Pure mathematics at Melbourne

Leonid Chekhov: Fool's crowns, discs, and volumes of moduli spaces

14  December

We consider "fool's crowns'' --- trumpets with n horocycle-decorated boundary cusps and discs—-ideal n-gones. Unlike the celebrated Mirzakhani's construction for moduli spaces of hyperbolic Riemann surfaces, the moduli spaces of both crowns and discs have infinite volumes when integrating against the invariant measure, and we need to introduce regularization, or action, to make these volumes finite. We use the variant of such an action that is independent on decorations introduced in arXiv:2411.03913 We evaluate the volumes of moduli spaces for arbitrary action coefficient \kappa and show that in the case \kappa=1, these volumes are expressed for arbitrary n in the form generalizing the one for Mirzhakhani's volumes, so it is natural to call this case the geometric case.  I will discuss a strong coupling limit (n \to \infty, \kappa/n \to const) in cases of the crown and disc and demonstrate the appearance of Schwarzian field theory in this limit. This is a joint work with Timothy Budd (Neijmegen).

Alexander Alexandrov:  From Topological Recursion to Matrix Models

28  November

Topological recursion originates from the solution of the Virasoro constraints for matrix models. This procedure provides surprisingly universal answers to many questions in enumerative geometry and mathematical physics. Now, with the help of x-y duality, we have a tool to return to the matrix models in cases where one side of the duality is trivial (or almost trivial). In my talk, I will describe this construction. This talk is based on a joint work with Boris Bychkov, Petr Dunin-Barkowski, Maxim Kazarian, and Sergey Shadrin.

Ethan Fursman: Breaking down representation theory with composition functors

17 October

Affine vertex algebras are algebraic objects which can be constructed from complex semisimple Lie algebras and are known to admit a rich representation theory. Quantum hamiltonian reduction (QHR) is a tool for passing from the (complicated) representation theory of an affine vertex algebra to the (simpler) representation theory of a related W-algebra. Other tools, such as inverse reduction, exist for passing back to representations of the original affine vertex algebra.

In recent work with David Ridout and Justine Fasquel, we have developed extensions of these tools, called composition functors. These can be used to study relaxed modules, twists by spectral flow automorphisms and certain important reducible but indecomposable modules. After giving a review of W-algebras, I will discuss our main result regarding how reduction acts on certain important classes of modules.

Alex Ghitza: Some kinds of Galois representations just don’t exist

10 October

Modularity statements (proved or conjectured) in number theory can be thought of as describing correspondences between two sets of different-looking objects.  An interesting special case is to show that when one of the sets is known to be empty, so is the other one.  I will discuss an instance of this, namely the non-existence of moderate-dimensional Galois representations modulo a small prime p that are only ramified at this prime.  The focus will be on providing some context for this result, including some of the algebraic and analytic number theoretic tools used in its proof.  (Joint work with Takuya Yamauchi, Tohoku University.)

Ae Ja Yee (Penn Sate): Partition Statistics and the Littlewood Decomposition

19  September

Integer partitions carry various interesting statistics. The most well-known combinatorial statistics are Dyson’s rank and crank.

In 1944, Dyson defined the rank of a partition claiming that this statistic combinatorially explains Ramanujan’s mod 5 and 7 partition congruences. Dyson’s claim was confirmed by Atkin and Swinnerton-Dyer in 1955. In the same paper, Dyson also conjectured the existence of another statistic for Ramanujan’s mod 11 congruence, namely crank, and this conjecture was settled by Andrews and Garvan in 1988.

While the rank and crank are combinatorial statistics, the proofs of Atkin–Swinnerton-Dyer and Andrews–Garvan are based on analytic methods. Thus, it was still sought to find a truly combinatorial proof for Ramanujan’s congruences.  In 1990, Garvan, Kim and Stanton found other cranks, which split the set of partitions into t equinumerous classes for t = 5, 7, 11, and thus gave a combinatorial account for the three congruences of Ramanujan.  In their proof, the Littlewood decomposition of partitions into t-core and t-quotient partitions is an essential component. Since then, their cranks along with Dyson’s rank and crank have been adopted to prove other partition congruences and refinements.

In this talk, I will discuss the Littlewood decomposition from a partition theory point of view and present some recent results on various partition statistics arising from the Littlewood decomposition.

Lior Yanovski: Higher Semi-additivity and Power Operations

12 September

The basis of commutative algebra is the structure of a commutative monoid.  Roughly, a rule for summing any finite family of elements in a way which is independent of parenthesization and order.  In this talk, I will discuss the notion of a "higher commutative monoid", where one can "integrate" families of elements indexed by more general homotopy types.  I will explain how such structures arise naturally in stable homotopy and category theory and outline some relations and applications to algebraic structures, such as lambda- and beta-rings.

Shelly Harvey: Obstructions to a knot being ribbon

5 September

Abstract:  A knot is a simple closed curve in R^3 - these play a very important role in 3- and 4-dimensional topology.  A slice knot is a knot which is the intersection of a knotted up 2-sphere in R^4 with R^3 x {0} (here we can think of the fourth dimension as time).  These turn out to play a special role in the study of 4-dimensional manifolds.  Slice knots are difficult to understand and the ones we understand best are ribbon knots -- they can be described as a knot that bounds a certain type of immersed disk in 3-dimensions!   One of the biggest open questions in low-dimensional topology (from the 60s) is whether every slice knot is ribbon, the so-called Ribbon-Slice conjecture.  This question is difficult for many reasons.  The first is that it is difficult to find examples of slice knots that don't come from ribbon knots.  The second is that we don't have very many ribbon obstructions that aren't also slice obstructions.  I will give a survey of some of the known ribbon obstructions to a knot (or link) being ribbon.

Campbell Wheeler:  Global regulators from quantum Chern-Simons theory

29 August

Towards the end of the 20th century, deep connections emerged between algebraic K-theory and the functional equations of polylogarithms. Notably, Bloch demonstrated that the algebraic structure of the third algebraic K-group of a number field is governed by the five-term relation of Euler’s dilogarithm function. This K-group also plays a fundamental role in the hyperbolic geometry of three-manifolds, particularly in relation to their classical Chern-Simons invariants. There are various realisations of these invariants, such as complex analytic, p-adic, or étale. In this talk, I will explore how the semi-classical limit of quantum Chern-Simons theory on three-manifolds—governed by the quantum dilogarithm—unifies all these realisations into a single and computable power series with coefficients in the underlying number field. As a consequence, we obtain a strong yet subtle integrality generalising work of Habiro. This is based on joint work with Garoufalidis, Scholze, and Zagier.

Simon Marshall:  Quantum ergodicity for sequences of locally symmetric spaces

August 22

Let X be a compact Riemannian manifold.   The quantum ergodicity theorem of Shnirelman, Zelditch, and Colin de Verdière is a result in semiclassical analysis which relates the dynamics of the geodesic flow on X, and the behaviour of Laplace eigenfunctions on X with large eigenvalue.  It states that if the geodesic flow is ergodic, then the Laplace eigenfunctions are quantum ergodic, meaning that a density 1 subsequence of them distribute their L^2 mass evenly over X.

In my talk, I will give some background on the quantum ergodicity theorem, then describe some extensions which apply to sequences of manifolds and regular graphs, rather than a single manifold.  Finally, I will describe a new result, joint with Brumley, Matz, and Peterson, which applies to sequences of locally symmetric spaces.  Our result relies on bounds for the volumes of intersecting shells in globally symmetric spaces, and new estimates for spherical functions on semisimple Lie groups.

Veronica Pasquarella:  Primitive invariants from laminations

11 August

Combining geometric group theory techniques with geometric topology tools, we show how primitive cohomologies provide useful insights towards unifying the mathematical formulation of Gromov-Witten invariants. In particular, we emphasise the role played by geodesic laminations in analysing such invariants for the case of complete intersections in projective space.

Matthew Emerton (Chicago):  An introduction to representations of p-adic groups in characteristic p

8 August

A lot of effort has been put into studying representations of groups like GL_n(F) for various natural choices of field F.  I will discuss some aspects of these problems, ultimately landing on the case of F = Q_p, with characteristic p coefficients, after first discussing the cases F = R or F_p with various coefficients.

Allen Knutson (Cornell):  Simplicial spheres from Richardson varieties

1 August

Algebraic geometers frequently consider "simple normal crossings divisors", hypersurfaces with in some sense the tamest possible singularities; in particular Hironaka's theorem lets one blow up any singular locus to an "sncd". Given an sncd one can associate, in a dual way, a simplicial complex. While these complexes can look like anything, there is a folk conjecture that if the sncd is anticanonical, its dual complex is nearly a sphere. I'll describe two ways to resolve the singularities of Schubert and more generally Richardson varieties, verifying the conjecture in one case and (for orbifold reasons) modifying it in the other.

David Ridout:  What is a vertex algebra?

23 May

Victor Kac is fond of recalling the following quote of I.M. Gelfand about vertex algebras:

“I am an old man, and I know that a definition cannot be so complicated.”

Well, the definition is complicated.  But, the reasons for the definition are not so complicated, if you look at any (non-trivial) example.  Unfortunately (for pure mathematicians), the best source of examples and understanding is physics, specifically conformal field theory.

I will go over the simplest non-trivial example with a view to explaining why the complicated definition is actually very natural.

Alexandria Rose (ANU):  The hyperbolic fractal uncertainty principle

16 May

The fractal uncertainty principle qualitatively states that a function and its Fourier transform cannot both concentrate on a (regular) fractal set.

Fractal uncertainty principles were first used to answer questions in quantum chaos about the behavior of chaotic billiards on negatively curved manifolds and to obtain estimates for the eigenfunctions of the Laplacian on such
manifolds.  Expressing these manifolds as a hyperbolic quotient, one sees that the group structure of the quotient generates a fractal set which captures the limiting behavior of chaotic trapped particles.

In this talk, we will explore a more generalized version of the fractal uncertainty principle in which the Fourier transform is replaced by a Fourier integral operator with a more general phase.  An important model case of special interest is the hyperbolic phase.  We prove new bounds for the hyperbolic fractal uncertainty principle that holds for fractal sets even in cases in which there is no usual (linear) fractal uncertainty principle, demonstrating the fundamental difference between the hyperbolic and the linear case.

Will Troiani: Algebraic geometry and linear logic

9 May

Geometry and logic have been intertwined in the study of mathematical truth since at least the time of Euclid. In the modern era, logical proofs are treated as formal mathematical objects and studied through denotational semantics alongside structures such as vector spaces and coalgebras. Constructive logics, particularly linear logic, are of contemporary interest due to their deep connections with computer science. In linear logic, an atomic formula X must be used exactly once, whereas the nonlinear formula !X may be reused freely and, in our model, is represented by the "space of proofs of X". We formalise this moduli space by defining !X as the Hilbert scheme of the projective scheme associated with X. This construction yields a novel geometric model of linear logic, and offers new insights into the relationships between geometry, logic, and computer science.

Peter McNamara: The spin Brauer category

2 May

We aim to study the representation theory of Lie algebras, in particular the special orthogonal Lie algebras. The aim is to give a graphical calculus for this monoidal category of representations. Time-permitting we will discuss the quantum case.

Dougal Davis: Hodge theory and unitary representations of Lie groups

11 May

Classifying the irreducible unitary representations of a non-compact Lie group is one of the oldest open problems in representation theory. The problem seeks to generalise the theory of Fourier series to functions with complicated symmetry and has close ties to both number theory and mathematical physics. In this talk, I will explain joint work with Kari Vilonen, in which we prove a conjecture of Schmid and Vilonen linking unitary representations to another beautiful story in mathematics: Hodge theory and the topology of algebraic varieties. I’ll explain how this turns the work of many mathematicians over the past century into sharp tools for understanding unitary representations and indicate how this has led to some rapid progress (joint with Lucas Mason-Brown) on previously intractable questions about them.

Aru Ray: How and why to slice knots in 4-manifolds

4 April

A knot in S³ is said to be slice in a 4-manifold M if it bounds an embedded disc in M – Int(B⁴}.  I will explain why this notion is interesting, by explaining its connection to a variety of questions about 4-manifolds. I will also introduce some constructions and obstructions, postponing the details to a follow-up talk in the Topology Seminar on Monday April 7th.

Friday March 28th - Peter Hall 162, 3:00pm - 4:00pm

Viveka Erlandsson (Bristol University): On Mirzakhani’s curve counting theorem and some applications

Given a hyperbolic surface of genus g, Mirzakhani studied growth of the number of simple geodesics of length bounded by L, as L grows, showing it is asymptotic to a constant times L^{6g-6}. In this talk I will discuss this result and give a brief overview of some generalizations and implication. For example, we will discuss why the same asymptotics hold also for other metrics, how one can obtain properties of “random” geodesics, as well as applications to square tiled surfaces.

Anand Deopurkar (ANU): The enumerative geometry of orbit closures

21 March

Take a degree d form in n variables.  What can we say about its GL(n) orbit?  In particular, what is its degree?  For n = 1 and 2, the answers were known by the year 1900.  For n = 3, Carel Faber and Paolo Aluffi found the answers around the year 2000.  I will discuss recent progress in higher dimensions on this and related questions, and the intricate geometry that seems to govern the answers.  Based on joint work with Anand Patel and Dennis Tseng.

Chiara Sava (Charles University Prague): The ∞-Dold-Kan correspondence via representation theory

14 March

In this talk we give a purely derivator theoretical reformulation and proof of a classic result of Happel and Ladkani, showing that it occurs uniformly across stable derivators and it is then independent of coefficients. The resulting equivalence provides a bridge between homotopy theory and representation theory: indeed, we explain how our result is a derivator-theoretic version of the ∞-Dold-Kan correspondence for bounded chain complexes.

Alexander Fish (Sydney): Sum-product phenomenon for sets of positive density in the integer lattice

7 March

We will present a refinement, developed in collaboration with Bjorklund, of Furstenberg's ergodic-theoretic approach tailored to addressing the problem of identifying  'twisted infinite patterns’ in positive-density subsets of the integer lattice. These patterns correspond to an infinite structure within the 'sum-products' formed by such sets. The talk is based on joint works with Bjorklund and Bulinski

Alex Küronya (Goethe University Frankfurt): Lattice polygons and finite generation of certain valuation semigroups

21 February

The main theme of the talk is the combinatorics of lattice
polygons and its relationship to the geometry of the associated toric
surfaces. We will focus on geometric  finiteness properties such as the
polyhedrality of the cone of curves and the finite generation of
valuation semigroups. The latter  is  a central (and wide open) 
question  in combinatorial algebraic geometry. This is an account of
joint work with Klaus Altmann, Christian Haase, Karin Schaller, and Lena
Walter.

Simone Cecchini (Texas A&M): Positive scalar curvature with point singularities

16 December

I will discuss obstructions to metrics of positive scalar curvature with uniformly Euclidean point singularities. This provides counterexamples to a conjecture by Schoen. I will also discuss the existence of metrics with uniformly Euclidean point singularities which cannot be smoothed by a geometric flow while preserving nonnegativity of the scalar curvature.

This is based on joint work with Georg Frenck and Rudi Zeidler.

Justin Sawon (UNC):  Lagrangian fibrations by Prym surfaces

6 December

Holomorphic symplectic manifolds (aka hyperkahler manifolds) are complex analogues of real symplectic manifolds. They have a rich geometric structure, though few compact examples are known. In this talk I will describe attempts to construct and classify holomorphic symplectic manifolds that also admit a holomorphic fibration. In particular, we will consider examples in four dimensions that are fibred by abelian surfaces known as Prym varieties.

Christoph Schweigert (Hamburg): Davydov-Yetter cohomology: some tools and some applications

15 November

We introduce Davydov-Yetter cohomology and determine
coefficients for this cohomology theory. We describe
applications to deformation theory and present
some tools for explicit computations of the cohomology
groups

John Huerta (ANU): Factorization algebras: A visual introduction

15 November

Factorization algebras are part of a mathemaical framework for quantum field theory developed by Costello and Gwilliam, but this talk will not be about quantum field theory. Instead, using pictures and intuitive topology, we will explain how factorization algebras encode some classic gadgets from algebra and topology: associative algebras, enveloping algebras, E_n-algebras (as proved by Lurie) and vertex algebras (as proved by Costello and Gwilliam). To conclude, we will touch on joint work with Rui Peixoto showing how factorization algebras can also encode vertex superalgebras.

Jack Hall (University of Melbourne): Full faithfulness for Deligne-Mumford stacks

25 October

In the mid-90s, Bondal and Orlov gave a simple criterion to check for when a functor between bounded derived categories of smooth projective complex varieties is fully-faithful. This type of result is frequently used to relate derived categories of smooth projective varieties with derived categories of moduli spaces of sheaves on them. I will describe an extension of Bondal-Orlov full faithfulness to smooth Deligne-Mumford stacks over fields of characteristic 0, which is a mild generalization of some recent work of Lim-Polischuk.

(SPMS) Adam Monteleone (University of Melbourne):  Fourier-Mukai Transforms and Derived Equivalences

11  October

The theory of derived categories, introduced by Grothendieck and Verdier, provides a powerful tool for studying the geometry of algebraic varieties via their categories of coherent sheaves. In 1981, Mukai introduced the Fourier-Mukai transform while studying abelian varieties, and it has since become a fundamental tool used in understanding when the derived categories of two varieties are equivalent.In this talk, I will setup the basic theory of derived categories, introducing the Fourier-Mukai transform and the Bondal-Orlov full faithfulness criterion. If time permits, I will discuss the role of the Fourier-Mukai transform in Kontsevich's homological mirror symmetry conjecture.

Chenyan Wu (University of Melbourne):  Explicit relation between invariants from Eisenstein series and theta lifts, with an application to Arthur packets

4 October

To a cuspidal automorphic representation of a classical group or a metaplectic group and a conjugate self-dual character, we associate an Eisenstein series and a family of representations that are called theta lifts. We establish a precise relation between the poles of the Eisenstein series and the lowest occurrence index among the theta lifts.

Thorsten Hertl (University of Melbourne):  Moduli spaces of Riemannian metrics with positive curvature

20 September

A manifold, which a priori a topological object, can be turned into a geometric object by equipping it with a Riemannian metric. This choice, however, is completely arbitrary, and one may very well wonder how geometric meaningful quantities, like the diameter, the volume, the total curvature or the geometric mass, depend on the Riemannian metric. Usually, these (global) quantities are invariant under the group of diffeomorphisms, so they should be studied as function on the quotient, which we refer to as the moduli space.

(SPMS) Chengjing Zhang (University of Melbourne):  An introduction to the theta correspondence

13 September

In 1964, to simplify Siegel’s analytical study of quadratic forms over integers, Weil generalised the oscillator representation in quantum mechanics to other settings and introduced an automorphic reincarnation of Jacobi’s theta function. In 1979, inspired by the classical invariant theory, Howe used the oscillator representation and the adelic theta function to construct the so-called theta correspondence.

There are two closely related versions of the theta correspondence: the local version is a ‘duality' between irreducible smooth representations of certain pairs of algebraic groups (for example, symplectic groups and orthogonal groups of even-dimensional quadratic spaces), while the global version is a method to transfer cuspidal automorphic representations between certain pairs of algebraic groups.

In this talk, I will discuss Weil’s generalisation of the oscillator representation and the theta function, and how they are used to construct the theta correspondence.

(SPMS)  Louie Bernhardt (University of Melbourne):  The Role of Curvature in Geometry

For thousands of years, the notion of curvature has been a touchstone of mathematics. From Archimedes to Einstein, curvature has not just been of intrinsic interest to mathematicians, but has fundamentally shaped the way we understand the (curvy) world around us. Nowadays, the idea of curvature lies in the disciplines of differential geometry and topology. These are large, active fields of research which interact with many areas of mathematics, physics, and other physical sciences. In this talk I will discuss the notion of curvature, and the important role it plays in the study of geometry. I will begin with a brief history of the field, touching in particular on Gauss' genius insights into the geometry of surfaces. Then, after a recap of some important concepts from differential geometry, I will discuss the relevance of curvature in modern mathematical research. This will include the study of general relativity, and of Riemannian manifolds with positive curvature.

Luca Cassia (University of Melbourne):  Virasoro constraints for β-ensembles and generalized Catalan numbers

30 August

Random matrix models involve integrals over spaces of matrices with various measures. The generating functions of correlators in these models often have a topological expansion that encodes information about enumerative geometric problems like map enumeration, Hurwitz theory, and intersection theory on moduli spaces. These models also satisfy Virasoro constraints linked to reparametrization invariance of the integrals, which can be expressed as linear differential equations for the generating function. In this talk, I will explore the connection between these aspects of random matrix models. Additionally, I will discuss a 1-parameter deformation, with the deformation parameter β related to the Virasoro algebra’s central charge, and I will show how one can derive a genus expansion of the deformed generating function, where the coefficients are polynomials in β, reducing to generalized Catalan numbers when β=1.

(SPMS)  Yifan Guo (Brown University):  Matrix ensembles and zeros of the Riemann zeta function

23 August

Odlyzko showed in 1987 that the distribution of the non-trivial zeros of the Riemann zeta function can be described by the spacings of eigenvalues of certain random matrices. These random matrices belong to a class called Gaussian ensembles. In this talk, I will introduce some key examples of Gaussian ensembles and the distribution of their eigenvalues. I will also attempt to link Gaussian ensembles to other well-known problems, such as the Riemann Hypothesis.

Jonathan Bowden (Regensburg):  The tight geography problem for high dimensional contact manifolds

16 August

Contact manifolds arise naturally in the context of classical mechanics as regular level sets of Hamiltonians in phase space satisfying certain convexity properties. More precisely, a contact structure is a totally non-integrable hyperplane field, with classical examples given by left invariant fields on certain Lie groups. Relatively recently  Borman-Eliashberg-Murphy proved an h-principle for so-called overtwisted contact structures, reducing the existence and classification problem to classical obstruction theory. This leads to the problem of studying contact manifolds that exhibit rigidity, so-called tight contact structures. In this talk I will report on some progress on a programme to utilise classical surgery theory together with analytic tools such as Floer homology to construct examples of such tight contact structures. (This is part of joint work with D. Crowley and J. Hammet).

(SMPS) Diarmuid Crowley (University of Melbourne):  An introduction to contact topology

9 August

In this talk I'll review the basic definitions in contact topology and discuss the notions of overtwistedness and tightness for contact structures.  Then I'll take a brief look at the hierarchy of tight contact structures, including the recent work of Bowden, Gironella and Moreno.  Finally, I'll conclude by indicating how bordism theory can be used to formulate and investigate the tight contact geography problem.

This is an introduction to Jonathan Bowden's talk on August 9th.  Good background for both talks is available in the Qanta article on the work of Bowden, Gironella and Moreno

https://www.quantamagazine.org/in-the-wild-west-of-geometry-mathematicians-redefine-the-sphere-20231107/

Yuxuan Li (University of Melbourne):  Aldous' spectral gap conjecture and its generalizations

2 August

Aldous' spectral gap conjecture asserts that for any finite graph \Gamma with vertex set [n]=\{1, 2, \ldots, n\}, the interchange process and the random walk on \Gamma, both continuous-time Markov chains, exhibit identical spectral gaps. Notably, the random walk with the state space [n] is a subprocess of the interchange process, which operates over the larger state space S_n. After nearly two decades of being unresolved, this conjecture was conclusively validated in 2010.

This presentation aims to introduce several equivalent formulations of Aldous' conjecture from diverse perspectives. Additionally, it will explore various generalizations of the conjecture, emphasizing insights drawn from algebraic graph theory.

(SPMS) David Lumsden (University of Melbourne):  Card shuffling, the symmetric group and spectral gaps

26 July

In this talk I will give an overview of some results related to shuffling cards viewed as a random walk on the symmetric group. In particular the Aldous spectral gap conjecture and results surrounding generating a permutation from random transpositions will be discussed.

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.

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.

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.

Coming soon