## Geometry and Topology

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A fundamental result in 3-dimensional contact topology due to Ding-Geiges tells us that any contact 3-manifold can be obtained via doing a surgery on a Legendrian link in the standard contact 3-sphere. So it's natural to ask how simple or complicated a surgery diagram could be for a given contact manifold? Contact surgery number is a measure of  this complexity. In this talk, I will discuss this notion of complexity along with some examples. This is joint work with Marc Kegel.

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This series of talks will discuss connections between Riemannian geometry and contact topology. Both structures have deep connections to the topology of 3-manifolds, but there has been little study of the interactions between them (at least the implications in contact topology). We will see that there are interesting connections between curvature and properties of contact structures. The talks will give a brief review of both Riemannian geometry and contact topology and then discuss various was one might try to connect them.

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I read Benoist's paper Convexes Divisibles IV (2006, Invent. Math.), and will talk about it. The main result is a striking structural theorem for triangles in the boundaries of 3-dimensional properly convex divisible domains O that are not strictly convex (which exist). These bound "flats" in O. Benoist shows that every Z^2 subgroup of the group G preserving O preserves a unique such triangle. Conversely, all such triangles are disjoint and any such triangle descends to either a torus or Klein bottle in the quotient M = O/G (and so must have many symmetries!).

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Clifford algebra was first developed to describe Maxwell's equations, but the subject has found applications in quantum mechanics, computer graphics, robotics, and even machine learning, way beyond its original purpose. In topology and geometry, Clifford algebra appears in the proofs of the celebrated Atiyah-Singer Index Theorem and Bott Periodicity; it is fundamental to the understanding of spin structures on Riemannian manifolds. Despite its algebraic nature, it somehow gives us the power to understand and manipulate geometry.

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The goal of this talk is to explore curve graphs, which are combinatorial tools that encode topological information about surfaces. We focus on variants of the fine curve graph of a surface. The fine curve graph has its vertices essential simple closed curves on the surface and its edges connect pairs of curves that are disjoint.

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Virtual knot theory is a variant of classical knot theory in which one allows a new type of crossing called a "virtual" crossing. It was originally developed by Louis Kauffman in order to study the Jones polynomial but has since developed into its own field and has genuine significance in low dimensional topology. One notable interpretation is that virtual knots are equivalent to knots in thickened surfaces. In this talk we'll introduce virtual knots and show why they are a natural extension of classical knots.

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This series of talks will discuss connections between Riemannian geometry and contact topology. Both structures have deep connections to the topology of 3-manifolds, but there has been little study of the interactions between them (at least the implications in contact topology). We will see that there are interesting connections between curvature and properties of contact structures. The talks will give a brief review of both Riemannian geometry and contact topology and then discuss various was one might try to connect them.

Series
Time
for
Location
Speaker
Organizer

This series of talks will discuss connections between Riemannian geometry and contact topology. Both structures have deep connections to the topology of 3-manifolds, but there has been little study of the interactions between them (at least the implications in contact topology). We will see that there are interesting connections between curvature and properties of contact structures. The talks will give a brief review of both Riemannian geometry and contact topology and then discuss various was one might try to connect them.

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This talk has two goals. The first is to talk through Keynes-Newton’s construction of minimal non-uniquely ergodic interval exchange transformations. The second is to explain why I’m talking about this in the student topology seminar.