This is a survey talk on the knot concordance group and the homology cobordism group.
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In this talk I will describe those linear subspaces of $\mathbf{R}^d$ which can be formed by taking the linear span of lattice points in a half-open parallelepiped. I will draw some connections between this problem and Keith Ball's cube slicing theorem, which states that the volume of any slice of the unit cube $[0,1]^d$ by a codimension-$k$ subspace is at most $2^{k/2}$.
Mapping classes are the natural topological symmetries of surfaces. Their study is often restricted to the orientation-preserving ones, which form a normal subgroup of index two in the group of all mapping classes. In this talk, we discuss orientation-reversing mapping classes. In particular, we show that Lehmer's question from 1933 on Mahler measures of integer polynomials can be reformulated purely in terms of a comparison between orientation-preserving and orientation-reversing mapping classes.
Unlike
symplectic structures in 4-manioflds, contact structures are abundant in
3-dimension. Martinet showed that there exists a contact structure on any
closed oriented 3-manifold. After that Lutz showed that there exist a contact
structure in each homotopy class of plane fields. In this talk, we will review
the theorems of Lutz and Martinet.
It is a classical theorem in algebraic topology that the loop space of a
suitable Lie group can be modeled by an infinite dimensional variety,
called the loop Grassmannian. It is also well known that there is an
algebraic analog of loop Grassmannians, known as the affine Grassmannian
of an algebraic groop (this is an ind-variety). I will explain how in
motivic homotopy theory, the topological result has the "expected"
analog: the Gm-loop space of a suitable algebraic group is
A^1-equivalent to the affine Grassmannian.
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.
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
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.
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
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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