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More details about the dissertation are given in the dissertation handbook .

Past dissertations can be viewed by members of the Mathematical Institute (log in required).

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## Dissertation Topics

## Department of Mathematics

These topics are also offered to students in MSc Mathematics.

For more information on any of these projects, please contact the project supervisor.

For more information, please email Dr Filippo Cagnetti or visit his staff profile

Key words: Isoperimetric Inequality, sets of finite perimeter.

For more information, please email Dr Miroslav Chlebík or visit his staff profile

Miroslav Chlebik Presentation [PDF 309.98KB]

Recommended modules: Functional Analysis, Partial Differential Equations

Key words: Hausdorff dimension, Lipschitz mappings, rectifiable sets, fractals

Recommended modules: Measure and Integration, Functional Analysis

Key words: curve length, Lipschitz curve, calculus of variations

For more information, please email Dr Antoine Dahlqvist or visit his staff profile

Antoine Dahlqvist - Random matrices and Free Probability [PDF 345.10KB]

Antoine Dahlqvist - Brownian queues [PDF 151.46KB]

For more information, please email Dr Masoumeh Dashti or visit her staff profile

Key words: probability metrics, rates of convergence, Bayesian inverse problems

Recommended modules: Introduction to Probability, Measure and Integration.

!$[1]$! Gibbs A. L. and Su F. E. (2002) On choosing and bounding probability metrics.

Key words: Inverse problems, Tikhonov regularization, Bayesian regularization

!$[2]$! Stuart A. (2010) Inverse problems: a Bayesian perspective, 19 , 451--559.

Key words: Navier-Stokes equations, Regularity theory

Recommended modules: Partial differential equations, Functional analysis, Measure and Integration.

For more information, please email Dr Nicos Georgiou or visit his staff profile

The project has several aspects suitable for various forms of an MSc thesis:

Supervisor: Dr. Nicos Georgiou

Helpful mathematical background: Random processes, Monte Carlo Simulations, Statistical Inference.

For more information, please email Dr Peter Giesl or visit his staff profile

Peter Giesl - Computational analysis of periodic orbits in nonsmooth differential [PDF 11.57KB]

Peter Giesl - Calculation of Contraction Metrics [PDF 16.77KB]

Peter Giesl Project 3 [PDF 92.74KB]

For more information, please email Prof. James Hirschfeld or visit his staff profile

James Hirschfeld Presentation 1 [PDF 36.89KB]

Key words: polynomial, algebraic geometry

Recommended modules: Coding Theory

!$[1]$! Reid, M. Undergraduate Algebraic Geometry, University Press, 1988.

!$[2]$! Semple, J. G. and Roth, L. Introduction to Algebraic Geometry, Oxford University Press, 1949

James Hirschfeld Presentation 2 [PDF 25.58KB]

Key words: algebraic curve, cubic, group

!$[1]$! Seidenberg, A. Elements of the Theory of Algebraic Curves Addison-Welsley 1968

!$[2]$! Clemens, C.H. A scrapbook for Complex Curve Theory Plenum Press 1980

James Hirschfeld Presentation 3 [PDF 26.96KB]

Key words: geometry, projective plane, finite field

!$[1]$! Dembowski, P. Finite Geometries, Springer Verlag, 1968

!$[2]$! Hirschfeld, J.W.P. Projective Geometries over a Finite Field Oxford University Press, 1998.

To send just the two messages YES and NO, the following encoding suffices: YES = 1, NO = 0:

For more information, please email Prof Istvan Kiss or visit his staff profile

Key words: inference, networks, simulation, probability, stochastic processes, Matlab/Python

References for all three projects:

!$[3]$! Britton, N. F. (2003) Essential Mathematical Biology: Infectious Diseases. London: Springer.

!$[16]$! A.L. Barabási (2016) Network science. Cambridge University Press.

For more information, please email Dr Konstantinos Koumatos or visit his staff profile

!$[2]$! B. Dacorogna, Direct methods in the calculus of variations, volume 78, Springer, 2007

!$[1]$! C. M. Dafermos, Hyperbolic conservation laws in continuum physics, Springer, 2010

!$[3]$! L. C. Evans, Partial Differential Equations, American Mathematical Society, 1998

For more information, please email [email protected] or visit her staff profile

For more information, please email Dr Omar Lakkis or visit his staff profile

Omar Lakkis Presentations [PDF 358.53KB]

For more information, please email Prof. Michael Melgaard or visit his staff profile

Among the many problems one can study, we give a short list:

- The atomic Schrödinger operator (Kato's theorem and all that);
- The periodic Schrödinger operator (describing crystals);
- Scattering properties of Schrödinger operators (describing collisions etc);
- Spectral and scattering properties of mesoscopic systems (quantum wires, dots etc);
- Phase space bounds (say, upper bounds on the number of energy levels) with applications, e.g., the Stability of Matter or Turbulence.

Key words: differential operators, spectral theory, scattering theory.

Recent results on rigorous QC are found in the references.

- As a first step towards systems subject to a magnetic field, Question I(i) is addressed for the unrestricted KS model, which is suited for the study of open shell molecular systems (i.e., systems with a odd number of electrons such as radicals, and systems with an even number of electrons whose ground state is not a spin singlet state). The aim is to consider the (standard and extended) local density approximation (LDA) to DFT.
- The spin-polarized KS models in the presence of an external magnetic field with constant direction are studied while taking into account a realistic local spin-density approximation, in short LSDA.

Key words: operator and spectral theory, semiclassical analysis, micro local analysis.

The Choquard equation in three dimensions reads:

For more information, please email Prof Veronica Sanz or visit her staff profile

PLEASE NOTE THAT PROF SANZ IS NOT AVIALABLE FOR PROJECT SUPERVISION IN 19/20

For more information, please email Dr Nick Simm or visit his staff profile

Simm: Random matrix theory and the Riemann zeta function [PDF 156.59KB]

Simm: Asymptotic analysis of integrals and applications [PDF 124.34KB]

For more information, please email Prof. Enrico Scalas or visit his staff profile

For more information, please email Dr Ali Taheri or visit his staff profile

Key words: Poisson integrals, Nevanlinna class, Non-tangential convergence, M&F Riesz theorem

Recommended modules: Complex Analysis, Functional Analysis, Measure Theory

!$[1]$! Real and Complex Analysis by Rudin

!$[2]$! Introduction to !$H^p$! spaces by Koosis

!$[3]$! Theory of !$H^p$! spaces by Duren

!$[4]$! Bounded Analytic Functions by Garnett.

Recommended modules: Complex Analysis, Functional Analysis, Measure and Integration

!$[1]$! Fourier Analysis, T.W. Koner, Cambridge University Press, 1986

!$[2]$! Real and Complex Analysis, W. Rudin, McGraw Hill, 1987

!$[3]$! Real Variable Methods in Harmonic Analysis, A. Torchinsky, Dover, 1986.

Key words: Young measures, Weak convergence, Div-Curl lemma

Recommended modules: Partial Differential Equations, Functional Analysis, Measure Theory

!$[1]$! Parameterised Measures and Variational Principles, P.Pedregal, Birkhäuser, 1997.

!$[2]$! Partial Differential Equations, L.C. Evans, AMS, 2010.

!$[3]$! Weak Convergence Methods in PDEs, L.C. Evans, AMS, 1988.

Key words: Harmonic maps, Dirichlet energy, Minimal connections, Singular cones.

!$[1]$! Infinite dimensional Morse theory by Chang

!$[2]$! Two reports on Harmonic maps by Eells and Lemaire

!$[3]$! Cartesian Currents in the Calculus of Variations by Giaquinta, Modica and Soucek.

For more information, please email Dr Chandrasekhar Venkataraman or visit his staff profile

!$[1]$! Turing, A. M. (1952). The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B

For more information, please email Dr Minmin Wang or visit her staff profile

Minmin Wang - Probabilistic and combinatorial analysis of coalescence [PDF 64.35KB]

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## Senior Thesis

This page is for Undergraduate Senior Theses. For Ph.D. Theses, see here .

## Dissertation examples

## Edinburgh Research Archive

## Mathematics thesis and dissertation collection

By Issue Date Authors Titles Subjects Publication Type Sponsor Supervisors

Search within this Collection:

## Recent Submissions

## (Oxford MSc) Maths and Foundations of Computer Science dissertation

- Happy International Womens Day
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- Official University of Edinburgh 2023 Applicant Thread
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- Statistics/data science degree apprenticeship help !!!!! Omg omg!!! Lord help me

- Computational methods, machine learning algorithms, and stochastic models in financial mathematics
- Asymptotic analysis of deep neural networks (e.g., law of large numbers, central limit theorems, and large deviation principles)
- Applications of deep learning to partial differential equation models
- Stochastic online algorithms for optimizing over high-dimensional computationally-intensive simulations
- Introduction to Machine Learning (Imperial College Math Dept., Spring 2016): Master/PhD level course on machine learning.
- Deep Learning (University of Illinois at Urbana-Champaign): Master/PhD level course on deep learning. Topics include convolution neural networks, recurrent neural networks, and deep reinforcement learning. Homeworks and projects on image classification, video recognition, and deep reinforcement learning. Training of deep learning models using TensorFlow and GPUs. Supported by a computational grant from a national supercomputer. Course website
- Deep Learning II (University of Illinois at Urbana-Champaign): Deep learning applications in (1) reinforcement learning, (2) image recognition, and (3) high-frequency models of financial markets. There will be a special focus on distributed training of deep learning models. Supported by a computational grant from Microsoft.
- "Continuous-time stochastic gradient descent for optimizing over the stationary distribution of stochastic differential equations" (with Z. Wang), 2022.
- "Global Convergence of the ODE Limit for Online Actor-Critic Algorithms in Reinforcement Learning" (with Z. Wang), 2021.
- "PDE-constrained Models with Neural Network Terms: Optimization and Global Convergence" (with J. MacArt and K. Spiliopoulos). Major Revision at Journal of Computational Physics , 2021.
- "Mean Field Analysis of Deep Neural Networks" (with K. Spiliopoulos). Mathematics of Operations Research , accepted 2020.
- "Online Adjoint Methods for Optimization of PDEs" (with K. Spiliopoulos). Applied Mathematics and Optimization , In Press, 2022.
- "Deep Learning Closure of the Navier-Stokes Equations for Transitional Flows" (with J. F. MacArt and M. Panesi). Proceedings of AIAA Scitech, 2022.
- "Deep Learning for Mortgage Risk" (with A. Sadhwani and K. Giesecke). Journal of Financial Econometrics , 2021.
- "Mean Field Analysis of Neural Networks: A Law of Large Numbers" (with K. Spiliopoulos). SIAM Journal on Applied Mathematics , 2020.
- "Inference for large financial systems" (with G. Schwenkler and K. Giesecke). Mathematical Finance , 2020.
- "Asymptotics of Reinforcement Learning with Neural Networks" (with K. Spiliopoulos). Accepted, Stochastic Systems , 2020.
- "DGM: A deep learning algorithm for solving partial differential equations" (with K. Spiliopoulos). Journal of Computational Physics , 2018.
- "Mean Field Analysis of Neural Networks: A Central Limit Theorem" (with K. Spiliopoulos). Stochastic Processes and their Applications , 2019.
- "Stochastic Gradient Descent in Continuous Time: A Central Limit Theorem" (with K. Spiliopoulos). Stochastic Systems , 2020.
- "Universal features of price formation in financial markets: perspectives from Deep Learning" (with Rama Cont). Media coverage: Risk Magazine. Quantitative Finance , 2019.
- "Stochastic Gradient Descent in Continuous Time" (with K. Spiliopoulos). SIAM Journal on Financial Mathematics , 8(1), 933–961, 2017.
- "DPM: A deep learning PDE augmentation method with application to large-eddy simulation" (with J. Freund and J. MacArt). Datasets. Journal of Computational Physics , 2020.
- "Embedded training of neural-network sub-grid-scale turbulence models" (with J. MacArt and J. Freund). Physical Review of Fluids , In Press, 2020.
- "Deep Learning for Limit Order Books" . Quantitative Finance , 2018.
- "Risk Analysis for Large Pools of Loans" (with K. Giesecke). Winner of SIAM Financial Mathematics and Engineering Conference Paper Prize. Management Science , 2018.
- "Deep Learning Models in Finance" . Invited article for SIAM News , June 2017.
- "Fluctuation Analysis for the Loss from Default" (with K. Giesecke and K. Spiliopoulos). Stochastic Processes and their Applications , (124): 2322-2362, 2014.
- "Large-scale Loan Portfolio Selection" (with K. Giesecke and G. Tsoukalas). Operations Research , 2016.
- "Large Portfolio Asymptotics for Loss From Default" (with K. Giesecke, K. Spiliopoulos, and R. Sowers). Mathematical Finance , 2015.
- "Optimization of Secondary-Air Addition in a Continuous One-Dimensional Spray Combustor" (with L. Rodriquez, A. Sideris, and W. Sirignano). Journal of Propulsion and Power , 2010.
- "A Forward-Backward Algorithm for Stochastic Control Problems" (with S. Ludwig, R. Huang, and G. Papanicolaou). Proceedings of the First International Conference on Operations Research and Enterprise Systems . Vilamoura, Portugal.
- Book Review on deep learning, SIAM Review , 60(3), September 2018.
- Blue Waters supercomputer, 44 million core hours, value of $860,631.
- Summit supercomputer, 120,000 GPU hours.
- Lei Fan (ISE, UIUC, 2021). PhD Thesis: "Machine Learning Methods for Pricing and Hedging Financial Derivatives." Job Placement: J.P. Morgan Systematic Trading.
- Xiaobo Dong (ISE, UIUC, 2021). PhD Thesis: "Deep Reinforcement Learning Models of High Frequency Financial Data." Job Placement: J.P. Morgan Machine Learning.
- Ziheng Wang (Math, Oxford, 2024). PhD Thesis: "Asymptotic Analysis of Deep Reinforcement Learning." PhD funded by HSBC.
- Filippo De Angelis (Math, Oxford, 2024). PhD Thesis on machine learning models and methods in financial mathematics. PhD funded by HSBC.
- Deqing Jiang (Math, Oxford, 2024). PhD Thesis on deep reinforcement learning and mathematical finance. PhD funded by Alan Turing Institute.
- Additional supervised research: Yunxiang Zhang (supervised undergraduate thesis; now a PhD student at Cornell ORIE), Giri Tarun, Abhinav (supervised MS thesis; now a PhD student at UIUC CS), and Rachneet Kaur (PhD candidate at UIUC).
- Google Deepmind, Paris, January 2022.
- Two Sigma Investments, New York City, January 2020.
- Maven Securities, London, October 2020.
- J.P. Morgan, New York City, August 2017. Deep Learning in Finance.
- Bank of England, London, May 2016. Machine Learning for Loan Risk.
- Winton Capital Management, London, January 2016. Modeling financial data with Neural Networks.
- Capital Fund Management-Imperial Workshop, London, September 2015. Risk Analysis for Loan Portfolios.
- Lending Club, San Francisco, June 2015. Risk Analysis for Loan Portfolios.
- Associate Editor, Mathematical Finance .
- Managing Editor, Quantitative Finance .
- Associate Editor, Journal of Dynamics and Games (an AIMS journal).
- Associate Editor, Special Issue of Management Science on Data Science.
- Currently, I am the director of the MSc program in Mathematical & Computational Finance at the University of Oxford.
- Co-organized an internship program for students with banks, investment companies, and hedge funds. (Participating companies include Citadel, JP Morgan, Nomura, BNP Paribas, Citibank, and EDF Trading.) The quantitative research conducted during the internship is part of the students' MSc thesis.
- Organized the "Careers in Quantitative Finance" seminar series where companies present internship and job opportunities to the students. Participants include JP Morgan, Nomura, Deutsche Bank, Citibank, EDF Trading, Mazars, and NatWest Markets (formerly Royal Bank of Scotland).
- Introduced new courses/topics on deep learning and high-performance computing to develop models on large-scale financial datasets.
- Increased the number of applicants by 200% to almost 750 total applicants per year and increased number of enrolled students by almost 30%.
- Invited to organize minisymposium at SIAM Annual Meeting, Toronto, 2020.
- Machine Learning minisymposium at SIAM Financial Mathematics Conference, Toronto, 2019.
- Machine Learning for Finance minisymposium at SIAM Financial Mathematics Conference, Austin, November 2016.
- Machine learning in Finance session at INFORMS Annual Meeting, Houston, October 2017.
- Financial engineering session at INFORMS Applied Probability Meeting, Northwestern University, July 2017.
- Machine Learning for Finance session at INFORMS Annual Meeting, Nashville, November 2016.
- Large-scale Portfolio Risk session at INFORMS Annual Meeting, Philadelphia, November 2015.
- Reviewer for SIAM Journal on Applied Mathematics, SIAM Journal on Financial Mathematics, Journal of Machine Learning Research, Constructive Approximation (Special Issue on Deep Learning), NeurIPS, Operations Research, Stochastic Systems, and Management Science.
- Isaac Newton Institute, University of Cambridge, November 2021.
- London Business School, December 2021.
- Department of Applied Mathematics, Brown University, November 2021.
- Department of Mathematics, UCLA, February 2021.
- SIAM Conference on Dynamical Systems, 2021. Invited Speaker.
- SIAM Conference on Financial Mathematics, 2021. Invited Speaker.
- Seminar, Department of Physics, University of Oxford, November 2020.
- NSF Workshop on Machine Learning in Transport Phenomena in Dallas, Texas, 2020. Distinguished Speaker.
- Department of Mathematics, University of Minnesota, February 2021.
- Two Sigma Investments, New York City, January 2020. Invited Seminar.
- Workshop on Machine Learning in Finance at the University of Toronto, 2019. Invited Speaker.
- Theory of Deep Learning Workshop, Alan Turing Institute, London, 2020. Invited Speaker.
- Department of Mathematics, UCLA, Colloquium, 2019. Deep Learning: Applications and Asymptotics.
- Department of Mathematics, University of Michigan, 2019.
- Department of Mathematics, Oxford University, Data Science Seminar, 2019.
- Department of Mathematics, Oxford University, Financial Mathematics Seminar, 2019.
- Department of Statistics, Carnegie Mellon University, January 2019.
- Department of Industrial Engineering & Operations Research, Columbia University, February 2019.
- Department of Management Science & Engineering, Stanford University, 2019.
- Department of Mathematics, Washington University in St. Louis, 2019.
- BIRS Workshop on Modern Challenges between Financial Mathematics, Financial Technology, and Financial Economics . Banff, Canada, 2020. Invited Talk.
- American Institute of Mathematics/NSF Workshop on "Deep Learning and PDEs", 2019.
- Department of Statistics, Purdue University, 2019.
- SIAM Conference on Financial Math, Toronto, June 2019.
- INFORMS Annual Meeting, Seattle, October 2019.
- Department of Mathematics, Univ. of Illinois Urbana Champaign, Sept. 2018. Mean Field Analysis of Neural Networks in Machine Learning.
- SIAM Annual Conference, July 2018.
- London Quantitative Finance Seminar, May 2018. Deep Learning Models of High Frequency Financial Data.
- Seminar at the Department of Applied Math at the University of Colorado Boulder, April 2018.
- Deep Learning conference at the National Center for Supercomputing Applications, 2017. Deep Learning for PDEs.
- Seminar at Princeton University, 2017. Machine Learning in Finance.
- J.P. Morgan, Machine Learning and AI Forum seminar, August 2017. Deep Learning in Finance.
- Seminar at Northwestern University, April 2017.
- Seminar at UIUC Machine Learning Seminar Series, March 2017.
- Seminar at UIUC Business School, February 2017. Deep Learning for Modeling Limit Order Books.
- SIAM Financial Mathematics Conference, Austin, Texas, November 2016.
- Seminar at London Business School, London, June 2016. Deep Learning for Modeling Limit Order Books.
- Seminar at Oxford University, May 2016. Deep Learning for Modeling Financial Data.
- Statistics Seminar at Imperial College, London, May 2016. Deep Learning for Modeling Financial Data.
- INFORMS Annual Meeting, Philadelphia, November 2015. Invited Speaker and Organizer of Large-scale Portfolio Risk Session.
- Finance and Stochastics Seminar at Imperial College, London, October 2015.
- London-Paris Bachelier Workshop on Mathematical Finance, London, September 2015. Invited Speaker.
- IPAM Workshop on Systemic Risk and Financial Networks, Los Angeles, 2015.
- SIAM Financial Mathematics and Engineering Meeting, Chicago, 2014. Invited Speaker.
- INFORMS Annual Meeting, San Francisco, 2014. Invited Speaker and Organizer of Financial Risks Session.
- Likelihood Estimation for Large Financial Systems. Joint Mathematics Meeting, Baltimore, 2014. Invited Speaker.
- Fluctuation Analysis for Loss from Default. INFORMS Annual Meeting, Minneapolis, 2013. Invited Speaker.
- Lecture on Subprime Crisis. For a general audience at Stanford University, 2013.
- Fifth Western Conference on Mathematical Finance, Stanford University, 2013. Invited Speaker.
- INFORMS Annual Meeting, Phoenix, October, 2012. Invited Speaker.
- Financial Mathematics Seminar, Stanford University, 2012. Invited Speaker.
- SIAM Financial Mathematics and Engineering Meeting, Minneapolis, 2012. Chair of Credit Risk Session.
- Annual Meeting of the Canadian Applied and Industrial Mathematics Society, Toronto, 2012. Invited Speaker.
- 5th Financial Risks International Forum, Paris, France, 2012.

## DPhil Thesis Title: Projective Fibrations in Weighted Scrolls [submitted]

## UNFORTUNATE SHORT STORY

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This can be either a mathematics dissertation or a statistics dissertation. ... Please note that Maths Part C students are only permitted to chose a maximum

Students normally prepare their dissertations during TT and the long ... A student's dissertation topic should be selected in consultation with their

Dissertation Topics · 1. R. Hill, A First Course on Coding Theory, Oxford, 1986; QE 1302 Hil. The course · 2. V.S. Pless, Introduction to the Theory of Error-

Theses, see here. So that Math Department senior theses can more easily benefit other undergraduate, we would like to exhibit more senior theses online (while

Dissertation examples. Listed below are some of the best examples of research projects and dissertations from undergraduate and taught postgraduate students at

This collection contains a selection of the latest doctoral theses completed at the School of Mathematics. Please note this is not a comprehensive record.

Hi, can anyone tell me what their dissertation topic for the maths and foundations of Computer Science masters was/is?

Associate Professor, Mathematics, University of Oxford, 2020- ... Introduction to Machine Learning (Imperial College Math Dept., Spring 2016): Master/PhD

Address: Mathematics Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, OX2 6GG, Oxford-UK. Email: last name at maths dot ox dot ac

Name, Dissertation Title, Advisor(s), First Employment ... action on Coherent Cohomology of Hilbert modular varieties, Prasanna & Bhatt, Oxford University.