Past Geometry and Analysis Seminar

3 March 2014
14:15
Giuseppe Tinaglia
Abstract
In this talk I will discuss results on the geometry of constant mean curvature (H\neq 0) disks embedded in R^3. Among other things I will prove radius and curvature estimates for such disks. It then follows from the radius estimate that the only complete, simply connected surface embedded in R^3 with constant mean curvature is the round sphere. This is joint work with Bill Meeks.
  • Geometry and Analysis Seminar
17 February 2014
14:15
Goncalo Oliveira
Abstract
The Monopole (Bogomolnyi) equations are Geometric PDEs in 3 dimensions. In this talk I shall introduce a generalization of the monopole equations to both Calabi Yau and G2 manifolds. I will motivate the possible relations of conjectural enumerative theories arising from "counting" monopoles and calibrated cycles of codimension 3. Then, I plan to state the existence of solutions and sketch how these examples are constructed.
  • Geometry and Analysis Seminar
3 February 2014
14:15
Tara Holm
Abstract
A folded symplectic form on a manifold is a closed 2-form with the mildest possible degeneracy along a hypersurface. A special class of folded symplectic manifolds are the origami manifolds. In the classical case, toric symplectic manifolds can classified by their moment polytope, and their topology (equivariant cohomology) can be read directly from the polytope. In this talk we examine the toric origami case: we will recall how toric origami manifolds can also be classified by their combinatorial moment data, and present some theorems, almost-theorems, and conjectures about the topology of toric origami manifolds.
  • Geometry and Analysis Seminar
27 January 2014
14:15
Brent Pym
Abstract

Noncommutative projective geometry is the study of quantum versions of projective space and other projective varieties.  Starting with the celebrated work of Artin, Tate and Van den Bergh on noncommutative projective planes, a substantial theory of noncommutative curves and surfaces has been developed, but the classification of noncommutative versions of projective three-space remains unknown.  I will explain how a portion of this classification can be obtained, via deformation quantization, from a corresponding classification of holomorphic foliations due to Cerveau and Lins Neto.  In algebraic terms, the result is an explicit description of the deformations of the polynomial ring in four variables as a graded Calabi--Yau algebra.

  • Geometry and Analysis Seminar
20 January 2014
14:15
Andrew Dancer
Abstract

We produce new families of steady and expanding Ricci solitons
that are not of Kahler type. In the steady case, the asymptotics are
a mixture of the Hamilton cigar and the Bryant soliton paraboloid
asymptotics. We obtain some examples of Ricci solitons on homeomorphic
but non-diffeomorphic spaces. We also find numerical evidence of solitons
with more complicated topology.

  • Geometry and Analysis Seminar
2 December 2013
14:00
Yanki Lekili
Abstract
We consider Fano threefolds on which SL(2,C) acts with a dense open orbit. This is a finite list of threefolds whose classification follows from the classical work of Mukai-Umemura and Nakano. Inside these threefolds, there sits a Lagrangian space form given as an orbit of SU(2). We prove this Lagrangian is non-displaceable by Hamiltonian isotopies via computing its Floer cohomology over a field of non-zero characteristic. The computation depends on certain counts of holomorphic disks with boundary on the Lagrangian, which we explicitly identify. This is joint work in progress with Jonny Evans.
  • Geometry and Analysis Seminar
25 November 2013
14:00
Kirill Krasnov
Abstract
I will define and describe in some details a large class of gauge theories in four dimensions. These theories admit a variational principle with the action a functional of only the gauge field. In particular, no metric appears in the Lagrangian or is used in the construction of the theory. The Euler-Lagrange equations are second order PDE's on the gauge field. When the gauge group is taken to be SO(3), a particular theory from this class can be seen to be (classically) equivalent to Einstein's General Relativity. All other points in the SO(3) theory space can be seen to describe "deformations" of General Relativity. These keep many of GR's properties intact, and may be important for quantum gravity. For larger gauge groups containing SO(3) as a subgroup, these theories can be seen to describe gravity plus Yang-Mills gauge fields, even though the associated geometry is much less understood in this case.
  • Geometry and Analysis Seminar

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