Geometry, Stochastics, & Dynamics. Celebrating 20 years of UK-Japan Winter Schools

Abstracts

Monday, 12 September

Kenji Fukaya (Stony-Brook): Lagrangian correspondence and Floer homology
Lagrangian correspondence is expected to play a role of ‘morphisms’ in the ‘category’
of symplectic manifolds, as proposed byWeinstein. In this talk I will explain its relation
to Floer theory together (possible) applications. The latter includes,
1) Study of Floer homology of symplectic quotient 2) Existence of generating function
for immersed Lagrangian submanifolds 3) Relation to gauge theory

Martin Hairer (ICL): TBA

Chris Budd (Bristol): The mathematics of climate change
Climate change is important, controversial, and the subject of huge debate. Much of our understanding of the future climate comes from the use of complex climate models based on mathematical and physical ideas.
In this talk, I will describe how these models work and the assumptions that go into them. I will discuss how reliable our predictions of climate change are, and show how mathematicians can give us insights into both past and future.

Yoshiaki Maeda (Tohoku/Keio): Geometry of Loop space and the fundamental group of contact manifolds
We study the diffeomorphism and isometry groups of manifolds Mp, p 2 Z, which are circle bundles over a closed 4n-dimensional integral symplectic manifold. Equivalently, Mp is a compact (4n+1)-dimensional contact manifold with closed Reeb orbits. We use Wodzicki-Chern-Simons forms to prove that 1(Diff(Mp) and 1(Isom(Mp)) are innite for jpj 0: For the Kodaira-Thurston manifold, we explicitly compute that this result holds for all p. We also give the rest examples of nonvanishing Wodzicki-Pontryagin forms.”

Tuesday, 12 September

Darryl Holm (ICL): A Stochastic Climate Change Model
A generic approach to stochastic climate modelling is developed for the example of an idealized Atmosphere-Ocean model that rests upon Hasselmann’s paradigm for stochas- tic climate models. Namely, stochasticity is incorporated into the fast moving atmo- spheric component of an idealised coupled model by means of stochastic Lie transport, while the slow moving ocean model remains deterministic. This is joint work with D Crisan and P Korn. A remarkable property of the model is that the dynamics of its higher moments are governed by deterministic equations obtained by replacing the drift velocity of the stochastic Lie transport vector field by its expected value.

Takashi Sakajo (kyoto): Topological flow data analysis – theory and applications
We construct a mathematical theory classifying topological structures of orbits gener- ated by structurally stable Hamiltonian vector fields, which is a model of two-dimensional incompressible fluid flows. Based on the classification theory, we can show that struc- turally stable Hamiltonian flows are in one-to-one correspondence with Reeb graphs, and their symbolic expressions, named COT representations. By using this theory, we then develop a new way of topological data analysis, which we call Topological Flow Data Analysis (TFDA). In the present talk, after the classification theory is presented, I will talk about the recent applications of TFDA to geophysical data in atmospheric sci- ence and oceanography. The talk is based on the joint works with T. Yokoyama (Gigu U), T. Uda (Tohoku U), M. Inatsu (Hokkaido U), S. Oishi (RIKEN) and K. Koga (Kyoto U).

Graeme Segal (Oxford): The smooth homotopy category
Ordinary homotopy theory describes how a space is connectedup globally, disregarding its local structure. The objects of the smooth homotopy category have a homotopy type, but also have local structure described by differential geometry. They are therefore a generalization of smooth manifolds. I shall describe the history
of the idea, some examples of the structure, and reasons for being interested in it — it is a kind of classical limit of noncommutative geometry, and hence a model for a quantum space-time.
Furthermore, the structure is described perturbatively both at short distances and at long distances, reflecting the infrared/ultraviolet behaviour of quantum field theory, and there is a surprising duality
between the two extremes.

Hiroshi Iritani (Tokyo): Equivariant quantum cohomology and Fourier transformation
We discuss the relationship between Fourier transformation and quantum cohomology of symplectic reductions, following the idea of C. Teleman. We also give applications of this idea to mirror symmetry and computation of quantum cohomology. This is based on joint work with Fumihiko Sanda and Yuki Koto.

Yota Samoto (Waseda): Mirror symmetry and Stokes structure
To describe the asymptotic behavior of solutions of linear differential equations in one complex variable, Deligne introduced the 3 notion of a Stokes filtered local system, or, Stokes structure. In the context of mirror symmetry conjecture, the notion can be used to relate the derived categories of coherent sheaves and the quantum cohomologies of complex manifolds. To give an equivariant analog of this application of Stokes structure to mirror symmetry, the speaker is trying to construct the theory of Stokes structure of linear
difference equations as an analog of the theory of Deligne. In this talk, after introductions to these topics, we shall explain this attempt and the result.

Wednesday, 14 September

Peter Topping (Warwick): Hamilton’s pinching conjecture
The Bonnet-Myers theorem tells us that a uniform positive lower bound on the Ricci curvature of a manifold has topological implications. Richard Hamilton proposed a scale-invariant version of this theorem. I will give an introduction to the problem and describe some of the interesting work that has been developed over
the years in an attempt to solve it. This year Hamilton’s conjecture has been solved as a result of new work of Deruelle-Schulze- Simon and of M.-C.Lee and myself, using Ricci flow. I will give an overview of the strategy and the innovations required to solve the conjecture.

Jonathan Fraser (St Andrews): Dimension interpolation in conformal dynamics
Dimension interpolation’ is the idea that by viewing two distinct notions of fractal di- mension (e.g. Hausdorff and box-counting dimension) as extremes in a carefully defined ’continua of dimensions’, one may gain a more nuanced understanding of the fractal ob- jects at hand. I will review recent developments in this area in the context of conformal dynamics.

Ben Lambert:  Alexandrov Immersed Mean Curvature Flow
In this talk I will introduce mean curvature flow with surgery and in particular introduce Andrews’ noncollapsing estimates. We will see that noncollapsing is a vital ingredient in the mean curvature
flow of embedded surfaces. Finally, we will see that all of these ideas may be extended beyond the embedded setting to allow surgery for Alexandrov immersed mean curvature flow.

Thursday, 15 September

Takashi Kumagai (Waseda): Anomalous diffusions and time fractional differential equations
Time fractional diffusion equations have been widely used to model anomalous diffu- sions exhibiting sub-diffusive behavior, due to particle sticking and trapping phenom- ena. In this talk, I will discuss how anomalous sub-diffusions and the corresponding time-fractional differential equations arise naturally as limits of random walks in ran- dom media. I will then present some results on the probabilistic representation to the solutions of time fractional Poisson equations and estimates of their fundamental solu- tions. This talk is based on joint works with Z.-Q. Chen (Washington), P. Kim (Seoul) and J. Wang (Fuzhou).
Roland Bauerschmidt (Cambridge): Log-Sobolev inequalities for Euclidean field theories and spin models
I will present an extension of the Bakry-Emery method for Log-Sobolev inequalities that applies to Euclidean field theories which are invariant measures of singular SPDEs. The method uses as input estimates on the renormalised potential which is the solution to Polchinski’s continuous renormalisation group equation. Examples where this applies include the sine-Gordon model (with mass term) and the ϕ4 models in d < 4 (uniformly in the volume up to the critical point), and also the near-critical Ising model in d > 4. This talk is based on joint works in Thierry Bodineau and Benoit Dagallier.

Seiichiro Kusuoka (kyoto): Construction of a non-Gaussian and rotation-invariant Φ − 4-measure and associ- ated flow on R3 through stochastic quantization
In this talk, we construct the Φ4-measure on R3 by approximations of interactions with localization and regularization. Here, we remark that for approximations, we do not apply scaling of a torus. As an advantage of our approximations, we can prove the rota- tion invariance of the Φ4-measure. To prove the convergence of the approximations, we apply the stochastic quantization and the methods of singular stochastic PDEs. This is a joint work with Sergio Albeverio. Asma Hassannezhad asma.hassannezhad@bristol.ac.uk A tour on Steklov eigenvalue problem We discuss the importance and the beauty of the Steklov eigenvalue problem and its connection to the Laplace eigenvalue problem. The talk will be a brief tour of some classic results and recent developments on the subject.

Terry Lyons (Oxford/Turing institute): TBA

Ajay Chandra (ICL): Paracontrolled calculus and regularity structures
We prove a general equivalence statement between the notions of models and modelled distributions over a regularity structure, and paracontrolled systems indexed by the reg- ularity structure. The construction of a modelled distribution from a paracontrolled system is explicit, and takes a particularly simple form in the case of the regularity structures introduced by Bruned, Hairer and Zambotti for the study of singular stochastic partial differential equations. This talk is based on a joint work with Ismae¨l Bailleul (Universite´ Rennes 1).

Masato Hoshino (Osaka): Paracontrolled calculus and regularity structures
We prove a general equivalence statement between the notions of models and modelled distributions over a regularity structure, and paracontrolled systems indexed by the regularity structure. The construction of a modelled distribution from a paracontrolled system is explicit, and takes a particularly simple form in the case of the regularity structures introduced by Bruned, Hairer and Zambotti for the study of singular stochastic partial differential equations. This talk is based on a joint work with Isma¨el Bailleul (Universit
´e Rennes 1).

Friday, 16 September

Mark Pollicott (Warwick): Zeta functions for closed geodesics
In 1956 Selberg defined a complex function of a single complex variable for closed Riemann surfaces of constant negative curvature by analogy to the famous Riemann zeta function in number theory. In Selberg’s zeta function the closed geodesics play the role of prime numbers in the Riemann zeta functions. Over the past decade there has been progress in understanding the more general setting of variable negative curvature or surfaces with boundary and I will describe some of these results. This talk is intended for
a general mathematical audience.

Asma Hassannezhad (Bristol): A tour on Steklov eigenvalue problem
We discuss the importance and the beauty of the Steklov eigenvalue problem and its connection to the Laplace eigenvalue problem. The talk will be a brief tour of some classic results and recent developments
on the subject.

Adam Harper (Warwick): Random multiplicative functions: progress and problems
A random multiplicative function is a random function on the natural numbers, that is constructed from a sequence of independent random variables in a way that respects the multiplicative structure.
These objects arise naturally in analytic number theory as models for things like Dirichlet characters, but can also be thought of simply as probabilistic objects with an interesting dependence structure. In this talk I will try to survey what we know about random multiplicative functions, and some open problems, in a way
that is (hopefully) accessible and interesting to number theorists, probabilists, and others.

Public Lectures at the Embassy of Japan (Entrance strictly by registration)

Martin Hairer (ICL): On Coin Tossing, Atoms and Forest Fires
We will encounter some of the mathematical objects arising naturally in probability theory, as well as some of their surprising properties. In particular, we will see how one of these objects was involved in the confirma- tion of the existence of atoms over 100 years ago and how new properties of related objects are still being discovered today.
Profile: Sir Martin Hairer KBE FRS is a professor of mathematics and currently holds a chair in probability and stochastic analysis at Imperial College London. Hairer works in probability theory, including stochastic dynamics and stochastic partial differential equations. He is a Fellow of the Royal Society, of the American
Mathematical Society, of the Academy of Sciences Leopoldina, and of the Austrian Academy of Sciences. His work was distinguished with a number of prizes, most notably the Fermat prize and a Fields Medal.

Hiroshi Ooguri (Caltech and Kavli IPMU): The Science of the Man from the 9 Dimensions
This consist of a short video talk, with a 30 minutes movie over wine. Should wifi allows, a Q&A
The Man from the 9 Dimensions” is a science movie on Superstring Theory, the leading candidate for the unified theory of forces and matters, including gravity. Professor Ooguri will explain the science
behind the movie, which takes us from the microscopic world of elementary particles to the macroscopic world of the universe, and to its beginning – the Big Bang.
Profile: Professor Hirosi Ooguri is a Japanese physicist, the Fred Kavli Professor for Theoretical Physics and Math- ematics and the Director of the Walter Burke Institute for Theoretical Physics at California Insitute of Tech- nology, and the Director of the Kavli Institute for the Physics and Mathematics of the Universe at the
Univer- sity of Tokyo. He received a Medal of Honor with Purple Ribbon from the Emperor of Japan and the Eisenbud Prize from the American Mathematical Society. He is a Fellow of the American Academy of Arts and Sciences.