Past Geometry and Analysis Seminar

21 May 2018
14:15
Momchil Konstantinov
Abstract

Lagrangian Floer cohomology can be enriched by using local coefficients to record some homotopy data about the boundaries of the holomorphic disks counted by the theory. In this talk I will explain how one can do this under the monotonicity assumption and when the Lagrangians are equipped with local systems of rank higher than one. The presence of holomorphic discs of Maslov index 2 poses a potential obstruction to such an extension. However, for an appropriate choice of local systems the obstruction might vanish and, if not,
one can always restrict to some natural unobstructed subcomplexes. I will showcase these constructions with some explicit calculations for the Chiang Lagrangian in CP^3 showing that it cannot be disjoined from RP^3 by a Hamiltonian isotopy, answering a question of Evans-Lekili. Time permitting, I will also discuss some work-in-progress on the topology of monotone Lagrangians in CP^3, part of which follows from more general joint work with Jack Smith on the topology of monotone Lagrangians of maximal Maslov number in
projective spaces.

 

  • Geometry and Analysis Seminar
14 May 2018
14:15
Arend Bayer
Abstract

Stability conditions on derived categories of algebraic varieties and their wall-crossings have given algebraic geometers a number of new tools to study the geometry of moduli spaces of stable sheaves. In work in progress with Macri, Lahoz, Nuer, Perry and Stellari, we are extending this toolkit to a the "relative" setting, i.e. for a family of varieties. Our construction comes with relative moduli spaces of stable objects; this gives additional ways of constructing new families of varieties from a given family, thereby potentially relating different moduli spaces of varieties.

 

  • Geometry and Analysis Seminar
30 April 2018
14:15
Kelli Francis-Staite
Abstract

A C^infinity scheme is a version of a scheme that uses a maximal spectrum. The category of C^infinity schemes contains the category of Manifolds as a full subcategory, as well as being closed under fibre products. In other words, this category is equipped to handle intersection singularities of smooth spaces.

While originally defined in the set up of Synthetic Differential Geometry, C^infinity schemes have more recently been used to describe derived manifolds, for example, the d-manifolds of Joyce. There are applications of this in Symplectic Geometry, such as the describing the moduli space of J-holomorphic forms.

In this talk, I will describe the category of C^infinity schemes, and how this idea can be extended to manifolds with corners. If time, I will mention the applications of this in derived geometry.

  • Geometry and Analysis Seminar
23 April 2018
14:15
Esther Cabezas-Rivas
Abstract

We study Brownian motion and stochastic parallel transport on Perelman's almost Ricci flat manifold,  whose dimension depends on a parameter $N$ unbounded from above. By taking suitable projections we construct sequences of space-time Brownian motion and stochastic parallel transport whose limit as $N\to \infty$ are the corresponding objects for the Ricci flow. In order to make precise this process of passing to the limit, we study the martingale problems for the Laplace operator on Perelman’s manifold and for the horizontal Laplacian on the corresponding orthonormal frame bundle.

As an application, we see how the characterizations of two-sided bounds on the Ricci curvature established by A. Naber applied to Perelman's manifold lead to the inequalities that characterize solutions of the Ricci flow discovered by Naber and Haslhofer.

This is joint work with Robert Haslhofer.

 

  • Geometry and Analysis Seminar
5 March 2018
14:15
Maxence Mayrand
Abstract

Symplectic reduction is the natural quotient construction for symplectic manifolds. Given a free and proper action of a Lie group G on a symplectic manifold M, this process produces a new symplectic manifold of dimension dim(M) - 2 dim(G). For non-free actions, however, the result is usually fairly singular. But Sjamaar-Lerman (1991) showed that the singularities can be understood quite precisely: symplectic reductions by non-free actions are partitioned into smooth symplectic manifolds, and these manifolds fit nicely together in the sense that they form a stratification.

Symplectic reduction has an analogue in hyperkähler geometry, which has been a very important tool for constructing new examples of these special manifolds. In this talk, I will explain how Sjamaar-Lerman’s results can be extended to this setting, namely, hyperkähler quotients by non-free actions are stratified
spaces whose strata are hyperkähler.

 

  • Geometry and Analysis Seminar
26 February 2018
14:15
Amihay Hanany
Abstract

3d N=4 supersymmetric gauge theories provide a method for constructing HyperK\”ahler singularities, known as the Coulomb branch.
This method is complementary to the more traditional way of construction using HyperK\”ahler quotients, known in physics as the “Higgs branch”.
Out of all possible gauge theories there is an interesting subclass of quiver varieties, where the Coulomb branch has been studied in some detail.
Some examples are moduli spaces of classical and exceptional instantons and closures of nilpotent orbits. An interesting feature of Coulomb and Higgs branches is the phenomenon of "3d mirror symmetry” where for a pair of gauge theories, the Higgs branch and Coulomb branch exchange.
There is a large class of “mirror pairs” which I will discuss in some detail.

A topic of recent interest is the notion of implosions. I will argue that there is a simple operation on the quiver which leads to implosion. In other words, given a quiver such that its Coulomb branch is moduli space A, a simple operation of the quiver (making a bouquet) provides the implosion of A.
This has been tested on closures of nilpotent orbits of A type and on nilpotent cones of orthogonal groups and found to agree with the expected results.
If time permits, I will discuss isometries of Coulomb branches

  • Geometry and Analysis Seminar
19 February 2018
14:15
Abstract

For many moduli problems, in order to construct a moduli space as a geometric invariant theory quotient, one needs to impose a notion of (semi)stability. Using recent results in non-reductive geometric invariant theory, we explain how to stratify certain moduli stacks in such a way that each stratum admits a coarse moduli space which is constructed as a geometric quotient of an action of a linear algebraic group with internally graded unipotent radical. As many stacks are
naturally filtered by quotient stacks, this involves describing how to stratify certain quotient stacks. Even for quotient stacks for reductive group actions, we see that non-reductive GIT is required to construct the coarse moduli spaces of the higher strata. We illustrate this point by studying the example of the moduli stack of coherent sheaves over a projective scheme. This is joint work with G. Berczi, J. Jackson and F. Kirwan.

  • Geometry and Analysis Seminar
12 February 2018
14:15
Paul Ziegler
Abstract

I will talk about recent work, joint with M. Gröchenig and D. Wyss, on two related results involving the cohomology of moduli spaces of Higgs bundles. The first is a positive answer to a conjecture of Hausel and Thaddeus which predicts the equality of suitably defined Hodge numbers of moduli spaces of Higgs bundles with SL(n)- and PGL(n)-structure. The second is a new proof of Ngô's geometric stabilization theorem which appears in the proof of the fundamental lemma. I will give an introduction to these theorems and outline our argument, which, inspired by work of Batyrev, proceeds by comparing the number of points of these moduli spaces over finite fields via p-adic integration.

 

  • Geometry and Analysis Seminar
5 February 2018
14:15
Ailsa Keating
Abstract

In real dimension two, the symplectic mapping class group of a surface agrees with its `classical' mapping class group, whose properties are well-understood. To what extend do these generalise to higher-dimensions? We consider specific pairs of symplectic manifolds (S, M), where S is a surface, together with collections of Lagrangian spheres in S and in M, say v_1, ...,v_k and V_1, ...,V_k, that have analogous intersection patterns, in a sense that we will make precise. Our main theorem is that any relation between the Dehn twists in the V_i must also hold between Dehn twists in the v_i. Time allowing, we will give some corollaries, such as embeddings of certain interesting groups into auto-equivalence groups of Fukaya categories.

 

  • Geometry and Analysis Seminar

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