# Weekly bulletin

## Week starting Monday 23 May 2022

### 10.30am 24 May - Partial Differential Equations and Analysis Seminar

Start time: 10.30am 24 May
Location: Zoom
Presenter(s): Malabika Pramanik (University of British Columbia)
Abstract:

Abstract:

Hausdorff dimension is a notion of size ubiquitous in geometric measure theory. A set of large Hausdorff dimension contains many points, so it is natural to expect that it should contain specific configurations of interest. Yet many existing results in the literature point to the contrary. In particular, there exist full-dimensional sets $K$ in the plane with the property that if a point ($x_1,x_2$) is in $K$, then no point of the form ($x_1,x_2 +t$) lies in $K$, for any $t \neq 0$.

A recent result of Kuca, Orponen and Sahlsten shows that every planar set of Hausdorff dimension sufficiently close to 2 contains a two-point configuration of the form $(x_1,x_2)+\{(0,0),(t,t_2)\}$ for some $t \neq 0$. This suggests that sets of sufficiently large Hausdorff dimension may contain patterns with “curvature”, suitably interpreted. In joint work with Benjamin Bruce, we obtain a characterization of smooth functions $\Phi : \mathbb{R} → \mathbb{R}^d$ such that every set of sufficiently high Hausdorff dimension in $d$-dimensional Euclidean space contains a two point configuration of the form $\{x, x + \Phi(t)\}$, for some $t$ with $\Phi(t) \neq 0$.

The Zoom link for this talk is available here. If you are not currently affiliated with the ANU, please contact Po-Lam Yung for access.

### 3pm 24 May - Algebra and Topology Seminar

Start time: 3pm 24 May
Location: Seminar Room 1.33 Hanna Neumann Building 145 Science Road Acton ACT 2601
Presenter(s): Bryden Cais (University of Arizona)
Abstract:

Abstract: Iwasawa theory is the study of the growth of arithmetic invariants in Galois extensions of global fields with Galois group a p-adic Lie group. Beginning with Iwasawa's seminal work in which he proved that the p-primary part of the class group in Z_p-extensions of number fields grows with striking and unexpected regularity, Iwasawa theory has become a central strand of modern number theory and arithmetic geometry. While the theory has traditionally focused on towers of number fields, the function field setting has been studied extensively, and has important applications to the theory of p-adic modular forms. This talk will introduce an exciting new kind of p-adic Iwasawa theory for towers of function fields over finite fields of characteristic p, and discuss some applications to problems and open conjectures in the field.

### 4pm 24 May - Mathematics and Computational Sciences Seminar

Start time: 4pm 24 May
Location: Seminar Room 1.33 Hanna Neumann Building #145 Science Road The Australian National University Canberra ACT 2600
Presenter(s): David Ham, Imperial College London
Abstract:

#### Abstract:

Simulating continuous systems modelled by PDEs underpins much of computational science and engineering. Each simulation is a complex combination of PDEs, parametrisations, discretisations, preconditioners and solvers. The precise combination that is optimal is different for each application and changes with the hardware, or as further advances in numerical mathematics are made. Many (possibly most) simulation challenges in science and engineering are actually inverse problems in which parameters are sought, sensitivities analysed and/or data assimilated.

Here I will present Firedrake, an automated system for generating numerical solutions to PDEs from a high level mathematical specification. I will examine some of the capabilities of the system before lifting the lid on the sequence of automated mathematical transformations that make it possible. I will also cover the interaction with dolfin-adjoint to produce gradients of solution functionals by solving the adjoint PDE.

#### Bio:

Dr David Ham is a reader in computational mathematics at Imperial College London. He studied mathematics and law at the ANU followed by a doctorate in numerical methods for ocean modelling at TU Delft in the Netherlands. He leads the Firedrake project and is a founding contributor to dolfin-adjoint. For the latter work he was jointly awarded the 2015 Wilkinson Prize for Numerical Software. He is the mathematics coordinator of the joint mathematics and computing programme at Imperial, and is chief executive editor of the European Geosciences Union journal Geoscientific Model Development. From April to August 2022 he is on sabbatical at the Research School of Earth Sciences at the ANU working with the G-ADOPT project to support the development of inverse models for geodynamics.

#### Zoom Details:

Topic: Mathematics and Computational Sciences Seminar Series
Time: This is a recurring meeting Meet anytime

Meeting ID: 880 8490 0678
Password: 971872
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+61861193900,,88084900678#,,,,0#,,971872# Australia

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+61 8 6119 3900 Australia
+61 8 7150 1149 Australia
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Meeting ID: 880 8490 0678
Password: 971872
Find your local number: https://anu.zoom.us/u/ksbYVntAY

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Meeting ID: 88084900678
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Join by Skype for Business
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### 11am 25 May - Partial Differential Equations and Analysis Seminar

Start time: 11am 25 May
Location: Seminar Room 1.33 / Zoom Hanna Neumann Building 145 Science Road Action 2601
Presenter(s): Pascal Auscher (Université Paris-Saclay)
Abstract:

Abstract: In this talk, I want to explain the yes answer to the title by looking at a class of elliptic linear problems of Dirichlet type. The strategy is one one could take using Green’s kernels or fundamental solutions and we show that it can still be used at an operator level and uniqueness depends on having enough solutions of a dual problem. This talk is based on joint works with Moritz Egert.

Bio: A professor at Paris-Sud University since 2002, Pascal Auscher is specialized in harmonic analysis and contributed both to the wavelet theory and partial differential equations. He was involved in proving the Kato conjecture. Along with his research, he headed the Laboratoire amiénois de mathématique fondamentale et appliquée (LAMFA) between 2000 and 2002. From 2006, he was in charge of assessing laboratories and training programs in his discipline on behalf of the French Ministry of Research before becoming (from 2007 to 2009) a scientific representative at the former AERES (agency for the evaluation of research and higher education. A member of the scientific board of the INSMI between 2010 and 2014, Pascal Auscher also coordinated an International Associated Laboratory with Australia, prior to becoming Director of the Institute in August 2017.

Join Zoom Meeting

https://anu.zoom.us/j/84195387287?pwd=aEtTZ2IxM1R0UE54RkFJMzJ1TEtZdz09

Meeting ID: 841 9538 7287

Password: 469891

One tap mobile

+61861193900,,84195387287#,,,,0#,,469891# Australia

+61871501149,,84195387287#,,,,0#,,469891# Australia

Dial by your location

+61 8 6119 3900 Australia

+61 8 7150 1149 Australia

+61 2 8015 6011 Australia

+61 3 7018 2005 Australia

+61 7 3185 3730 Australia

Meeting ID: 841 9538 7287

Password: 469891

Find your local number: https://anu.zoom.us/u/kbxquR6KDS

Or an H.323/SIP room system:

Dial: +61262227588 (AUCX)

Meeting ID: 84195387287

H323/SIP Password: 469891

Join by Skype for Business

https://anu.zoom.us/skype/84195387287

### 9am 27 May

Start time: 9am 27 May
Location: ANU Sports Centre 19 North Rd, Acton ACT 2601 The Australian National University
Abstract:

Young maths enthusiasts from across the region are invited to battle it out in a test of mind and muscles at ANU Maths Day.

In teams of five, year 12 students take part in mathematical challenges designed to build teamwork skills and promote a love of maths. The day is divided into four challenges:

• a group of story problems
• a swiss 'find the rule' problem
• a cross number puzzle
• and the day’s highlight, the relay – a test of brainpower and physical stamina.

ANU Maths Day has been running in the ACT since 1982. The University's competition includes contestants from schools in Canberra, Wollongong, the South Coast, Southern Highlands and Sydney. The day gives the students a chance to meet another 160 like-minded young people who are into mathematics, which is particularly important for people from smaller schools.

ANU Maths Day isn’t all about being the cleverest or the brightest. Activities are formulated to ensure that everyone can have a go and have fun while taking part in some healthy and rigorous competition.

## Cost:

\$80 per team. Each school can only enter 1 registration for a team of 5. The second team registration will be added onto the waiting list.

## Sample questions & solutions

Questions Solutions
2012 ANU Maths Day Swiss Contest (PDF 106KB)
2012 ANU Maths Day Relay Contest (PDF 95KB) 2012 ANU Maths Day Relay Contest Answer Sheet (PDF 51KB)
2012 ANU Maths Day Cross Contest (PDF 106 KB)
2012 ANU Maths Day Group Contest (PDF 67KB)

2012 ANU Maths Day Group Contest Answer Sheet (PDF 52KB)