# What is ... ? Seminar

The What is ... ? Seminar'' (WiSe) is a weekly Zoom mathematics seminar. The goal is to explain interesting things to each other in a casual manner. The name is inspired by the What is ... column of the Notices of the American Mathematical Society. The format follows the tradition of Sydney Informal Friday Seminar.

## Rules:

• Any question/additional explanation from the audience is allowed (in fact, encouraged).

• Ego out the (Zoom) window.

• Examples, examples, examples.

• All participants are strongly encouraged to have their videos on during Zoom talks.

The seminar is intended to be conversational, so come ready to engage.

• Date and time: We will have no more WiSe talks in Semester 1, 2021. Have a nice end-of semester and break! We plan to resume meeting regularly in August. Regular meeting time for Semester 2, 2021 will be Fridays at 11 am Brisbane time.

• Organisers: Anna Puskás (a.puskas@uq.edu.au), Masoud Kamgarpour

• Email List: If you want to be added to the email list, please fill out this form. Emails will come from Anna.

## Guidance for Speakers:

• You should have a basic website so that participants can learn something about you. It is possible to set up a website in a few hours (e.g. the present website).

• As much as possible, avoid slide talks. The preferred method of delivery is writing on a tablet (e.g. iPad). If you don't have one, consider borrowing one for your talk.

• Connect with both your computer and tablet so that we can see you and your writing. Here is an excellent example with notes.

• Please send the notes for your talk to Anna beforehand so he can post it online. Include the references you found helpful while preparing your talk.

• Suggested timing is 50 minutes of talk, 10 minutes break, then another 50 minutes.

• Have mercy. Go slow! If necessary, we will organise additional talks for you.

## Semester II, 2021

Date and Time: Usually Fridays at 11 am Brisbane time

Friday, August 27, 10 am: Alex Weekes (Assistant Prof, University of Saskatchewan): What is the affine Grassmannian?

Abstract: Affine Grassmannians are infinite-dimensional spaces which play an important role in geometric representation theory. One part of the richness of these spaces is that they can defined in several seemingly distinct ways: via loop groups, via a moduli space of principal bundles, via Kac-Moody groups, or via lattices. In this talk we'll overview the definition of the affine Grassmannian, with some motivation from number theory, and discuss a few examples which relate back to (possibly) more familiar spaces like the nilpotent cone. Finally, if time permits, we'll touch on more advanced topics such as the geometric Satake equivalence. Notes.

September 10, 11am: Matthew Spong (Postdoc, UQ): What is equivariant elliptic cohomology?

Abstract: Elliptic cohomology was introduced in the late 1980s following Witten's results about the index theory of families of differential operators on free loop spaces. In a certain sense it is an approximation to the K-theory of the free loop space. The first equivariant version of the theory was constructed in 1994 by Grojnowski, who made comments about its mysterious relationship to the representation theory of loop groups. In this talk, we outline a construction of equivariant elliptic cohomology whose main ingredient is the loop group equivariant K-theory of the free loop space. The construction is based on a recent construction of Kitchloo. Notes.

September 24, 11am: Matthew Spong (Postdoc, UQ): What is... (non-equivariant) elliptic cohomology?

Abstract: In this talk we will begin with a sketch of what elliptic cohomology is really about. Thus we will introduce the concept of a genus, which is an invariant of manifolds which are equipped with extra structure, and from there we will define an elliptic genus. We then aim to briefly describe the role of elliptic genera in elliptic cohomology, and to sketch the relationship to index theory on free loop spaces. If time permits, we will finally describe a version of elliptic cohomology which was constructed in terms of the K-theory of free loop spaces by Kitchloo and Morava. Notes. See also the AMS Notices article What is... an Elliptic Genus?

Monday, October 11, 6pm: Maryam Khaqan (Postdoc, Stockholm University): What is moonshine?

Abstract: Moonshine began as a series of numerical coincidences connecting finite groups to modular forms but has since evolved into a rich theory that sheds light on the underlying algebraic structures that these coincidences reflect. In this talk, I will give a brief history of moonshine, describe some of the existing examples of the phenomenon in the literature, and discuss how my work fits into the story.

October 22, 11 am: Madeline Nurcombe (PhD, UQ): What is Kazhdan-Lusztig theory?

Abstract: In 1979, Kazhdan and Lusztig introduced a new basis for the Hecke algebra of a Coxeter group, related to the standard basis by polynomial coefficients. These polynomials relate diverse areas in Lie Theory, such as Verma modules of semisimple Lie algebras, Schubert varieties in algebraic geometry, and primitive ideals of enveloping algebras, leading to a new topic called Kazhdan-Lusztig theory. In this talk, I will focus on the Kazhdan-Lusztig basis in the simpler case of the Hecke algebra of the symmetric group, giving some necessary background information on the symmetric group, Bruhat order and Hecke algebra. I will then relate this to the more general case of the Hecke algebra of a Coxeter group. Notes.

November 5, 11 am: Madeline Nurcombe (PhD, UQ): What is Kazhdan-Lusztig theory? - Part II

Abstract: see above. Typed notes, Handwritten notes.

## Semester I, 2020

Thursday, June 4: Marielle Ong (PhD UPenn), What is a Spectral Curve?

Abstract: A Higgs bundle is a pair consisting of a holomorphic vector bundle and a matrix-valued 1-form, called a Higgs field. Every Higgs field can be associated to a spectral curve , which is the zero locus of its characteristic polynomial, and a line bundle of its eigenvectors. Conversely, one can recover a Higgs bundle from its spectral data up to isomorphism. This spectral data correspondence is the heart behind the study of Higgs bundles, and the Hitchin fibration. Notes, expository article

Thursday, June 11: Grace Garden (Phd Sydney), What is Dynamics on the Character Variety?

Abstract: The dynamics of character varieties is a new topic of study that has emerged in recent years where the action of the mapping class group on the character variety is examined. This began mostly with the work of Goldman (2003) with the study of the mapping class group action on the SL(2, C)-character variety of the once-punctured torus. We first introduce the concepts of the mapping class group and character variety of a surface with a specific focus on examples of the torus and once-punctured torus. We then present some of the initial results of Goldman (2003) and, time permitting, will discuss current work that extends the action. Notes.

Thursday, June 18: Anna Puskas (Faculty UQ), What is the Double Affine Hecke Algebra?

Abstract: Double affine Hecke algebras were introduced by Cherednik, who used them to prove the Macdonald constant term conjecture. In this talk, we will define a double affine Hecke algebra and consider how the definition is built using different presentations of affine objects. We will review finite and affine Hecke algebras along the way. We will discuss the polynomial representation, Macdonald polynomials, connections to double cosets of p-adic groups and related identities. Notes, solutions to exercises.

Thursday, June 25 (special time: 11am-1pm): Valentin Buciumas (Postdoc UQ) What is the Local Langlands correspondence for GL(2,Q_p)?

Abstract: The Langlands program is a set of influential conjectures relating number theory, representation theory and geometry. The purpose of this talk will be to briefly explain a small part of this program: the Local Langlands correspondence for GL_2 over Q_p. More precisely we'll talk about the representation theory of GL_2 over Q_p and how it relates to Weil-Deligne representations from algebraic number theory. Time permitting, we will talk about L- and \epsilon- factors and some motivation coming from the theory of L-functions, or about the Satake isomorphism and the LLC. Notes, expository article.

Thursday, July 2: Nick Bridger (MPhil UQ) What is the Character Variety?

Abstract: The G-character variety is an affine GIT quotient by the conjugation action. Roughly speaking, its points can be thought of as homomorphisms in to G modulo conjugation. In this talk, we will define what this means more precisely and look at some examples. The topology of character varieties of surface groups is so far best understood in Type A and other reductive groups of small rank, but more generally determining its topology is largely still an open problem. We will discuss how one can determine interesting topological information by counting points over finite fields using the character table of G, we will look at an example of this for GL_2. Notes.

Thursday, July 9: Seamus Albion (MPhil UQ) What is Dyson's Conjecture?

Abstract: Freeman J. Dyson (1923--2020) made many important contributions to physics and mathematics. In this talk I will discuss the story behind a remarkable conjecture made by Dyson in the first of his series of papers "Statistical Theory of Energy Levels of Complex Systems". Known as Dyson's constant term conjecture, it simply asserts that the constant term of a certain Laurent polynomial is equal to a multinomial coefficient. This conjecture was almost immediately proved independently by K.G. Wilson and J. Gunson in 1962, with a short elementary proof by I.J. Good following in 1970. Despite how seemingly simple Dyson's conjecture was shown to be, much fascinating mathematics has been developed in the quest for generalisations. Beginning with the motivation for Dyson's original conjecture, I will survey the mathematical developments inspired by Dyson's work, culminating in the Macdonald conjectures: a deep generalisation to root systems posed by Macdonald in 1982. Notes.

Thursday, July 16 (special time: 9am-11am): Alex Dunn (Postdoc Caltech) What is a Maass Form?

Abstract: In fact, actually a small part of this talk will be about Maass forms (real analytic automorphic eigenfunctions of hyperbolic Laplace operator). I plan to discuss L-functions and the Riemann hypothesis in general, and try to state the adelic version of Ramanujan—Petersson conjecture in GL_2(A_Q). Notes.

## Semester II, 2020

During the second part of semester 2 we meet Fridays at 11am unless otherwise noted.

Thursday, August 6: Anna Romanov (Postdoc Sydney) What is the Local Langlands correspondence for GL(2,R)?

Abstract: The local Langlands correspondence describes representations of a reductive algebraic group G over a local field F in terms of a purely arithmetic object (the Weil-Deligne group, depending only on F) and a purely algebraic object (the complex dual group, depending only on G). This correspondence provides a deep link between representation theory and number theory. A few weeks ago, Valentin described the correspondence for us in the special case of the group GL_2 and the local field Q_p. In this talk, I’ll discuss in the archimedean case of Valentin’s example: the group GL_2 and the field R. I will approach this topic from the perspective of a representation theorist, and explain why this correspondence is interesting for those who care about representations of real reductive Lie groups. Notes.

Thursday, August 13: Chris Raymond (PhD UQ) What is a Wakimoto module?

Abstract: This talk will be a discussion of Minoru Wakimoto's 1986 paper "Fock representations of the affine Lie algebra A_1^(1) ". In this paper, Wakimoto presents a construction of a two-parameter family of representations of A_1^(1), now known as Wakimoto modules. These representations can be thought of as an affine generalisation of the differential operator realisation of A_1. Wakimoto modules provide explicit constructions of irreducible non-integrable modules, as well as modules over A_1^(1) at the so-called critical level. I will give a historical overview of Wakimoto's construction, and outline some of the broader impact of the paper. Notes.

Thursday, August 20: Joshua Ciappara (PhD Sydney) What is modular representation theory?

Abstract: The goal of this talk is broadly to introduce and explore some of the distinctive features of the representation theory of algebraic groups in characteristic p > 0. We will emphasize the case of SL_2, which is both well suited to explicit calculations and revealing of deep theoretical patterns. Topics of discussion will include Chevalley's theorem on simple modules, the Steinberg tensor product theorem, and (if time permits) Frobenius kernels. Little or no prior knowledge of geometric representation theory will be necessary to follow the talk. Notes, Exercises.

Friday, August 28 (new time: 11am): Masoud Kamgarpour (Faculty UQ) What is a hypergeometric local systems?

Abstract: Hypergeometric functions have a long and celebrated history going back to the works of Wallis, Newton, Euler, Gauss, Kummer, …. The geometry underpinning hypergeometric functions emerged from Riemann’s study of the local system of solutions of the Euler--Gauss hypergeometric differential equation. Using the remarkable properties of this local system, Riemann was able to give a conceptual explanation for hypergeometric identities of Gauss and Kummer. The goal of this talk is to give an introduction to these circle of ideas. Notes.

Friday, September 4: James Stanfield (PhD UQ) What is the Calabi Conjecture?

Abstract: In 1954, Eugenio Calabi posed a question about the existence of certain Riemannian metrics with prescribed curvature information on complex manifolds. This revealed itself to be a very hard question to answer, as the problem reduced to proving the existence and uniqueness of solutions to a complex Monge–Ampère equation, a highly non-linear PDE. In 1978, a proof of Calabi's conjecture was published by Shing-Tung Yau. This was a huge triumph for the blossoming field of geometric analysis, and lead to fascinating results in many other fields. It was largely for this that Yau became a Fields medallist in 1982. Our goal in this talk will be to try and understand the precise statement of Calabi's conjecture, its history, some of its implications, and part of its proof. Notes.

Friday September 11: Behrouz Taji (Faculty Sydney) What is Uniformisation?

Abstract: The aim of this talk is to discuss higher dimensional analogues of Riemann Uniformization Theorem. More precisely I will address the following question: What sort of topological constraints (numerical to be more precise) on a compact complex manifold forces the universal cover to be one of the following three: unit ball, projective space and complex space? The key turns out to be the so-called Einstein -or more generally Hermitian-Yang-Mills- metrics and their topological manifestations via Bogomolov-Gieseker and Miyaoka-Yau discriminants. Notes.

Friday September 18: Alexander He (PhD UQ) What is the Graph Isomorphism Theorem?

Abstract: Consider the following fundamental problem: given two n-vertex graphs X and Y, determine whether X and Y are isomorphic. It is natural to ask whether there is a polynomial-time (read: fast) algorithm to solve this problem. Although there is some indirect evidence that suggests that the graph isomorphism problem is not that difficult, an exponential-time (read: slow) algorithm from 1983 stood for over three decades as the fastest known algorithm. This algorithm was finally superseded in 2015, when László Babai announced that he had discovered an “almost polynomial-time” (or more precisely, quasipolynomial-time) algorithm for the graph isomorphism problem. In this talk, I will discuss the significance of this result, and outline some of the ideas involved in Babai’s algorithm. Notes. Supplemental materials.

Friday September 25: Ramiro Lafuente (Faculty UQ) What is Geometric Invariant Theory?

Abstract: In this talk we will discuss the origins and key principles of geometric invariant theory (GIT), putting ourselves into the broader context of group actions and their corresponding quotients. I will not assume any serious knowledge of algebraic geometry, and will rather aim for a more example-oriented presentation. If time permits, I will also briefly discuss the more modern connections that GIT has with symplectic geometry, gauge theory, and other related areas. Notes.

Friday October 1: No seminar. Come see us at the conference on integrable systems.

Friday October 9: Seamus Albion (MPhil UQ) What is the Honey Comb Model?

Abstract: In 1912 Hermann Weyl posed the following question: Given a pair of Hermitian matrices A and B, what are the possible eigenvalues of the sum A+B? Weyl and others made partial progress on the problem before, in 1962, Alfred Horn conjectured a complete family of necessary conditions on the eigenvalues of A+B. Horn's conjecture was finally proven correct by Knutson and Tao in 1998 using a new family of combinatorial objects they called honeycombs. I define honeycombs and discuss how they are related to Weyl's original problem, touching on ideas in representation theory and combinatorics along the way. Notes.

Friday October 16: GyeongHyeon Nam (PhD UQ) What is the Birch and Swinnerton-Dyer Conjecture?

Abstract: In this talk, we will discuss the history and some well-known results about the Birch and Swinnerton-Dyer (BSD) conjecture. There are many open questions related to the BSD conjecture, in this time, we will focus on the average rank of elliptic curves. Goldfeld and Katz-Sarnak conjectured that the average rank of elliptic curves might be 1/2. Some upper bounds of the average rank of elliptic curves was found by Brumer, Heath and Young with assumption that BSD and general Riemann hypothesis are true. But recently, Bhargava and Shankar found some upper bounds without using BSD and GRH. I will introduce their idea that how they calculated upper bounds with using n-Selmer groups and n-coverings of elliptic curves. Notes.

Friday October 23: Owen Colman (PhD Melbourne) What is Hodge theory?

Abstract: Hodge theory is an application of analysis to the study of compact Riemannian manifolds. Differential forms satisfying certain differential equations (being both closed and co-closed) are called harmonic, and one shows that every de Rham cohomology class is represented by a unique harmonic form. If one restricts attention to the class of complex projective manifolds, then the interaction of harmonicity and the complex structure gives rise to the so-called Hodge decomposition, which is of great significance in algebraic geometry.

This talk is in two parts. In the first part, I will outline an approach to the analytical aspects of Hodge theory via the heat equation. In the second part, I will state the Hodge decomposition theorem for compact Kähler manifolds, and discuss the Kähler identities and how Hodge's theorem follows from them. Notes.

Friday October 30: Linyuan Liu (Postdoc, Institute For Advanced Studies Princeton) What is mixed Hodge theory?

Abstract: In last week’s talk, we have seen that the complexed valued cohomology groups of every compact Kähler manifold admit a natural Hodge structure. In particular, this result applies to smooth complex projective varieties. In order to generate this theory to non-projective or non-smooth varieties, we need to define something called "mixed Hodge structure". To do this, I will start with some linear algebras, namely the Hodge filtration and the weight filtration. Then I will define what a mixed Hodge structure is, and state Deligne’s famous theorem on the existence of a natural mixed Hodge structure for a quasi-projective variety. I will also give some examples to illustrate this result. Notes.

Thursday November 12 (at the usual 11am): Joe Baine (PhD University of Sydney) What is a Soergel bimodule?

Abstract: Soergel bimodules are algebraic structures sitting at the nexus of representation theory and geometry. In this talk we will introduce Soergel’s classical approach to Soergel bimodules, and the diagrammatic approach due to Elias and Williamson. We will state many of their remarkable properties, like the Soergel Categorification theorem. However, this talk will emphasise that with minimal prerequisite knowledge one can develop a working understanding of Soergel bimodules using the diagrammatic calculus. There is no assumed knowledge for this talk.

Friday November 20 (special time 1pm): Cailan Li (PhD Columbia University) What is categorification?

Abstract: Categorification arose from a dream of Crane and Frenkel in their mission to find interesting examples of 4d TQFTs. In this talk, we first discuss the origins of categorification and then give a reasonable definition for what it means to "categorify" an object. The remainder of the talk will be spent on one such example: the (weak) categorification of the Fock space representation of the Heisenberg (lie) algebra. Notes.

Friday November 27 (special time 3pm): Stephen Lynch (Postdoc University of Tübingen) What is Thurston Geometrization?

Abstract: The uniformisation theorem for surfaces says that every closed two-manifold is a quotient of one of three constant curvature spaces. Thurston's geometrisation conjecture is an analogous result in three dimensions; it asserts that there are eight 'model geometries' from which every closed three-manifold can be constructed. By the combined works of Hamilton and Perelman, Thurston's conjecture could be proven using the Ricci flow. This is a nonlinear diffusion process that deforms Riemannian manifolds with pointwise speed determined by their curvature. After stating the geometrisation conjecture more precisely, we will discuss the major developments in Ricci flow that led to its resolution. Notes

Friday December 4 : Zhaoting Wei (Faculty Texas A&M University-Commerce) What is the Kobayashi-Hitchin correspondence?

Abstract: The Kobayashi-Hitchin correspondence gives a one-one correspondence between isomorphic classes of stable holomorphic vector bundles over a compact complex manifold and gauge equivalent classes of Hermitian-Einstein connections of vector bundles on the same manifold. In my talk I will explain these concepts and show the ideas of the proof. To illustrate the idea I will largely work on the simple case that the base manifold is a Riemann surface. Notes.

## Semester I, 2021

Date and Time: Usually Thursdays at 10 am Brisbane Time

Friday, March 5, 10am (Unusual day!): Rohin Berichon (PhD, UQ): What is the exotic structure on R^4?

Abstract: A classical question in differential topology asks how many distinct differentiable structures exist on a certain topological manifold. Remarkably, there is a unique differentiable structure on Euclidean spaces of dimensions not equal to 4, but uncountably many on Euclidean 4-space. In this presentation, we discuss the multiple constructions for exotic structures on 4 dimensional Euclidean space, and how to produce an uncountable family of these exotic structures. Notes.

Thursday, March 11: Benjamin Gammage (Postdoc, Harvard): What is Mirror Symmetry?

Abstract: Mirror symmetry predicts that a Kähler manifold X (near a certain scaling limit) admits a dual space X^ so that symplectic invariants of X are equal to algebraic invariants of X^. We will begin by reviewing the Fukaya category of Lagrangian submanifolds of X, focusing on the case when X is a Stein manifold, and then describe the homological mirror symmetry conjecture that the Fukaya category of X is equal to the category of coherent sheaves on X^. If time permits, we will explain how to prove this conjecture. Notes.

Friday, March 19, 9am (Unusual day and time!): Alexander Stokes (Postdoc, University College London): What is an integrable difference equation?

Abstract: An interesting feature of the field of integrable systems in general is that there is no single definition (applicable to all contexts) of what integrability is, but “you know it when you see it”, so much work in this area relates to defining or describing integrability in different classes of systems.

This is especially so in the theory of discrete integrable systems, and in this talk we will present some novel definitions of certain classes of integrable difference equations, emphasising how they are formulated in parallel with the classical differential case.

A particularly beautiful feature of the discrete case is that integrability can be described in terms of a wide range of concepts, varying from analytic measures of entropy to the geometry of complex algebraic surfaces associated with affine Weyl groups.

We will see definitions of integrability for lattice equations, for second-order equations defining birational mappings of the plane, and a particularly beautiful way of defining discrete analogues of the Painlevé differential equations. Notes.

Thursday, April 1, 10am: Alex Weekes (Postdoc, University of British Columbia): What is a Coulomb branch?

Abstract: As hinted at in their name, Coulomb branches come from physics: they are spaces which physicists associate to certain quantum field theories. But it so happens that many spaces of mathematical interest arise as Coulomb branches, which are especially important in representation theory and in the study of integrable systems.

As with many constructions in quantum field theory, a precise mathematical definition of Coulomb branches was difficult to achieve. Fortunately for us this was accomplished in recent work of Braverman, Finkelberg and Nakajima (BFN), who provide a rigorous definition in a large family of cases.

In this talk we will take a look at the BFN construction of Coulomb branches, making stops along the way to see some of the interesting spaces that arise. Notes.

April 8: No seminar, mid-semester break.

Thursday, April 15: Anna Romanov (Postdoc, Sydney): What is a Hecke algebra?

Abstract: If you hang around representation theory circles, you have probably heard a definition of a Hecke algebra. (For example, if you attended Anna Puskas’s WiSe talk last June.) If you hang around representation theory circles a lot, you have probably heard several definitions of a Hecke algebra. If you are like me, you may have found this confusing. In this talk, we will explore a few definitions of Hecke algebras. I will try to explain why they arise naturally in the representation theory of groups, and how the different definitions are related. We’ll also take a detour into Gelfand pairs, and explain how these fit into the story. Notes.

Thursday, April 22: Ian Whitehead (Assistant Professor, Swarthmore College): What is an Apollonian Packing?

Abstract: Fix four mutually tangent circles in the plane. Fill in the spaces between these circles with additional tangent circles. By repeating this process ad infinitum, on smaller and smaller scales, we obtain an Apollonian circle packing. In this talk I will sketch a proof of Descartes' theorem on circle configurations, and introduce a group which acts on packings in two different ways, with a subtle duality between them. If time allows, I will also talk about my own recent work relating packings to Kac-Moody root systems. This connection is via a four-variable generating function for curvatures that appear in an Apollonian packing, which is essentially a character for a rank 4 indefinite Kac-Moody root system. I will discuss its domain of convergence, the Tits cone of the root system, which inherits the rich geometry of Apollonian packings. Slides.

Thursday, April 29, 3pm (Unusual time!): Sebastian Heller (Priv.-Doz., Gottfried Wilhelm Leibniz Universität Hannover): What is a hyperkähler manifold?

Abstract: A hyperkähler structure is a geometric structure which occurs naturally in different fields such as algebraic geometry, theoretical physics and Riemannian geometry. For differential geometers, a hyperkähler manifold is a Riemannian manifold with three anti-commuting, parallel and orthogonal complex structures. The most prominent examples – Calabi-Yau manifolds – play an important role in string theory.

After discussing the definition and first properties of hyperkähler manifolds, we will explain some examples in detail. These examples are either constructed as hyperkähler quotients by adapting the symplectic reduction method to the Kähler forms or as the space of real holomorphic sections of the associated twistor spaces. If time permits, we will end the talk by referring to current research results. Notes.

Thursday, May 6, 11am (Unusual time!): Emily Thompson (PhD, Monash University): What is a hyperbolic knot?

Abstract: One of the major advances in modern knot theory is the result of William Thurston that classifies all knots as one of three types: a torus knot, a satellite knot, or a hyperbolic knot. When a knot is hyperbolic, we can apply tools and results from hyperbolic geometry to study it. But what is a hyperbolic knot?!

In the first half of this talk we will discuss some general knot theory, the upper half space model of hyperbolic space, and what makes a knot hyperbolic. In the second half we will carefully step through the decomposition of the figure-8 knot complement into two ideal tetrahedra and use this decomposition to prove that the figure-8 knot is hyperbolic. Notes.

Thursday, May 13: Steven Rayan (Assoc. Prof., University of Saskatchewan): What is a hyperpolygon? See also the AMS Notices What Is...? article (by Steven Rayan and Laura P. Schaposnik)

Abstract: Hyperpolygons are geometric objects originating in representation theory and, in particular, act as a bridge between a number of important geometric and representation-theoretic moduli spaces. Given this role, hyperpolygons interact naturally with a number of other notions that have been presented in this series, including Higgs bundles, character varieties, hyperkähler geometry, nonabelian Hodge theory, integrable systems, mirror symmetry, and Coulomb branches, to name a few. In the first part of the talk, we will review the construction of a Nakajima quiver variety, of which hyperpolygon space is a particular instance. In the second half of the talk, we will focus on the connections that hyperpolygons have with the various other notions from this series, which include a number of recent, interesting results. Notes.

Friday, May 21 4pm (Unusual day and time!): Pengfei Huang (Postdoc, Universität Heidelberg): What is nonabelian Hodge theory?

Abstract: Nonabelian Hodge theory can be thought as nonabelian analogue of (abelian) Hodge theory by replacing the abelian (coefficient) groups into nonabelian (coefficient) groups. This is mainly due to the celebrated work of Donaldson, Corlette, Hitchin, and Simpson, which gives us a correspondence between local systems and Higgs bundles. More precisely, the nonabelian Hodge theory gives an equivalence between the category of reductive representations of the fundamental group, the category of semisimple flat bundles, and the category of polystable Higgs bundles with vanishing rational Chern classes, through pluri-harmonic metrics. Moreover, such an equivalence of categories is functorial, and preserves tensor products, direct sums, and duals. In moduli viewpoint, this theory indicates that, the moduli space of irreducible representations (called character variety, or Betti moduli space), as a smooth affine variety, is complex analytic isomorphic to the moduli space of irreducible flat bundles (called de Rham moduli space), which is a smooth Stein manifold (in the sense of analytic topology), and is real analytic isomorphic to the moduli space of stable Higgs bundles (called Dolbeault moduli space), which is a smooth quasi-projective variety. All of these objects can be generalized to a family of flat λ-connections parametrized by λ ∈ C, a notion introduced by Deligne, further studied by Simpson, and Mochizuki.

In this talk, I will begin with a quick review of (abelian) Hodge theory as the motivation of this theory. Then I will introduce this theory precisely from an analytic viewpoint by introducing the work of Donaldson, Corlette, Hitchin, Simpson, and Mochizuki on the existence of pluri-harmonic metrics. Then I will talk about this theory from the moduli viewpoint. A good reference of this theory is a survey paper by S. Rayan and A. Garcı́a-Raboso ( “Introduction to nonabelian Hodge theory: flat connections, Higgs bundles, and complex variations of Hodge structure, Fields Inst. Monogr. 34 (2015), 131-171.”), you can also take the first chapter of my thesis as a reference. Notes.