3) vector calculus, Green's formula, (normal, tangent/vectors, parametrisation of surfaces and
curves.)
We propose 4 (2+2) sets of homework. The suggested text book is L.C. Evans, Partial Differential
Equations, AMS Graduate Studies in Mathematics.
Advanced PDE II: Hyperbolic Equations Synopsys -- Pieter Blue and John Ball
Credits: (15 credits)
The course will cover linear, semilinear and quasilinear hyperbolic equations from a rigorous perspective, and will include: The linear wave equation in bounded and unbounded domains, Huygen’s principle, weak solutions and associated semigroup. Semilinear wave equations, local and global existence via variation of constants formula and energy estimates. Finite-time blow-up. Introduction to quasilinear systems and more detailed treatment of scalar case. Prerequisites: some knowledge of Sobolev spaces and elementary functional analysis an advantage. Lecturers: John Ball and Pieter Blue.
COURSE PROPOSAL: MODERN DEVELOPMENTS IN FOURIER ANALYSIS
Instructor: Dr. Jonathan Hickman Email: TBAOffice: TBA Course website: TBA
Syllabus: Recently, there have been a number of remarkable developments in euclidean harmonicanalysis, related to the Fourier restriction conjecture. Broadly speaking, one is interested in studyingfunctions f whose Fourier transform is supported in a neighbourhood of a submanifold of Rn, suchas a paraboloid or a cone or a sphere. Such situations arise naturally in PDE, as well as in harmonicanalysis and analytic number theory.
One of the goals of this course is to understand the so-called decoupling inequalities of Wolff [4]and Bourgain–Demeter [2]. The idea is that whilst f may be difficult to analyse, it can be broken upas a sum of pieces fθ which are much easier to understand (in particular, the pieces fθ are localisedin frequency to small regions where the submanifold is essentially flat). The key question is then tounderstand how the various fθ interact with one another. In decoupling theory this is achieved viaa norm inequality of the form
(1) ‖f‖Lp(Rn) /(∑
θ
‖fθ‖2Lp(Rn)
)1/2.
The key feature of (1) is that an `2 expression appears on the right-hand side, rather than the trivial`1 expression given by the triangle inequality; this crucially takes into account complex destructiveinterference patterns between the different fθ.
Decoupling theory has had a profound impact on a wide range of (ostensibly) distinct areas ofmathematical analysis. A large portion of the course will investigate applications.
Possible topics include:
• Fourier analysis philosophy and uncertainty principle heuristics.• Multilinear harmonic analysis: the Bennett–Carbery–Tao theorem via induction-on-scale
[1].• The Bourgain–Guth method for estimating oscillatory integral operators [3].• Proof of the `2-decoupling theorem of Bourgain–Demeter [2].• Relation to incidence geometry.• Applications of decoupling to PDE: Strichartz estimates on the torus, spectral theory, local
smoothing for the wave equation.• Applications of decoupling to harmonic analysis: Bochner–Riesz means, Fourier restriction,Lp-Sobolev and maximal bounds for generalised Radon transforms.
• Applications of decoupling to analytic number theory: diophantine equations, the proof ofthe Vinogradov mean value theorem, Weyl sum bounds, the Lindelof hypothesis.
• Variable coefficient extensions and analysis on manifolds.
Schedule: The class would meet for a 2 hour lecture (including a short break) once a week for aperiod of 10 - 11 weeks. This would be supplemented with additional contact hours and meetingsfor discussion between students.
Textbooks: The topic of this course is a very recent development and no textbooks are available.A comprehensive bibliography will be made available to the students on the course webpage.
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Assessment: Whilst the initial lectures will be given by the instructor, the students will be askedto chose a topic to present later as the course progresses. A list of topics will be prepared and madeavailable at the outset of the course, with the option for students to suggest their own (relevant)topic if they so wish. The syllabus is particularly conducive to this approach: we hope to explorevarious applications of the decoupling theory and each application should fill 1 - 2 sessions. Supportfor the students will be provided through office hours. The students will also be asked to preparea latex write up of the topic they present. Ideally, these reports will be compiled at the end of thecourse and will be made publicly available and are likely to prove a valuable reference for the widermathematical community.
Prerequisites: A modest background in functional analysis and measure theory: Lp spaces, in-terpolation of operators, Holder and Minkowski inequalities, etc. Elementary theory of the Fouriertransform: Schwartz functions, Fourier inversion, Plancherel’s theorem. The course will aim to de-velop some basic understanding of the Fourier transform at a heuristic level and should be accessibleto students working in pure analysis in a broad sense.
References
[1] J. Bennett, A. Carbery, and T. Tao. On the multilinear restriction and Kakeya conjectures. Acta Math.,196(2):261–302, 2006.
[2] J. Bourgain and C. Demeter. The proof of the l2 decoupling conjecture. Ann. of Math. (2), 182(1):351–389, 2015.
[3] J. Bourgain and L. Guth. Bounds on oscillatory integral operators based on multilinear estimates. Geom. Funct.Anal., 21(6):1239–1295, 2011.
[4] T. Wolff. Local smoothing type estimates on Lp for large p. Geom. Funct. Anal., 10(5):1237–1288, 2000.
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Stochastic networks in the presence of heavy tails. Sergey Foss, HWU Overview: I am interested in the tail asymptotics for the probability of unusual behaviour of a stochastic system/network, that may course its damage/collapse. In the classical large deviations theory, one assumes that random characteristics of a system have `light-tailed' distributions, i.e. distributions with finite exponential moments. In such systems, an unusual behaviour may be detected well in advance and is caused by a long chain of `small unusual events'. It has been notices in the 1990's that characteristics of modern stochastic system have much heavier tail distributions, like Pareto/power or Weibull tails, and that the main cause for a large fluctuation is a single unusual and highly unpredictable event. In the first part of the course, I intend to introduce basic elements of the modern theory of heavy-tailed distributions. This part is based on the book ``Introduction to heavy-tailed and subexponential distributions'' by S Foss, D Korshunov and S Zachary, Springer (2011 and 2013). In the second part, I plan to analyse ``overflow'' probabilities in various communication and queueing systems, including single- and multi-server queues, queues in tandem, generalised Jackson networks, polling models, multiple-access systems. This part is based on research papers published within the last 20 years and on lecture notes. ======================= The course may be of 15 credits.
Course proposer: Istvan Gyongy
Title:
Stochastic PDEs, Applications and Numerics
Summary:
In the first part of the lectures some examples of stochastic PDEs (SPDEs), arising in
physics, population genetics and engineering, will be presented.
Then basic methods of solving SPDEs will be discussed and the main results of
the L_2 and L_p theories for SPDEs will be summarised.
In the second part of the lectures methods for solving SPDEs numerically will be studied,
theorems on accuracy of numerical schemes will be proved.
Applications in population genetics and stochastic filtering will be discussed.
Syllabus:
I. Main results on solvability of linear SPDEs
1. Introduction: Examples of stochastic PDEs (SPDEs) arising in applications,
SPDEs in nonlinear filtering and in population genetics
2. Stochastic processes with values in Sobolev spaces, and Ito formulas for their functions
3. Existence and uniqueness theorems in Sobolev spaces for SPDEs in the whole Euclidean
space.
4. Stochastic Fubini theorem and Ito-Wentzell formula. Feynman-Kac formulas for PDEs
and SPDEs
II. Numerical schemes for PDEs and SPDEs of parabolic type
1. Spatial discretisation, rate of convergence, accelerated schemes
2. Time discretisation, accuracy of implicit and explicit methods
3. Fully discretised schemes
4. Localisation error
5. Splitting up approximations, accelerated splitting up methods
6. Wong-Zakai approximations for SPDEs
Aims: -to show that SPDEs arise in solving real world problems
-to provide the students with concepts, methods and results to understand SPDEs,
-to provide the students with basic methods of solving PDEs and SPDEs nuimerically
Outcomes: -familiarity with random models and their connections with SPDEs
-basic knowledge of concepts, techniques and results of the theory of SPDEs and
their numerics
-familiarity with applications of SPDEs in physics, engineering, financial industry
and biology.
Prerequisites: -Basics of probability theory and stochastic analysis
-Elements of functional analysis
Heiko Gimperlein Course Title: Numerical Analysis of Partial Differential Equations Format: 10 lectures of 2 hours each, additional student lectures The format of the course is highly interactive, where students and upcoming seminar talks determine the content of the lectures. Credits: 15 for students giving a lecture, less for participation or active contribution to tutorials Aim: This course aims to give an introduction to current topics and techniques in the numerical analysis of partial differential equations, with a focus on the underlying linear and nonlinear analysis. After the course the student should know key ideas in a broad range of topics, as they are relevant to the research in their own area or in relevant seminar talks in numerical analysis. Prerequisites: a previous course in either PDE or their numerical analysis Contents: We cover some essential basic and advanced topics in the numerical analysis of PDEs. In particular, we expect to touch on the following topics: * Basics I: Numerical methods, such as finite differences, finite elements, finite volume methods, boundary elements, time-stepping schemes * Basics II: Relevant topics in analysis, such as approximation properties of functions, Sobolev spaces and functional analysis * Finite element methods for elliptic problems: Conforming variational and mixed methods, error analysis, adaptive methods * Non-conforming and non-standard methods * Finite elements for the Stokes problem, analysis and stabilisation * Heat and wave equations: time-stepping schemes and their analysis * Fast solvers: review of numerical linear algebra, preconditioning, multigrid methods * Applications in computational mechanics, fluid dynamics or biology Some references: * D. Braess, Finite elements: Theory, fast solvers, and applications in solid mechanics, Cambridge University Press * H. Gimperlein, Interface and contact problems, lecture notes * Y. Saad, Iterative methods for sparse linear systems, SIAM * E.P. Stephan, Theory of approximation methods, lecture notes Student talks: Interested students will give a 60-minute lecture on a topic of their choice, ideally a topic related to their research interests.
Additive Combinatorics Course Proposal
April 23, 2019
Overview
Additive combinatorics is a growing field of mathematics that concerns itselfwith questions regarding the additive structure of subsets of abelian groups.The results and techniques from additive combinatorics have given a new per-spective on many important problems in harmonic analysis; conversely, additivecombinatorics has shown itself to be an excellent setting in which to apply theresults and techniques of harmonic analysis.
An archetypal result in additive combinatorics is Szemeredi’s theorem onarithmetic progressions:
Thoerem (Szemeredi’s Theorem on Arithmetic Progressions). Let δ > 0 and
let k ≥ 3 be an integer. Then there exists an integer N0(δ, k) such that the
following holds: Suppose that N ≥ N0 and that A ⊂ {1, . . . , N} is such that
the cardinality of A is at least δN . Then A must contain a k-term arithmetic
progression.
This result has been used to study the Kakeya problem in harmonic anal-ysis. The Kakeya problem asks: given a Borel subset K ⊂ R
n that containsa unit-length line segment in every direction, is it possible to obtain a lowerbound on the dimension of K? One strategy used by Bourgain to study thisproblem has been to slice K with parallel hyperplanes. Ultimately, the strategyrelies on being able to find a three-term arithmetic progression among a finiteset obtained from K by slicing with parallel hyperplanes and discretizing. Amodified version of Szemeredi’s theorem can be used to obtain a lower boundon the Hausdorff dimension. This strategy gives the best-known bounds in highdimensions.
Conversely, the techniques of harmonic analysis are useful for studying prob-lems in additive combinatorics. Consider Szemeredi’s theorem with k = 3. Thisspecial case of Szemeredi’s theorem is sometimes known as Roth’s theorem.One of the standard proofs of Roth’s theorem is based on the discrete Fouriertransform: if the nonzero Fourier coefficients of the indicator function of A aresmall, then a simple Plancherel-Parseval argument can be used to show that Acontains a three-term arithmetic progression; otherwise, A will be concentratedon a large arithmetic progression contained in {1, . . . , N}, and the argument can
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be repeated with a larger value for δ. More generally, one proof of Szemeredi’stheorem, due to Gowers, follows the same general outline as the proof of Roth’stheorem, but the Fourier analysis is replaced by higher-order Fourier analysis.
Prerequisites
Students will be expected to have basic knowledge of Fourier analysis on Eu-clidean spaces: students should be generally familiar with how Fourier series aredefined and know facts like the Fourier inversion theorem and Plancherel’s theo-rem. No knowledge of the general theory of Fourier analysis on locally compactabelian groups will be required.
Some basic familiarity with elementary number theory and the theory ofabelian groups is useful, but not necessary. An undergraduate algebra course ismore than sufficient.
Course Structure and Assessment
This advanced MIGSAA course would be a 15-credit course. Lectures will be2 hours long, with a short break, and will take place once per week over a 10to 11 week period. The first segment of the course will consist of lectures ontopics such as those listed in the overview. Time permitting, the application ofthe Hardy-Littlewood Circle method to Waring’s problem will also be discussed.The second segment of the course will consist of student presentations on topicssimilar to those covered in the course. Potential examples for presentation topicsinclude the Croot-Lev-Pach/Ellenberg-Gijswijt solution to the capset problem,improvements to the bounds on Roth’s theorem, or the construction of Shmerkinof a subset of R of Fourier dimension 1 that does not contain a three-termarithmetic progression. Assessment will consist of the presentations given in thesecond half of the course, as well as a brief written report on the presentationtopic.
Textbooks
The general reference that will be used for most of the course topics is Taoand Vu’s Additive Combinatorics. For the application of additive combinatoricsto the Kakeya problem, we will use Mattila’s Fourier Analysis and Hausdorff
Dimension. For topics on additive number theory, the reference book will beNathanson’s Additive Number Theory: The Classical Bases.
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Convergence of Probability Measures and Its Applications
Our goal in this course is to cover the techniques of weak convergence of probability measures and get familiar with applications of these techniques in various fields. The course will be delivered by Dr. Burak Buke from University of Edinburgh in 2-hour blocks every week. The classes will be delivered in traditional lecture style, but in the last two weeks, the students will be asked to present their projects. Depending on the number of students enrolled in the course, the assessment will be carried as either individual or group projects that focus on how weak convergence techniques are used for different applications. The provisional syllabus for the course is as follows:
1. Basics of Weak Convergence a. Measures on Metric Spaces b. Properties of Weak Convergence c. Prokhorov’s Theorem
2. The Space of Continuous Functions C a. Uniform topology on C b. Criteria for Compactness and Tightness c. Donsker’s Theorem and Brownian Motion
3. The Space of Right-Continuous Functions (Space D) a. Four Skorokhod Topologies b. Tightness in D c. Sequences with Continuous Limits, C-Tightness
4. Convergence of Markov Processes a. Characterization of Markov Processes b. Stone’s Lemma for Birth-Death processes c. Convergence of Generators
5. Martingale Methods for Weak Convergence a. Basic properties of Martingales, Quadratic Variations and Basic Inequalities b. Martingale Central Limit Theorem c. Martingale Problems for Markov Processes
6. Applications a. Manufacturing and Service Industry b. Telecommunication Networks c. Mathematical Finance d. Chemical Reactions e. Mathematical Biology f. Statistical Physics and Mean Field Limits
The regularity structures introduced in [Hai14], inspired from rough paths[Lyo91, Gub04], have been able to solve several singular stochastic partial differentialequations (SPDEs). These equations are mainly of the form
(∂tui −∆ui) =M∑j=1
F ji (u,∇u)ξj , 1 ≤ i ≤ N, 1 ≤ j ≤M, (0.1)
where the ξj are space-time noises and the F ji are non-linearities depending on
the solution u and its derivatives. In this course, we will see how to solve thesesingular equations using this theory. We will first recall the basic definitions of aregularity structure and see different examples and applications. The main part ofthe course will be dedicated to the study of singular SPDEs but the main idea is toview regularity structures as a computational tool. We will also give a summary ofsome recent advances in this field explaining how these equations can be solved in amodern way using a black box introduced in [BHZ19, CH16, BCCH17].
References
[BCCH17] Y. Bruned, A. Chandra, I. Chevyrev, and M. Hairer. Renormalising SPDEsin regularity structures. ArXiv e-prints (2017). arXiv:1711.10239.
[BHZ19] Y. Bruned, M. Hairer, and L. Zambotti. Algebraic renormalisation ofregularity structures. Invent. Math. 215, no. 3, (2019), 1039–1156. arXiv:1610.08468. doi:10.1007/s00222-018-0841-x.
[CH16] A. Chandra and M. Hairer. An analytic BPHZ theorem for regularitystructures. ArXiv e-prints (2016). arXiv:1612.08138.
[Gub04] M. Gubinelli. Controlling rough paths. Journal of Functional Analysis 216,no. 1, (2004), 86 – 140. doi:10.1016/j.jfa.2004.01.002.
[Hai14] M. Hairer. A theory of regularity structures. Invent. Math. 198, no. 2, (2014),269–504. arXiv:1303.5113. doi:10.1007/s00222-014-0505-4.
[Lyo91] T. Lyons. On the non-existence of path integrals. Proceedings of the RoyalSociety of London. Series A: Mathematical and Physical Sciences 432, no. 1885,(1991), 281–290. arXiv:http://rspa.royalsocietypublishing.org/content/432/1885/281.full.pdf+html. doi:10.1098/rspa.1991.0017.
