Past Junior Geometry Seminar

A natural energy arising in the modelling of thin ferromagnetic films and liquid crystals is the Dirichlet energy, along with a penalisation term which favours a particular configuration. In this talk, I present recent work which establishes a quantitative analysis for sequences of minimisers to both the Baby-Skyrme model and two regularisation arising from geometry and physics respectively.

Riemannian manifolds whose holonomy group lies inside the exceptional Lie group G_2 are called G_2-manifolds. These manifolds have several interesting properties (they are Ricci-flat) and are of interest in geometric analysis as well as in mathematical physics and other fields. In this seminar, we will give an introduction to the theory of G_2-manifolds and discuss an associated non-linear Laplacian-type operator whose kernel essentially determines whether a compact manifold is G_2. We will present uniqueness and existence results for the Poisson’s equation of this operator on homogeneous and cohomogeneity-one manifolds.

The ε-energy is a regularisation of the Dirichlet energy introduced by Tobias Lamm. Like the famous Sacks-Uhlenbeck regularisation this greatly improves the existence and regularity theory. When we take the limit of a sequence of ε-harmonic maps with the parameter ε decreasing to 0 these converge, in the standard bubbling sense, to harmonic maps, which we hope to extract information about. I will talk about some recent results for these sequences, being when we might hope to have no loss of energy and no neck forming and what sort of harmonic maps we can obtain in the limit.

Given a cyclic subgroup G of GL(2,C) acting on C^2, it was first noticed by Wunram in the 80s that there is a correspondence between certain special representations of G and the exceptional curves appearing in the minimal resolution Y of the surface singularity C^2/G. In modern terms, this was reformulated by Ishii and Ueda as the existence of a fully faithful functor from the derived category of sheaves of Y to the G-equivariant derived category of C^2. In this talk, I will describe a mirror symmetric interpretation of this which exhibits the fully faithful inclusion in algebraic geometry as a sequence of positive Lefschetz stabilizations in symplectic geometry.

Asymptotically Locally Flat (ALF) Ricci-flat metrics are expected to model certain long-time singularities in four-dimensional Ricci flow, so understanding their stability is essential. In this talk, I will discuss that conformally Kähler, non-hyperkähler Ricci-flat ALF metrics are dynamically unstable under Ricci flow. Our work establishes three key tools in this setting: a Fredholm theory for the Laplacian on ALF metrics, the preservation of the ALF structure along the Ricci flow, and an extension of Perelman’s λ-functional to ALF metrics. This is joint work with Tristan Ozuch.

Hamilton’s Ricci flow is a widely studied tool of geometric analysis, with a variety of applications. It is sometimes possible to obtain existence results for Ricci flows coming out of singular spaces, which leads to the question of uniqueness in these cases. In this talk, we will discuss a new result on uniqueness of Ricci flows coming out of Reifenberg Alexandrov spaces, and give some indication of the methods used in the proof.

 Hyper-Kähler manifolds are rigid geometric structures. They have three different symplectic and complex structures, in direct analogy with the quaternions. Being Ricci-flat, they solve the vacuum Einstein equations, and so there has been considerable interest among physicists to explicitly construct such spaces. We will discuss in detail the examples arising from the Gibbons-Hawking ansatz. These give concrete descriptions of the metric, giving many examples to work with. They also lead to the generalised classification as hyper-Kähler quotients by P.B. Kronheimer, with one such space for each finite subgroup of SU(2). Finally, we will look at the McKay correspondence, relating the finite subgroups of SU(2) with the simple Lie algebras of type A,D,E.

Lagrangian mean curvature flow (LMCF) is a way to deform Lagrangian submanifolds inside a Calabi-Yau manifold according to the negative gradient of the area functional. There are influential conjectures about LMCF due to Thomas-Yau and Joyce, describing the long-time behaviour of the flow, singularity formation, and how one may flow past singularities. In this talk, we will show how to flow past a conically singular Lagrangian by gluing in expanders asymptotic to the cone, generalizing an earlier result by Begley-Moore. We solve the problem by a direct P.D.E.-based approach, along the lines of recent work by Lira-Mazzeo-Pluda-Saez on the network flow. The main technical ingredient we use is the notion of manifolds with corners and a-corners, as introduced by Joyce following earlier work of Melrose.

In this talk, we discuss nearly G2 structures, which define positive Einstein metrics, and are, up to scale, critical points of a geometric flow called (modified) Laplacian co-flow. We will discuss a recent joint work with Jason Lotay showing that many of these nearly G2 critical points are unstable for the flow. 

In this talk I will describe a possible strategy to obtain new solution to LMCF in the Kummer K3 surface by a fixed point argument. The key idea is that the regions where curvature concentrates in the Kummer K3 surface are modeled on the Eguchi-Hanson space. 

For some values of degrees d=(d_1,...,d_c), we construct a compactification of a Hilbert scheme of complete intersections of type d. We present both a quotient and a direct construction. Then we work towards the construction of a quasiprojective coarse moduli space of smooth complete intersections via Geometric Invariant Theory.

Riemannian manifolds with holonomy G_2 form an exceptional class of Ricci-flat manifolds occurring only in dimension 7. In the compact setting, their moduli spaces are known to be smooth (unobstructed), finite-dimensional, and to carry a natural Riemannian structure induced by the L^2-metric; but besides this very little is known about the global properties of G_2-moduli spaces. In this talk, I will review the basics of G_2-geometry and present new results concerning the distance theory and the geometry of the moduli spaces.
 

We develop a new construction of complete non-compact 8-manifolds with holonomy equal to $\Spin(7)$. As a consequence of the holonomy reduction, these manifolds are Ricci-flat. These metrics are built on the total spaces of principal $T^2$-bundles over asymptotically conical Calabi Yau manifolds. The resulting metrics have a new geometry at infinity that we call asymptotically $T^2$-fibred conical ($AT^2C$) and which generalizes to higher dimensions the ALG metrics of 4-dimensional hyperkähler geometry. We use the construction to produce infinite diffeomorphism types of $AT^2C$ $\Spin(7)$-manifolds and to produce the first known example of complete toric $\Spin(7)$-manifold.

For some values of degrees d=(d_1,...,d_c), we construct a compactification of a Hilbert scheme of complete intersections of type d. We present both a quotient and a direct construction. Then we work towards the construction of a quasiprojective coarse moduli space of smooth complete intersections via Geometric Invariant Theory.

The Hilbert scheme of d-points on a smooth surface is a well-studied object that still enjoys relatively large interest. We generalize Aldo Conca's Canonical Hilbert-Burch matrices and obtain explicit families of d-points. We show that such descriptions give us Białynicki-Birula cells of the Hilbert scheme for any choice of one-dimensional torus, thus describing the punctual component. This can be potentially applied to the study of singularities of the nested Hilbert scheme of points.

Gotzmann's regularity and persistence theorems provide tools which allow us to find explicit equations for the Hilbert scheme Hilb_P(P^n). A natural next step is to generalise these results to the multigraded Hilbert scheme Hilb_P(X) of a smooth projective toric variety X. In 2003 Maclagan and Smith generalise Gotzmann's regularity theorem to this case. We present new persistence type results for the product of two projective spaces, and time permitting discuss how these may be applied to a more general smooth projective toric variety.

Richard Hamilton introduced the Ricci flow as a way to study the Poincaré conjecture, which says that every simply connected, compact three-manifold is homeomorphic to the three-sphere. In this talk, we will introduce the Ricci flow in a way that is accessible to anyone with basic knowledge of Riemannian geometry. We will give some examples, discuss finite time singularities, and give an application to a theorem of Hamilton which says that every compact Riemannian 3-manifold with positive Ricci curvature admits a metric of constant positive sectional curvature.

Two areas of current research in Mathematical Gauge Theory are the study of higher dimensional instantons on manifolds with special holonomy (for example, Calabi-Yau three folds, G₂ and Spin(7) manifolds), and low dimensional gauge theories (for example the Kapustin-Witten, Haydys-Witten and ADHM Seiberg-Witten equations). A common feature of these two sets of theories is that the moduli spaces of solutions are in general not compact. In both cases, compactness issues arise because of solutions to a certain non-linear equation called the Fueter equation. In this talk, I'll explain how this non compactness gives a relationship between these high and low dimensional gauge theories.

Alan Weinstein, introduced the concept of "fat bundle" as a tool to understand when the total space of a fiber bundle with totally geodesic fibers allows a metric with positive sectional curvature. 

In recent times, certain weaker notions than the condition of having a metric with positive sectional curvature have been studied due to the apparent scarcity of spaces that meet this condition. Positive k^th-intermediate Ricci curvature (Ric_k > 0) on a Riemannian manifold Mⁿ is a condition that bridges the gap between positive sectional curvature and positive Ricci curvature. Indeed, when k = 1, this condition corresponds to positive sectional curvature, and when k = n−1, it corresponds to positive Ricci curvature. 

In this talk, I will discuss an ongoing project with Miguel Domínguez Vázquez, David González-Álvaro, and Jason DeVito, which aims to create new examples of compact Riemannian manifolds with Ric₂ > 0. We achieve this by employing a certain generalisation of the "fat bundle" concept.

Given a non-orientable Lagrangian surface L in a symplectic 4-manifold, how far
can the cohomology class of the symplectic form be deformed before there is no
longer a Lagrangian isotopic to L? I will properly introduce this and a
related question, both of which are less interesting for orientable
Lagrangians due to topological conditions. The majority of this talk will be
an exposition on Evans' 2020 work in which he solves this deformation
question for a particular Klein bottle. The proof employs the heavy machinery
of symplectic field theory and more classical pseudoholomorphic
curve theory to severely constrain the topology and intersection properties of
the limits of certain pseudoholomorphic curves under a process called
neck-stretching. The treatment of SFT-related material will be light and focus
mainly on how one can use the compactness theorem to prove interesting things.

In work of Haiden-Katzarkov-Konsevich-Pandit (HKKP), a canonical filtration, labeled by sequences of real numbers, of a semistable quiver representation or vector bundle on a curve is defined. The HKKP filtration is a purely algebraic object that depends only on a poset, yet it governs the asymptotic behaviour of a natural gradient flow in the space of metrics of the object. 

In this talk, we show that the HKKP filtration can be recovered from the stack of semistable objects, thus generalising the HKKP filtration to other moduli problems of non-linear origin. In particular, we will make sense of the notion of a filtration labelled by sequence of numbers for a point of an algebraic stack.

We give an introduction to the subject of higher geometry, by giving many examples of higher geometric objects, and looking at their properties. These include examples of 2-rings, 2-vector spaces, and 2-vector bundles. We show how these concepts help solve problems in ordinary geometry, as one of the many motivations of the subject. We assume no prerequisites on the subject, and the talk should be applicable to both differential and algebraic geometry.

Minimal submanifolds are the critical points of the volume functional. If the second derivative of the volume is nonnegative, we say that such a minimal submanifold is stable.

After reviewing some basics of minimal submanifolds in a generic Riemannian manifold, I will give some motivations behind the Lawson--Simons conjecture, which claims that there are no stable minimal submanifolds in 1/4-pinched spheres. Finally, I will discuss my recent work with Giada Franz on the nonexistence of stable minimal submanifolds in conformal pinched spheres.

Cayley submanifolds in Spin(7) geometry are an analogue and generalisation of complex submanifolds in Kähler geometry. In this talk we provide a glimpse into calibrated geometry, which encompasses both of these, and how it ties into the study of manifolds of special holonomy. We then focus on the deformation theory of compact and conically singular Cayleys. Finally we explain how to remove conical singularities via a gluing construction.

Associated to any smooth projective curve C is its degree d Jacobian variety, parametrising isomorphism classes of degree d line bundles on C. Letting the curve vary as well, one is led to the universal Jacobian stack. This stack admits several compactifications over the stack of marked stable curves, depending on the choice of a stability condition. In this talk I will introduce these compactified universal Jacobians, and explain how their moduli spaces can be constructed using Geometric Invariant Theory (GIT). This talk is based on arXiv:2210.11457.

The blow-up B of a scheme X in a closed subscheme Z enjoys the universal property that for any scheme X' over X such that the pullback of Z to X' is an effective Cartier divisor, there is a unique morphism of X' into B over X. It is well-known that the blow-up commutes along flat base change.

In this talk, I will discuss a derived enhancement B' of B, namely the derived blow-up, which enjoys a universal property against all schemes over X, satisfies arbitrary (derived) base-change, and contains B as a closed subscheme. To this end, we will need some elements from derived algebraic geometry, which I will review along the way. This will allow us to construct the derived blow-up as the projective spectrum of the derived Rees algebra, and state its functor of points in terms of virtual Cartier divisors, using Weil restrictions.

This is based on ongoing joint work with Adeel Khan and David Rydh.

In the 1980s, gauge theory was used to provide new invariants (up to
diffeomorphism) of orientable four dimensional manifolds, by counting
solutions of certain equations up to to a choice of gauge. More
recently, similar techniques have been used to study manifolds of
different dimensions, most notably on Spin(7) and G_2 manifolds. Using
dimensional reduction, one can find candidates for gauge theoretic
equations on manifolds of lower dimension. The talk will give an
overview of gauge theory in the 4 and 8 dimensional cases, and how
gauge theory on Spin(7) manifolds could be used to develop a gauge
theory on 5 dimensional manifolds.

In 2008 Pila and Zannier used a Theorem coming from Logic, proven by Pila and Wilkie, to give a new proof of the Manin-Mumford Conjecture, creating a new, powerful way to prove Theorems in Diophantine Geometry. The Pila-Wilkie Theorem gives an upper bound on the number of rational points on analytic varieties which are not algebraic; this bound usually contradicts a Galois-theoretic bound obtained by arithmetic considerations. We show how this technique can be applied to the following problem of Lang: given an irreducible polynomial f(x,y) in C[x,y], if for infinitely many pairs of roots of unity (a,b) we have f(a,b)=0, then f(x,y) is either of the form x^my^n-c or x^m-cy^n for c a root of unity.

In the '80s, Gromov proved that sequences of Riemannian manifold with a lower bound on the Ricci curvature and an upper bound on the dimension are precompact in the measured Gromov--Hausdorff topology (mGH for short). Since then, much attention has been given to the limits of such sequences, called Ricci limit spaces. A way to study these limits is to introduce a synthetic definition of Ricci curvature lower bounds and dimension upper bounds. A synthetic definition should not rely on an underlying smooth structure and should be stable when passing to the limit in the mGH topology. In this talk, I will briefly introduce CD spaces, which are a generalization of Ricci limit spaces.