AS 12 SS 12 AS 11 SS 11 AS 10 SS 10 AS 09 SS 09