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Series: Geometry Topology Seminar

Among n-dimensional regions with fixed volume, which one hasthe least boundary? This question is known as an isoperimetricproblem; its nature depends on what is meant by a "region". I willdiscuss variations of an isoperimetric problem known as thegeneralized Cartan-Hadamard conjecture: If Ω is a region in acomplete, simply connected n-manifold with curvature bounded above byκ ≤ 0, then does it have the least boundary when the curvature equalsκ and Ω is round? This conjecture was proven when n = 2 by Weil andBol; when n = 3 by Kleiner, and when n = 4 and κ = 0 by Croke. Injoint work with Benoit Kloeckner, we generalize Croke's result to mostof the case κ < 0, and we establish a theorem for κ > 0. It was originally inspired by the problem of finding the optimal shape of aplanet to maximize gravity at a single point, such as the place wherethe Little Prince stands on his own small planet.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Recently, a "symplectic duality" between D-modules on certainpairs of algebraic symplectic manifolds was discovered, generalizingclassic work of Beilinson-Ginzburg-Soergel in geometric representationtheory. I will discuss how such dual spaces (some known and some new) arisenaturally in supersymmetric gauge theory in three dimensions.

Series: Geometry Topology Seminar

Groups, rings, modules, and compact Hausdorff spaces have underlying sets ("forgetting" structure) and admit "free" constructions. Moreover, each type of object is completely characterized by the shadow of this free-forgetful duality cast on the category of sets, and this syntactic encoding provides formulas for direct and inverse limits. After we describe a typical encounter with adjunctions, monads, and their algebras, we introduce a new "homotopy coherent" version of this adjoint duality together with a graphical calculus that is used to define a homotopy coherent algebra in quite general contexts, such as appear in abstract homotopy theory or derived algebraic geometry.

Series: Geometry Topology Seminar

The Lickorish-Wallace theorem states that every closed, connected, orientable three-manifold can be expressed as surgery on a link in the three-sphere (i.e., remove a neighborhood of a disjoint union of embedded $S^1$'s from $S^3$ and re-glue). It is natural to ask which three-manifolds can be obtained by surgery on a single knot in the three-sphere. We discuss a new way to obstruct integer homology spheres from being surgery on a knot and give some examples. This is joint work with Jennifer Hom and Cagri Karakurt.

Series: Geometry Topology Seminar

I will sketch how to detect nontrivial higher homotopy groups of the space of complete nonnegatively curved metrics on an open manifold.

Series: Geometry Topology Seminar

The algebraic K-theory of the sphere spectrum, K(S), encodes significant information in both homotopy theory and differential topology. In order to understand K(S), one can apply the techniques of chromatic homotopy theory in an attempt to approximate K(S) by certain localizations K(L_n S). The L_n S are in turn approximated by the Johnson-Wilson spectra E(n) = BP[v_n^{-1}], and it is not unreasonable to expect to be able to compute K(BP). This would lead inductively to information about K(E(n)) via the conjectural fiber sequence K(BP) --> K(BP) --> K(E(n)). In this talk, I will explain the basics of the K-theory of ring spectra, define the ring spectra of interest, and construct some actual localization sequences in their K-theory. I will then use trace methods to show that it the actual fiber of K(BP) --> K(E(n)) differs from K(BP), meaning that the situation is more complicated than was originally hoped. All this is joint work with Ben Antieau and Tobias Barthel.

Series: Geometry Topology Seminar

We prove that the torsion of any smooth closed curve in Euclidean space which bounds a simply connected locally convex surface vanishes at least 4 times (vanishing of torsion means that the first 3 derivatives of the curve are linearly dependent). This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in 3-space. Furthermore, our result generalizes the 4 vertex theorem of Sedykh for convex space curves, and thus constitutes a far reaching extension of the classical 4 vertex theorem for planar curves. The proof follows from an extensive study of the structure of convex caps in a locally convex surface.

Series: Geometry Topology Seminar

These are two half an hour talks.Evgeny's abstract: The most useful approach to a classication of 3-manifolds is the complexity theory foundedby S. Matveev. Unfortunately, exact values of complexity are known for few infinite seriesof 3-manifold only. We present the results on complexity for two infinite series of hyperbolic3-manifolds with boundary.Andrei's abstract: We define coordinates on virtual braid groups. We prove that these coordinates are faithful invariants of virtual braids on two strings, and present evidence that they are also very powerful invariants for general virtual braids.The talk is based on the joint work with V.Bardakov and B.Wiest.

Series: Geometry Topology Seminar

Construction of pseudo-Anosov elements of mapping class groups of surfaces is a non-trivial task. In 1988, Penner gave a very general construction of pseudo-Anosov mapping classes, and he conjectured that all pseudo-Anosov mapping classes arise this way up to finite power. This conjecture was known to be true on some simple surfaces, including the torus. In joint work with Hyunshik Shin, we resolve this conjecture for all surfaces.