Geometry and Topology

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Alesker has introduced the notion of a smooth valuation on a smooth manifold M. This is a special kind of set function, defined on sufficiently regular compact subsets A of M, extending the corresponding idea from classical convexity theory. Formally, a smooth valuation is a kind of curvature integral; informally, it is a sum of Euler characteristics of intersections of A with a collection of objects B. Smooth valuations admit a natural multiplication, again due to Alesker.

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A fundamental result in equivariant homotopy theory due to Elmendorf states that the homotopy theory of G-spaces, with w.e.s measured on all fixed points, is equivalent to the homotopy theory of G-coefficient systems in spaces, with w.e.s measured at each level of the system. Furthermore, Elmendorf’s result is rather robust: analogue results can be shown to hold for, among others, the categories of (topological) categories and operads. However, it has been known for some time that in the G-operad case such a result
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Boothby Wang fibrations are historically important examples of contact manifolds and it turns out that we can equip these contact manifolds with extra structures, namely K-contact structures. Based on the study of the relation of these examples and the regularity properties of the corresponding Reeb vector fields, works of Boothby, Wang, Thomas and Rukimbira gives a classification of K-contact structures.
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A knot is a simple closed curve in the 3-space. Knots appeared as one of the first objects of study in topology. At first knot theory was rather isolated in mathematics. Lately due to newly discovered invariants and newly established connections to other branches of mathematics, knot theory has become an attractive and fertile area where many interesting, intriguing ideas collide. In this talk we discuss a new class of knot invariants coming out of the Jones polynomial and an algebra of surfaces based on knots (skein algebra) which has connections to many
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We explain the (classical) transverse Markov Theorem which relates transverse links in the tight three sphere to classical braid closures. We review an invariant of such transverse links coming from knot Floer homology and discuss some applications which appear in the literature.
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We already know that the Euclidean unit ball is at the center of the Banach-Mazur compactum, however its structure is still being explored to this day. In 1987, Szarek and Talagrand proved that the maximum distance $R_{\infty} ^n$ between an arbitrary $n$-dimensional normed space and $\ell _{\infty} ^n$, or equivalently the maximum distance between a symmetric convex body in $\mathbb{R} ^n$ and the $n$-dimensional unit cube is bounded above by $c n^{7/8}$. In this talk, we will discuss a related paper by A.

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Many problems in algebraic geometry involve counting solutions to geometric problems. The number of intersection points of two projective planar curves and the number of lines on a cubic surface are two classical problems in this enumerative geometry. Using A1-homotopy theory, one can gain new insights to old enumerative problems. We will outline some results in A1-enumerative geometry, including the speaker’s current work on Bézout’s Theorem.
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Let K be a link braided about an open book (B,p) supporting a contact manifold (Y,x). K and B are naturally transverse links. We prove that the hat version of the transverse link invariant defined by Baldwin, Vela-Vick and Vertesi is non-zero for the union of K with B. As an application, we prove that the transverse invariant of any braid having fractional Dehn twist coefficient greater than one is non-zero. This generalizes a theorem of Plamenevskaya for classical braid closures.

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