# Titles & Abstracts

Depository of PRCM Titles and Abstracts.

Name | Affiliation | Session | Title | Abstract |

Brett Parker | Australian National University | Algebraic and symplectic geometry | Gromov--Witten invariants of log Calabi--Yau 3-folds are holomorphic Lagrangian correspondences | Motivated by geometric quantisation, Alan Weinstein famously proposed using Lagrangian correspondences as morphisms in a symplectic category. Analytic difficulties plague this idea in the smooth setting, but a holomorphic version of Weinstein's symplectic category overcomes such difficulties. Moreover, Gromov--Witten invariants of log Calabi--Yau 3-folds are naturally encoded as holomorphic Lagrangian correspondences. |

Cheol Hyun Cho | Seoul National University | Algebraic and symplectic geometry | Floer theory for the variation operator of an isolated singularity | The variation operator in singularity theory maps relative homology cycles to compact cycles in the Milnor fiber using the monodromy. We construct its symplectic analogue for an isolated singularity. We define a new Floer cohomology, called monodromy Lagrangian Floer cohomology, which provides categorifications of the standard theorems on the variation operator and the Seifert form. The key ingredients are a special class in the symplectic cohomo |

Huai-liang Chang | HKUST | Algebraic and Symplectic geometry | High genus Gromov Witten invariants via Mixed Spin P field theory | Study of high genus Gromov Witten invariants for compact Calabi Yau threefold is achieved by Mixed Spin P (MSP) fields theory. It encodes all the phase transition process and has many applications. One consequence is BCOV conjecture. Another is CY-LG correspondence. I will introduce MSP theory and some of its applications. |

Hiroshi Iritani | Kyoto University | Algebraic and symplectic geometry | Quantum cohomology of GIT quotients and blowups | We explain a D-module version of Teleman's conjecture, which relates the quantum cohomology of a GIT quotient to the equivariant quantum cohomology of the original manifold via Fourier transformation. This framework allows us to prove a decomposition theorem for quantum cohomology under blowups. |

Jianxun Hu | Sun Yat-Sen University | Algebraic and symplectic geometry | Some recent progress on Gamma conjectures | Gamma conjectures, proposed by V. Golyshev, S. Galkin and H. Iritani, consists of conjecture O, Gamma conjecture I and II. Previous answers are affirmative. In this talk, I will talk about some counter-examples to conjecture O and Gamma conjecture I. This talk is based on a joint work with S. Galkin, H. Iritani, H. Ke, C. Li and Z. Su. |

Mohammed Abouzaid | Stanford University | Algebraic and symplectic geometry | Morse theory and stable homotopy | Stable homotopy theory was born in an attempt to devise computable algebraic invariants of topological spaces, which take the form of cohomology theories. I will present joint work with Andrew Blumberg which goes back to the roots of the subject, and combines Morse's insight that the study of critical points of function sheds light on topology, with Pontryagin and Thom's isomorphism between homotopy and bordism groups, to provide a Morse model for the category of spectra. |

Shuai Guo | Peking University | Algebraic and symplectic geometry | Structures for higher genus Gromov-Witten invariants of Calabi-Yau threefolds | In this talk, I will first review the conjectural structures for the Gromov-Witten invariants of Calabi-Yau threefolds proposed by physists. Then I will try to explain how they solve the generating function by using these conjectures for one-parameter models, especially for the quintic threefold. In the end, I will introduce the recent mathematical progresses to BCOV’s conjectures. |

Alexandre Ern | ENPC & INRIA | Computational mathematics | Some recent results on the finite element approximation of Maxwell's equations | We present some recent results on the finite element approximation of Maxwell's equations: an asymptotically-optimal error estimate for the problem posed in the frequency domain, and the proof of spectral correctness for the eigenvalue problem approximated using discontinuous Galerkin. |

Lina Zhao | City University of Hong Kong | Computational mathematics | A staggered mixed method for the biharmonic problem | In this talk, a staggered mixed method based on the first-order system will be presented. The proposed method hinges on carefully balancing the involved finite element spaces. It supports polygonal meshes of arbitrary shapes and is free of stabilization. All the variables converge optimally with respect to the polynomial order. In particular, superconvergence will also be proved, which motivates the definition of the postprocessing. Several numer |

Pouria Behnoudfar | Mineral Resources, Commonwealth Scientific and Industrial Research Organisation (CSIRO) | Computational mathematics | Computational mathematics | We introduce novel high-order methods for first-order time-dependent partial differential equations. Unlike traditional methods like Runge-Kutta and backward differentiation formulae, ours offer explicit control over numerical dissipation. A single parameter governs dissipation in high-frequency regions. Our technique accommodates arbitrary high-order accuracy and resolves $k>1, k\in \mathbb{N}$ matrix problems, yielding $(3/2k)^{th}$-order for even $k$ and $(3/2k+1/2)^{th}$-order for odd $k$. The stability region remains unchanged regardless of accuracy order and provides A-stability or L-stability. |

Quanling Deng | Australian National University | Computational mathematics | Some Recent Advances in Numerical Spectral Approximation of Elliptic Operators | I will present some recent breakthroughs including (a) the reduction of stiffness (characterised by condition numbers) and improved accuracy in the high-frequency region by SoftFEM (w/ A. Ern), softIGA (w/ P. Behnoudfar and V. Calo), and generalised SoftFEM (w/ J. Chen and V. Calo); (b) gauranteed eigenvalue bounds for Schrödinger equation by Crouzeix-Raviart FEM (w/ C. Carstensen). If time allows, I will present neural network based approximatio |

Yasu Hiraoka | Kyoto University | Computational mathematics | Single-cell trajectory inference framework based on entropic Gaussian mixture optimal transport | We present scEGOT, a comprehensive single-cell trajectory inference framework based on entropic Gaussian mixture optimal transport. The main advantage of scEGOT allows us to go back and forth between continuous and discrete problems, and it provides versatile trajectory inference methods at a low computational cost. Applied to the human primordial germ cell-like cell (PGCLC) induction system, scEGOT identified the PGCLC progenitor population with |

Akhilesh Prasad | Indian Institute of Technology (Indian School of Mines) Dhanbad, India | Contributed talk | Pseudo-differential operator associated with Mehler-Fock transform | In this paper, study related to $\mu^{th}$ order Mehler-Fock transform ($\mu$MFT) is carried out. Boundedness of translation and convolution operators in Lebesgue space are obtained. Continuity of $\mu$MFT in Lebesgue as well as some test function spaces are discussed. Further, pseudo-differential operator (p.d.o.) associated to $\mu$MFT is defined and studied its continuity over certain function spaces. |

Amine Sbai | Hassan First University of Settat, Morocco | Contributed talk | Boundary controllability for degenerate/singular wave equations with a drift term | We deal with the boundary null controllability for a degenerate wave equation in non divergence form, with a singular potential and a drift term. In particular, we provide conditions for the controllability of the solution of the associated Cauchy problem at a sufficiently large time. |

David Ompong | Charles Darwin University | Contributed talk | Reflective Practice: a means to engagement mathematics students | Reflective practice is a critical component of effective teaching for teachers to engage diverse student cohorts effectively. This study uses action research to carry out a critical self-reflection of a teacher in a first-year engineering mathematics unit at Charles Darwin University. The data is collected via videotape transcripts, critical incident reports, and student comments. After applying thematic analysis, the findings reveal that the teacher has to adopt a slower-paced teaching strategy, promote active student participation, and help students understand the relevance of mathematics in engineering. |

Harikesh | University of the Sunshine Coast | Contributed talk | Enhancing the Accuracy of Mapping Wildfire Susceptibility through the integration of Remote Sensing Data and Artificial Intelligence (AI) | Wildfires are a significant environmental hazard that poses threats to ecosystems, biodiversity, human settlements, and economic activities worldwide. Traditional methods of Wildfire prediction often lack precision and timeliness, hindering proactive prevention and mitigation efforts. In recent years, Machine Learning (ML) techniques have emerged as promising tools for improving Wildfire prediction by leveraging historical fire events and other r |

John Bailie | University of Auckland | Contributed talk | Bifurcations of resonance tongues in periodically forced vector fields: a case study of vertical mixing in the North Atlantic | We investigate a periodically forced planar vector field for temperature and salinity in the North Atlantic. The interaction between an intrinsic oscillation and the forcing gives rise to dynamics on an invariant torus. We present an algorithm for computing the associated rotation number, and use it to determine the structure of resonance tongues in a parameter plane. We also explain how this structure changes with a third parameter at generic bifurcations of resonance tongues. |

Musashi Koyama | Australian National University | Contributed talk | Computing Vietoris-Rips persistent homology efficiently for Euclidean point-clouds | Vietoris-Rips persistent homology (VRPH) is widely used in the field of Topological Data Analysis. However it suffers from being prohibitively expensive to compute. Here, we present new methods and techniques to compute VRPH efficiently. |

TrungTin Nguyen | School of Mathematics and Physics, The University of Queensland | Contributed talk | Demystifying parameter estimation in Gaussian mixtures of experts | Understanding the parameter estimation of mixtures of experts has remained a long-standing open problem in machine learning and statistics. This is mainly due to the inclusion of covariates in the gating functions and expert networks, which leads to their intrinsic interaction via some partial differential equations with respect to their parameters. We address these issues by designing novel Voronoi loss functions among the parameters to accurately capture the heterogeneity of parameter estimation rates and to empirically verify our theoretical results. |

Beomjun Choi | Pohang University of Sciecne and Technology | Differential geometry | Gradient conjecture of Thom for geometric evolution equations | R. Thom conjectured, for a given convergent gradient flow of analytic potential, the direction of secant line to its limit converges. This is a next-order question following the seminal works by Lojasiewicz and Simon, and was proved by Kurdyka-Mostowski-Parusinski for classical gradient flow. In this talk, we discuss a joint work with P.-K. Hung where we prove the conjecture and identify the rate of convergence for the class of geometric equations considered by Simon. |

Gerhard Huisken | University of Tuebingen | Differential geometry | Curvature flows and geometric inequalities | Parabolic geometric evolution equations such as mean curvature flow and inverse mean curvature flow of hypersurfaces can be used to prove sharp inequalities of isoperimetric type in Riemannian manifolds satisfying natural bounds on their curvature tensor. The lecture describes recent new estimates and their applications in geometry and mathematical relativity. |

Leon Simon | Stanford University | Differential geometry | Singularities of minimal submanifolds | A quick survey of known results about the structure of singularities of minimal surfaces will be given, including a discussion of key open problems. The talk will conclude with a description of recent advances. |

Mat Langford | Australian National University | Differential Geometry | ancient solutions to curvature flows in low dimensions | we present a survey of the landscape of ancient solutions to geometric flows in low dimensions, and pose a number of open problems and conjectures. |

Matthew Gursky | University of Notre Dame | Differential geometry | Some rigidity results for Poincare-Einstein manifolds | In this talk I will review some rigidity results for Poincare-Einstein manifolds, focusing on four dimensions. |

Valentina Wheeler | University of Wollongong | Differential geometry | Concentration-compactness for curvature flow of arbitrarily high order | We extend Struwe and Kuwert-Sch\"atzle's concentration-compactness method for the analysis of geometric evolution equations to flows of arbitrarily high order, with the geometric polyharmonic heat flow (GPHF) of surfaces, a generalisation of surface diffusion flow, as exemplar. For the (GPHF) we apply the technique to deduce localised energy and interior estimates, a concentration-compactness alternative, pointwise curvature estimates, a gap theorem, and study the blowup at a singular time. This gives general information on the behaviour of the flow for any initial data. Applying this for initial data satisfying $||A^o||_2^2 < \varepsilon$ where $\varepsilon$ is a universal constant, we perform global analysis to obtain exponentially fast full convergence of the flow in the smooth topology to a standard round sphere. This is joint work with James McCoy, Scott Parkins, and Glen Wheeler. |

Xuezhang Chen | Nanjing University | Differential geometry | Sobolev trace inequality, biharmonic Poisson kernel and Green function associated to conformal boundary operators | Conformal boundary operators associated to the Paneitz operator have attracted a lot of attention. We focus on two related topics: One is to establish a sharp Sobolev trace inequality on a three-ball, which is equivalent to a sharp Sobolev inequality on a two-sphere involving the fractional GJMS operator $P_3$; the other is to classify nonnegative solutions of a biharmonic equation on the half-space and unit ball with proper pair of conformal boundary operators, where the notions of biharmonic Poisson kernel and Green function are introduced. This is jointly with Shihong Zhang. |

Yng-Ing Lee | National Taiwan University | Differential geometry | The existence of minimal Lagrangians through deformation | In this talk, I will discuss the study of the existence of minimal Lagrangians/special Lagrangians through deformation. One can study the problem in a fixed ambient manifold in which Lagrangian mean curvature flow or other flows are such examples. Or one can change the ambient manifold and study the corresponding deformation problems. Some old results and possible new approaches will be reported. |

Yoshihiko Matsumoto | Osaka University | Differential geometry | Harmonic maps from the product of the hyperbolic planes to the hyperbolic space | I will talk about an existence result for the asymptotic Dirichlet problem for harmonic maps from the product of the two hyperbolic planes targeted to the hyperbolic space, where the Dirichlet data is given on the corner of the product. Some subtleties regarding generalization to higher-dimensional cases will also be discussed. This is based on joint work with Kazuo Akutagawa. |

Yuya Takeuchi | University of Tsukuba | Differential geometry | Spectral analysis of the CR Paneitz operator | The CR Paneitz operator, a CR invariant fourth-order linear differential operator, plays a crucial role in three-dimensional CR geometry. It is deeply connected to global embeddability, the CR positive mass theorem, and the logarithmic singularity of the Szegő kernel. In this talk, I will discuss recent progress on the spectrum of the CR Paneitz operator. Specifically, I will focus on differences in its nature depending on whether it is embeddable or not. |

Thao Thuan Vu Ho | Monash University | Differential geometry, mathematical physics | Differential geometry, mathematical physics | A Hamiltonian reduction approach is defined, studied, and finally used to derive asymptotic models of internal wave propagation in density stratified fluids in two-dimensional domains. Beginning with the general Hamiltonian formalismof Benjamin (1986 J. Fluid Mech. 165 445‚Äì74) for an ideal, stably stratified Euler fluid, the corresponding structure is systematically reduced to the setup of two homogeneous fluids under gravity, separated by an interface and confined between two infinite horizontal plates. |

Anthony Carbery | University of Edinburgh | Harmonic analysis | Weighted inequalities for the Fourier extension operator | The Fourier extension operator represents a superposition of plane waves travelling in different directions, and which therefore admits constructive and destructive interference. The overall aim is to understand the net size and structure of the superposition, and in particular to ascertain whether the interference pattern necessarily contains remnants of linear structure. This we aim to do by means of weighted integral inequalities for the Fourier extension operator which we shall describe. We will note several good reasons for asking this sort of question, and present some results (joint work with Hong Wang and Marina Iliopoulou) in this direction. There is unexpected interplay with questions of a geometric/combinatorial nature such as: what is the largest number of unit squares in an $N \times N$ grid which one can colour black so that no strip of width one meets more than $O(1)$ of these coloured squares? |

Anthony Dooley | University of Technology Sydney | Harmonic analysis | The Kirillov character formula, wrapping maps and $e$-functions | Kirillov’s character formula gives an expression for the character of an irreducible representation of a Lie group in terms of the (Euclidean) Fourier transform of its associated coadjoint orbit. Wildberger and I re-interpreted this using the wrapping map, which allows one to transfer Ad-invariant distributions from the Lie algebra to the Lie group, as a convolution homomorphism. In this talk, I will describe how the theory works for compact symmetric pairs (G,K). The convolution of $K$-invariant distributions needs to be twisted by the so-called $e$-function, and one then retrieves the characters of $G/K$ as limits of generalised Bessel functions. |

Ji Li | Macquarie University | Harmonic analysis | Schatten Properties of Calderon–Zygmund Singular Integral Commutator on stratified Lie groups | We provide full characterisation of the Schatten properties of $[M_b,T]$, the commutator of Calder\'{o}n--Zygmund singular integral $T$ with symbol $b$ $(M_bf(x):=b(x)f(x))$ on stratified Lie groups $\mathcal G$. We show that, when $p$ is larger than the homogeneous dimension $\mathbb Q$ of $\mathcal G$, the Schatten $\mathcal L_p$ norm of the commutator is equivalent to the Besov semi-norm $B_{p}^{\mathbb Q/p}$ of the function $b$; but when $p\ |

Sanghyuk Lee | Seoul National University | Harmonic analysis | Endpoint eigenfunction bounds for the Hermite operator | This talk concerns $L^2$--$L^q$ bounds on the Hermite spectral projection operator $\Pi_\lambda$ in $\mathbb R^d$, which is the projection onto the eigenspace spanned by the eigenfunctions with eigenvalue $\lambda$ of the Hermite operator $-\Delta+ |x|^2.$ The bounds on $\Pi_\lambda$ not only provide $L^q$ estimates for the eigenfunctions but also have applications to various related problems such as the Bochner--Riesz problem and the unique continuation problem for the heat operator. For $d\ge 2$, the optimal $L^2$--$L^{q}$ bound on $\Pi_\lambda$ has been known except for the endpoint case $q_\circ=\frac{2(d+3)}{d+1}$. However, the endpoint $L^2$--$L^{q_\circ}$ bound has remained unsettled for a long time. We prove this missing endpoint case for every $d\ge 3$. Our result is built on a new phenomenon: the improvement of bounds due to asymmetric localization near the sphere $\{x: |x|=\sqrt\lambda\}$. |

Gabor Lugosi | ICREA & Pompeu Fabra University | Mathematical data science | Network archaeology: an overview of mathematical models | Large networks that change dynamically over time are ubiquitous in various areas such as social networks, and epidemiology. These networks are often modeled by random dynamics which, despite being relatively simple, give a quite accurate macroscopic description of real networks. "Network archaeology" is an area of combinatorial statistics in which one studies statistical problems of inferring the past properties of such growing networks. In this talk we review some models and recent results. |

Nikita Zhivotovsky | Berkeley University | Mathematical data science | Improving Risk Bounds with Unbounded Losses via Data-Dependent Priors | In this talk, we revisit sequential linear regression, classification, and logistic regression, using scenarios where design vectors are known in advance but unordered. We discuss how this allows us to convert bounds into statistical ones with random design without additional assumptions. Using the exponential weights algorithm and data-dependent priors, we manage unbounded norms of optimal solutions. We show our classification regret bounds depend only on dimension and rounds, not on design vectors or norms. For linear regression, we extend to sparse cases with bounds based on response magnitudes. We argue these bounds are unique to this setting and unattainable in worst-case setups, offering polynomial-time algorithms involving log-concave sampling. |

Peter Bartlett | Google DeepMind and UC Berkeley | Mathematical data science | Improving optimization efficiency by choosing the step size too large for gradient descent | Simple gradient descent algorithms are ubiquitous in machine learning. Although these methods are traditionally viewed as a time discretization of gradient flow, in practice, large step sizes---large enough to cause oscillation of the loss---exhibit performance advantages. We study gradient descent in logistic regression with a constant step size that is so large that the loss initially oscillates. We show the benefits of this initial oscillatory phase, achieving a loss of 1/T^2 in T steps, where a step size small enough to ensure a monotonic decrease of the loss cannot do better than 1/T. We show similar benefits in a nonlinear setting. Based on joint work with Jingfeng Wu, Matus Telgarsky and Bin Yu |

Subhro Ghosh | National University of Singapore | Mathematical data science | The unreasonable effectiveness of negative association | In 1960, Wigner published an article famously titled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences”. In this talk we will, in a small way, follow the spirit of Wigner’s coinage, and explore the unreasonable effectiveness of negatively associated (i.e., self-repelling) stochastic systems far beyond their context of origin. As a particular class of such models, determinantal processes (a.k.a. DPPs) originated in quantum and statistical physics, but have emerged in recent years to be a powerful toolbox for many fundamental learning problems. In this talk, we aim to explore the breadth and depth of these applications. On one hand, we will explore a class of Gaussian 2 of 2 DPPs and the novel stochastic geometry of their parameter modulation, and their applications to the study of directionality in data and dimension reduction. At the other end, we will consider the fundamental paradigm of stochastic gradient descent, where we leverage connections with orthogonal polynomials to design a minibatch sampling technique based on data-sensitive DPPs ; with provable guarantees for a faster convergence exponent compared to traditional sampling. Principally based on the following works (and their follow-ups). [1] Gaussian determinantal processes: A new model for directionality in data, with P. Rigollet, Proceedings of the National Academy of Sciences, vol. 117, no. 24 (2020), pp. 13207--13213 (PNAS Direct Submission) [2] Determinantal point processes based on orthogonal polynomials for sampling minibatches in SGD, with R. Bardenet and M. Lin Advances in Neural Information Processing Systems 34 (Spotlight at NeurIPS 2021) |

Tselil Schramm | Stanford University | Mathematical data science | spectral clustering in high-dimensional Gaussian mixture block models | The Gaussian mixture block model is a simple generative model for networks: to generate a sample, we associate each node with a latent feature vector sampled from a mixture of Gaussians, and we add an edge between nodes if and only if their feature vectors are sufficiently similar. In this talk I will discuss recent results on the performance of spectral clustering algorithms on networks sampled from high-dimensional Gaussian mixture block models |

C. K. Raju | Indian Institute of Education | Mathematics Without Borders | Personal title: Honorary Professor, Title of talk: In defence of decolonisation of mathematics | A report on 15 years of experiments teaching decolonised math: calculus (in 5 universities, 3 countries, UG+PG), geometry etc. (in secondary schools), and arithmetic and calendar (primary level). Decolonisation challenges received history of math (axiomatic proofs from politics of Crusading church, not Euclid, calculus stolen from India in 16th C., Newton etc. didn't even understand how to sum its infinite series). It also challenges the philosophy (axiomatic proofs inferior like unreal 'real' numbers). Decolonisation makes math easy and secular. Full abstract: https://bit.ly/ckr- |

C. K. Raju | Indian Institute of Education | Mathematics Without Borders | Rajju ganita | Rajju ganita fundamentally challenges axiomatic geometry (not actually found in the 'Euclid' book): it accepts BOTH empirically manifest and deductive reasoning as means of proof, as does science. An angle defined using a curved arc can't be measured with compass-box instruments, all of which a string replaces. The kamal measures angles in space to a fraction of a degree. This is a pre-requisite to teach the Indian calendar, essential for the economy and festivals, instead of the alienating and ruinous Gregorian calendar. Full abstract: https://bit.ly/ckr-2 |

Cris Edmonds-Wathen | Charles Darwin University | Mathematics Without Borders | Mathematical expression in Indigenous languages: Different solutions to common problems | Around the world, mathematical expression in different languages varies greatly depending on the culture and on the grammar of the language. I describe work in progress to use diverse Australian Indigenous languages for teaching and learning mathematics in the early years of school. Looking what is common in mathematical expression in different languages and what varies between them can help deepen our understanding of the relationship between ma |

Francis Su | Harvey Mudd College & Sydney Mathematical Research Institute | Mathematics Without Borders | The future of maths demands a better purpose for maths | People often view mathematics as important for career, economic, or technological advancement. As maths researchers and educators, we have a moral obligation to present a more dignifying, uplifting, and socially beneficial purpose for maths: one that promotes human flourishing. I offer some thoughts on how we do so, in a world rapidly being transformed by AI and other 'maths assistants', where even the purest maths can find unsavory applications. |

Hongzhang Xu | Australian National University | Mathematics Without Borders | Qualitative mathematics, Indigenous mathematics and their connections | This talk refutes some misunderstandings and fallacious beliefs concerning Indigenous mathematics. In the dominant western narrative, numbers are the origin of mathematics. This norm overshadows the richness and diversity of qualitative, or non-numeric, mathematical practices that are well-developed in most societies and subcultures, and has contributed to the marginalization and devaluation of Indigenous mathematical knowledge systems. By connecting qualitative mathematics, we showcase the mathematics applied by Australian First Nations people in cosmological construct, kinship rules and way-finding techniques to illuminate the cultural, social, and historical contexts that shape Indigenous and other non-Western mathematical practices. |

Jared Field | University of Melbourne | Mathematics Without Borders | Gamilaraay kinship revisited | 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. In this talk 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. In particular, I will show how Gamilaraay kinship reduces incidence of recessive diseases on the one hand, while i |

Jason John Sharples | University of New South Wales | Mathematics Without Borders | Understanding bushfire dynamics | In this talk I will discuss how various models have been used to help us understand bushfire behaviour and to predict its propagation. Starting with basic concepts, I will discuss how empirical knowledge has been combined with geometric notions to model the spread of bushfires across a landscape. I will also touch on some of the more recent developments in bushfire research and describe some of the more sophisticated mathematical models that are being considered in this context. Finally, I will discuss the alignment between Western scientific and Indigenous perspectives on bushfire and how it can be used as a topic to motivate and enhance mathematics education, as well as broader reconciliation. |

Kay Owens | Charles Sturt University | Mathematics Without Borders | Mathematics Without Borders: Expanding Relationships and Noticing | When you are out of your comfort zone, you begin to notice different things, take risks and make sense of it by forming new connections. To make sense of school mathematics, many children need to make links to their cultural experiences. To make sense of the mathematics of different cultures, we need to think differently about mathematics. Ethnomathematics research requires a study of both the relationships of culture and of mathematics, noticing them, often through systematic work. By crossing the borders of academic mathematics, examples of mathematical, cultural relationships are provided. |

Khalid Khan | Charles Darwin University | Mathematics Without Borders | Teaching Mathematics in the era of Artificial Intelligence | This presentation will address theoretical and pedagogical issues related to mathematics teaching and discuss their relevance in the age of AI. Integrating Artificial Intelligence (AI) in mathematics education presents opportunities and challenges. I will explore the significance of contexts and human agency within the framework of utilising artificial intelligence and mathematics and examine some questions in which learning as an exchange between technology and the human intellect needs to be seen. |

Marcy Robertson | University of Melbourne | Mathematics Without Borders | Mathematics As Culture | How do we evaluate the quality of a mathematician? I'll discuss some obstructions to adapting to mathematical culture (as a young mathematician) and things that have worked/not worked when trying to change the culture (as a not so young mathematician). |

Martin Nakata | James Cook University | Mathematics Without Borders | Indigenous academic performance in school-based mathematics | This keynote is based on a longitudinal project led by Prof Nakata and his STEM team in collaboration with schools in remote regions of Australia. It is part of the broader project of the ARC-COE for Australian Biodiversity and Heritage. The partnership with schools sought to test whether improvements in the academic performance of Indigenous students could be achieved through innovations in the design and delivery of the mathematics curriculum. Central to the innovation was providing timely and detailed information to teachers on the learning challenges individual students were having with the required tasks. |

Raynier Tutuo | The University of the South Pacific | Mathematics Without Borders | Exploring the development and use of culturally derived teaching ideas in mathematics: A Solomon Islands case study | This study provided opportunity for a small number of mathematics teachers to develop and implement ethnomathematics activities at the junior secondary level in Solomon Islands. This study investigated the use of ethnomathematical activities as a means of looking at how it impacts students learning in the junior secondary level in Solomon Islands. |

Rowena Ball | Australian National University | Mathematics Without Borders | Equity and diversity OF mathematics | In what sense is Western mathematics Western? How did European mathematics colonise the minds and curriculums of the whole world, to the exclusion or suppression of non-Western, Indigenous, and vernacular mathematics? In this talk I shall describe our research and teaching initiative to decolonise, strengthen and grow mathematics as a discipline, and attract and retain students from under-represented groups, through studies of cross-cultural mathematics and truth-telling in mathematics history. We link achieving a diversity of people IN mathematics to a genuine diversity OF mathematics. |

Sophie Raynor | James Cook University | Mathematics Without Borders | Where is the maths? | I will give some personal reflections on mathematical participation, fulfilment and achievement. And what the abstract question of “where?” can bring to conversations on mathematics, its value, its teaching and its problems. |

Feimin Huang | Chinese Academy of Sciences | Non-Linear PDE | Large time behavior of strong solutions for stochastic Burgers equation with transport noise | We consider the large time behavior of strong solutions to the stochastic Burgers equation with transport noise. It is well known that both the rarefaction wave and viscous shock wave are time-asymptotically stable for deterministic Burgers equation since the pioneer work of A. Ilin and O. Oleinik in 1964. However, the stability of these wave patterns under stochastic perturbation is not known until now. We give a definite answer to the stability problem of the rarefaction and viscous shock waves for the 1-d stochastic Burgers equation with transport noise. That is, the rarefaction wave is still stable under white noise perturbation and the viscous shock is not stable yet. Moreover, a time-convergence rate toward the rarefaction wave is obtained. To get the desired decay rate, an important inequality (denoted by Area Inequality) is derived. This inequality plays essential role in the proof, and may have applications in the related problems for both the stochastic and deterministic PDEs. |

Florica-Corina Cirstea | The University of Sydney | Non-Linear PDE | Nonlinear elliptic equations with Hardy potential and a gradient-dependent nonlinearity | In this talk, we present new existence and classification results for the positive solutions to nonlinear elliptic equations with a gradient-dependent nonlinearity in D-\{0\}, where D is a bounded domain containing zero. We will explore some the behaviour of the positive solutions near zero and using a dynamical systems approach, we prove the existence of positive radial solutions manifesting these possible profiles. This is joint work with Dr Ma |

Jungcheng Wei | Chinese University of Hong Kong | Non-Linear PDE | Nonexistence of Type II blowups for energy critical nonlinear heat equation in large dimensions | We show that all blowups are Type I for the energy critical Fujita heat equation in dimensions 7 or higher. The proof is built on several key ingredients: first we perform tangent flow analysis and study bubbling formation in this process; next we give a second order bubbling analysis in the multiplicity one case, where we use a reverse inner-outer gluing mechanism; finally, in the higher multiplicity case (bubbling tower/cluster), we develop Schoen's Harnack inequality and obtain next order estimates in Pohozaev identities for critical parabolic flows. (Joint work with Kelei Wang.) |

Kazuhiro Ishige | the University of Tokyo | Non-Linear PDE | Non-preservation of concavity properties by the Dirichlet heat flow on Riemannian manifolds | In this talk we discuss the preservation of concavity properties by the Dirichlet heat flow. In particular, we prove that no concavity properties are preserved by the Dirichlet heat flow in a convex domain of a Riemannian manifold unless the sectional curvature vanishes everywhere on the convex domain. |

Kenji Nakanishi | Kyoto University | Non-Linear PDE | Global wellposedness of general evolution PDE on the Fourier half space | This is joint work with Baoxiang Wang (Jimei & Peking). We study the Cauchy problem for very general PDEs on the whole Euclidean space written by derivatives, Fourier multipliers and analytic nonlinearity. Examples include nonlinear hyperbolic, parabolic and dispersive equations, the Navier-Stokes and the Euler equations, as well as classically ill-posed ones such as the backward heat equation. We construct a function space of distributions for the Fourier transform supported on the half space, where the global wellposendess holds without size restriction. The Fourier support requires the initial data to be complex valued, but they may be very rough, grow polynomially at infinity, and contain mixture of various periods. The global wellposedness holds even if the solutions blow up in the classical sense, and even if the nonlinearity has singularities and the initial data are beyond its radius of convergence. |

Seung-Yeal Ha | Seoul National University | Non-Linear PDE | A mean-field approach for the asymptotic tracking of continuum target clouds | In this talk, we propose a new coupled kinetic system arising from the asymptotic tracking of a continuum target cloud, and study its asymptotic tracking property. For the proposed kinetic system, we present an energy functional which is monotonic and distance between particle trajectories corresponding to kinetic equations for target, and tracking ensembles tend to zero asymptotically under a suitable sufficient framework. The framework is formulated in terms of system parameters and initial data. This is a joint work with Hyunjin Ahn (Myongji Univ. Korea) |

Shih-Hsien Yu | Academia Sinica | Non-Linear PDE | Boltzmann equation and fluid dynamics | In this talk one gives descriptions of the Boltzmann equation in terms of its microscopic and macroscopic natures and relates those natures to viscous fluids and relavent singular layers. The key tool to establish the connection is the pointwise structure of a Green's function of a linearized Boltzmann equation. The development of the Green's function is based on spectral informations and a mixture lemma to convert microscopic regularity to macr |

Cale Rankin | Monash University | Non-linear PDEs | The Monopolist''s Problem:: Regularity and Structure Results for a New Free Boundary Problem | The Monopolist's problem is a simple model from economics which displays rich mathematical behaviour and lies at the intersection of optimal transport, free boundary problems, and convex analysis. Mathematically one aims to minimize a uniformly convex Lagrangian, however restricted to the space of convex functions. The requirement that the minimization take place over the space of convex functions leads to a free boundary between the regions of strict and nonstrict convexity and in these regions the solution displays qualitatively different behaviour. In this talk I discuss joint work-in-progress with Robert McCann and Kelvin Shuangjian Zhang in which we prove results on the configuration of the different domains, regularity of the free boundary, and completely describe the solution in the case of most interest to applications. |

Stephen Muirhead | University of Melbourne | Probability | Probability | We consider the event that a stationary Gaussian field (SGF) on R^d or Z^d stays positive on some given large domain; such events are called `persistence‚Äô or `hard wall‚Äô events. Under certain conditions, if one conditions a SGF on this event, one expects the field to be propelled to a high level, a phenomenon known as `entropic repulsion'. So far this has only been proven rigorously for a handful of SGFs, starting with the Gaussian free field. We‚Äôll discuss joint work with Naomi Feldheim and Ohad Feldheim that establishes entropic repulsion for a wide class of SGFs. |

Allan Sly | Princeton University | Probability theory | Rotationally invariant first passage percolation: Scaling relations and Chaos | While believed to be in the KPZ universiality class, the lack of an integrable structure has made analysis of first passage percolation particularly challenging. For rotationally invariant first passage percolation on the plane we prove a version of the scaling relations between the passage times fluctuation and transversal fluctuations of geodesics and give the first improvement on the Benjamini, Kalai and Schramm variance bound. |

Fima Klebaner | Monash University | Probability Theory | Probability Theory | We study how populations emerge when starting with just a few individuals, maybe only one, and then growing to its (large) carrying capacity. We prove an old conjecture and suggest new approximations. The talk is based on a number of papers with Andrew Barbour, Pavel Chigansky, Peter Jagers, Kais Hamza and Haya Kaspi. |

Hao Shen | University of Wisconsin-Madison | Probability theory | Stochastic quantization of Yang-Mills | We will discuss the stochastic Yang-Mills flow, which is the deterministic Yang-Mills flow driven by a (very singular) space-time white noise. It turns out that due to singularity, even construction of local solutions is challenging. We will discuss our construction for a trivial bundle over 2 and 3 dimensional tori, but starting with a gentle introduction to Stochastic PDE. In the end, I will also discuss the meaning of "gauge equivalence” and “ |

Nathan Ross | University of Melbourne | Probability theory | Gaussian random field approximation via Stein's method, with applications to wide random neural networks | We describe a general technique to derive bounds for Gaussian random field approximation with respect to a Wasserstein transport distance in function space, equipped with the supremum metric. The technique combines Stein’s method and infinite dimensional Gaussian smoothing, and we apply it to derive bounds on Gaussian approximations of wide random neural networks of any depth. The bounds are explicit in the widths and natural parameters of the n |

Pierre Nolin | City University of Hong Kong | Probability theory | Monochromatic exponents for two-dimensional percolation | We discuss monochromatic arm events for Bernoulli percolation in 2D, deriving an exact expression for the celebrated backbone exponent. Contrary to previously-known arm exponents for 2D percolation (all rational), it has a transcendental value. We use techniques developed to compute the conformal radii of random domains defined by SLE curves, based on the coupling between SLE and LQG, and using crucially input from Liouville CFT. Based on a joint work with Wei Qian, Xin Sun and Zijie Zhuang. |

Rongfeng Sun | National University of Singapore | Probability theory | Universality of the Brownian net | The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from every point in space and time, while the Brownian net is an extension that also allows branching. We show here that the Brownian net is the universal scaling limit of one-dimensional branching-coalescing random walks with weak binary branching and arbitrary increment distributions that have finite (3+Œµ)-th moment. This gives the first example in the domain of attraction of the Brownian net where paths can cross without coalescing. |

Binyong Sun | Zhejiang University | Representation Theory | Representation theory | Inspired by the study of automorphic forms, Arthur suggested the existence of certain collections of representations of real reductive groups. Arthur's desired representations, the special unipotent representations, were defined by Barbasch-Vogan and Adams-Barbasch-Vogan. Starting with some basic notions of Lie group representations, we will explain Adams-Barbasch-Vogan's definition and the construction of all special unipotent representations of classical groups by Howe duality. This is a report on a joint work with Dan Barbasch, Jia-Jun Ma and Chen-Bo Zhu. |

Bryan Wang Peng Jun | National University of Singapore | Representation Theory | BZSV duality and generalised Whittaker representations | Recent work of Ben-Zvi, Sakellaridis and Venkatesh (BZSV) predicts the existence of a duality in the relative Langlands program (BZSV-duality), indexed by a class of Hamiltonian spaces with reductive group action. I will first give a motivated introduction to BZSV-duality, focusing on a local incarnation of BZSV-duality in the setting of branching problems in representation theory. Then, I will present a new family of examples of BZSV-duality, whose proof proceeds via earlier results of Gomez and Zhu on theta correspondence of generalised Whittaker models. This is based on my thesis work supervised by Wee Teck Gan. |

Chaoping Dong | Soochow University | Representation Theory | Understanding the Dirac series of real reductive Lie groups | Dirac operator was raised by Parthasarathy in 1972 to give geometric construction of discrete series. The notion of Dirac cohomology, which sharpens the Dirac operator inequality, was given by Vogan in 1997. Five years later, the Vogan conjecture was proven by Huang and Pandzic. After that, it is an open problem to classify the Dirac series (irreducible unitary representations with non-zero Dirac cohomology). This talk will summarize our understanding of this problem. |

Dougal Davis | University of Melbourne | Representation Theory | Mixed Hodge modules and unitary representations of real groups | I will discuss recent progress (joint with Kari Vilonen) on the old problem of determining the unitary representations of a reductive Lie group. The headline result (originally conjectured by Schmid and Vilonen) is that every irreducible representation with real infinitesimal character has a canonical filtration, the Hodge filtration, which (a) has excellent properties and (b) detects whether the representation is unitary. If time permits, I will discuss an application (joint in preparation with Lucas Mason-Brown) to the structure and unitarity of the unipotent representations constructed via the orbit method. |

Emile Okada | National University of Singapore | Representation theory | Harmonic analytic and spectral characterisations of Arthur packets | In their study of the unitary dual for complex reductive groups, Barbasch and Vogan gave a characterisation of the unipotent Arthur packets using a microlocal invariant called the wavefront set. In this talk I will present joint results with Maxim Gurevich in which we pursue this approach for classical p-adic groups. We find that the packets defined microlocally are a union of Arthur packets which can be characterised by a surprising spectral property, namely the existence of an invariant vector for a (not necessarily hyperspecial) maximal compact subgroup. |

Shilin Yu | Xiamen University | Representation Theory | Extended duality map and unipotent representations | I will define a generalization of various duality maps between nilpotent orbits of a complex semisimple Lie group and its Langlands dual group, found by Barbasch-Vogan, Lusztig, Spaltenstein and Sommers, and explain how it is related to the study of unipotent representations of semisimple groups. It is based on the joint work with Lucas Mason-Brown and Dmytro Matvieievskyi (arXiv:2309.14853). |

Ting Xue | University of Melbourne | Representation theory | Springer theory for symmetric spaces | The Springer theory relates nilpotent orbits in the Lie algebra of a reductive algebraic group to irreducible representations of Coxeter groups. We discuss a Springer theory for symmetric spaces and its applications. We describe the character sheaves arising in this setting, where irreducible representations of Hecke algebras enter the description. This is based on joint work with Vilonen and partly with Grinberg. |

Yoshiki Oshima | The University of Tokyo | Representation Theory | Discrete branching laws of derived functor modules | We consider the restriction of unitary representations of real reductive groups to their subgroups. In particular, we study the restriction of Zuckerman's derived functor modules for symmetric pairs assuming that it is discretely decomposable in the sense of Kobayashi. In this talk, we would like to discuss how to obtain explicit branching formulas and observe some relationship with the projection of coadjoint orbits. |