Is Dijkstra’s Algorithm Optimal?’ – A Basser-SMRI Joint Seminar by Robert Tarjan
Monday 1 December 2025
Speaker: Robert Tarjan, Princeton University
Abstract: Dijkstra’s algorithm is a classic algorithm for doing route planning. Given a starting location it finds shortest paths from to all other reachable locations using the greedy method. Not only does it find shortest paths, it finds these in increasing order by length. A natural question is whether this algorithm is best possible. The answer depends on exactly how one poses the question. The talk will cover recent work by the speaker and his colleagues that gives the answer “yes” and briefly examine work by others that gives the answer “no.”
About the speaker: Robert Tarjan is the James S. McDonnell Distinguished University Professor of Computer Science at Princeton University. He has held academic positions at Cornell, Berkeley, Stanford, and NYU, and industrial research positions at Bell Labs, NEC, HP, Microsoft, and Intertrust Technologies. He has invented or co-invented many of the most efficient known data structures and graph algorithms. He was awarded the first Nevanlinna Prize from the International Mathematical Union in 1982 for “for outstanding contributions to mathematical aspects of information science,” the Turing Award in 1986 with John Hopcroft for “fundamental achievements in the design and analysis of algorithms and data structures,” and the Paris Kanellakis Award in Theory and Practice in 1999 with Daniel Sleator for the invention of splay trees. He is a member of the U.S. National Academy of Sciences, the U. S. National Academy of Engineering, the American Academy of Arts and Sciences, and the American Philosophical Society.
Informal Friday Seminar, ‘Universal Algorithmic Intelligence’
Friday 14 & Monday 17 November 2025
Speaker: Marcus Hutter, Australian National University/ Google DeepMind
Abstract: There is great interest in understanding and constructing generally intelligent systems approaching and ultimately exceeding human intelligence. Universal AI is such a mathematical theory of machine super-intelligence. More precisely, AIXI is an elegant parameter-free theory of an optimal reinforcement learning agent embedded in an arbitrary unknown environment that possesses essentially all aspects of rational intelligence. The theory reduces all conceptual AI problems to pure computational questions. After a brief discussion of its philosophical, mathematical, and computational ingredients, I will give a formal definition and measure of intelligence, which is maximized by AIXI. AIXI can be viewed as the most powerful Bayes-optimal sequential decision maker, for which I will present general optimality results. This also motivates some variations such as knowledge-seeking and optimistic agents, and feature reinforcement learning. Finally I present some recent approximations, implementations, and applications of this modern top-down approach to AI.
About the speaker: Marcus Hutter is Senior Researcher at DeepMind and Professor in the RSCS at the Australian National University. He received his PhD and BSc in physics from the LMU in Munich and a Habilitation, MSc, and BSc in informatics from the TU Munich. Since 2000, his research at IDSIA and ANU and DeepMind has centered around the information-theoretic foundations of inductive reasoning and reinforcement learning, which has resulted in 200+ publications and several awards. His books on “Universal Artificial Intelligence” develop the first sound and complete theory of super-intelligent machines (ASI). He also runs the Human Knowledge Compression Contest (500’000€ H-prize). See http://www.hutter1.net/ for further information.
Mini Workshop, ‘Celebrating Women in Analysis and Partial Differential Equations’
Monday 1 September 2025, 9 am – 5 pm AEST
Details: The aim of this one-day mini workshop was to celebrate the achievements of women in Analysis and Partial Differential Equations. The event brought together specialists, early career researchers, and PhD students working in related fields, both internationally and locally from the University of Sydney. We hope this workshop will provide a welcoming and inclusive platform to present research, foster collaboration, and spark new research initiatives among participants.
The mini workshop was supported by the School of Mathematics and Statistics and the Sydney Mathematical Research Institute (SMRI). Staff, visitors and students were warmly invited to attend. Organised by Jiakun Liu.
Geometry and Topology Seminar: Straightening structures on surfaces
Robert Tang, Xi’an Jiaotong-Liverpool University
Wednesday 21st August 2025
Abstract: Triangulations are ubiquitous throughout low-dimensional topology and geometry. It is a well-known result that any two triangulations of a topological surface are related by a finite sequence of flips. The analogous result is also known to be true for geometric triangulations for surfaces equipped with certain geometric structures, for example, Euclidean cone metrics or hyperbolic surfaces with a fixed vertex set. However, the prevailing methods for studying triangulations and flip sequences, in the geometric setting, tend to be very specific to the type of geometry involved. In this talk, I will introduce straightening structures on surfaces. This provides an axiomatic framework which models the behaviour of geodesic paths on non-positively curved surfaces. Moreover, it permits a notion of ‘straight triangulation’, generalising the geometric triangulations from the Euclidean or hyperbolic settings. Our main result is that the ‘straight flip graph’ associated to a straightening structure is non-empty, connected, and quasi-isometrically embedded in the flip graph of the underlying topological surface. I will also describe other classes of geometric structures that give rise to straightening structures. This is joint work with Valentina Disarlo. Watch on YouTube.
Geometry and Topology Seminar: Small Triangulations of 4-Manifolds, the 4-Manifold Census, and 2-Knots
Rhuaidi Burke, University of Queensland
Wednesday 13th August 2025
Abstract: We present a framework to classify PL-types of large censuses of triangulated 4-manifolds, which we use to classify the PL-types of all triangulated orientable 4-manifolds with up to 6 pentachora. This is successful except for triangulations homeomorphic to the 4-sphere, complex projective plane, and a particular rational homology sphere, where we find at most four, three, and two PL-types respectively. We conjecture that they are all standard. In the first half of this talk, I will discuss some of the ideas behind these results. In the second half I will focus on the combinatorial structure of the triangulations still resisting classification, and how these relate to ongoing work concerned with 2-knots — which we consider interesting in its own right.
Geometry and Topology Seminar: The existence of constrained Willmore surfaces in R^3 and R^4
Ross Ogilvie, University of Mannheim
Wednesday 6th August 2025
Abstract: The Willmore energy of a immersion of a closed surface is the integral of the square of its mean curvature. This is a measure of how far a surface is from being a sphere. It is a conformal invariant. A constrained Willmore surface is a critical point of this energy functional under deformations that preserve the conformal class of the surface. By describing a surface in terms of “holomorphic” data, the Kodaira and Weierstrass representations, and formulating a a corresponding “weak” problem, we were able to take limits of sequences of immersions and prove the existence of minimizers in each conformal class. Watch on YouTube.
Seminar: Dedekind’s lemma, Galois connections and pseudoreflection groups
Matthew Dyer, University of Notre Dame
Thursday 31st July 2025
Abstract: The first part of this talk will discuss generalizations of Dedekind’s lemma on linear independence of characters. The second part will be concerned with generalities on Galois connections which underlie many interesting applications of results in the first part. As an illustrative application, the final part will describe a distinguished role for “pseudoreflection groups” amongst all groups acting faithfully and linearly on vector spaces. Watch on YouTube.
Geometry and Topology Seminar: New geometric theorems about Fluids and Conics
Albert Chern, UC San Diego
Thursday 17th July 2025
Abstract: This talk consists of two recent work, one about topological analysis on fluid dynamics, and the other about Penrose’s 8-conic theorem. Vorticity formulation is a widespread description in fluid mechanics. However, its applicability has been limited to simply-connected domains. We show that on non-simply-connected domains, fluid’s cohomology component plays an important role and can interact with fluids. This interaction corresponds to a new equation of motion and new conservation laws which can be viewed as Casimir invariants in Hamiltonian formulation of fluid dynamics. The new equation allows us to construct new analytical solutions to Euler’s equation in terms of the Hilbert transforms on complex manifolds. In the special case of point vortices on surfaces, the configuration can be viewed as divisors on a Riemann surface. In this fluid analogy of the theory of algebraic curve, the cohomology equation can be phrased elegantly in terms of the divisor class group realized in the Jacobi variety.
The second part of the talk is about an incidence theorem about conics in double contact, first discovered by Sir Roger Penrose in his undergraduate years but remained unpublished until recently revealed in interviews. The theorem includes as special cases many well-known theorems of projective geometry as well as in metric geometries. Through collaboration with a few projective geometry enthusiasts and Penrose himself, the theorem is further developed. I will show four proof sketches.
Random Polytopes Seminar: The diameter of polytopes and the Hirsch conjecture (1/2)
Geordie Williamson, The University of Sydney
Friday 20th June 2025
Abstract: Can one reasonably bound the diameter of the graph of a simple polytope? This is a fascinating open problem in combinatorial geometry. I’ll outline what is known, and in particular sketch Santos’ remarkable 2012 counter-example to the 50 year-old Hirsch conjecture. With the DeepMind team, we recently attacked this problem using machine learning. We made some progress, but this is not the focus of the talk. I want to communicate a simple problem where the right idea could be revolutionary.
Random Polytopes Seminar: A short survey on random polytopes (2/2)
Renjie Feng, The University of Sydney
Friday 20th June 2025
Abstract: Suppose we take random points in the plane from a Gaussian and look at the polytope they generate. What can I say about it? What is its expected volume, number of vertices, number of edges etc? This talk will provide an introduction to these kinds of questions. The focus is on many points in a fixed dimension, but some striking results where one allows the dimension to grow will also be touched upon.
Special Seminar: Modern Perspectives in Representation Theory, Wild character varieties and braid varieties (6/6)
Masoud Kamgarpour, University of Queensland
Thursday 12th June 2025
Special Seminar: Modern Perspectives in Representation Theory, Singularities of orbit closures in loop spaces of symmetric varieties (5/6)
Tsao-Hsien Chen, University of Minnesota
Tuesday 10th June 2025
Special Seminar: Modern Perspectives in Representation Theory, Hodge theory and unitary representations of real reductive groups (4/6)
Dougal Davis, The University of Melbourne
Tuesday 10th June 2025
Special Seminar: Modern Perspectives in Representation Theory, Catalan from the DAHA lens (3/6)
Arun Ram, The University of Melbourne
Tuesday 10th June 2025
Special Seminar for Modern Perspectives in Representation Theory (2/6)
Calvin Yost-Wolff, The University of Michigan
Wednesday 28th May 2025
Special Seminar for Modern Perspectives in Representation Theory (1/6)
Robert Cass, The University of Michigan
Wednesday 28th May 2025
Short Course: Tensor products and highest weight structures
A short course by Jonathan Gruber
Abstract: In this series of talks, I will discuss connections and interactions between two types of structures that are ubiquitous in Lie theory and representation theory. The first is the formalism of highest weight categories, which provides an axiomatic framework for studying categories with “highest weight modules” (or “standard modules”). The second is the theory of monoidal categories, i.e. of categories equipped with a tensor product bifunctor.
The primordial example of a highest weight category with a monoidal structure is the category of rational representations of a reductive algebraic group, and here the highest weight structure interacts with the monoidal structure by way of the fact that tensor products of standard modules admit filtrations whose successive subquotients are standard modules. This kind of interaction can surprisingly be observed in many other examples, and I will give an explanation for this phenomenon via a monoidal enhancement of Brundan-Stroppel’s semi infinite Ringel duality. As applications, I will present solutions to two open problems: One concerns the existence of monoidal structures on categories of representations of affine Lie algebras at positive levels; the other concerns the existence of highest weight structures on monoidal abelian envelopes of certain “interpolation tensor categories”. All of this is based on joint work with Johannes Flake.
The program for the lecture series is as follows:
Wednesday 22 January 2025 Lecture 1: In the first talk, I will mostly discuss the motivating examples (algebraic groups, affine Lie algebras, interpolation categories), explain how they fit into the framework of “lower finite” or “upper finite” highest weight categories, and state the two aforementioned open problems.
Thursday 23 January 2025 Lecture 2: In the second talk, I will explain how lower finite and upper finite highest weight categories are related via Ringel duality and how a monoidal structure on a lower finite highest weight category gives rise to a monoidal structure on the Ringel dual upper finite highest weight category. This allows us to solve our first open problem: We define a canonical monoidal structure on a parabolic version of the BGG category O for an affine Lie algebra at positive level, and we construct a monoidal functor (a “Kazhdan-Lusztig correspondence”) to a category of representations of a quantum group at a root of unity.
Wednesday 29 January 2025 Lecture 3: The third and fourth lecture will be devoted to explaining the converse, that is, how a monoidal structure on an upper finite highest weight category gives rise to a monoidal structure on the Ringel dual lower finite highest weight category. As an application, we show that a large class of interpolation tensor categories (defined by Knop, generalizing a construction of Deligne) can be embedded as categories of tilting objects in monoidal lower finite highest weight categories. This gives a uniform explanation for the previously mysterious observation that many “abelian envelopes” of these interpolation categories are highest weight categories.
Thursday 30 January 2025 Lecture 4: (As above) The third and fourth lecture will be devoted to explaining the converse, that is, how a monoidal structure on an upper finite highest weight category gives rise to a monoidal structure on the Ringel dual lower finite highest weight category. As an application, we show that a large class of interpolation tensor categories (defined by Knop, generalizing a construction of Deligne) can be embedded as categories of tilting objects in monoidal lower finite highest weight categories. This gives a uniform explanation for the previously mysterious observation that many “abelian envelopes” of these interpolation categories are highest weight categories.
Short Course: Affine Grassmannians and Bun_G: A Parahoric Perspective
A short course by Jiuzu Hong
Abstract: Affine Grassmannians and Bun_G play a central role in geometric representation theory, geometric Langlands and algebraic geometry. Parahoric group schemes originated from Bruhat-Tits theory, and their global counterparts over algebraic curves called parahoric Bruhat-Tits group schemes were introduced more recently by Pappas-Rapoport and Heinloth.
All partial affine flag varieties can be viewed as affine Grassmannians of parahoric group schemes. This perspective has advantage of globalizing their geometry over an algebraic curve in the style of Beilinson-Drinfeld Grassmannians. This approach has led to interesting applications, including Zhu’s proof of the coherence conjecture of Pappas and Rapoport, and the determination of smooth loci of Schubert varieites in twisted affine Grassmannians by Besson and myself.
Moreover, the moduli of bundles over parahoric Bruhat-Tits group schemes generalizes parabolic Bun_G and Prym varieties. In fact, their non-abelian theta functions can be identified with (twisted) conformal blocks. This is my recent work joint with Damiolini, building on my earlier works with Kumar.
Friday 7 February 2025 Lecture 1: Affine Grassmannians and Schubert varieties
Friday 14 February 2025 Lecture 2: Loop groups and parahoric group schemes
Friday 21 February 2025 Lecture 3: Global Schubert varieties
Friday 28 February 2025 Lecture 4: Line bundles on moduli of parahoric bundles