Past Geometry and Analysis Seminar

The notion of a higher Segal object was introduces by Dyckerhoff and Kapranov as a general framework for studying (higher) associativity inherent
in a wide range of mathematical objects. Most of the examples are related to Hall algebra type constructions, which include quantum groups. We describe a construction that assigns to a simplicial object S a datum H(S)  which is naturally interpreted as a "d-lax A-infinity algebra” precisely when S is a (d+1)-Segal object. This extends the extensively studied d=2 case.

The first examples of complete holonomy G2 metrics were constructed by Bryant-Salamon and are thus of central importance in geometry, but also in physics, appearing for example in the work of Atiyah-Witten, Acharya-Witten and Acharya-Gukov.   I will describe joint work in progress with Spiro Karigiannis which realises Bryant-Salamon manifolds in dimension 7 as coassociative fibrations.  In particular, I will discuss the relationship of this study to gravitational instantons, conical singularities, and to recent work of Donaldson and Joyce-Karigiannis.

Moduli spaces of polarised varieties (varieties together with an ample line bundle) are not Hausdorff in general. A basic goal of algebraic geometry is to construct a Hausdorff moduli space of some nice class of polarised varieties. I will discuss how one can achieve this goal using the theory of canonical Kähler metrics. In addition I will discuss some fundamental properties of this moduli space, for example the existence of a Weil-Petersson type Kähler metric. This is joint work with Philipp Naumann.

Partition Lie algebras are generalisations of rational differential graded Lie algebras which, by a recent result of Mathew and myself, govern the formal deformation theory of algebro-geometric objects in finite and mixed characteristic. In this talk, we will take a closer look at these new gadgets and discuss some of their applications in algebra and topology

The stack of Higgs bundles of a given rank and degree over a non-singular projective curve can be stratified in two ways: according to its Higgs Harder-Narasimhan type (its instability type) and according to the Harder-Narasimhan type of the underlying vector bundle (instability type of the underlying bundle). The semistable stratum is an open stratum of the former and admits a coarse moduli space, namely the moduli space of semistable Higgs bundles. It can be constructed using Geometric Invariant Theory (GIT) and is a widely studied moduli space due to its rich geometric structure.

In this talk I will explain how recent advances in Non-Reductive GIT can be used to refine the Higgs Harder-Narasimhan and Harder-Narasimhan stratifications in such a way that each refined stratum admits a coarse moduli space. I will explicitly describe these refined stratifications and their intersection in the case of rank 2 Higgs bundles, and discuss the topology and geometry of the corresponding moduli spaces

In this talk we discuss new effective methods to test pairwise containment of arbitrary (possibly singular) subvarieties of any smooth projective toric variety and to determine algebraic multiplicity without working in local rings. These methods may be implemented without using Gröbner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used. The methods arise from techniques developed to compute the Segre class s(X,Y) of X in Y for X and Y arbitrary subschemes of some smooth projective toric variety T. In particular, this work also gives an explicit method to compute these Segre classes and other associated objects such as the Fulton-MacPherson intersection product of projective varieties.
These algorithms are implemented in Macaulay2 and have been found to be effective on a variety of examples. This is joint work with Corey Harris (University of Oslo).

I will describe joint work with Bruno Premoselli which gives a new existence theorem for negatively curved Einstein 4-manifolds, which are obtained by smoothing the singularities of hyperbolic cone metrics. Let (M_k) be a sequence of compact 4-manifolds and let g_k be a hyperbolic cone metric on M_k with cone angle \alpha (independent of k) along a smooth surface S_k. We make the following assumptions:

1. The injectivity radius i(k) of M_k tends to infinity (where in defining injectivity radius we ignore those geodesics which hit the cone singularity)

2. The normal injectivity radius of S_k is at least i(k)/2.

3. The area of the singular locii satisfy A(S_k)\leq C \exp(5 i(k)/2) for some C independent of k.

When these assumptions hold, we prove that for all large k, M_k carries a smooth Einstein metric of negative curvature. The proof involves a gluing theorem and a parameter dependent implicit function theorem (where k is the parameter). As I will explain, negative curvature plays an essential role in the proof. (For those who may be aware of our arxiv preprint, https://arxiv.org/abs/1802.00608 [arxiv.org], the work
I will describe has a new feature, namely we now treat all cone angles, and not just those which are greater than 2\pi. This gives lots more examples of Einstein 4-manifolds.)

I will discuss results relating different partially wrapped Fukaya categories.  These include a K\"unneth formula, a `stop removal' result relating partially wrapped Fukaya categories relative to different stops, and a gluing formula for wrapped Fukaya categories.  The techniques also lead to generation results for Weinstein manifolds and for Lefschetz fibrations.  The methods are mainly geometric, and the key underlying Floer theoretic fact is an exact triangle in the Fukaya category associated to Lagrangian surgery along a short Reeb chord at infinity.  This is joint work with Sheel Ganatra and Vivek Shende.

In this talk, we will explain how to construct embedded closed Lagrangian submanifolds in mirror quintic threefolds using tropical curves and the toric degeneration technique. As an example, we will illustrate the construction for tropical curves that contribute to the Gromov–Witten invariant of the line class of the quintic threefold. The construction will in turn provide many homologous and non-Hamiltonian isotopic Lagrangian
rational homology spheres, and a geometric interpretation of the multiplicity of a tropical curve as the weight of a Lagrangian. This is a joint work with Helge Ruddat.

In this talk, we are going to talk about the Type I singularity on 4-dimensional manifolds foliated by homogeneous S3 evolving under the Ricci
flow. We review the study on rotationally symmetric manifolds done by Angenent and Isenberg as well as by Isenberg, Knopf and Sesum. In the latter, a global frame for the tangent bundle, called the Milnor frame, was used to set up the problem. We shall look at the symmetries of the manifold, derived from Lie groups and its ansatz metrics, and this global tangent bundle frame developed by Milnor and Bianchi. Numerical simulations of the Ricci flow on these manifolds are done, following the work by Garfinkle and Isenberg, providing insight and conjectures for the main problem. Some analytic results will be proven for the manifolds S1×S3 and S4 using maximum principles from parabolic PDE theory and some sufficiency conditions for a neckpinch singularity will be provided. Finally, a problem from general relativity with similar metric symmetries but endowed on a manifold with differenttopology, the Taub-Bolt and Taub-NUT metrics, will be discussed.

We consider the two-dimensional Euler flow for an incompressible fluid confined to a smooth domain. We construct smooth solutions with concentrated vorticities around $k$ points which evolve according to the Hamiltonian system for the Kirkhoff-Routh energy,  using an outer-inner solution gluing approach. The asymptotically singular profile  around each point resembles a scaled finite mass solution of Liouville's equation.
We also discuss the {\em vortex filament conjecture} for the three-dimensional case. This is joint work with Juan D\'avila, Monica Musso and Juncheng Wei.

After discussing a general excision technique for constructing canonical orientations for moduli spaces that derive from an elliptic equation, I shall
explain how to carry out this program in the case of G2-instantons and the 7-dimensional real Dirac operator. In many ways our approach can
be regarded as a categorification of the Atiyah-Singer index theorem. (Based on joint work with Dominic Joyce.)

This talk is based on joint work with Dominic Joyce and Markus Upmeier. Issues we'd like to talk about are a) the orientability of moduli spaces that
appear in various gauge-theoretic problems; and b) how to orient those moduli spaces if they are orientable. We begin with briefly mentioning backgrounds and motivation, and recall basics in gauge theory such as the Atiyah-Hitchin-Singer complex and the Kuranishi model by taking the anti-self-dual instanton moduli space as an example. We then describe the orientability and canonical orientations of the anti-self-dual instanton moduli space, and other
gauge-theoretic moduli spaces which turn up in current research interests.

Einstein metrics are fixed points (up to scaling) of Hamilton's Ricci flow. A natural question to ask is whether a given metric is stable in the sense that the flow returns to the Einstein metric under a small perturbation. I'll give a brief survey of this area focussing on the case when the Einstein constant is positive. An interesting class of metrics where this question is not completely resolved are the compact symmetric spaces. I'll report on some recent progress with Tommy Murphy and James Waldron where we have been able to use a criterion due to Kroencke to show the Kaehler-Einstein metric on some Grassmannians and the bi-invariant metric on the Lie group G_2 are unstable.

Amplituhedra are mathematical objects generalising the notion of polytopes into the Grassmannian. Proposed as a geometric construction encoding scattering amplitudes in the four-dimensional maximally supersymmetric Yang-Mills theory, they are mathematically interesting objects on their own. In my talk I strengthen the relation between scattering amplitudes and geometry by linking the amplituhedron to the Jeffrey-Kirwan residue, a powerful concept in symplectic and algebraic geometry. I focus on a particular class of amplituhedra in any dimension, namely cyclic polytopes, and their even-dimensional
conjugates. I show how the Jeffrey-Kirwan residue prescription allows to extract the correct amplituhedron canonical differential form in all these cases. Notably, this also naturally exposes the rich combinatorial structures of amplituhedra, such as their regular triangulations

The goal of the talk will be to introduce a class of functions that answer a question in topology, can be computed via analytic methods more common in the theory of dynamical systems, and in the end turn out to enjoy beautiful modular properties of the type first observed by Ramanujan. If time permits, we will discuss connections with vertex algebras and physics of BPS states which play an important role, but will be hidden "under the hood" in much of the talk.

Introduced by Konno, hyperpolygon spaces are examples of Nakajima quiver varieties.  The simplest of these is a noncompact complex surface admitting the structure of a gravitational instanton, and therefore fits nicely into the Kronheimer-Nakajima classification of complete ALE hyperkaehler 4-manifolds, which is a geometric realization of the McKay correspondence for finite subgroups of SU(2).  For more general hyperpolygon spaces, we can speculate on how
this classification might be extended by studying the geometry of hyperpolygons at "infinity". This is ongoing work with Hartmut Weiss.

We describe the moduli space of distributions in terms of Grothendieck’s Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety.

 We study codimension one holomorphic distributions on projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2. We show how the connectedness of the curves in the singular sets of foliations is an integrable phenomenon. This part of the  talk  is work joint with  M. Jardim(Unicamp) and O. Calvo-Andrade(Cimat).

We also study foliations by curves via the investigation  of their  singular schemes and  conormal  sheaves and we provide a classification  of foliations of degree at most 3 with  conormal  sheaves locally free.  Foliations of degrees  1 and 2 are aways given by a global intersection of two codimension one distributions. In the classification of degree 3 appear Legendrian foliations, foliations whose  conormal sheaves are instantons and other ” exceptional”
type examples. This part of the  talk   is  work joint with  M. Jardim(Unicamp) and S. Marchesi(Unicamp).

The Beilinson-Drinfeld Grassmannian, which classifies a G-bundle trivialised away from a finite set of points on a curve, is one of the basic objects in the geometric Langlands programme. Similar construction in higher dimensions in the algebraic and analytic settings are not very interesting because of Hartogs' theorem. In this talk I will discuss a differentiable version. I will also explain a theory of D-modules on differentiable spaces and use it
to define differentiable chiral and factorisation algebras. By linearising the Grassmannian we get examples of differentiable chiral algebras. This is joint work with Dennis Borisov.

In recent joint work with Lorenzo Foscolo and Johannes Nordstr\”om we gave an analytic construction of large families of complete circle-invariant $G_2$
holonomy metrics on the total space of circle bundles over a complete noncompact Calabi—Yau 3-fold with asymptotically conical geometry. The
asymptotic models for the geometry of these $G_2$ metrics are circle bundles with fibres of constant length $l$, so-called asymptotically local conical
(ALC) geometry. These ALC $G_2$ metrics can Gromov—Hausdorff collapse with bounded curvature to the given asymptotically conical Calabi—Yau 3-fold as the fibre length $l$ goes to $0$. A natural question is: what happens to these families of $G_2$ metrics as we try to make $l$ large? In general the answer to this question is not known, but in cases with sufficient symmetry we have recently been able to give a complete picture.  

We give an overview of all these results and discuss some analogies with the class of asymptotically locally flat (ALF) hyperkaehler 4-manifolds. In
particular we suggest that a particular $G_2$ metric we construct should be regarded as a $G_2$ analogue of the Euclidean Taub—NUT metric on the complex plane.

In this talk, we consider the question of exact (boundary) controllability of wave equations: whether one can steer their solutions from any initial state to any final state using appropriate boundary data. In particular, we discuss new and fully general results for linear wave equations on time-dependent domains with moving boundaries. We also discuss the novel geometric Carleman estimates that are the main tools for proving these controllability results

A C^k-extension of a smooth and connected Lorentzian manifold (M,g) is an isometric embedding of M into a proper subset of a connected Lorentzian manifold (N,h) of the same dimension, where the Lorentzian metric h is C^k regular. If no such extension exists, then we say that (M,g) is C^k-inextendible. The study of low-regularity inextendibility criteria for Lorentzian manifolds is motivated by the strong cosmic censorship conjecture in general relativity.

The Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a C^2 regular metric. In this talk I will describe how one
proves the stronger statement that the maximal analytic Schwarzschild spacetime is inextendible as a Lorentzian manifold with a continuous metric.

I will explain our recent description of the fundamental degrees of freedom underlying a generalized Kahler structure. For a usual Kahler
structure, it is well-known that the geometry is determined by a complex structure, a Kahler class, and the choice of a positive(1,1)-form in this class, which depends locally on only a single real-valued function: the Kahler potential. Such a description for generalized Kahler geometry has been sought since it was discovered in1984. We show that a generalized Kahler structure of symplectic type is determined by a pair of holomorphic Poisson manifolds, a
holomorphic symplectic Morita equivalence between them, and the choice of a positive Lagrangian brane bisection, which depends locally on
only a single real-valued function, which we call the generalized Kahler potential. To solve the problem we make use of, and generalize,
two main tools: the first is the notion of symplectic Morita equivalence, developed by Weinstein and Xu to study Poisson manifolds;
the second is Donaldson's interpretation of a Kahler metric as a real Lagrangian submanifold in a deformation of the holomorphic cotangent bundle.

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.