Past

In this talk we will discuss some classical and recent results on local limits of random graphs. It is well known that the limiting object of the local structure of the classical Erdos-Renyi random graph is a Galton-Watson tree. This can nicely be formalised in the language of Benjamini-Schramm or Aldous-Steele local weak convergence. Regarding local limits of sparse random planar graphs, there is a smooth transition from a Galton-Watson tree to a Skeleton tree. This talk is based on joint work with Michael Missethan.

Plebanski generating functions give a compact encoding of the geometry of self-dual Ricci-flat space-times or hyper-Kahler spaces.  They have applications as generating functions for BPS/DT/Gromov-Witten invariants.  We first show that Plebanski's first fundamental form also provides a generating function for the gravitational MHV amplitude.  We then obtain these Plebanski generating functions from the corresponding twistor spaces as the value of the action of new sigma models for holomorphic curves in twistor space.   
In four-dimensions, perturbations of the hyperk¨ahler structure corresponding to positive helicity gravitons. The sigma model’s perturbation theory gives rise to a sum of tree diagrams for the gravity MHV amplitude observed previously in the literature, and their summation via a matrix tree theorem gives a first-principles derivation of Hodges’ determinant formula directly from general relativity. We generalise the twistor sigma model to higher-degree (defined in the first instance with a cosmological constant), giving a new generating principle for the full tree-level graviton S-matrix in general with or without  cosmological constant.  This is joint work with Tim Adamo and Atul Sharma in https://arxiv.org/abs/2103.16984.  

Abstract. The aim of the course is to present several results on extensions of functions. Among the most important are Kirszbraun's and Whitney's theorems.
They provide powerful technical tools in many problems of analysis. One way to view these theorems is that they show that there exists an interpolation
of data with certain properties. In this context they are useful in computer science, e.g. in clustering of data (see e.g. [26, 23]) and in dimension reduction (see e.g. [15]).

1. Syllabus
Lecture 1. McShane's theorem [25], Kirszbraun's theorem [18, 31, 35], Kneser- Poulsen conjecture [19, 29, 16].
Lecture 2. Whitney's covering and associated partition of unity, Whitney's ex-tension theorem [37, 12, 33].
Lecture 3. Whitney's theorem { minimal Lipschitz extensions [22].
Lecture 4. Ball's extension theorem, Markov type and cotype [6].

2. Required mathematical background
Markov chains, Hilbert spaces, Banach spaces, metric spaces, Zorn lemma

3. Reading list
The reading list consists of all the papers cited above, lecture notes [27], and parts of books [36, 8].

4. Assesment
Students will be encouraged to give a short talk on a topic related to the content of the course. Suggested topics include:
(1) Brehm's theorem [10],
(2) continuity of Kirszbraun's extension theorem [20],
(3) Kirszbraun's theorem for Alexandrov spaces [21, 1],
(4) two-dimensional Kneser-Poulsen conjecture [9],
(5) origami [11],
(6) absolutely minimising Lipschitz extensions and innity Laplacian [17, 32,
34, 2, 3, 5, 4],
(7) Fenchel duality and Fitzpatrich functions [30, 7],
(8) sharp form of Whitney's extension theorem [13],
(9) Whitney's extension theorem for Cm [14],
(10) Markov type and cotype calculation [27, 6, 28], 

(11) extending Lipschitz functions via random metric partitions [24, 27].

References
1. S. Alexander, V. Kapovitch, and A. Petrunin, Alexandrov meets Kirszbraun, 2017.
2. G. Aronsson, Minimization problems for the functional supx F(x; f(x); f0(x)), Ark. Mat. 6 (1965), no. 1, 33{53.
3. , Minimization problems for the functional supx F(x; f(x); f0(x))(ii), Ark. Mat. 6 (1966), no. 4-5, 409{431.
4. , Extension of functions satisfying lipschitz conditions, Ark. Mat. 6 (1967), no. 6, 551{561.
5. , Minimization problems for the functional supx F(x; f(x); f0(x))(iii), Ark. Mat. 7 (1969), no. 6, 509{512.
6. K. Ball, Markov chains, Riesz transforms and Lipschitz maps, Geometric & Functional Analysis GAFA 2 (1992), no. 2, 137{172.
7. H. Bauschke, Fenchel duality, Fitzpatrick functions and the extension of rmly nonexpansive mappings, Proceedings of the American Mathematical Society 135 (2007), no. 1, 135{139. MR 2280182
8. Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, Colloquium publications (American Mathematical Society) ; v. 48, American Mathematical Society, Providence, R.I., 2000 (eng).
9. K. Bezdek and R. Connelly, Pushing disks apart { the Kneser-Poulsen conjecture in the plane, Journal fur die reine und angewandte Mathematik (2002), no. 553, 221 { 236.
10. U. Brehm, Extensions of distance reducing mappings to piecewise congruent mappings on Rm, J. Geom. 16 (1981), no. 2, 187{193. MR 642266
11. B. Dacorogna, P. Marcellini, and E. Paolini, Lipschitz-continuous local isometric immersions: rigid maps and origami, Journal de Mathematiques Pures et Appliques 90 (2008), no. 1, 66 { 81.
12. L. C. Evans and R. F. Gariepy, Measure theory and ne properties of functions; Rev. ed., Textbooks in mathematics, ch. 6, CRC Press, Oakville, 2015.
13. C. L. Feerman, A sharp form of Whitney's extension theorem, Annals of Mathematics 161 (2005), no. 1, 509{577. MR 2150391
14. , Whitney's extension problem for Cm, Annals of Mathematics 164 (2006), no. 1, 313{359. MR 2233850
15. L.-A. Gottlieb and R. Krauthgamer, A nonlinear approach to dimension reduction, Weizmann Institute of Science.
16. M. Gromov, Monotonicity of the volume of intersection of balls, Geometrical Aspects of Functional Analysis (Berlin, Heidelberg) (J. Lindenstrauss and V. D. Milman, eds.), Springer Berlin Heidelberg, 1987, pp. 1{4.
17. R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Archive for Rational Mechanics and Analysis 123 (1993), no. 1, 51{74.
18. M. Kirszbraun,  Uber die zusammenziehende und Lipschitzsche Transformationen, Fundamenta Mathematicae 22 (1934), no. 1, 77{108 (ger).
19. M. Kneser, Einige Bemerkungen uber das Minkowskische Flachenma, Archiv der Mathematik 6 (1955), no. 5, 382{390.
20. E. Kopecka, Bootstrapping Kirszbraun's extension theorem, Fund. Math. 217 (2012), no. 1, 13{19. MR 2914919
21. U. Lang and V. Schroeder, Kirszbraun's theorem and metric spaces of bounded curvature, Geometric & Functional Analysis GAFA 7 (1997), no. 3, 535{560. MR 1466337
22. E. Le Gruyer, Minimal Lipschitz extensions to dierentiable functions dened on a Hilbert space, Geometric and Functional Analysis 19 (2009), no. 4, 1101{1118. MR 2570317
23. J. Lee, Jl lemma and Kirszbraun's extension theorem, 2020, Sublinear Algorithms for Big Data Lectues Notes, Brown University.
24. J. R. Lee and A. Naor, Extending Lipschitz functions via random metric partitions, Inventiones mathematicae 160 (2005), no. 1, 59{95.
25. E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12, 837{842. MR 1562984
26. A. Naor, Probabilistic clustering of high dimensional norms, pp. 690{709. 

27. , Metric embeddings and Lipschitz extensions, Princeton University, Lecture Notes, 2015.
28. A. Naor, Y. Peres, O. Schramm, and S. Sheeld, Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces, Duke Math. J. 134 (2006), no. 1, 165{197.
29. E. T. Poulsen, Problem 10, Mathematica Scandinavica 2 (1954), 346.
30. S. Reich and S. Simons, Fenchel duality, Fitzpatrick functions and the Kirszbraun{Valentine extension theorem, Proceedings of the American Mathematical Society 133 (2005), no. 9, 2657{2660. MR 2146211
31. I. J. Schoenberg, On a Theorem of Kirzbraun and Valentine, The American Mathematical Monthly 60 (1953), no. 9, 620{622. MR 0058232
32. S. Sheeld and C. K. Smart, Vector-valued optimal Lipschitz extensions, Communications on Pure and Applied Mathematics 65 (2012), no. 1, 128{154. MR 2846639
33. E. Stein, Singular integrals and dierentiability properties of functions, ch. 6, Princeton University Press, 1970.
34. P. V. Than, Extensions lipschitziennes minimales, Ph.D. thesis, INSA de Rennes, 2015.
35. F. A. Valentine, A Lipschitz condition preserving extension for a vector function, Amer. J. Math. 67 (1945), 83{93. MR 0011702
36. J. H. Wells and L. R. Williams, Embeddings and extensions in analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete ; Bd. 84, Springer-Verlag, Berlin, 1975 (eng).
37. H. Whitney, Analytic extensions of dierentiable functions dened in closed sets, Transactions of the American Mathematical Society 36 (1934), no. 1, 63{89. MR 1501735 

University of Oxford, Mathematical Institute and St John's College, Oxford, United Kingdom
E-mail address: @email

Over the past few decades, there has been a lot of interest in partial sums of Dirichlet characters. Montgomery and Vaughan showed that these character sums remain a constant size on average and, as a result, a lot of work has been done on the distribution of the maximum. In this talk, we will investigate the distribution of these character sums themselves, with the main goal being to describe the limiting distribution as the prime modulus approaches infinity. This is motivated by Kowalski and Sawin’s work on Kloosterman paths.
 

Let S be an orientable surface of finite type. Using Pho-On's infinite unicorn paths, we prove the hyperfiniteness of the orbit equivalence relation coming from the action of the mapping class group of S on the Gromov boundary of the arc graph of S. This is joint work with Marcin Sabok.

We give a brief introduction to Floer homotopy, from the Seiberg-Witten point of view.  We will then discuss Manolescu's version of finite-dimensional approximation for rational homology spheres.  We prove that a version of finite-dimensional approximation for the Seiberg-Witten equations associates equivariant spectra to a large class of three-manifolds.  In the process we will also associate, to a cobordism of three-manifolds, a map between spectra.  We give some applications to intersection forms of four-manifolds with boundary. This is joint work with Hirofumi Sasahira. 

In this session we will host a Q&A with current researchers who have recently gone through successful applications as well as more senior staff who have been on interview panels and hiring committees for postdoctoral positions in mathematics. The session will be a chance to get varied perspectives on the application process and find out about the different types of academic positions to apply for.

The panel members will be Candy Bowtell, Luci Basualdo Bonatto, Mohit Dalwadi, Ben Fehrman and Frances Kirwan. 

In a broad class of gravity theories, the equations of motion for vacuum compactifications give a curvature bound on the Ricci tensor minus a multiple of the Hessian of the warping function. Using results in so-called Bakry-Émery geometry, I will show how to put rigorous general bounds on the KK scale in gravity compactifications in terms of the reduced Planck mass or the internal diameter.
If time permits, I will reexamine in this light the local behavior in type IIA for the class of supersymmetric solutions most promising for scale separation. It turns out that the local O6-plane behavior cannot be smoothed out as in other local examples; it generically turns into a formal partially smeared O4.

Anabelian geometry asks how much we can say about a variety from its fundamental group. In 1997 Shinichi Mochizuki, using p-adic hodge theory, proved a fundamental anabelian result for the case of p-adic fields. In my talk I will discuss representation theoretical data which can be reconstructed from an absolute Galois group of a field, and also types of representations that cannot be constructed solely from a Galois group. I will also sketch how these types of ideas can potentially give many new results about p-adic Galois representations.

Geometrical questions commonly arise in clinical practice: for example, what is the optimal shape for a particular medical device? or what shapes of anatomical structures are indicative of pathological events? In this talk we explore two disparate clinical applications of geometrical underpinning: (A) how to design the optimal device for kidney stone removal surgery? and (B) what blood vessel shapes are associated with biomechanical failure? (A) Flexible ureteroscopy is a minimally invasive treatment for the removal of kidney stones by irrigating dust-like stone fragments with a saline solution. Finding the optimal ureteroscope tip shape for efficient flushing of stone fragments is a pertinent but complex question. We represent the renal pelvis (the main hollow cavity within the kidney) as a 2D cavity and employ adjoint-based shape optimisation to identify tip geometries that shrink the size of recirculation zones thereby reducing stone washout times. (B) The aorta is the largest blood vessel in the body, with an archetypal arched “candy-cane” shape and is responsible for transporting blood from the heart to the rest of the body. Aortic dissection, in which the inner layer of the aorta tears, can lead to frank rupture and is often rapidly fatal. Accurate clinical assessment of dissection risk from a CT scan of a patient’s thorax is paramount to patient survival. We apply statistical shape analysis, coupled with hemodynamic simulations, to identify pathological shape features of the aortic arch and to elucidate mechanistic underpinnings of aortic dissection.

Abstract, lecture notes and the manuscript for Lecture 1

References
[1] F. Bethuel, H. Brezis, F. Helein, Ginzburg-Landau vortices, Birkhauser, Boston, 1994.
[2] H. Brezis, L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries,
Selecta Math. (N.S.) 1 (1995), 197{263.
[3] R. Ignat, R.L. Jerrard, Renormalized energy between vortices in some Ginzburg-Landau models
on 2-dimensional Riemannian manifolds, Arch. Ration. Mech. Anal. 239 (2021), 1577{1666.
[4] R. Ignat, L. Nguyen, V. Slastikov, A. Zarnescu, On the uniqueness of minimisers of Ginzburg-
Landau functionals, Ann. Sci. Ec. Norm. Super. 53 (2020), 589{613.
[5] R.L. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals, SIAM J. Math. Anal.
30 (1999), 721-746.
[6] R.L. Jerrard, H.M. Soner, The Jacobian and the Ginzburg-Landau energy, Calc. Var. PDE 14
(2002), 151-191.
[7] E. Sandier, Lower bounds for the energy of unit vector elds and applications J. Funct. Anal.
152 (1998), 379-403.
[8] E. Sandier, S. Serfaty, Vortices in the magnetic Ginzburg-Landau model, Birkhauser, 2007.

We survey some of the applications of generalized indiscernible sequences, both in model theory and in structural Ramsey theory.  Given structures $A$ and $B$, a semi-retraction is a pair of  quantifier-free type respecting maps $f: A \rightarrow B$ and $g: B \rightarrow A$ such that $g \circ f: A \rightarrow A$ is quantifier-free type preserving, i.e. an embedding.  In the case that $A$ and $B$ are locally finite ordered structures, if $A$ is a semi-retraction of $B$ and the age of $B$ has the Ramsey property, then the age of $A$ has the Ramsey property.

We introduce a new method for high-dimensional, online changepoint detection in settings where a $p$-variate Gaussian data stream may undergo a change in mean. The procedure works by performing likelihood ratio tests against simple alternatives of different scales in each coordinate, and then aggregating test statistics across scales and coordinates. The algorithm is online in the sense that both its storage requirements and worst-case computational complexity per new observation are independent of the number of previous observations. We prove that the patience, or average run length under the null, of our procedure is at least at the desired nominal level, and provide guarantees on its response delay under the alternative that depend on the sparsity of the vector of mean change. Simulations confirm the practical effectiveness of our proposal, which is implemented in the R package 'ocd', and we also demonstrate its utility on a seismology data set.

I will discuss the following papers in my talk:
(1) Protecting consumers from collusive prices due to AI, 2020 with E. Calvano, V. Denicolò, J. Harrington, S.  Pastorello.  Nov 27, 2020, SCIENCE, cover featured article.
(2) Artificial intelligence, algorithmic pricing and collusion, 2020 with E. Calvano, V. Denicolò, S. Pastorello. AMERICAN ECONOMIC REVIEW,  Oct. 2020.
(3) Algorithmic Collusion with Imperfect Monitoring, 2021, with E. Calvano, V. Denicolò, S.  Pastorello

Controlling film flows has long been a central target for fluid dynamicists due to its numerous applications, in fields from heat exchangers to biochemical recovery, to semiconductor manufacture. However, despite its significance in the literature, most analyses have focussed on the “forward” problem: what effect a given control has on the flow. Often these problems are already complex, incorporating the - generally multiphysical - interplay of hydrodynamic phenomena with the mechanism of control. Indeed, many systems still defy meaningful agreement between models and experiments.
 
The inverse problem - determining a suitable control scheme for producing a specified flow - is considerably harder, and much more computationally expensive (often involving thousands of calculations of the forward problem). Performing such calculations for the full Navier-Stokes problem is generally prohibitive.

We examine the use of electric fields as a control mechanism. Solving the forward problem involves deriving a low-order model that turns out to be accurate even deep into the shortwave regime. We show that the weakly-nonlinear problem is Kuramoto-Sivashinsky-like, allowing for greater analytical traction. The fully nonlinear problem can be solved numerically via the use of a rapid solver, enabling solution of both the forward and adjoint problems on sub-second timescales, allowing for both terminal and regulation optimal control studies to be implemented. Finally, we examine the feasibility of controlling direct numerical simulations using these techniques.

In this talk, we discuss the feasibility of algorithms based on deep artificial neural networks (DNN) for the solution of high-dimensional PDEs, such as those arising from stochastic control and games. In the first part, we show that in certain cases, DNNs can break the curse of dimensionality in representing high-dimensional value functions of stochastic control problems. We then exploit policy iteration to reduce the associated nonlinear PDEs into a sequence of linear PDEs, which are then further approximated via a multilayer feedforward neural network ansatz. We establish that in suitable settings the numerical solutions and their derivatives converge globally, and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration. Numerical experiments on Zermelo's navigation problem and on consensus control of interacting particle systems are presented to demonstrate the effectiveness of the method. This is joint work with Kazufumi Ito, Christoph Reisinger and Wolfgang Stockinger.

The conformal dimension of a hyperbolic group is a powerful numeric quasi-isometry invariant associated to its boundary.

As an invariant it is finer than the topological dimension and allows us to distinguish between groups with homeomorphic boundaries.

I will start by talking about what conformal geometry even is, before discussing how this connects to studying the boundaries of hyperbolic groups.

I will probably end by saying how jolly hard it is to compute.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

We present an exponential improvement to the lower bound on diagonal Ramsey numbers for any fixed number of colors greater than two.
This is a joint work with David Conlon.
 

In 1972 Robert May argued that (generic) complex systems become unstable to small displacements from equilibria as the system complexity increases. His analytical model and outlook was linear. I will talk about a “minimal” non-linear extension of May’s model – a nonlinear autonomous system of N ≫ 1 degrees of freedom randomly coupled by both relaxational (’gradient’) and non-relaxational (’solenoidal’) random interactions. With the increasing interaction strength such systems undergo an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically non-trivial regime where equilibria are on average exponentially abundant, but typically all of them are unstable, unless the dynamics is purely gradient. When the interaction strength increases even further the stable equilibria eventually become on average exponentially abundant unless the interaction is purely solenoidal. One can investigate these transitions with the help of the Kac-Rice formula for counting zeros of random functions and theory of random matrices applied to the real elliptic ensemble with some of the mathematical problems remaining open. This talk is based on collaborative work with Gerard Ben Arous and Yan Fyodorov.

How does a colony of ants find the shortest path between its nest and a source of food without any means of communication other than the pheromones each ant leave behind itself?
In this joint work with Daniel Kious (Bath) and Bruno Schapira (Marseille), we introduce a new probabilistic model for this phenomenon. In this model, the nest and the source of food are two marked nodes in a finite graph. Ants perform successive random walks from the nest to the food, and the distribution of the $n$th walk depends on the trajectories of the $(n-1)$ previous walks through some linear reinforcement mechanism.
Using stochastic approximation methods, couplings with Pólya urns, and the electric conductances method for random walks on graphs (which I will explain on some simple examples), we prove that, depending on the exact reinforcement rule, the ants may or may not always find the shortest path(s) between their nest and the source food.