Seminars

Location: Online via Zoom
Abstract: This is a joint work with Samuel Jeralds. Let 𝔤 be an affine Kac-Moody Lie algebra and let λ, µ be two dominant integral weights for 𝔤. We prove that under some mild restriction, for any positive root β, V(λ) ⊗ V(µ) contains V(λ + µ – β) as a component, where V(λ) denotes the integrable highest weight (irreducible) 𝔤-module with highest weight λ. This extends the corresponding result by Kumar from the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras. One crucial ingredient in the proof is the action of Virasoro algebra via the Goddard-Kent-Olive construction on the tensor product V(λ) ⊗ V(µ). Then, we prove the corresponding geometric results including the higher cohomology vanishing on the 𝒢-Schubert varieties in the product partial flag variety 𝒢/𝒫 × 𝒢/𝒫 with coefficients in certain sheaves coming from the ideal sheaves of 𝒢-sub Schubert varieties. This allows us to prove the surjectivity of the Gaussian map.
Seminar notes: PDF document
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Abstract: Over the past two decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete but there is an infinite series of groups in dimension 4 – the symplectically primitive but complex imprimitive groups – and 10 exceptional groups up to dimension 10, for which it is still open. Recently, we have proven that for all but possibly 39 cases in the remaining infinite series there is no symplectic resolution. We have thereby reduced the classification problem to finitely many open cases. We do not expect any of the remaining cases to admit a symplectic resolution. This is joint work with Gwyn Bellamy and Johannes Schmitt.

Abstract: We describe a K-theoretic version of the equivariant constructible derived category. We state (with evidence!) a ‘geometric Satake’ conjecture relating its value on the affine Grassmannian to representations of the Langlands dual group.
About the speaker: Gus Lonergan is the Chief Mathematician at A Priori Investment Management LLC. He was previously a L.E. Dickson Instructor in the mathematics department at the University of Chicago. He is interested in representation theory.
Lonergan completed his PhD at MIT under Roman Bezrukavnikov; the thesis was an attempt to understand mod p phenomena in algebraic topology in the context of geometric representation theory. He attended Cambridge University for his undergraduate and master’s degree. He plays a little music on the side.
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Location: Online via Zoom
Abstract: We introduce the notion of a symmetric quiver as provided by Derksen and Weyman in 2002 and discuss symmetric degenerations in this context (which correspond to orbit closure relations in the symmetric representation variety). After motivating our particular interest in the latter by presenting connections to group actions in algebraic Lie Theory, we explain our main questions: are symmetric degenerations induced by “usual” degenerations in the representation variety of the underlying quiver? We look at (counter)examples and recent results. This is joint work with Giovanni Cerulli Irelli.
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Location: The Quadrangle S227 (University of Sydney staff, students and affiliates only) and via Zoom
Abstract: In the 1920s, building on Fermat’s Last Theorem, Mordell conjectured that the set of rational points of any smooth projective curve of genus at least two, over any number field, is finite. In the 1960s, Shafarevich turned this into a purely algebro-geometric conjecture involving families of smooth projective curves. Parshin, Arakelov and Faltings settled this conjecture by showing that the base spaces of such families are in some sense hyperbolic, as long as there is some variation in the algebraic structure of the fibers. Inspired by recent advances in birational geometry, Kebekus and Kovacs conjectured that these hyperbolicity type properties should hold for a vast class of projective families, with fibers of arbitrary dimension. In this talk I will discuss this conjecture and my solution to it. I will also talk about more recent progress in this area, based on a joint work with Kovacs (University of Washington).
Seminar slides: PDF document

Abstract: In the first part of the talk, I will review Grobner degenerations of matrix Schubert varieties, following Knutson et al and others. I will interpret this in terms of quantum integrable systems and discuss how this construction is not entirely satisfactory and needs to be generalized. Then I will provide such a generalization; we’ll work with the “lower-upper scheme”; one component of which is closely related to the commuting variety. We’ll discuss applications to the latter.

Abstract: A group is said to have polynomial representation growth if the sequence enumerating the isomorphism classes of finite dimensional irreducible representations according to their dimension is polynomially bounded. The representation zeta function of such group is the associated Dirichlet generating series. In this talk I will focus on representation zeta functions of arithmetic groups and their properties. I will explain the ideas behind a proof of a variant of the Larsen-Lubotzky conjecture on the representation growth of arithmetic lattices in high rank semisimple Lie groups (joint with Nir Avni, Benjamin Klopsch and Christopher Voll) and analogous results for arithmetic groups of type A_2 in positive characteristic (joint with Amritanshu Prasad and Pooja Singla).
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Abstract: This talk will be in two parts.
I will outline joint work with Daryl Cooper concerning the space of holonomies of properly convex real projective structures on manifolds whose fundamental group satisfies a few natural properties. This generalises previous work by Benoist for closed manifolds. A key example, computed with Joan Porti, is used to illustrate the main results.
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Abstract: This talk will be in two parts.
The first part will be introductory, and will address the question:
Given an ordinary differential equation (ODE) with certain physical/geometric properties (for example a preserved measure, first and/or second integrals), how can one preserve these properties under discretization?
The second part of the talk will cover some more recent work, and address the question:
How can one deduce hard to find properties of an ODE from its discretization?
Bio: Reinout Quispel was an undergraduate at the University of Utrecht (the Netherlands) for nine years, before obtaining a PhD on the discretization of soliton theory from Leiden University in 1983. He moved to Australia for a three-year position in 1986 and is still there 35 years later. His main areas of expertise are in integrable systems and in the geometric numerical integration of differential equations. He was awarded the Onsager Professorship and Medal by the Norwegian University of Science and Technology (NTNU) in 2013.
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Abstract: In the first half of the talk we shall introduce the notion of Lie superalgebras, and then give a quick outline of the classification of finite-dimensional complex simple Lie superalgebras.
In the second part of the talk we shall discuss the representation theory of these Lie superalgebras and explain the irreducible character problem in the BGG category. Our main focus will be on our computation of the irreducible characters for two of the exceptional Lie superalgebras. This part is based on recent joint works with C.-W. Chen, L. Li, and W. Wang.
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Abstract: Welded tangles are knotted surfaces in R^4. Bar-Natan and Dancso described a class of welded tangles which have “foamed vertices” where one allows surfaces to merge and split. The resulting welded tangled foams carry an algebraic structure, similar to the planar algebras of Jones, called a circuit algebra. In joint work with Dancso and Halacheva we provide a one-to-one correspondence between circuit algebras and a form of rigid tensor category called “wheeled props.” This is a higher dimensional version of the well-known algebraic classification of planar algebras as certain pivotal categories.
This classification allows us to connect these “welded tangled foams” to the Kashiwara-Vergne conjecture in Lie theory. In work in progress, we show that the group of homotopy automorphisms of the (rational completion of) the wheeled prop of welded foams is isomorphic to the group of symmetries KV, which acts on the solutions to the Kashiwara-Vergne conjecture. Moreover, we explain how this approach illuminates the close relationship between the group KV and the pro-unipotent Grothendieck–Teichmueller group.
Bio: Marcy Robertson obtained her PhD in Algebraic Topology from the University of Illinois at Chicago in 2010. From there she worked in Canada, France and her native United States before settling down in Australia 2015. She is now a Senior Lecturer of Pure Mathematics at the University of Melbourne.
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Abstract: The asymptotic approach is suggested for the description of interacting surface and internal oceanic solitary waves. This approach allows one to describe a stationary moving wave patterns consisting of two plane solitary waves moving at an angle to each other. The results obtained within the approximate asymptotic theory is validated by comparison with the exact two-soliton solution of the Kadomtsev-Petviashvili equation.
The suggested approach is equally applicable to a wide class of non-integrable equations too. As an example, the asymptotic theory is applied to the description of wave patterns in the 2D Benjamin-Ono equation describing internal waves in the infinitely deep ocean containing a relatively thin one of the layers.
Bio: Yury Stepanyants graduated in 1973 with the HD of MSc Diploma from the Gorky State University (Russia) and started to work as the Engineer with the Research Radiophysical Institute in Gorky. He proceeded his career with the Institute of Applied Physics of the Russian Academy of Sciences (Nizhny Novgorod) from 1977 to 1997. In 1983 Yury obtained a PhD in Physical Oceanography, and in 1992 he obtained a degree of Doctor of Sciences in Geophysics. After immigration in Australia in 1998, Yury worked for 12 years as the Senior Research Scientist with the Australian Nuclear Science and Technology Organisation in Sydney. Since July 2009 he holds a position of Full Professor at the University of Southern Queensland in Toowoomba, Australia. Yury has published more than 100 journal papers, three books, several review papers and has obtained three patents.
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Piggott Abstract: A program of research, started in the 1980s, seeks to classify the groups that can be presented by various classes of length-reducing rewriting systems. We discuss the resolution of one part of the program (joint work with Andy Eisenberg (Temple University), and recent related work with Murray Elder (UTS).
Elder Abstract: The growth function of a finitely generated group is a powerful and well-studied invariant. Gromov’s celebrated theorem states that a group has a polynomial growth function if and only if the group is ‘virtually nilpotent’. Of interest is a variant called the ‘geodesic growth function’ which counts the number of minimal-length words in a group with respect to some finite generating set. I will explain progress made towards an analogue of Gromov’s theorem in this case.
I will start by defining all of the terms used in this abstract (finitely generated group; growth function; virtual property of a group; nilpotent) and then give some details of the recent progress made. The talk is based on the papers arxiv.org/abs/1009.5051, arxiv.org/abs/1908.07294 and arxiv.org/abs/2007.06834 by myself, Alex Bishop, Martin Brisdon, José Burillo and Zoran Šunić.
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Abstract: Traditional Indigenous marriage rules have been studied extensively since the mid 1800s. Despite this, they have historically been cast aside as having very little utility. Here, I will walk through some of the interesting mathematics of the Gamilaraay system and show that, instead, they are in fact a very clever construction.
Indeed, the Gamilaraay system dynamically trades off kin avoidance to minimise incidence of recessive diseases against pairwise cooperation, as understood formally through Hamilton’s rule.
Bio:Jared Field completed his undergraduate studies at the University of Sydney in Mathematics and French literature, before reading for a DPhil in Mathematical Biology at Balliol College, Oxford. He is now a McKenzie Fellow in the School of Mathematics and Statistics at the University of Melbourne, with broad interests at the intersection of mathematics, evolution and ecology.

Abstract: The theory of complex representations of p-adic groups can feel very technical and unwelcoming, but its central role in the conjectural local Langlands correspondence has pushed us to pursue its understanding.
In this talk, I will aim to share the spirit of, and open questions in, the representation theory of G, through the lens of restricting these representations to maximal compact open subgroups.
Our point of departure: the Bruhat-Tits building of G, a 50-year-old combinatorial and geometric object that continues to reveal secrets about the structure and representation theory of G today.
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Abstract: In 1983, Gerd Faltings proved the Mordell conjecture stating that curves of
genus at least two have only finitely many rational points. This can be understood as
the statement that most polynomial equations (in a precise sense)

f(x,y)=0

of degree at least 4 have at most finitely many solutions. However, the effective
version of this problem, that of constructing an algorithm for listing all rational
solutions, is still unresolved. To get a sense of the difficulty, recall how long it
took to prove that there are no solutions to

x^n+y^n=1

other than the obvious ones. In this talk, I will survey some of the recent progress on
an approach to this problem that proceeds by encoding rational solutions into arithmetic
principal bundles and studying their moduli in a manner reminiscent of geometric gauge
theory.
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Abstract: Higher-rank graphs, or k-graphs, are higher-dimensional analogues of directed graphs.
Each k-graph can be described in terms of a coloured graph, called its skeleton, and some factorisation rules that describe how 2-coloured paths pair up into commuting squares. C*-algebras of k-graphs generalise Cuntz-Krieger algebras, and have been the subject of sustained interest essentially because questions about crossed products of C*-algebras by higher-rank free abelian groups are hard, and k-graph algebras constitute a comparably tractable class of examples that could point the way to general theorems.
A particularly obstinate question in this vein is that of determining the K-theory of a k-graph algebra, or even just whether the K-theory depends on the factorisation rules, or only on the skeleton. I’ll outline some joint work with James Fletcher and Elizabeth Gillaspy that uses a homotopy argument to establish a surprising link between this question and the question of connectedness (or otherwise) of the space of solutions to a Yang-Baxter-like equation. I won’t assume any background about C*-algebras, k-graphs, or the Yang-Baxter equations, and all are welcome—and people who might know about connectedness (or otherwise) of the spaces of solutions to Yang-Baxter-like equations are especially welcome!

Abstract: A group G acting faithfully by homeomorphisms of the Cantor set is called piecewise full if any homeomorphism assembled piecewise from elements of G is itself an element of G.
They first appeared in the work of Giordano, Putnam and Skau in the context of Cantor minimal systems. Recently they have received significant attention as a source of new examples of finitely generated infinite simple groups. I will present a number of results about these groups obtained in joint work with Alejandra Garrido and Colin Reid.

Abstract: The aim of these two talks is to give an overview of the geometric aka stacky approach to various cohomology theories for schemes: de Rham, Hodge, crystalline and prismatic (due to Simpson and later Drinfel’d). The basic observation is that interesting cohomology theories for schemes can be realised as the (humble) coherent cohomology an associated stack. Interesting aspects of the cohomology theories e.g. comparison theorems, theories of coefficients, perfectness etc can then be naturally expressed and proven in terms of the geometry of the associated stacks.
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Abstract: We explain an equivalence of categories between a module category of quiver Hecke algebras associated with the Kronecker quiver and a category of equivariant perverse coherent sheaves on the nilpotent cone of type A. This provides a link between two different monoidal categorifications of the open quantum unipotent cell of affine type A₁, one given by Kang–Kashiwara–Kim–Oh–Park in terms of quiver Hecke algebras, the other given by Cautis–Williams in terms of equivariant perverse coherent sheaves on affine Grassmannians. The first part of the talk will be devoted to introduction to quiver Hecke algebras and categorification of quantum cluster algebras. The main result will be explained in the second part. This is a joint work with Michela Varagnolo and Eric Vasserot.
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Abstract: Autoequivalences of triangulated categories are an interesting and understudied class of groups. In large part due to the development of the theory of Bridgeland stability conditions, there are suggestive parallels between these groups and mapping class groups of surfaces. The goal of this talk will be to explain how some of the geometric group theory which appears in the study of mapping class groups also arises in the study of triangulated autoequivalences.

Abstract: There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio: Trace(ρ(g)) / dim(ρ), for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.
Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank.
Rank suggests a new organization of representations based on the very few “Small” ones. This stands in contrast to Harish-Chandra’s “philosophy of cusp forms”, which is (since the 60s) the main organization principle, and is based on the (huge collection) of “Large” representations.
This talk will discuss the notion of rank for the group GLₙ over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks.
This is joint work with Roger Howe (Yale and Texas A&M). The numerics for this work was carried with Steve Goldstein (Madison) and John Cannon (Sydney).

Abstract: This is joint work with Justin Hilburn. We will explain a theorem showing that D-modules on the Tate vector space of Laurent series are equivalent to ind-coherent sheaves on the space of rank 1 de Rham local systems on the punctured disc equipped with a flat section. Time permitting, we will also describe an application of this result in the global setting. Our results may be understood as a geometric refinement of Tate’s ideas in the setting of harmonic analysis. They also may be understood as a proof of a strong form of the 3d mirror symmetry conjectures in a special case.
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Abstract: In this talk I will state a conjecture which predicts that the derived category of smooth representations of a p-adic split reductive group admits a fully faithful embedding into the derived category of (quasi-)coherent sheaves on a stack of L-parameters. We will make the conjecture precise in the case of the principal block of GLn and relate it to the construction of a family of representations on the stack of L-parameters that interpolates a modified version of the local Langlands correspondence. The existence of this family is suggested by the work of Helm and Emerton–Helm. I will explain why the derived tensor product with this “Emerton–Helm family” should realize the expected embedding of derived categories and discuss some explicit examples.
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Abstract: This talk is based on joint work with Benson and Etingof. We say that a symmetric tensor category is incompressible if there is no symmetric tensor functor from this category to a smaller tensor category. Our main result is a construction of new examples of incompressible tensor categories in positive characteristic.
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Abstract: I will discuss some of the structures suggested by the physics of supersymmetric gauge theory of relevance to the geometric Langlands program. The discussion will include defects of various dimensions in field theory, with an emphasis on the role of boundary conditions and how they lead to the suggestion of a duality between hamiltonian actions of Langlands dual groups.
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Abstract: These lectures will cover some basic results about derived categories of coherent sheaves (e.g. the structure of the derived category of a curve, Fourier-Mukai transforms and how to construct them, tilting bundles, auto-equivalence groups, perhaps spaces of stability conditions). I will assume that the audience is vaguely familiar with the definition of derived and triangulated categories, but I will spend a fair amount of time in the first few lectures trying to give some intuitive feel for these general constructions, and explaining how one makes calculations in practice. I will also need to assume some familiarity with basic algebraic geometry and sheaf theory.

Abstract: Statistical science has a 200-year history of advances in theory and application. Data science is a relatively newly defined area of enquiry developing from the explosion in the ubiquitous collection of data. The interplay between these fields, and their interactions with science, are a topic of lively discussion among statisticians. This talk will overview some of the current research in statistical science that is motivated by new developments in data science.
Bio: Nancy Reid is University Professor and Canada Research Chair in Statistical Theory and Applications at the University of Toronto. Her research interests include statistical theory, likelihood inference, design of studies, and statistical science in public policy. Her main research contributions have been to the field of theoretical statistics. Professor Reid is a Fellow of the Royal Society, the Royal Society of Canada, the American Association for the Advancement of Science, and a Foreign Associate of the National Academy of Sciences. In 2014 she was appointed Officer of the Order of Canada.