Titles and Abstracts
Keith Ball (University of Warwick)
The probabilistic character of convex domains
Over the last two decades it has become clear that convex domains exhibit
properties that we would expect of the joint densities of independent randomvariables. The developments have been guided by 3 conjectures made25-30 years ago by Kannan, Lovász and Simonovits, Bourgain and myself.
I will explain how these conjectures t together and roughly what is known.

Jacob Bedrossian (University of Maryland)
Lower bounds on the top Lyapunov exponent of stochastic systems

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We will discuss recent progress on estimating Lyapunov exponents ofstochastic models and their application to Eulerian and Lagrangian chaos instochastic uid mechanics. For Eulerian chaos, we discuss our recently
introduced methods for obtaining strictly positive lower bounds on the top
Lyapunov exponent of high-dimensional, stochastic di erential equations
such as the weakly-damped Lorenz-96 (L96) model or Galerkin truncationsof the 2d Navier-Stokes equations. For Lagrangian chaos we discuss ourearlier work on positive Lyapunov exponents and almost-sure exponential
mixing of scalars by the stochastic Navier-Stokes equations. Thesehallmarks of chaos have long been observed, however, no mathematical
proof had been made for either deterministic or stochastic forcing.
The work on Eulerian chaos combines (A) a new identity connecting theLyapunov exponents to a Fisher information of the stationary measure of theMarkov process tracking tangent directions (the so-called “projectiveprocess”); and (B) an L^1-based hypoelliptic regularity estimate to show that
this Fisher information is an upper bound on some fractional regularity. For
L96 and GNSE, we then further reduce the lower bound to proving the
projective process satis es Hörmander’s condition and introduce a
computational framework for checking this condition in certain systems. The
work on Lagrangian chaos involves an adaptation of the à la Furstenberg
method followed by large-deviation-type estimates to study the two-pointdynamics. All of the work discussed is joint with Alex Blumenthal and Sam
Punshon-Smith.

Kevin Buzzard (Imperial College London)
The rise of formalism in mathematics
Special lecture on proof formalisation for ordinary mathematicians
Formalism is the art of writing down what you actually mean. Mathematics
has a rich history of formalisation: Euclid, Russell–Whitehead and Bourbaki
all tried it. This century Avigad, Hales and Gonthier have shown us that there
is another way. Now a new generation of young people are formalisingalgebra, analysis, category theory, combinatorics, geometry, number theory,
topology and more at Masters level and beyond, this time using a computer.
Lean’s mathematics library mathlib contains nearly a million lines of free andopen source code corresponding to proofs of over 80,000 theorems such as
the fundamental theorem of Galois theory, and it is growing fast. AIs trainedon the library have solved IMO problems by themselves. What is happening?
This is not about making sure the papers are right. This is not about making
a computer program which will print out a one billion line proof of the Birch
and Swinnerton-Dyer conjecture using only the axioms of mathematics. This
is not about extracting the beauty from a proof and leaving only the directed
acyclic graph. This is about developing computer tools which have thepotential to help researchers and PhD students in new ways. Much remains
to be done. I will give an overview of the area.

Alison Etheridge (University of Oxford)
The motion of hybrid zones (and how to stop them)
Mathematical models play a fundamental role in theoretical population
genetics and, in turn, population genetics provides a wealth of mathematicalchallenges. In this lecture we illustrate this by using a mathematical
caricature of the evolution of genetic types in a spatially distributed
population to demonstrate the role that the shape of the domain inhabited bya species can play in mediating the interplay between natural selection,
spatial structure, and (if time permits) so-called random genetic drift (therandomness due to reproduction in a nite population).

Tadahisa Funaki (University of Tokyo)
Hydrodynamic limit and stochastic PDEs related to interface motion

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We consider several types of interacting particle systems at microscopic
level, in which particles move performing random walks with or without
creation and annihilation depending on the situation. From these systems,
via the hydrodynamic space-time scaling limit or its nonlinear uctuation

limit, we derive three di erent objects at macroscopic level: the motion bymean curvature arising in phase separation phenomena, Stefan freeboundary problem describing segregation of species, and coupled KPZequation which is a system of singular stochastic PDEs. These are all related
to the problem of interface motion. The Boltzmann-Gibbs principle plays afundamental role. We also touch the discrete Schauder estimate.

Craig B. Gentry (TripleBlind, New York)
Homomorphic Encryption
Is it possible to delegate processing of data without giving away access toit? For example, can I query a search engine, and get a useful response,
without telling the search engine what I am searching for? Can I send myencrypted nancial information to an online tax service, and get back anencrypted completed tax form?
Using an “ordinary” encryption system, it is virtually impossible for someone
without the secret decryption key to manipulate the underlying encrypted
data in any useful way. However, some encryption systems are"homomorphic" or "malleable". In a homomorphic encryption system, thedecryption function is a homomorphism that commutes with operations likeaddition and multiplication. This homomorphic property allows anyone tomanipulate (in a meaningful way) what is encrypted, without knowing thesecret key, by operating on ciphertexts.
This talk will survey basic concepts in modern cryptography and complexity
theory, including how to prove the security of a system by reducing it to the(assumed) computational infeasibility of well-known mathematical problems,such as factoring large integers. And it will highlight the main ideas behindrecent "fully homomorphic encryption" systems, which allow arbitrarily
complex functions to be computed on data while it remains encrypted.

Jan S Hesthaven (EPFL)
Structure Preserving Model Order Reduction for HamiltonianSystemsfffiflWe discuss the recent developments of projection-based model order
reduction (MOR) techniques targeting Hamiltonian problems. Hamilton’s
principle completely characterizes many high-dimensional models inmathematical physics, resulting in rich geometric structures, with examples
in uid dynamics, quantum mechanics, optical systems, and epidemiologicalmodels. MOR reduces the computational burden associated with theapproximation of complex systems by introducing low-dimensionalsurrogate models, enabling e cient multiquery numerical simulations.
However, standard reduction approaches do not guarantee the conservationof the delicate dynamics of Hamiltonian problems, resulting in reduced
models plagued by instability or accuracy loss over time.
By approaching the reduction process from the geometric perspective of
symplectic manifolds, the resulting reduced models inherit stability andconservation properties of the high-dimensional formulations.
In this presentation we discuss the construction of reduced order models
which preserve the symplectic structure through careful construction of thebasis itself and extensions to non-linear basis for problems which exhibit low
linear reducibility. We shall also brie y discuss the treatment of problems
posed on non-canonical form.
The performance of this class of methods is illustrated through a number ofcomputational examples.

Nicholas J. Higham (University of Manchester)
Numerical Stability of Algorithms at Extreme Scale and Low
Precisions

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As computer architectures evolve and the exascale era approaches, we aresolving larger and larger problems. At the same time, much modern
hardware provides oating-point arithmetic in half, single, and double
precision formats, and to make the most of the hardware we need to exploit
the di erent precisions. How large can we take the dimension n in matrix
computations and still obtain solutions of acceptable accuracy? Standard
rounding error bounds are proportional to p(n)u, with p growing at least
linearly with n. We are at the stage where these rounding error bounds are
not able to guarantee any accuracy or stability in the computed results for
extreme-scale or low-accuracy computations. We explain how rounding
error bounds with much smaller constants can be obtained. The key ideas
are to exploit the use of blocked algorithms, which break the data into
blocks of size b and lead to a reduction in the error constants by a factor bor more; to take account of architectural features such as extended precision
registers and fused multiply – add operations; and to carry out probabilistic
rounding error analysis, which provides error constants that are the squareroots of those of the worst-case bounds. Combining these di erent
considerations provides new understanding of the limits of what we cancompute at extreme scale and low precision in numerical linear algebra.

Hyeonbae Kang (Inha University)
Quantitative analysis of eld concentration in presence of closely
located inclusions of high contrast
In composites consisting of inclusions and a matrix of di erent materials,
some inclusions are located closely to each other. If the material property ofinclusions is of high contrast with that of the matrix, eld concentration
occurs in the narrow region between closely located inclusions.
Understanding the eld concentration quantitatively is important in thetheory of composites and imaging since it represent stress or eldenhancement. Last thirty years or so have witnessed signi cant progress inanalyzing this phenomena of eld concentration: optimal estimates andasymptotic characterization capturing the eld concentration have been
derived in the contexts of the conductivity equation (or anti-plane elasticity),
the Lamé system of linear elasticity, the Stokes system. In this talk, we
review some of them in a coherent manner.

Sylvie Méléard (École Polytechnique Paris)
Multiscale eco-evolutionary models: from individuals to populations
Motivated by recent biological experiments, we emphasize the e ects ofsmall and random populations on long time population dynamics. We willquantify such e ects on macroscopic approximations. The individual
behaviors are described by the mean of a stochastic measure-valuedprocess. We study di erent long time asymptotic behaviors depending on
the assumptions on mutation size and frequency and on horizontaltransmission rate. In some cases, simulations indicate that these models
should exhibit surprising asymptotic behaviors such as cyclic behaviors. We
explore these behaviors on a simple model where population and time sizes
are on a log-scale. Explicit criteria are given to characterize the possible
asymptotic behaviors. The impact of the time and size scales onmacroscopic approximations is also investigated, leading to Hamilton-Jacobi
equations.

Laure Saint-Raymond (IHES)
Dynamics of dilute gases: a statistical approach

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The evolution of a gas can be described by di erent models depending on
the observation scale. A natural question, raised by Hilbert in his sixth
problem, is whether these models provide consistent predictions. In
particular, for dilute gases, it is expected that continuum laws of kinetic

theory can be obtained directly from molecular dynamics governed by
Newton's fundamental principle.
In the case of hard sphere gases, Lanford showed that the Boltzmann
equation emerges as the law of large numbers in the low density limit, atleast for very short times. The objective of this talk is to present this limiting
process, including recent results on the uctuations.

David Silver (Deep Mind, London)
Simulation-Based Search
Planning is one of the oldest and most important problems in arti cialintelligence. Simulation-based search algorithms such as AlphaZero haveachieved superhuman performance in chess and Go and are now used inreal-world applications from chemistry to quantum computing. In this talk we
present a uni ed framework for understanding a wide variety of simulationbased search algorithms, including AlphaZero. We conclude by presenting
recent results using these algorithms.

Eric Vanden-Eijnden (Courant Institute, New York University)
Enhancing Markov Chain Monte Carlo Sampling Methods withDeep Learning
Sampling high-dimensional probability distributions is a common task incomputational chemistry, Bayesian inference, etc. Markov Chain Monte
Carlo (MCMC) is the method of choice to perform these calculations, but it isoften plagued by slow convergence properties. I will discuss how methods
from deep learning (DL) can help enhance the performance of MCMC via afeedback loop in which we simultaneously use DL to learn better samplers
based e.g. on generative models, and MCMC to obtain the data for thetraining of these models. I will illustrate these techniques via several
examples, including the sampling of random elds, the calculation ofreaction paths in metastable systems and the calculation of free energies
and Bayes factors.

Vlad Vicol (Courant Institute, New York University)

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Formation and development of singularities for the compressible
Euler equations

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We consider the compressible Euler equations of uid dynamics, in multiple
space dimensions. In this talk we discuss our program of completely
describing the formation and development of stable singularities, fromsmooth initial conditions. The questions we address are: given smooth initialconditions, precisely how does the rst singularity arise? is the mechanism
stable? how can one geometrically characterize the preshock (the boundaryof the space-time set on which the solution remains smooth)? precisely how
does the entropy producing shock wave instantaneously develop from thepreshock? does uniqueness hold once the shock has formed? do othersingularities instantaneously arise after the preshock? In this level of detailthe problem was previously open even in one space dimension. We discussa sequence of joint works with Steve Shkoller, Tristan Buckmaster, andTheodore Drivas, in which we have developed a multidimensional theory toanswer the above questions.