Past Junior Geometry and Topology Seminar

https://teams.microsoft.com/l/meetup-join/19%3ameeting_ZGRiMTM1ZjQtZWNi…

I will introduce the notion of moduli spaces of curves and specifically genus 0 curves. They are in general not compact, and we will discuss the most common way to compactify them. In particular, I will try to explain the construction of Mbar_{0,5}, together with how to classify the boundary, how it is related to a moduli space of tropical curves, and how to do intersection theory on this space.

Invariants counting sheaves on Calabi--Yau 4-folds are obtained by virtual integrals over moduli spaces. These are expressed in terms of virtual fundamental classes, which conjecturally fit into
a wall-crossing framework proposed by Joyce. I will review the construction of vertex algebras in terms of which one can express the WCF.  I describe how to use  them to obtain explicit results for Hilbert schemes of points. As a consequence, I reduce multiple conjectures to a technical proof of the WCF. Surprisingly, one gets a complete correspondence between invariants of Hilbert schemes of CY 4-folds and elliptic surfaces.
 

Link: https://teams.microsoft.com/l/meetup-join/19%3ameeting_ZGRiMTM1ZjQtZWNi…

Totally geodesic submanifolds are perhaps one of the easiest types of submanifolds of Riemannian manifolds one can study, since a maximal totally geodesic submanifold is completely determined by any one of its points and the tangent space at that point. It comes as a bit of a surprise then that classification of such submanifolds — up to an ambient isometry — is a nightmarish and widely open question, even on such a manageable and well-understood class of Riemannian manifolds as symmetric spaces.

We will discuss the theory of totally geodesic submanifolds of symmetric spaces and see that any maximal such submanifold is homogeneous and thus can be completely encoded by some Lie algebraic data called a 'Lie triple'. We will then talk about the duality between symmetric spaces of compact and noncompact type and discover that there is a one-to-one correspondence between totally geodesic submanifolds of a symmetric space and its dual. Finally, we will touch on the known classification in rank one symmetric spaces, namely in spheres and projective/hyperbolic spaces over real normed division algebras. Time permitting, I will demonstrate how all this business comes in handy in other geometric problems on symmetric spaces, e. g. in classification of isometric cohomogeneity one actions.

Link: https://teams.microsoft.com/l/meetup-join/19%3ameeting_ZGRiMTM1ZjQtZWNi…

The purpose of this talk is to provide a quick introduction to the buzzwords in the title. Then I'll discuss some (mostly unexplored) conjectures and thoughts.

The cohomology of a manifold classifies geometric structures over it. One instance of this principle is the classification of line bundles via Chern classes. The classifying space BG associated to a (Lie) group G is a simplicial manifold which encodes the group structure. Its cohomology hence classifies geometric objects over G which play well with its multiplication. These are known as characteristic classes, and yield invariants of G-principal bundles.
I will introduce multiplicative gerbes and show how they realise classes in H^4(BG) when G is compact. Along the way, we will meet different versions of Lie group cohomology, smooth 2-groups and a few spectral sequences.

Link: https://teams.microsoft.com/l/meetup-join/19%3ameeting_ZGRiMTM1ZjQtZWNi…

A graded unipotent group U is a unipotent group with a 1PS of automorphisms C^* -- > Aut(U), such that the this 1PS acts on the Lie(U) with all weights positive. Let \hat U be the semi-direct product of U with this 1PS. Let \hat U act linearly on (X,L), a projective variety with a very ample line bundle. With the condition `semistability coincides with stability', and after suitable twist of rational characters, the \hat U-linearisation has a projective geometric quotient, and the invariants are finitely generated. This is a result from \emph{Geometric invariant theory for graded unipotent groups and applications} by G Bérczi, B Doran, T Hawes, F Kirwan, 2018.

Link: https://teams.microsoft.com/l/meetup-join/19%3ameeting_NzU0ODY5MTUtMzUz…

A dagger category is a category where for every morphism f:x --> y there is a chosen adjoint f*:y --> x, as for example in the category of Hilbert spaces. I will explain this definition in elementary terms and give a few example. The only prerequisites for this talk are the notion of category, functor, and Hilbert space.

Dagger categories are a great categorical framework for some concepts from functional analysis such as C*-algebras and they also allow us to state Atiyah's definition unitary topological field theories in categorical lanugage. There is however a problem with dagger categories: they are what category theorists like to call 'evil'. This is isn't really meant as a moral judgement, it just means that many ways of thinking about ordinary categories don't quite translate to dagger categories.

For example, not every fully faithful and essentially surjective dagger functor is also a dagger equivalence. I will present a notion of 'indefinite completion' that I came up with to describe dagger categories in less 'evil' terms. (Those of you who know Karoubi completion will see a lot of similarities.) I'll also explain how this can be used to compute categories of dagger functors, and more specifically groupoids of unitary TFTs.

It is a well-known fact that conformal structures on Riemann surfaces are in 1:1 correspondence with complex structures, but have you ever wondered whether this is just a fluke in 2 dimensions? In this talk, I will explain the concept of Penrose's "non-linear graviton", a fancy name for the twistor space of a hyperkahler manifold and one of the major historical achievements of Oxford maths. The twistor correspondence associates points of the hyperkahler manifold with certain holomorphic rational curves embedded in twistor space. We will see how information of the hyperkahler metric can be encoded purely in the complex structure on twistor space, giving a partial but welcome generalization of the 2-dimensional "fluke". Then I will outline a recently found Dolbeault-framework for the metric's reconstruction from local representatives of this complex structure. This provides an explicit integral formula for Kahler forms and consequently for the hyperkahler metric in terms of holomorphic data on twistor space. Finally, time permitting, I will discuss some interesting applications to (some or all of) PDEs, hyperkahler quotients, and the physics of "quantum gravity".
 

Following Grothendieck, periods can be interpreted as numbers arising as coefficients of a comparison isomorphism between two cohomology theories. Due to the influence of the “yoga of motives” these numbers are omnipresent in arithmetic algebraic geometry. The first part of the talk will be a crash course on how to study periods, as well as the action of the motivic Galois group on them, via an elementary category of realizations. In the second part, we will see how one uses this framework to study Feynman integrals -- an interesting family of periods arising in quantum field theory. We will finish with a brief overview of some of the recent work in algebraic geometry inspired by the study of periods arising in physics.

In this talk I will briefly sketch the philosophy and methods in which derived enhancements of classical moduli problems are produced. I will then discuss the character variety and distinguish two of its enhancements; one of these will represent a derived moduli stack for local systems. Lastly, I will mention how variations of this moduli space have been represented in number theoretic and rigid analytic contexts.

Quiver varieties are one of the main objects of study in Geometric Representation Theory. Defined by Nakajima in 1994, there has been a lot of research on them, but there is still a lot to be yet discovered, especially about their geometry. In this seminar, I will talk about their first use in Geometric Representation Theory as providing geometric representations of symmetric Kac-Moody Lie algebras.

Email @email to get a link to the Jitsi meeting room (it is included in the weekly announcements).

Here, a connection is a algebraic structure that is weaker than an algebra and stronger than a module. I will define this structure and give examples. I will then define the quantum product and explain how connections capture important properties of this product. I will finish by stating a new result which describes how this extends to equivariant Floer cohomology. No knowledge of symplectic topology will be assumed in this talk.
 

This talk aims to provide a simple introduction on how to probe the
explicit geometry of certain moduli schemes arising in enumerative
geometry. Stable pairs, introduced by Pandharipande and Thomas in 2009, offer a curve-counting theory which is tamer than the Hilbert scheme of
curves used in Donaldson-Thomas theory. In particular, they exclude
curves with zero-dimensional or embedded components.

Ribbons are non-reduced schemes of dimension one, whose non-reduced
structure has multiplicity two in a precise sense. Following Ferrand, Banica, and Forster, there are several results on how to construct
ribbons (and higher non-reduced structures) from the data of line
bundles on a reduced scheme. With this approach, we can consider stable
pairs whose underlying curve is a ribbon: the remaining data is
determined by allowing devenerations of the line bundle defining the
double structure.

In 2013, Bhargava-Shankar proved that (in a suitable sense) the average rank of elliptic curves over Q is bounded above by 1.5, a landmark result which earned Bhargava the Fields medal. Later Bhargava-Gross proved similar results for hyperelliptic curves, and Poonen-Stoll deduced that most hyperelliptic curves of genus g>1 have very few rational points. The goal of my talk is to explain how simple curve singularities and simple Lie algebras come into the picture, via a modified Grothendieck-Brieskorn correspondence.

Moreover, I’ll explain how this viewpoint leads to new results on the arithmetic of curves in families, specifically for certain families of non-hyperelliptic genus 3 curves.

The Toda integrable system was originally designed as a specific model for lattice field theories. Following Kostant's insights, we will explain how it naturally arises from the representation theory of Lie algebras, and present some more recent work relating it to cotangent bundles of Lie groups and the topology of Affine Grassmannians.

In the talk we will define higher K-groups, and explain some of their relations to number theory

A big problem in Riemannian geometry is the search for a "best possible" Riemannian metric on a given compact smooth manifold. When the manifold is complex, one very nice metric we could look for is a Kahler-Einstein metric. For compact Kahler manifolds with non-positive first chern class, these were proven to always exist by Aubin and Yau in the 70's. However, the case of positive first chern class is much more delicate, and there are non-trivial obstructions to existence. It wasn't until this decade that a complete abstract characterisation of Kahler-Einstein metrics became available, in the form of K-stability. This is a purely algebro-geometric stability condition, whose equivalence to the existence of a Kahler-Einstein metric in the Fano case is analogous to the Hitchin-Kobayashi correspondence for vector bundles. In this talk, I will cover the definition of K-stability, its relation to Kahler-Einstein metrics, and (time permitting) give some examples of how K-stability is verified or disproved in practice.

G. Dimitrov and L. Katzarkov introduced in their paper from 2016 the counting of non-commutative curves and their (semi-)stability using T. Bridgeland's stability conditions on triangulated categories. To some degree one could think of this as the non-commutative analog of Gromov-Witten theory. However, its full meaning has not yet been fully discovered. For example there seems to be a relation to proving Markov's conjecture. 

For the talk, I will go over the definitions of stability conditions, non-commutative curves and their counting. After developing some tools relying on working with exceptional collections, I will consider the derived category of representations on the acyclic triangular quiver and will talk about the explicit computation of the invariants for this example.

Recently, Joyce constructed a Ringel-Hall style graded vertex algebra on the homology of moduli stacks of objects in certain categories of algebro-geometric and representation-theoretic origin. The construction is most natural for 2n-Calabi-Yau categories. We present this construction and explain the geometric reason why it exists. If time permits, we will explain how to compute the homology of the moduli stack of objects in the derived category of a smooth complex projective variety and to identify it with a lattice-type vertex algebra.

The homotopy bordism category hCob_d has as objects closed (d-1)-manifolds and as morphisms diffeomorphism classes of d-dimensional bordisms. This is a simplified version of the topologically enriched bordism category Cob_d whose classifying space B(Cob_d) been completely determined by Galatius-Madsen-Tillmann-Weiss in 2006. In comparison, little is known about the classifying space B(hCob_d).

In the first part of the talk I will give an introduction to bordism categories and their classifying spaces. In the second part I will identify B(hCob_1) showing, in particular, that the rational cohomology ring of hCob_1 is polynomial on classes \kappa_i in degrees 2i+2 for all i>=1. The seemingly simpler category hCob_1 hence has a more complicated classifying space than Cob_1.

Springer theory is an important branch of geometric representation theory. It is a beautiful interplay between combinatorics, geometry and representation theory.
It started with Springer correspondence, which yields geometric construction of irreducible representations of symmetric groups, and Ginzburg's construction of universal enveloping algebra U(sl_n).

Here I will present a view of two-row Springer theory of type A (thus looking at nilpotent elements with two Jordan blocks) from a scope of a symplectic topologist (hence the title), that yields connections between symplectic-topological invariants and link invariants (Floer homology and Khovanov homology) and connections to representation theory (Fukaya category and parabolic category O), thus summarising results by Abouzaid,
Seidel, Smith and Mak on the subject.

Arakelov geometry studies schemes X over ℤ, together with the Hermitian complex geometry of X(ℂ).
Most notably, it has been used to give a proof of Mordell's conjecture (Faltings's Theorem) by Paul Vojta; curves of genus greater than 1 have at most finitely many rational points.
In this talk, we'll introduce some of the ideas behind Arakelov theory, and show how many results in Araklev theory are analogous—with additional structure—to classic results such as intersection theory and Riemann Roch.

A cohomology class on the diffeomorphism group Diff(M) of a manifold M

can be thought of as a characteristic class for smooth M-bundles.
I will survey a technique for producing examples of such classes,
and then explain how the signature (of 4-manifolds) provides an
obstruction to this technique in dimension 3.

I will define Miller-Morita-Mumford classes and explain how we can
think of them as coming from classes on the cobordism category.
Madsen and Weiss showed that for a surface S of genus g all cohomology
classes
of the mapping class group MCG(S) (of degree < 2(g-2)/3) are MMM-classes.
This technique has been successfully ported to higher even dimensions d= 2n,
but it cannot possibly work in odd dimensions:
a theorem of Ebert says that for d=3 all MMM-classes are trivial.
In the second part of my talk I will sketch a new proof of (a part of)
Ebert's theorem.
I first recall the definition of the signature sign(W) of a 4 manifold W,
and some of its properties, such as additivity with respect to gluing.
Using the signature and an idea from the world of 1-2-3-TQFTs,
I then go on to define a 'central extension' of the three dimensional
cobordism category.
This central extension corresponds to a 2-cocycle on the 3d cobordism
category,
and we will see that the construction implies that the associated MMM-class
has to vanish on all 3-dimensional manifold bundles.

Topological quantum field theories (TQFTs) are an extensively studied scheme for constructing invariants of manifolds, inspired by physics. In this talk, we will discuss a particular flavour of TQFT, where we equip our manifolds with principal bundles for some finite group. After introducing TQFTs and this particular flavour, I will discuss games one can play with these TQFTs, and a possible strategy for classifying equivariant TQFTs in three dimensions. 

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.