Events

Nikon Kurnosov, University College London

I will discuss examples of irreducible holomorphic symplectic manifolds, which are essentially hyperkahler in a Kahler case. However, without the Kahler assumption holomorphically symplectic manifolds are not necessarily hyperkahler. Higher-dimensional simply-connected non-Kahler holomorphically symplectic manifolds have been constructed by Guan and by Bogomolov. I will talk about the differences and similiarities in the geometry of hyperkahler and BG-manifolds and about ideas which can lead to the proof of the finiteness of the deformation types in any dimension.

MC 5417

Note special time for this talk.

Andriy Haydys, Free University of Brussels

The Alexander polynomial is an invariant of links which appears in various places in low dimensional topology. In this talk I shall focus on a relation between the Alexander polynomial and the moduli space of flat stable PSL(2,C) connections on closed three-manifolds. Surprisingly, it turns out that the Alexander polynomial carries some information about the blow up behavior of the sequences of flat stable PSL(2,C)-connections.

MC 5417

Note special time for this talk.

Jun-Yong Park, University of Melbourne

I will begin by reviewing a classical algorithm of Tate with some explicit polynomial calculations. Combining this with twisted stable maps theory leads us to the height moduli of rational points of fixed stacky height on the fine modular curve Mbar_{1,1} over global function fields. We will then compute arithmetic invariants of elliptic surfaces moduli via topological methods and give applications to counting elliptic curves over Fq(t).

MC 5417

Steve Rayan, University of Saskatchewan

Finite quotient singularities play a distinguished role in mathematics, intertwining algebraic geometry, hyperkähler geometry, representation theory, and integrable systems.  I will explain the correspondences at play here and how they culminate in Nakajima quiver varieties, which are of intense interest in geometric representation theory and physics.  I will explain some recent work of G. Bellamy, A. Craw, T. Schedler, H. Weiss, and myself in which we show that, remarkably, all of the resolutions of a particular finite quotient singularity are realized by a certain Nakajima quiver variety, namely that of the 4-pointed star-shaped quiver.  I will place this work in the wider context of the search for McKay-type correspondences for finite subgroups of SL(n,C) on the one hand, and of the construction of finite-dimensional approximations to meromorphic Hitchin systems and their hyperkähler metrics on the other hand.  The Hitchin system perspective draws upon my joint work with J. Fisher L. Schaponsik, respectively. Time permitting, I will speculate upon the symplectic duality of Higgs and Coulomb branches in this setting.

MC 5417

Lorenzo Fatibene, University of Torino

We relax spin structures to discuss Dirac fields in interaction with gravity and their conservation laws. That is an example of a more general framework, called gauge natural field theories, which accommodates theories which are generally covariant and they have an additional gauge symmetry. The framework is geometrically expressed and it accommodates a geometric formulation of both variational calculus and conservation laws. We shall also briefly compare with other frameworks such as the double cover of frame bundles.

MC 5479

Jesse Madnick, University of Oregon

In C^n, mean curvature flow preserves the class of Lagrangian submanifolds, a fact known as "Lagrangian mean curvature flow" (LMCF). As LMCF typically forms finite-time singularities, it is of interest to understand the blowup models of such singularities, as well as the soliton solutions.

In this talk, we'll consider the mean curvature flow of Lagrangians that are cohomogeneity-one under the action of a compact Lie group. Interestingly, each such Lagrangian lies in a level set \mu^{-1}(c) of the moment map \mu, and mean curvature flow preserves this containment. Using this, we'll classify all cohomogeneity-one shrinking, expanding, and translating solitons. Further, in the zero level set \mu^{-1}(0), we'll classify the Type I and Type II blowup models of cohomogeneity-one LMCF singularities.

Finally, given any cohomogeneity-one special Lagrangian in \mu^{-1}(0), we'll show that it arises as a Type II blowup, thereby yielding infinitely many new singularity models. This is joint work with Albert Wood.

MC 5417

Jesse Madnick, University of Oregon

In C^n, mean curvature flow preserves the class of Lagrangian submanifolds, a fact known as "Lagrangian mean curvature flow" (LMCF).  As LMCF typically forms finite-time singularities, it is of interest to understand the blowup models of such singularities, as well as the soliton solutions.

In this talk, we'll consider the mean curvature flow of Lagrangians that are cohomogeneity-one under the action of a compact Lie group.  Interestingly, each such Lagrangian lies in a level set \mu^{-1}(c) of the moment map \mu, and mean curvature flow preserves this containment.  Using this, we'll classify all cohomogeneity-one shrinking, expanding, and translating solitons.  Further, in the zero level set \mu^{-1}(0), we'll classify the Type I and Type II blowup models of cohomogeneity-one LMCF singularities.

Finally, given any cohomogeneity-one special Lagrangian in \mu^{-1}(0), we'll show that it arises as a Type II blowup, thereby yielding infinitely many new singularity models.  This is joint work with Albert Wood.

MC 5417

Stephen Melczer, Department of Combinatorics & Optimization, University of Waterloo

The field of analytic combinatorics adapts techniques from complex analysis, algebraic and differential geometry, topology, and computer algebra to create algorithmic methods for the study of combinatorial objects. Classical analytic combinatorics derives the asymptotic behaviour of a combinatorial sequence by manipulating univariate Cauchy integral representations. The newer area of analytic combinatorics in several variables (ACSV) attempts to generalize this approach to multivariate sequences and limit theorems. In this talk we describe the basics of analytic combinatorics and ACSV before discussing how (new modifications of) results from stratified Morse theory help characterize Cauchy integral deformations that lead to asymptotic expansions in the multivariate setting. We then show how combining Morse-theoretic decompositions with effective algorithms for numeric analytic continuation from computer algebra allow for the computation of certain homology coefficients which seem very difficult to compute using topology alone. Open topological problems in this area will be discussed, along with examples from a range of applications including queuing theory, the complexity of biological networks, algebraic statistics, and the analysis of algorithms.

QNC 2501

Giovanni Russo, Florida International University

Nearly Kahler manifolds are Riemannian spaces equipped with an almost Hermitian structure of special type. In dimension six, nearly Kahler metrics are Einstein with positive scalar curvature, and have interesting connections with G2 and spin geometry. At present there are very few compact examples, which are either homogeneous or of cohomogeneity one.

In this talk we explain a theory of nearly Kahler six-manifolds admitting a two-torus symmetry. The torus-action yields a multi-moment map, which we use as a Morse function to understand the structure of the whole manifold. In particular, we show how the local geometry of a nearly Kahler six-manifold can be recovered from three-dimensional data, and discuss connections with GKM theory.

QNC 2501

Mykola Matviichuk, Imperial College London

Poisson brackets are a crucial concept bridging classical mechanics and quantum mechanics. In this talk, I will present a novel approach to constructing holomorphic Poisson brackets that satisfy the log symplectic non-degeneracy condition and have a geometric connection to elliptic curves. This method relies on the deformation theory and involves combinatorics of graphs with decorations. I will demonstrate its effectiveness by creating new examples, as far as my knowledge extends, on complex projective bundles over a polydisc. Additionally, I will revisit and rediscover the Feigin-Odesski log symplectic brackets on projective spaces.

QNC 2501