Seminars

Past seminars

• 2024, Semester 1

  Alex Sherman (University of Sydney): How to fit a supergroup into a finite group

  17 May

  I will discuss recent developments which demonstrate surprisingly close analogies between supergroups over the complex numbers and finite groups in characteristic p>0 (modular representation theory).  Namely, for supergroups we are able to define analogues of Sylow subgroups, p-subgroups, and elementary abelian p-groups. Local representation theory is a powerful technique for studying modular representations of finite groups. I will explain how our theory allows us to apply local representation theory to the super setting, and prove several important results including a projectivity criterion and a description of the cohomological support variety.

  This is part of joint work with Julia Pevtsova, Vera Serganova, and Dmitry Vaintrob.

  June Park (University of Melbourne): Totality of rational points on modular curves over function fields

  10 May

  People want to count elliptic curves over global fields such as the field Q of rational numbers or the field F_q(t) of rational functions over the finite field F_q. To this end, we consider the fact that each E/K corresponds to a K-rational point on the fine moduli stack Mbar_{1, 1} of stable elliptic curves, which in turn corresponds to a rational curve on Mbar_{1, 1}. In this talk, I will explain the exact counting formula as well as basic generalities, relevant tools and ideas.

  Will Donovan (Tsinghua University): McKay correspondence and toric geometry

  26 April

  Finite subgroups of the matrix group SU(2) may be studied algebraically via their representations. They may also be studied geometrically via two-dimensional complex manifolds naturally associated to them. The McKay correspondence is a general phenomenon which, in particular, explains how these two approaches relate. I'll introduce this using diagrammatics from toric geometry, indicate how the correspondence generalizes to higher dimensions, and discuss open questions and current projects.

  Behrouz Taji (University of New South Wales): Boundedness problems: algebraic geometry meets arithmetic

  Tuesday 23 April

  In the 1960’s Shafarevich asked a simple question: Do families of curves of genus at least 2 have a finite number of deformation classes? Soon after Parshin gave an affirmative answer to this question and furthermore showed that this finiteness property is precisely the geometric underpinning of Mordell’s famous conjecture regarding finiteness of rational points for projective curves of genus at least 2  (over number fields). It was this very observation that led Faltings to his complete proof of Mordell’s conjecture in 1980s. For higher dimensional analogues of curves of high genus, Shafarevich’s problem was settled in 2010 by Kovács and Lieblich, partially thanks to multiple advances in moduli theory of so-called stable varieties. In this talk I will present this circle of ideas and then focus on our recent solution to Shafarevich’s question for other higher dimensional (non-stable) varieties, e.g. the Calabi-Yau case. This is based on joint work with Kenneth Ascher (UC Irvine).

  Jieru Zhu (University of Queensland): Tensor representations for the Drinfeld double of the Taft algebra

  12 April

  The Drinfeld double of the Talf algebra often serves as a common example of non-semisimple Hopf algebras, and is related to Lie theory being a quotient of the small quantum group. It is also a ribbon Hopf algebra where its module category is a ribbon category. We show that the braid group action on the tensor representation, introduced by Ram-Leduc, factors through the Temperley-Lieb algebra and induces an isomorphism with the centralizer algebra. This is under the assumption that the number of tensors is small. Further work includes studying a modular Schur-Weyl duality, as well as the action of the Karoubi envelope of the Temperley-Lieb category. This is joint work with Benkart-Biswal-Kirkman-Nguyen.

  Aravind Asok (University of Southern California): Motivic homotopy theory: what is it good for?

  27 March

  I will try to explain some aspects of motivic homotopy theory, culminating with a discussion of recent progress and applications to problems about when holomorphic vector bundles on complex affine varieties admit algebraic structures. This talk is based on joint work with Tom Bachmann, Jean Fasel and Mike Hopkins.

  Scott Mullane (University of Melbourne): Teichmüller dynamics and the moduli space of curves

  21 February

  Integrating a differential on a Riemann surface allows the pair to be expressed as a collection of polygons in the plane with parallel side identifications. The action of GL(2,R) on the plane extends naturally to these polygons, and the orbits of the action, originally considered for their dynamical importance, have unexpected algebraic properties. In this talk, we'll introduce these ideas and discuss ways that this new perspective can be applied to questions on the birational geometry of moduli spaces of curves.

  Arunima Ray (Max-Planck Institute Bonn): Knots, links, and 4-dimensional spaces

  20 February

  Manifolds are fundamental objects in topology since they locally model Euclidean space. A central problem of interest is the classification of low-dimensional manifolds, especially those of dimension four. I will explain how powerful techniques from high-dimensional manifold topology, such as surgery theory, can be useful in this context, e.g. in Freedman's proof of the 4-dimensional Poincare conjecture. The key step involves replacing a given map from a surface to a 4-dimensional manifold by an embedding. Remarkably, important open problems in 4-dimensional manifold topology have equivalent formulations in terms of properties of knots and links in 3-dimensional space.

  Dougal Davis (University of Melbourne): Hodge theory and unitary representations of real groups

  20 February

  A classical open problem in representation theory is to determine the set of irreducible unitary representations of a non-compact Lie group. This has proven difficult partly due to a lack of tools to control the key property of unitarity. In this talk, I will discuss a new such tool based on Hodge theory. Hodge theory has its roots in the study of the cohomology of complex algebraic varieties and plays a major role in modern algebraic geometry. The main result of this talk (joint work with Kari Vilonen) is that Hodge theory also controls unitary representations for the main class of interesting Lie groups, the real groups. If time permits, I will sketch how this new perspective leads to a simple proof of unitarity for a wide class of new representations, called rigid unipotent, which are conjectured to play a pivotal role in the full classification (joint work in preparation with Lucas Mason-Brown).

  Lior Yanovski (Hebrew University Jerusalem): Descent in algebraic K-theory and the telescope conjecture

  19 February

  Algebraic K-theory is a fundamental invariant of rings and categories with applications to various fields of mathematics including number theory, differential topology, and algebraic geometry. It is however very hard to compute, largely because algebraic K-theory fails to have good descent properties. I.e., it does not satisfy a good "local-to-global principle" on the source of the construction. Stable homotopy theory suggests a different ''local-to-global principle'' on the target of the construction, such that, remarkably, the local pieces of algebraic K-theory satisfy much better descent properties on the source. In this talk, I will discuss a recent work of Ben-Moshe, Carmeli, Shclank, and myself showing that these local pieces of algebraic K-theory satisfy in fact much stronger ''higher descent'' properties. I will also outline an application of these results to the resolution of the celebrated long-standing telescope conjecture by Burklund, Hahn, Levy, and Schlank, and the relationship to the Ausoni-Rognes redshift conjecture. I will review the relevant background material, so no prior knowledge of stable homotopy theory is required.

  Toni Annala (IAS, Princeton): Homotopy theory of varieties

  19 February

  In topology, homotopy theory is an important tool for understanding cohomology groups of topological spaces. Morel and Voevodsky set up an analogous theory in algebraic geometry, essentially by declaring homotopies to be parameterized by the affine line A^1. This theory is called A^1-homotopy theory, and it was used in a crucial way in Voevodsky's proof of the Milnor and Bloch-Kato conjectures. Unfortunately, A^1-homotopy theory has a flaw built into its heart: by design, the affine line A^1 is contractible in A^1-homotopy theory, meaning that A^1-homotopy theory can only treat cohomology theories for which the cohomology groups of A^1 are isomorphic to those of a point. Many important cohomology theories (prismatic cohomology, algebraic K-theory) violate this assumption, and this fact has sparked efforts to discover another, more fundamental form of homotopy theory of varieties. I will describe my program, together with Marc Hoyois and Ryomei Iwasa, whose purpose is to tackle exactly this issue by constructing "the correct" stable homotopy theory of varieties.

  Sabino Di Trani (University of Rome): On irreducible representations in exterior algebra and their multiplicities

  16 February

  Let g be a simple Lie algebra over C. The adjoint action of g on itself induces a structure of a g- representation on Λg, the exterior algebra  over g.   During the talk I will provide an overview of known results and open problems concerning irreducible representations appearing in Λg and their graded multiplicities.

  Giovanni Inchiostro (University of Washington): Wall crossings and moduli spaces

  7 February

  Moduli spaces provide an important tool to study how certain objects, say Riemann surfaces, relate to each other. I will start by discussing a few moduli spaces of compact and open Riemann surfaces, compare them, and explain generalizations to studying arbitrary dimensional spaces.

  Anne Dranowski (University of Southern California): Refinements in representation theory

  7 February

  In representation theory, groups are studied via their actions. An action is a way of associating to the elements of a group symmetries of a vector space. Symmetries are classified with the help of eigenbases. Good eigenbases are those which are compatible with certain natural filtrations. Perfect eigenbases are those which are approximately permutated by generators of the group. We present three examples of perfect bases constructed using very different methods (geometric, algebraic) as well as original results and works in progress comparing them.

  Dr Jian Wang (University of North Carolina): Mathematical theory of internal waves

  6 February

  Internal waves are a central topic in oceanography and the theory of rotating fluids. They are gravity waves in density-stratified fluids. In a two-dimensional aquarium, the velocity of linear internal waves can concentrate on certain attractors. Locations of internal wave attractors are related to periodic orbits of homeomorphisms of the circle, given by a nonlinear "chess billiard" dynamical system. This relation provides a surprising "quantum--classical correspondence" in fluid dynamics. In this talk, I will explain connections between homeomorphisms of circles, spectral theory, and internal wave dynamics. This talk is based on joint work with Semyon Dyatlov and Maciej Zworski.

  Slava Futorny (SUSTech Shenzhen): Representations of Lie algebras of vector fields on algebraic varieties

  1 February

  Derivations of a ring of polynomial functions on affine algebraic varieties is a rich source of simple infinite-dimensional Lie algebras. We will discuss the state of the art of the representation theory of these Lie algebras.

  Erik Carlsson (UC Davis): A descent basis for the Garcia-Procesi module

  25 January

  The Tanisaki ideal encodes the generators and relations in the cohomology ring of the regular nilpotent Springer fiber, whose graded character with respect to a certain action of the symmetric group was famously shown by Springer to be the Hall-Littlewood polynomial. The corresponding quotient ring was studied as a module by Garsia and Procesi, who gave an explicit basis of monomials, which is a subset of the Artin basis of the coinvariant algebra. I'll present a new basis consisting of certain Garsia-Stanton descent monomials, as well as some connections to the Garsia-Haiman ring, which encodes the modified Macdonald polynomial. This is joint work with Ray Chou.

• 2023, Semester 2

  J.S. Lemay (Macquarie University)

  21 November

  This talk will be an introduction to differential/tangent categories and how they link to differential geometry, algebraic geometry, operads, etc. This talk will be introductory and should be accessible to students who want to know about category theory.

  Miles Reid (University of Warwick): Fano 3-folds and the Graded Ring Database

  14 November, 3.15–4.5pm

  Fano varieties occupy the "positive curvature" niche in the classification of varieties. They include many of the familiar varieties such as projective space, low degree hypersurfaces and homogeneous spaces used in representation theory of algebraic groups.

  Fano varieties are a key area of progress in current algebraic geometry. I will give a colloquial presentation of the general state of the subject, and a few nice special cases.

  It goes without saying that the modern theory goes beyond the simple first cases. The Graded Ring Database (work of Gavin Brown and Al Kasprzyk) contains lists of candidate constructions of Fano 3-folds. A few hundred cases can be settled either by explicit constructions or by impossibility proofs, and while it is hard to reach convincing conclusions in the majority of cases GRDB provides a huge source of challenging research problems.

  Yalong Cao (RIKEN): Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds

  10 November

  BPS invariants were introduced by Gopakumar-Vafa on Calabi-Yau 3-folds, Klemm-Pandharipande on CY 4-folds and Pandharipande-Zinger on CY 5-folds. They are conjectured to be integers (proven in many cases) and have correspondence with Gromov-Witten invariants. On holomorphic symplectic 4-folds, (ordinary) GW and hence BPS invariants vanish, one can consider reduced GW invariants which are usually nontrivial rational numbers. In this talk, we will introduce BPS invariants for such a reduced theory. Joint works with Georg Oberdieck and Yukinobu Toda.

  Justine Fasquel (University of Melbourne): Building blocks for W-algebras

  3 November

  W-algebras are a large family of vertex algebras associated to nilpotent orbits of simple Lie algebras. For classical Lie algebras, they are parametrized by certain partitions. Among the W-algebras of type sl(n) those with nilpotent orbits corresponding to hook partitions (m,1,1,…) of n are the most understood ones. In this talk, we will show that in fact any W-algebras of type sl(n) should be expressed by using several hook-type W-algebras. We will illustrate with examples in small ranks. It’s a work in progress with T. Creutzig, A. Linshaw and N. Nakatsuka.

  Valeriia Starichkova (UNSW Canberra): Primes in short intervals

  20 October

  This talk is inspired by my main thesis project on primes in short intervals. I would like to talk about the main ingredients used in the last works in the area which involve sieves, zero-density estimates and some other combinatorial ideas. In particular, we will introduce and talk about sieve methods (such as the linear sieve and Harman's sieve), which play an important role in multiplicative number theory.

  Christian Haesemeyer (University of Melbourne): K-theory of singularities, revisited

  13 October

  Algebraic K-theory is an invariant that reflects algebraic and geometric information in a complicated mix. It has been known since the inception of the field that while K-theory behaves like a (co)homology theory in the regular case, homotopy invariance can fail in the presence of singularities. T Vorst conjectured that this failure can be used to detect singularities over fields; this conjecture has been proved in both characteristic zero and positive characteristic using trace methods, and the algebraic geometry of varieties. I will discuss this - by now classical - work, and then talk about work in progress with Weibel trying to understand failure of homotopy invariance as an expression of (homological) algebraic instead of (algebraic) geometric properties.

  Bryce Kerr (UNSW Canberra): Quantitative local to global principles and the sum product problem

  2 October

  I’ll describe some joint work with Jorge Mello and Igor Shparlinski which makes progress on some additive combinatorics problems in prime fields by applying quantitative local-to-global principles.

  Nora Ganter (University of Melbourne): An introduction to Grothendieck duality

  22 September

  I will speak about joint work with Simon Willerton and explain how Grothendieck duality, in its simplest setting, naturally arises from Fourier-Mukai theory.

  Brian Krummel (University of Melbourne): Analysis of singularities of area minimising currents

  15 September

  In his monumental work in the early 1980s, Almgren showed that the singular set of an n-dimensional locally area minimizing submanifold T has Hausdorff dimension at most n - 2.  We will discuss a new approach to this problem (joint work with Neshan Wickramasekera) in which we first prove certain regularity properties such as uniqueness of tangent cones at H^(n - 2)-a.e. singular point of T.  We then prove results about the fine structure of the singular set, namely that the singular set is an n-2-dimensional countably rectifiable set and T is asymptotic to a unique homogeneous multi-valued harmonic function at H^(n - 2)-a.e. branch point of T.

  Joint Pure Maths/Mathematical Physics seminar, Evan Williams Theatre – Ole Warnaar (University of Queensland): Virtual Koornwinder integrals

  8 September

  Virtual Koornwinder integrals are deformations of integrals over classical group characters that can be used to obtain combinatorial expressions for characters of affine Lie algebras. In this talk I will first describe the main ingredients of the classical theory and its connection to Gelfand pairs, and then discuss generalisations and applications to characters of affine Lie algebras.

  Gufang Zhao (University of Melbourne): Towards a cohomological field theory via Lagrangian correspondences

  1 September

  This talk aims to propose a construction of a cohomological field theory using the category of Lagrangian correspondences, and the bi-category of matrix factorizations. A motivating example comes from the GIT quotient of a vector space by an abelian group, in the presence of an equivariant regular function (potential).  In the example, virtual counts of quasimaps from prestable curves to the critical locus of the potential are defined, drawing ideas from the theory of gauged linear sigma models as well as recent developments in shifted symplectic geometry and the Donaldson-Thomas theory of Calabi-Yau 4-folds. Examples of virtual counts arising from quivers with potentials, based on work in collaboration with Yalong Cao, are discussed.

  Yau Wing Li (University of Melbourne): Endoscopy for affine Hecke categories

  25 August

  Affine Hecke categories are categorifications of Iwahori-Hecke algebras, which are essential in the classification of irreducible representations of loop group LG with Iwahori-fixed vectors. The affine Hecke category has a monodromic counterpart, which contains sheaves with prescribed monodromy under the left and right actions of the maximal torus. We show that the neutral block of this monoidal category is equivalent to the neutral block of the affine Hecke category (with trivial torus monodromy) for the endoscopic group H.

  Lisa Carbone (Rutgers University): Lie group analogs for infinite dimensional Lie algebras

  22 August

  We discuss the question of associating analogs of Lie groups to certain classes of infinite dimensional Lie algebras. We are particularly interested in Kac-Moody algebras, which are the infinite dimensional analogs of simple Lie algebras, and further generalizations of Kac-Moody algebras known as Borcherds algebras. The Monster Lie algebra is an example of a Borcherds algebra and it admits an action of the Monster finite simple group. We discuss recent developments in the construction of a Lie group analog for the Monster Lie algebra.

  Angus McAndrew (Australian National University): A descent theorem for K3 surfaces

  11 August

  Descent problems have fascinated mathematicians since ancient times. A modern descent question asks for the field of definition of a given algebraic variety, i.e. whether there is a criterion for when it can be descended from a field to a smaller one. A theorem of Grothendieck gives an answer to this question in the case of abelian varieties and transcendental field extensions. We will discuss a general conjecture inspired by this, and prove it in the case of K3 surfaces, under some hypotheses. The proof uses Madapusi-Pera's work on the Kuga-Satake construction.

  Daniele Celoria (University of Melbourne): The 3D index and Dehn surgery

  4 August

  After giving a general introduction to Dimofte-Gaiotto-Gukov's 3D index for cusped hyperbolic 3-manifolds, we'll dive into some of its relations with basic hypergeometric series. Then we'll describe an ongoing effort to prove how the 3D index changes under Dehn surgery. This is work in progress with Profs C. Hodgson and H. Rubinstein.

• 2023, Semester 1

  Chris Kottke (New College of Florida): Bigerbes

  2 June

  Gerbes are geometric objects on a space which represent degree 3 integer cohomology, in the same way that complex line bundles (classified by the Chern class) represent degree 2 integer cohomology. Higher versions of gerbes, representing cohomology classes of degree 4 and up, are typically complicated by the need to use higher categorical concepts in their definition. In contrast, bigerbes and their higher analogues admit a simple, geometric, non-higher-categorical description, and provide a satisfactory account of the relationship between so-called `string structures' on a manifold and `fusion spin structures' on its loop space. This is based on joint work with Richard Melrose.

  Victor Turchin (Kansas State University): Spaces of higher dimensional knots

  19 May

  (Partially based on joint work with Arone, Ducoulombier, Fresse, Willwacher)

  A higher dimensional knot is a smooth embedding S^m --> R^n. It is well-known that the problem of classification of classical knots m=1, n=3 is very hard and has not been fully done, though many efficient techniques to distinguish knots appeared over the years. In contrast with the classical case, in higher dimensions and assuming the complement is simply connected i.e., n-m>2, A. Haefliger showed in 1966 that the isotopy classes of knots form a finitely generated abelian group of rank at most one, which is a finite torsion in most of the cases. I will review some of Haefliger's results and will also speak about new techniques that allowed one to make the rational homotopy groups of such knot spaces fully computable.

  Volker Schlue (University of Melbourne)" A scattering theory for wave equations with homogeneous asymptotics

  12 May

  For the classical wave equation, a scattering theory was developed by Friedlander in the 80s. Its applicability is limited to linear equations and excludes data that is relevant in many physical situations. In this talk I will present recent progress on a scattering theory for non-linear wave equations which arise in the description of gravitational waves. In these settings Huygen's principle fails, and instead solutions display homogeneous asymptotics. I will present results which give a construction of global solutions from scattering data, even in settings when Huygen's principle fails, and relate their behaviour to the presence of masses, and charges, in the data. This is joint work with Hans Lindblad.

  Melissa Lee (Monash University): A wander through some intriguing problems in finite group theory

  5 May

  In this seminar I will discuss a number of problems that have caught the attention of myself and my collaborators over the past couple of years. I will talk about the classification of extremely primitive groups, which have been studied since the 1920s, how we might recognise certain families of groups according to their prime graphs and about some "hot off the press" research into computing with the Monster group. In addition to the recent developments on these problems, there are still some significant open questions, so I hope there will be something for everyone.

  Hui Gao (Southern University of Science and Technology, Shenzhen): Integral p-adic Hodge theory

  28 April

  In complex geometry, one uses Hodge structures to encode the linear algebraic structures of singular and de Rham cohomologies. In this talk, we construct a category of Breuil-Kisin G_K-modules to encode the (semi-) linear algebraic structures of the integral p-adic cohomologies recently developed by Bhatt--Morrow--Scholze and Bhatt--Scholze. These modules also classify integral semi-stable Galois representations.

  Moritz Doll (University of Melbourne): An overview of Weyl laws

  21 April

  Given a positive elliptic differential operator with discrete spectrum, we consider the asymptotics of the eigenvalue counting function. In 1911 Hermann Weyl calculated the leading order behavior of the counting function in the case of the Laplacian on bounded domains. We will give an overview of the various results of sharp remainder estimates and improved remainder estimates under various geometric conditions in the case of both compact and non-compact manifolds.

  Masoud Kamgarpour (University of Queensland): Geometry of representation spaces

  14 April

  The space of representations is a fundamental object with deep relationship to Higgs bundles, connections, and Yang—Mills equations. Thanks to the work of Hausel and collaborators, much is known about the geometry of this space "in type A" i.e., for representations into the group GL_n. However, if we consider representations into groups of more general type (required for applications to Langlands duality and mirror symmetry), then very little is known about the geometry of representation spaces. I will discuss how one can use Deligne—Lusztig theory to get a handle on these spaces and compute some of their invariants.

  Based on work in progress with my students Gyeonghyeon Nam and Bailey Whitbread.

  Lance Gurney (University of Melbourne): A tour through cohomology theories for algebraic varieties

  31 March

  In 1949 Andre Weil conjectured the existence of a cohomology theory for algebraic varieties with certain formal properties. By the mid 1960s Alexander Grothendieck had constructed many such cohomology theories: ℓ-adic étale, crystalline, algebraic de Rham, and today we have even more. For fifty years one of the central problems of arithmetic geometry has been to try to understand exactly how these cohomology theories are all related and, just maybe, to find a universal one.

  In this talk I'll give a brief tour through some of these cohomology theories and their fascinating relationships before describing some recent advances.

  The majority of the talk will be accessible to a general audience.

  Guillaume Laplante-Anfossi (University of Melbourne): Convex polytopes and higher categories

  24 March

  The theory of categories could be described as the study of directed graphs, given by objects and maps between them: sets and functions, vector spaces and linear maps, groups and group homomorphisms... The theory of higher categories, and more precisely the theory of strict n-categories, is in this respect the study of higher dimensional shapes called pasting diagrams. These are directed graphs, together with higher dimensional cells: 2-arrows between 1-paths, 3-arrows between 2-paths, and so on. These shapes, first studied by the Australian school -notably R. Street and M. Johnson- need to satisfy a certain loop-free condition in order to describe faithfully n-categorical composition. In a 1991 « list of results » paper, Kapranov and Voevodsky proposed to study pasting scheme structures on polytopes. They conjectured that a frame which is in generic position with respect to a polytope provides an example of a pasting scheme. We show that this is actually false, by exhibiting a certain choice of frame for which the 5-simplex is not loop free. This result, obtained in current joint work with Arnau Padrol, Eva Philippe and Anibal M. Medina-Mardones, suggests the existence of deep and yet to be discovered links between discrete geometry and higher category theory.

  Changlong Zhong (State University of New York at Albany): K-theory stable basis of Springer Resolutions

  17 March

  Stable bases (for cohomology, K-theory and elliptic cohomology) are introduced by Okounkov and his collaborators. They were used to study actions of various quantum groups on quantum cohomology theories. They are also related with various classes defined in algebraic geometry, namely, the Chern-Schwartz-MacPherson classes and the motivic Chern classes. In this talk I will introduce the K-theory stable basis of Springer resolutions. This is joint work with Changjian Su and Gufang Zhao.

  Amnon Neeman (Australian National University): Vanishing negative K-theory and bounded t-structures

  3 March

  We will begin with a gentle reminder of algebraic K-theory, and of a few classical, vanishing results for negative K-theory. The talk will then focus on a striking 2019 article by Antieau, Gepner and Heller - it turns out that there are K-theoretic obstructions to the existence of bounded t-structures. And we will illustrate what a bounded t-structure is by working through an example. The result suggests many questions. A few have already been answered, but many remain open. We will concentrate on the many possible directions for future research.

• 2019, Semester 1

  Coming soon