The Random Matrix Theory group in the School of Mathematics and Statistics at the University of Melbourne invites applications to fill two PhD positions. What is Random Matrix Theory? Random matrices are matrices that are randomly distributed along a chosen probability weight. It is like tossing a coin where you get a matrix instead of heads or tails. Like for ordinary matrices you can ask for their eigenvalues, eigenvectors, determinants, traces etc. All these linear algebra objects become also random and create certain statistics.
The remarkable phenomenon is that most quantities become independent of the chosen probability distribution when sending the matrix dimension to infinity. This is the reason why applications of random matrices can be found in quantum field theories and quantum chaos, quantum information theory and disordered systems, nuclear physics and condensed matter theory, number theory and combinatorics, enumerative geometry and harmonic analysis of matrix groups, statistics of time series analysis, description of signal transmission in wireless telecommunication, machine learning of neural networks and many more. One speaks of universality which can be seen as generalisations of the concept of central limit theorems in probability theory. Openings: (I) The first position is 1 PhD student position in "Optimal Rate of Convergences of Spectral Observables" You will work on a project that will explore the spectral statistics of eigenvalues and singular values of random matrices in the limit of large matrix dimensions. Our focus lies especially on the so-called local spectral statistics where one wants to understand the fluctuations of the eigenvalues on the scale of the mean level spacing between consecutive eigenvalues. We are interested in the leading order corrections of these local observables. Those strongly depend on the preparation of the spectrum called unfolding, which is usually a non-linear rescaling of the spectrum. This procedure of unfolding should make the observables comparable. It explains, for instance, why the hydrogen atom in a strong magnetic field shares the same spectral statistics (not the same spectrum!) with a irregularly shaped drum or with a particular random matrix. The problem is that the unfolding is not uniquely given and we want to investigate conditions such that the correction of the leading order in the limit of large matrix dimensions become particularly small. PhD Studies: You will carry out these studies fully at the University of Melbourne. Usually these studies will take 3-3.5 years. You will have two PhD supervisors Peter Forrester and Mario Kieburg. You will get a scholarship 31,000 AUD per annum, tax free, and have not to pay any enrolment fee. You can supplement this by part-time tutoring and exam marking employment within our School. Additionally, you will be expected to visit conferences and workshops and give presentations. Essential Criteria: - Master in Mathematics, Mathematical Physics or related areas
- Scored 80% or above in the overall grade
- Good analytical and/or numerical skills
- Good team work skills
- Proficiency in written and spoken English (eg. IELTS score of 6.5 in each of the four categories)
Desirable Criteria: - Knowledge in Complex Analysis, Asymptotic Analysis, Differential Equations and/or Probability/Measure Theory
- Experience in writing with LaTex
Contact Details and Application Details: Send to m.kieburg@unimelb.edu.au and/or pjforr@unimelb.edu.au the following documents: - cover letter
- CV (with possible list of publications)
- relevant certificates and a transcript of your grades (especially the one of the Master studies)
- list of at least two referees that have agreed to write letters of recommendation
- statement of motivation
(II) The second position is 1 PhD student position in "Critical Phenomena in Complex and Real Spectra" You will work on an analytical project that studies the complex eigenvalues of a non-Hermitian random matrix. Non-Hermitian matrices have generally complex eigenvalues. It happens that the probability density concentrates those eigenvalues in a well-defined area of the complex plane when the matrix dimension tends to infinity. Like a physical gas it will condensate. Therefore, this region where the eigenvalues accumulate is also called droplet. With the help of complex analysis, especially potential theoretic tools, you will investigate how specific deformations of the probability weight of the random matrix affect such droplets. PhD Studies: You will carry out these studies partially at the University of Melbourne/ Australia (2.5-3 years) and at KU Leuven/Belgium (1-1.5 years) as it is part of an International Training Research Group. Hence, in total these studies will take 3.5-4 years. The University of Melbourne will be your home university and KU Leuven be the hosting one. In the end you will be awarded two PhD's one from each university. You will have two PhD supervisors Mario Kieburg (University of Melbourne) and Arno Kuijlaars (KU Leuven). You will get a scholarship of approximately 31,000 AUD per annum (University of Melbourne), tax free, where you can supplement this by part-time tutoring and exam marking employment within our School. The scholarship at KU Leuven will be roughly 31,000 EUR per annum after tax deduction. Moreover, we provide additional funding for moving between the two universities. Additionally, you will be expected to visit conferences and workshops and give presentations as well as to carry out lecture duties as a tutor during your time at KU Leuven. Essential Criteria: - Master in Mathematics, Mathematical Physics or related areas
- Scored 80% or above in the overall grade
- Good analytical and/or proving skills
- Good team work skills
- Proficiency in written and spoken English (eg. IELTS score of 6.5 in each of the four categories)
Desirable Criteria: - Knowledge in Complex Analysis, Asymptotic Analysis, Differential Equations and/or Probability/Measure Theory
- Experience in writing with LaTex
- Experience as a tutor
Contact Details and Application Details: Send to m.kieburg@unimelb.edu.au and/or arno.kuijlaars@kuleuven.be the following documents: - cover letter
- CV (with possible list of publications)
- relevant certificates and a transcript of your grades (especially the one of the Master studies)
- list of at least two referees that have agreed to write letters of recommendation
- statement of motivation
Evaluation of the applications will start at 1st of December 2021 and the process will continue until the positions are filled.An Equal Opportunity employer. |

**Contact:**Mario Kieburg**Email:****Postal Mail:**- School of Mathematics and Statistics

University of Melbourne

Parkville VIC 3010

Australia

- School of Mathematics and Statistics
**Web Page:**http://www.ms.unimelb.edu.au/

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