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Casson invariant is defined for the class of oriented integral homology 3-spheres. It satisfies certain properties, and reduce to Rohlin invariant after mod 2. We will define Casson invariant as half of the algebraic intersection number of irreducible representation spaces (space consists of representations of fundamental group to SU(2)), and then prove this definition satisfies the expected properties.
We define the notion of a knot type having Legendrian large cables and
show that having this property implies that the knot type is not uniformly thick.
Moreover, there are solid tori in this knot type that do not thicken to a solid torus
with integer sloped boundary torus, and that exhibit new phenomena; specifically,
they have virtually overtwisted contact structures. We then show that there exists
an infinite family of ribbon knots that have Legendrian large cables. These knots fail
In this talk, I will discuss progress in our understanding of Legendrian surfaces. First, I will present a new construction of Legendrian surfaces and a direct description for their moduli space of microlocal sheaves. This Legendrian invariant will connect to classical incidence problems in algebraic geometry and the study of flag varieties, which we will study in detail. There will be several examples during the talk and, in the end, I will indicate the relation of this theory to the study of framed local systems on a surface.
The notion of an acylindrically hyperbolic group was introduced by Osin as a generalization of non-elementary hyperbolic and relative hyperbolic groups. Ex- amples of acylindrically hyperbolic groups can be found in mapping class groups, outer automorphism groups of free groups, 3-manifold groups, etc. Interesting properties of acylindrically hyperbolic groups can be proved by applying techniques such as Monod-Shalom rigidity theory, group theoretic Dehn filling, and small cancellation theory.
We formulate a geometric nonlinear theory of the mechanics of accretion. In this theory the material manifold of an accreting body is represented by a time-dependent Riemannian manifold with a time-independent metric that at each point depends on the state of deformation at that point at its time of attachment to the body, and on the way the new material isadded to the body. We study the incompatibilities induced by accretion through the analysis of the material metric and its curvature in relation to the foliated structure of the accreted body.
In the setup of classical knot theory---the study of embeddings of the circle into S^3---we recall two examples of classical knot invariants: the Alexander polynomial and the Seifert form.
In this talk, we will examine the relationship between homotopy, topological isotopy, and smooth isotopy of surfaces in 4-manifolds. In particular, we will discuss how to produce (1) examples of topologically but not smoothly isotopic spheres, and (2) a smooth isotopy from a homotopy, under special circumstances (i.e. Gabai's recent work on the ``4D Lightbulb Theorem").
It is well known that two knots in S^3 are ambiently isotopic if and only if there is an orientation preserving automorphism of S^3 carrying one knot to the other. In this talk, we will examine a family of smooth 4-manifolds in which the analogue of this fact does not hold, i.e. each manifold contains a pair of smoothly embedded, homotopic 2-spheres that are related by a diffeomorphism, but are not smoothly isotopic.
An element of the braid group can be visualized as a collection of n strings that are braided (like a hair braid). Braid groups are ubiquitous in mathematics in science, as they record the motions of a number of points in the plane. These points can be autonomous vehicles, particles in a 2-dimensional medium, or roots of a polynomial. We will give an introduction to and a survey of braid groups, and discuss what is known about homomorphisms between braid groups.
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