Past Junior Geometry and Topology Seminar

The world of quantum invariants began in 1983 with the discovery of the Jones polynomial. Later on, Reshetikhin and Turaev developed an algebraic machinery that provides knot invariants. This algebraic construction leads to a sequence of quantum generalisations of this invariant, called coloured Jones polynomials. The original Jones polynomial can be defined by so called skein relations. However, unlike other classical invariants for knots like the Alexander polynomial, its relation to the topology of the complement is still a mysterious and deep question. On the topological side, R. Lawrence defined a sequence of braid group representations on the homology of coverings of configuration spaces. Then, based on her work, Bigelow gave a topological model for the Jones polynomial, as a graded intersection pairing between certain homology classes. We aim to create a bridge between these theories, which interplays between representation theory and low dimensional topology. We describe the Bigelow-Lawrence model, emphasising the construction of the homology classes. Then, we show that the sequence of coloured Jones polynomials can be seen through the same formalism, as topological intersection pairings of homology classes in coverings of the configuration space in the punctured disc.

My goal for the talk is to give a "from the ground-up" introduction to symplectic topology. We will cover the Darboux lemma, pseudo-holomorphic curves, Gromov-Witten invariants, quantum cohomology and Floer cohomology.

Gauge-theoretic invariants such as Donaldson or Seiberg–Witten invariants of 4-manifolds, Casson invariants of 3-manifolds, Donaldson–Thomas invariants of Calabi–Yau 3- and 4-folds, and putative Donaldson–Segal invariants of G_2 manifolds are defined by constructing a moduli space of solutions to an elliptic PDE as a (derived) manifold and integrating the (virtual) fundamental class against cohomology classes. For a moduli space to have a (virtual) fundamental class it must be compact, oriented, and (quasi-)smooth. We first describe a general framework for addressing orientability of gauge-theoretic moduli spaces due to Joyce–Tanaka–Upmeier. We then show that the moduli stack of perfect complexes of coherent sheaves on a Calabi–Yau 4-fold X is a homotopy-theoretic group completion of the topological realisation of the moduli stack of algebraic vector bundles on X. This allows one to extend orientations on the locus of algebraic vector bundles to the boundary of the (compact) moduli space of coherent sheaves using the universal property of homotopy-theoretic group completions. This is a necessary step in constructing Donaldson–Thomas invariants of Calabi–Yau 4-folds. This is joint work with Yalong Cao and Dominic Joyce.

Multiplicative preprojective algebras (MPAs) were originally defined by Crawley-Boevey and Shaw to encode solutions of the Deligne-Simpson problem as irreducible representations. 
MPAs have recently appeared in the literature from different perspectives including Fukaya categories of plumbed cotangent bundles (Etgü and Lekili) and, similarly, microlocal sheaves 
on rational curves (Bezrukavnikov and Kapronov.) After some motivation, I'll suggest a purely algebraic approach to study these algebras. Namely, I'll outline a proof that MPAs are 
2-Calabi-Yau if Q contains a cycle and an inductive argument to reduce to the case of the cycle itself.

In this talk, I will sketch a geometrically flavoured proof of the 
Madsen-Weiss theorem based on work by Eliashberg-Galatius-Mishachev.
In order to prove the triviality of appropriate relative bordism groups, 
in a first step a variant of the wrinkling theorem shows
that one can reduce to consider fold maps (with additional structure). 
In a subsequent step, a geometric version of the Harer stability
theorem is used to get rid of the folds via surgery. I will focus on 
this second step.

Standard representation theory transforms groups=algebra into vector spaces = (linear) algebra. The modern approach, geometric representation theory constructs geometric objects from algebra and captures various algebraic representations through geometric gadgets/invariants on these objects. This field started with celebrated Borel-Weil-Bott and Beilinson-Bernstein theorems but equally is in rapid expansion nowadays. I will start from the very beginnings of this field and try to get to the recent developments (time permitting).

After considering motivations in symplectic geometry, I’ll give a summary of $C^\infty$-Algebraic Geometry and how to extend these concepts to manifolds with corners. 

I will describe a family of 2d TQFTs, due to Moore-Tachikawa, which take values in a category whose objects are Lie groups and whose morphisms are holomorphic symplectic varieties. They link many interesting aspects of geometry, such as moduli spaces of solutions to Nahm equations, hyperkähler reduction, and geometric invariant theory.

I will give an overview of the Nielsen-Thurston theory of the mapping class group and its connection to hyperbolic geometry and dynamics. Time permitting, I will discuss the surface entropy conjecture and a theorem of Hamenstadt on entropies of `generic' elements of the mapping class group. No prior knowledge of the concepts involved is required.

Since differentiation generally lowers exponents, it is straightforward that the space of Laurent polynomials $\mathbb{C}[x, x^{-1}]$ is a finitely generated module over the ring of differential operators $\mathbb{C}[x, \mathrm{d}/\mathrm{d}x]$. This innocent looking fact has been vastly generalized to a statement about holonomic D-modules, using the beautiful theory of b-functions (or Bernstein—Sato polynomials). I will give an overview of the classical theory before discussing some recent developments concerning a $p$-adic analytic analogue, which is joint work with Thomas Bitoun.

The Strominger-Yau-Zaslow (SYZ) conjecture postulates that mirror dual Calabi-Yau manifolds carry dual special Lagrangian fibrations. Within the study of Mirror Symmetry the SYZ conjecture has provided a particularly fruitful point of convergence of ideas from Riemannian, Symplectic, Tropical, and Algebraic geometry over the last twenty years. I will attempt to provide a brief overview of this aspect of Mirror Symmetry.

It is shown by Kashiwara and Schapira (1980s) that for every constructible sheaf on a smooth manifold, one can construct a closed conic Lagrangian subset of its cotangent bundle, called the microsupport of the sheaf. This eventually led to the equivalence of the category of constructible sheaves on a manifold and the Fukaya category of its cotangent bundle by the work of Nadler and Zaslow (2006), and Ganatra, Pardon, and Shende (2018) for partially wrapped Fukaya categories. One can try to generalise this and conjecture that Fukaya category of a Weinstein manifold can be given by constructible (microlocal) sheaves associated to its skeleton. In this talk, I will explain these concepts and confirm the conjecture for a family of Weinstein manifolds which are certain quotients of A_n-Milnor fibres. I will outline the computation of their wrapped Fukaya categories and microlocal sheaves on their skeleta, called pinwheels.

The usual finite dimensional Grassmannians are well known to be classifying spaces for vector bundles. It is maybe a less known fact that one has certain natural connections on the Stiefel bundles over them, which also have a universality property. I will show how these connections are constructed and explain how this viewpoint can be used to rediscover Chern-Weil theory. Finally, we will see how a certain stabilized version of this, called the restricted Grassmannian, admits a similar construction, which can be used to show that it is a smooth classifying space for differential K-theory.

Lagrangian Floer homology has been used by Ozsvath and Szabo to define a package of three-manifold invariants known as Heegaard Floer homology. I will give an introduction to the topic.

TQFTs lie at the intersection of maths and theoretical physics. Topologically, they are a recipe for calculating an invariant of manifolds by cutting them into elementary pieces; physically, they describe the evolution of the state of a particle. These two viewpoints allow physical intuition to be harnessed to shed light on topological problems, including understanding the topology of 4-manifolds and calculating geometric invariants using topology.

Recent results have provided classifications of certain types of TQFTs as algebraic structures. After reviewing the definition of TQFTs and giving some diagrammatic examples, I will give informal arguments as to how these classifications arise. Finally, I will show that in many cases these algebras are in fact free, and give an explicit classification of them in this case.
 

The aim of this talk is to tell the story of Non-Abelian Hodge Theory for curves. The starting point is the space of representations of the fundamental group of a compact Riemann surface. This space can be endowed with the structure of a complex algebraic variety in three different ways, giving rise to three non-algebraically isomorphic moduli spaces called the Betti, de Rham and Dolbeault moduli spaces respectively. 

After defining and outlining the construction of these three moduli spaces, I will describe the (non-algebraic) correspondences between them, collectively known as Non-Abelian Hodge Theory. Finally, we will see how the rich structure of the Dolbeault moduli space can be used to shed light on the topology of the space of representations.

Morse theory explores the topology of a smooth manifold $M$ by looking at the local behaviour of a fixed smooth function $f : M \to \mathbb{R}$. In this talk, I will explain how we can construct ordinary homology by looking at the flow of $\nabla f$ on the manifold. The talk should serve as an introduction to Morse theory for those new to the subject. At the end, I will state a new(ish) proof of the functoriality of Morse homology.

Conical symplectic resolutions are one of the main objects in the contemporary mix of algebraic geometry and representation theory, 

known as geometric representation theory. They cover many interesting families of objects such as quiver varieties and hypertoric

varieties, and some simpler such as Springer resolutions. The last findings [Braverman, Finkelberg, Nakajima] say that they arise

as Higgs/Coulomb moduli spaces, coming from physics. Most of the gadgets attached to conical symplectic resolutions are rather

algebraic, such as their quatizations and $\mathcal{O}$-categories. We are rather interested in the symplectic topology of them, in particular 

finding smooth exact Lagrangians that appear in the central fiber of the (defining) resolution, as they are objects of the Fukaya category.

This talk will be a general introduction to perverse sheaves and their applications to the study of algebraic varieties, with a view towards enumerative geometry. It is aimed at non-experts.

We will start by considering constructible sheaves and local systems, and how they relate to the notion of stratification: this offers some insight in the relationship with intersection cohomology, which perverse sheaves generalise in a precise sense.

We will then introduce some technical notions, like t-structures, perversities, and intermediate extensions, in order to define perverse sheaves and explore their properties.

Time permitting, we will consider the relevant example of nearby and vanishing cycle functors associated with a critical locus, their relationship with the (hyper)-cohomology of the Milnor fibre and how this is exploited to define refined enumerative invariants in Donaldson-Thomas theory.

Equivariant cohomology is adapted from ordinary cohomology to better capture the action of a group on a topological space. In Floer theory, given an autonomous Hamiltonian, there is a natural action of the circle on 1-periodic flowlines given by time translation. Combining these two ideas leads to the definition of  $S^1$-equivariant symplectic cohomology. In this talk, I will introduce these ideas and explain how they are related. I will not assume prior knowledge of Floer theory.

The Witten-Reshetikhin-Turaev invariant Z(X,K) of a closed oriented three-manifold X containing a knot K, was originally introduced by Witten in order to extend the Jones polynomial of knots  in terms of Chern-Simons theory. Classically, the Jones polynomial is defined for a knot inside the three-sphere in  a combinatorial manner. In Witten's approach, the Jones polynomial J(K) emerge as the expectation value of a certain observable in Chern-Simons theory, which makes sense when K is embedded in any closed oriented three-manifold X. Moreover; he proposed that these invariants should be extendable to so-called topological quantum field theories (TQFT's). There is a catch; Witten's ideas relied on Feynman path integrals, which made them unrigorous from a mathematical point of view. However; TQFT's extending the Jones polynomial were subsequently constructed mathematically through combinatorial means by Reshetikhin and Turaev. In this talk, I shall expand slightly on the historical motivation of WRT invariants, introduce the formalism of TQFT's, and present some of the open problems concerning WRT invariants. The guiding motif will be the analogy between TQFT and quantum field theory.

The Grojnowski-Nakajima theorem states that the direct sum of the homologies of the Hilbert schemes on n points on an algebraic surface is an irreducible highest weight representation of an infinite-dimensional Heisenberg superalgebra. We present an idea to rederive the Grojnowski-Nakajima theorem using Halpern-Leistner's categorical Kirwan surjectivity theorem and Joyce's theorem that the homology of a moduli space of sheaves is a vertex algebra. We compute the homology of the moduli stack of perfect complexes of coherent sheaves on a smooth quasi-projective variety X, identify it as a (modified) lattice vertex algebra on the Lawson homology of X, and explain its relevance to the aforementioned problem.

Given an endomorphism f:X --> X of a 'dualisable' object in a symmetric monoidal category, one can define its trace Tr(f). It turns out that the trace is 'universal' among the scalars we can produce from f. To prove this we will think of the 1d framed bordism category as the 'walking dualisable object' (using the cobordism hypothesis) and then apply the Yoneda lemma.
Employing similar techniques we can define 'hermitian' objects (generalising hermitian vector spaces) and prove that there is a 1-1 correspondence between Hermitian structures on a fixed object X and self-adjoint automorphisms of X. If time permits I will sketch how this relates to hermitian K-theory.

While all results of the talk hold for infinity-categories, they work equally well for ordinary categories. Therefore no knowledge of higher category theory is needed to follow the talk.