We discuss the problem of effective approximation of real and p-adic points on affine quadrics and cubics by integral points. The case of quadrics in four variables is used to construct optimally efficient universal quantum gates.

Perfect graphs are a class of graphs that behave particularly well with respect to coloring. In the 1960's Claude Berge made two conjectures about this class of graphs, that motivated a great deal of research, and by now they have both been solved.

The following remained open however: design a combinatorial algorithm that produces an optimal coloring of a perfect graph. Recently, we were able to make progress on this question, and we will discuss it in this talk. Last year, in joint work with Lo, Maffray, Trotignon and Vuskovic we were able to construct such an algorithm under the additional assumption that the input graph is square-free (contains no induced four-cycle). More recently, together with Lagoutte, Seymour and Spirkl, we solved another case of the problem, when the clique number of the input graph is fixed (and not part of the input).Over the last 10 years, the cost of sequencing the human genome has come down to around $1,000 per person. Human genomic data is a gold-mine of information, potentially unlocking the secrets to human health and longevity. As a society, we face ethical and privacy questions related to how to handle human genomic data. Should it be aggregated and made available for medical research? What are the risks to individual's privacy? This talk will describe a mathematical solution for securely handling computation on genomic data, and highlight the results of a recent international contest in this area. The solution uses "Homomorphic Encryption", based on hard problems in number theory related to lattices. This application highlights the importance of a new class of hard problems in number theory to be solved.

A set of lines in **R**^{d} is called equiangular if the angles between any two of them are the same. The problem of estimating the size of the maximum family of equiangular lines has had a long history since being posed by van Lint and Seidel in 1966. A closely related notion is that of a spherical code, which is a collection *C* of unit vectors in **R**^{d} such that *x·y∈L* for any distinct *x,y* in *C* and some set of real numbers *L*. Spherical codes have been extensively studied since their introduction in the 1970's by Delsarte, Goethals and Seidel. Despite a lot of attention in the last forty years, there are still many open interesting questions about equiangular lines and spherical codes. In this talk we report recent progress on some of them. Joint work with I. Balla, F. Drexler and P. Keevash.

An important problem in number theory is to understand the rational solutions to algebraic equations. One of the first non-trivial examples, cubics in two variables, leads to the theory of so-called elliptic curves. The famous Birch-Swinnerton-Dyer conjecture, one of the Clay Millenium Problems, predicts a relation between the rational points on an elliptic curve and a certain complex-analytic function, the L-function on an elliptic curve. In my talk, I will give an overview of the conjecture and of some new results establishing the conjecture in special cases.

"Symbolic dynamics" is a powerful technique for describing the combinatorial structure of large collections of orbits of dynamical systems with "chaotic" behaviour. I will describe this technique, and will report on recent advances on the question what sort of "chaos" is needed to this method to succeed. The talk is meant for a general audience, including people with little or no background in dynamical systems.

The Classification of Finite Simple Groups (CFSG) is a monumental achievement and a seemingly indispensable tool in modern finite group theory. By now there are a few results which can be used to bypass this tool in a number of cases, most notably a theorem of Larsen and Pink which describes the structure of finite linear groups of bounded dimension over finite fields.

In a few cases more ad hoc arguments can be used to delete the use of CFSG from the proofs of significant results. The talk will hopefully discuss a very recent example due to the speaker: how to obtain a CFSG-free version of Babai's quasipolynomial Graph Isomorphism algorithm by proving a weird lemma about permutation groups. The proof is being checked.Expander graphs in general, and Ramanujan graphs, in particular, have played a major role in computer science in the last 4 decades and more recently also in pure math. In recent years a high dimensional theory of expanders is emerging. A notion of topological expanders was defined by Gromov who proved that the complete d-dimensional simplical complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1. This question was answered recently (by T. Kaufman, D. Kazdhan and A. Lubotzky for d=2 and by T. Kaufman and S. Evra for general d) by showing that the d-skeleton of (d+1)-dimensional Ramanujan complexes provide such topological expanders. We will describe these developments and the general area of high dimensional expanders.

There is a lot of interest regarding the growth of invariants of chains of finite index subgroups, e.g. the growth of Betti numbers, rank, deficiency and so on. In this talk I will consider the growth of another invariant: the size of the torsion subgroup in homology. I will focus on two main classes of groups where there has been recent progress: amenable groups (joint with Kar and Kropholler) and right angled groups (joint work with Abert and Gelander). The main tools are from combinatorial group theory and the notion of combinatorial cost.

Recent work by M. Demers and B. Fernandez shows that the push-forward measure of open interval maps with indifferent fixed points at the origin converges to the point mass supported at the origin; in this context, an 'open' interval intermittent map is a map with a 'non-small' hole (roughly, a cylinder) that does not contain any neighborhood of the origin. In work in progress, restricting the study to piecewise linear maps with indifferent fixed points, we obtain a non trivial limit distribution under a different normalization of the push-forward transfer operator (and of some type of average). It is seems very likely that that the non-trivial limit can be used in the study of statistical properties of the open system. The topic of the talk develops on a question of R. Zweimueller.

In this talk, I will present a new class of generalized optimal transport costs. I will discuss in particular several applications of these transport costs to the study of concentration of measure properties of discrete probability measures. Based on joint works with C. Roberto, P-M Samson, Y. Shu, and P. Tetali.

In 1851, Carl Jacobi made the experimental observation that all integers are sums of seven non-negative cubes, with precisely 17 exceptions, the largest of which is 454. Building on previous work by Maillet, Landau, Dickson, Linnik, Watson, Bombieri, Ramaré, Elkies and many others, we complete the proof of Jacobi's observation.

Using an *L*^{1} localization argument, we prove that in metric measure spaces satisfying lower Ricci curvature bounds (more precisely *RCD*^{*}*(K,N)* or more generally essentially non branching *CD*^{*}*(K,N)*) the classical Lévy-Gromov isoperimetric inequality holds with the associated rigidity and almost rigidity statements.

Consider two quasirandomness properties for graphs: one based on eigenvalues (spectral expansion), and another based on discrepancy (epsilon-regularity). We know from the classic result of Chung-Graham-Wilson that these two quasirandomness notions are equivalent for dense graphs. However, for sparse graphs, they are not equivalent. We discuss what happens for Cayley graphs, and more generally, vertex-transitive graphs.

Structurable algebras form a class of non-associative algebras defined over fields of characteristic different from 2 and 3. These algebras have been introduced by Bruce Allison in 1978 in order to construct isotropic simple Lie algebras using a generalization of the so-called Tits-Kantor-Koecher construction. Because of the connections between simple linear algebraic groups and simple Lie algebras, it is not surprising that there is also a deep connection between structurable algebras and linear algebraic groups.

Central simple structurable algebras have been classified, first in characteristic 0 by Allison in his paper from 1978 (albeit with a gap), and in arbitrary characteristic different from 2, 3 or 5, by Smirnov in 1992. The classification is still open for fields of characteristic 5, but when the algebra is a division algebra, we have been able to extend the classification to include characteristic 5.This result in itself is perhaps only a detail hardly worth mentioning, but it fits in a much larger framework, connecting structurable algebras, Moufang sets, algebraic groups and various different Lie algebras. In my talk, I will try to explain these connections and indicate why characteristic 5 causes so many difficulties. This is joint work with L. Boelaert and A. Stavrova.The Brascamp-Lieb inequality generalizes many important inequalities in analysis, including the Höolder, Loomis-Whitney, and Young convolution inequalities. Sharp constants for such inequalities have a long history and have only been determined in a few cases. We investigate the stability and regularity of the sharp constant as a function of the implicit parameters. The focus of the talk will be a local-boundedness result with implications for certain nonlinear generalizations arising in PDE. This is joint work with Jonathan Bennett, Neal Bez, and Sanghyuk Lee.

In this talk I will discuss some recent joint work with T. Hartnick (Technion) and F. Pogorzelski (Technion) on the (spherical) diffraction of quasicrystals (model sets) in lcsc (not necessarily abelian/amenable) groups.

The limit distribution of primitive rational points on expanding horospheres on SL(*n*,**Z**)\SL(*n*, **R**) has been derived in the recent work by M. Einsiedler, S. Mozes, N. Shah and U. Shapira. For *n=3*, in my joint project with Jens Marklof, we prove the effective equidistribution of q-primitive points on expansion horospheres as square-free *q→∞*.

Given a pair of graphs G and H, the Ramsey number R(G,H) is the smallest N such that every red-blue coloring of the edges of the complete graph on N vertices contains a red copy of G or a blue copy of H. If a graph G is connected, it is well known that R(G,H) is at least (|G|-1)(c(H)-1) + s(H), where c(H) is the chromatic number of H and s(H) is the size of the smallest color class in a c(H)-coloring of H. A graph G is called H-good if R(G,H) = (|G|-1)(c(H)-1) + s(H). The notion of Ramsey goodness was introduced by Burr and Erdös in 1983 and has been extensively studied since then.

We will show that the path P is H-good for any graph H with |P|≥4|H|. For graphs with c(G)≥4, this confirms in a strong form a conjecture of Allen, Brightwell, and Skokan. Some results about Ramsey goodness of cycles will be presented as well.This is joint work with Sudakov.The pseudo-marginal algorithm is a popular variant of the Metropolis-Hastings scheme which allows us to sample asymptotically from a target probability density when we are only able to estimate unbiasedly an un-normalized version of it. It has found numerous applications in Bayesian statistics as there are many scenarios where the likelihood function is intractable but can be estimated unbiasedly using Monte Carlo samples. In this Bayesian context, we propose a modification of the pseudo-marginal algorithm, termed the correlated pseudo-marginal algorithm, which is based on a novel log-likelihood ratio estimator, computed using the difference of two positively correlated log-likelihood estimators. We establish a Central Limit Theorem for this estimator and, by combining this result with the Bernstein-von Mises theorem; we show that a re-scaled version of the resulting correlated pseudo-marginal process converges weakly to the so-called penalty method when the number of data goes to infinity. In our numerical examples, the efficiency of computations is increased relative to the standard pseudo-marginal method by several orders of magnitude for large data sets. Joint work with George Deligiannidis (Oxford) and Mike Pitt (Kings)

Cut and project sets give a way of defining discrete point patterns through a data of a linear subspace and an acceptance strip. The points in the pattern are obtained through first intersecting the integer lattice with the acceptance strip (cut) and then projecting these intersection points to the subspace (project). We establish a connection between finite patches in cut and project sets and an action of a toral rotation defined by the cut and project data. We then investigate the implications of this observation in both tiling theory and Diophantine approximation.

The talk is based on several works, joint with Alan Haynes, Antoine Julien, Lorenzo Sadun and James Walton.
To any *k*-partition (*D*_{1}, …, *D*_{k}) of a domain Ω is associated an energy which is the maximum of the first Dirichlet-Laplacian eigenvalue on each subdomain *D*_{j}:

The *r-neighbour bootstrap process* on a graph G starts with an initial set of 'infected' vertices and, at each step of the process, a healthy vertex becomes infected if it has at least r infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of G becomes infected during the process, then we say that the initial set *percolates*.

I will explain recent joint work with Catharina Stroppel, which establishes a relationship between quiver Schur algebras and unipotent l-modular representations of p-adic general linear groups.

Given a family F of elliptic curves defined over Q, we are interested in the set H(Y) of curves E in F, of positive rank, and for which the minimum of the canonical heights of non-torsion rational points on E is bounded by some parameter Y. When one can show that this set is finite, it is natural to investigate statistical properties of arithmetic objects attached to elliptic curves in the set H(Y). We will describe some problems related to this, and will state some results in the case of families of quadratic twists of a fixed elliptic curve.

The Teichmüller harmonic map flow is a gradient flow for the harmonic map energy of maps from a closed surface to a general closed Riemannian target manifold of any dimension, where both the map and the domain metric are allowed to evolve. It was introduced by M. Rupflin and P. Topping in 2012. The objective of the flow is to find branched minimal immersions on a given surface.

I will give some background on the flow and then describe some recent work with Rupflin and Topping. In particular we show that if the flow exists for all times then in a certain sense the maps (sub-)converge to a collection of branched minimal immersions with no loss of energy (even when allowing for degeneration of the metric at infinity). We also construct an example of a smooth flow where the image of the limit maps is disconnected.I will introduce the Abelian sandpile model, and report on recent results on some of its critical exponents. These concern the toppling probability, the avalanche radius, and the avalanche cluster size. Joint work with S. Bhupatiraju and J. Hanson.

Markov partitions introduced by Sinai and Adler and Weiss are a tool that enables transferring questions about ergodic theory of Anosov Diffeomorphisms into questions about Topological Markov Shifts and Markov Chains. This talk will be about a reverse reasoning, that gives a construction of *C ^{1}* conservative (satisfy Poincare's recurrence) Anosov Diffeomorphism of

This talk is a report on joint work with Nathan Broomhead and David Ploog. The notion of a discrete derived category was first introduced by Vossieck, who classified the algebras admitting such a derived category. Due to their tangible nature, discrete derived categories provide a natural laboratory in which to study concretely many aspects of homological algebra. Unfortunately, Vossieck's definition hinges on the existence of a bounded t-structure, which some triangulated categories do not possess. Examples include triangulated categories generated by 'negative spherical objects', which occur in the context of higher cluster categories of type A infinity. In this talk, we compare and contrast different aspects of discrete triangulated categories with a view toward a good working definition of such a category. This talk will not assume much background in homological algebra and representation theory and will aim to present an overview of the questions.

A classical problem in analytic number theory is to understand the distribution of interesting sequences, like the primes, in different arithmetic progressions. Sometimes it suffices to have results on average over many different progressions, as in the Bombieri-Vinogradov or Barban-Davenport-Halberstam theorems. I will try to describe some new results giving lower bounds for the Barban-Davenport-Halberstam ''variance'' in progressions, on a wide range (previously only known assuming the Riemann Hypothesis). The proofs exploit a connection between this variance and the minor arc contribution in the Hardy-Littlewood circle method. This is joint work with K. Soundararajan.

A famous theorem of Frankl and Rödl bounds the size of families from P[n] with a single forbidden intersection. Given a collection of vectors V={v_{i}: i ∈ [n]} in **R**^{D}, define the V-intersection of two sets A, B in [n] by v_{V}(A,B) := ∑_{i∈A∩B} v_{i}. Extending the Frankl-Rödl theorem, for a large collection of V we prove upper bounds on the size of families of sets with a forbidden V-intersection. A special case of our main theorem solves a conjecture of Kalai.

In the middle of the 18th century, Georges Louis Leclerc, Comte de Buffon, asked how often a needle, dropped on a floor of wooden planks, would fall across a join. In doing so, he gave rise to the development, in the late 19-th and early 20-th centuries, to the study of Stochastic (or Integral, depending on one’s vocabulary) Geometry. One of the most important results therein was the celebrated Kinematic Fundamental Formula, or KFF, which describes intersections of far more general random objects in far more general spaces.

The mathematical constructs that appear in the KFF - most commonly known as Minkowski functionals - reappeared in the mid 20-th century, when Hermann Weyl derived his celebrated Tube Formula, from which an entire research area, now differentially topological in nature, grew.In the late 20-th and (very) early 21-st centuries, all of these objects reappeared in what initially seemed to be a completely different setting, the study of Gaussian random processes and, in particular, their applications. However, the connection to earlier theory was peripheral to the main results here. A decade ago, Jonathan Taylor realised that there was a deep connection between the classical results and the new ones, and formulated the first version of the Gaussian Kinematic Formula. The GKF has revolutionalised much current research in Gaussian processes, as well as providing extremely elegant connections between Probability, Geometry, and Topology.This talk will be about the GKF, its family history, importance, applications, and maybe even 10 minutes about its proof.The representations of a finite group over a field of positive characteristic p are closely related on the one hand to its representations over the complex numbers (via p-modular reduction) and on the other hand to the embeddings of and relationships between the p-subgroups of the finite group (p-local structure). There are many long-standing conjectures which make these relationships precise. In my talk, I will focus on the so-called finiteness conjectures. The ultimate resolution of these seems to be distant, but recent developments in the modular representation theory of finite simple groups have led to strong new partial results.

The classical Problem of Plateau asks to find a disc-type surface of minimal area with prescribed Jordan boundary. This problem was first solved by Douglas and Rado in the setting of Euclidean space and then extended by Morrey to the setting of Riemannian manifolds. I will first present a generalization of this problem and its solution to the setting of locally compact metric spaces and discuss regularity properties under mild conditions on the ambient metric space. I will then outline some applications of the existence and regularity results to problems in metric geometry and geometric group theory, in particular to the large scale geometry of Riemannian manifolds and singular metric spaces and to the problem of characterizing non-positive curvature by an isoperimetric inequality for curves. As will become evident, passage to a metric setting is sometimes needed even if one is only interested in a smooth setting. The talk is based on joint work with Alexander Lytchak.

After a brief review of classical additive problems involving prime numbers, we will discuss some geometric analogues. We will see that this leads naturally to interesting questions in algebraic and arithmetic geometry, and we will discuss some results in this direction.

The dimer model on a finite bipartite planar graph is a uniformly chosen perfect matching of its vertices. It is a classical model of statistical mechanics, going back to work of Kasteleyn and Temeperley/Fisher in the 1960s who computed its partition function.

After giving an overview of the main questions and the state of the art on this topic, I will discuss some recent joint work with Benoit Laslier and Gourab Ray, where we prove that subject to planar boundary conditions, the fluctuations are described by a universal and conformally invariant object, known as the Gaussian free field. A main novelty in our approach is that we rely in a minimal way only on the exactly solvable character of the model: a key idea is to relate this problem to the so-called imaginary geometry of Miller and Sheffield, where Schramm-Loewner Evolution curves are viewed as flow lines of the Gaussian free field. The robustness inherent in this approach yields universality in several other cases.What are the underlying mechanisms for robustly chaotic behavior in smooth dynamics? In addressing this question, I'll focus on the study of diffeomorphisms of a compact manifold, where "chaotic" means "mixing" and and "robustly" means "stable under smooth perturbations." I'll describe recent advances in constructing and using tools called "blenders" to produce stably chaotic behavior with arbitrarily little effort.

The modulus of curve families is a powerful tool in the study of quasiconformal and related mappings. Quantitative lower bounds for the modulus can often be obtained if the considered curve family is of a special form, for instance if it constitutes a nice foliation, or if it consists of all possible curves connecting two nondegenerate continua. In this talk I will discuss a collection of curves that is not of this form, namely a family of planar curves that contains an isometric copy of every rectifiable curve in **R**^{2} of length one.

The definition of a 'cellular algebra' was given by Graham and Lehrer in 1996 and it has been shown by East, Wilcox and others that various semigroup algebras are cellular. Each cellular algebra possesses an anti-isomorphism, however for many semigroup algebras, no such natural anti-isomorphism exists: one may consider, for example, the transformation monoid whose elements consist of all maps from a set {1,2,...,n} to itself. For these algebras, it is helpful to use the definition of a 'standardly based algebra' given by Du and Rui in 1998. These algebras also possess a distinguished basis which gives information about their representation theory, but they do not require the existence of an anti-isomorphism.

In this talk we shall define and discuss semigroup algebras and standard bases and give some examples of the former that possess the latter.In this talk, I will discuss some recent developments on some enumeration problems in extremal combinatorics. Among others, I will discuss two problems asked by Cameron and Erdös: one on counting the number of maximal sum-free sets of integers, and the other one on counting the number of sets without any arithmetic progressions of fixed length.

I will discuss a construction of finite 'geometric' random graphs. This construction is motivated by the study of random walks on infinite groups, but might be interesting on its own right. I will briefly mention connections to other topics, including the Poisson boundary, Sznitman's random interlacements, and long range percolation. Paper available at http://arxiv.org/abs/1506.02697

There is a natural action of the group SL_{2}(**R**) on the moduli space of translation surfaces. The horocycle flow is the action of the one-parameter subgroup consisting of matrices of the form

1 | t |

0 | 1 |

The Serre-Faltings-Livne method concerns the comparison of two 2-dimensional 2-adic Galois representations of the absolute Galois group *G _{K}* of a number field

It is known for almost fifty years that geodesic flows on compact manifolds with negative sectional curvature are chaotic: the time-*t* maps are Bernoulli for all *t≠0*. In a joint work with Ledrappier and Sarig we show that this also holds for geodesic flows on surfaces with nonpositive and non-identically zero curvature and, more generally, for "most"' three-dimensional Reeb flows. A main ingredient in the proof is the construction of a symbolic model for non-uniformly hyperbolic three-dimensional flows, developed in a joint work with Sarig. The talk will consist on a discussion of both works.

A conjecture of Yau states that infinitely many distinct minimal hypersurfaces exist inside any closed ambient Riemannian manifold. A result of Marques and Neves (2014) has confirmed this to be the case when the ambient manifold has positive Ricci curvature, moreover that each of these hypersurfaces comes with a natural bound on its index. Therefore it seems reasonable to try to classify minimal hypersurfaces with respect to index. We will give an overview of some recent results concerning the relationship between topology, geometry and the Morse index of minimal hypersurfaces in Riemannian manifolds.

We consider the problem of minimising the number of edges contained in triangles, among n-vertex graphs with a given number of edges. A natural example of a graph with few edges in triangles is formed by adding a clique to the smaller side of a complete bipartite graph. Füredi and Maleki proved an asymptotically tight lower bound on the number of edges in triangles among n-vertex graph with e edges. They conjectured that the minimisers belong to the described family of examples or are subgraphs of such graphs. We prove their conjecture for sufficiently large n, thus providing an exact formula for the minimum number of edges in triangles.

This is joint work with V. Gruslys.The Kronecker problem asks for an algorithmic understanding of the coefficients arising in the decomposition of the tensor product of two simple modules for the symmetric group.

We provide an algorithm for calculating Kronecker coefficients labelled by so-called “co-Pieri triples” of partitions. This, in some sense, solves half of the Kronecker problem.This is joint work with Maud De Visscher and John Enyang.We consider the model problem of a discrete elliptic equation with independent and identically distributed conductances, and establish a path-wise theory of fluctuations in stochastic homogenization. We identify a single quantity, which we call the corrected energy density of the corrector, that drives the fluctuations in stochastic homogenization in the following sense. On the one hand, when properly rescaled, this quantity satisfies a functional central limit theorem, and converges to a Gaussian white noise. On the other hand, the fluctuations of the corrector and the fluctuations of the solution of the stochastic PDE (that is, the solution of the discrete elliptic equation with random coefficients) are characterized at leading order by the fluctuations of this corrected energy density. As a consequence, when properly rescaled, the solution satisfies a functional central limit theorem for d≥2, and the corrector converges to (a variant of) the (whole-space massless) Gaussian free field for d>2. Compared to previous contributions, our approach, based on the corrected energy density, unravels the complete structure of fluctuations. It holds for d≥2, yields the first path-wise results, quantifies the CLT in Wasserstein distance, and only relies on arguments that extend to the continuum setting.

A projective variety X defined over the rational numbers is said to satisfy the Hasse principle if the presence of an adelic point on X guarantees the presence of a rational point. We give a programme for establishing Hasse principle for hypersurfaces of degree at least 4. The key ingredient is Kloosterman type extra averaging in conjunction with the van der Corput differencing applied to estimate the ''minor arc contribution'' in the Hardy-Littlewood circle method. We show the utility of this approach by improving upon current bounds in the quartic case. This is a joint work with Oscar Marmon.

A celebrated result of Brooks says that, for a wide class of Riemannian manifolds, the bottom of the spectrum of the Laplacian on a regular cover is equal to the bottom of the spectrum of the base if and only if the covering group is amenable. In the case where the base manifold is a quotient of a simply connected manifold with pinched negative curvatures by a convex co-compact group, we will give an analogous result for the critical exponents of this group and a normal subgroup. A notable feature in each case is that, while the quantities we consider depend on the geometry, whether or not we have equality only depends on a combinatorial property of the covering group.This is joint work with Richard Sharp.

Width questions concern covering a given group G with elements from a normal subset, using products of minimal length. For example can we write every group element as a product of two or perhaps three elements from a fixed conjugacy class? In particular we will look at what is known when we consider classes of involutions.

In this talk we study estimates from below for the *L*^{2} norm of the *s*-dimensional Riesz transform, with kernel *x*/|*x*|^{s+1} for *s*∈(*d*−1,*d*), of general Borel measures in **R**^{d}. These estimates allow to establish an equivalence between the capacity γ_{s} associated with the *s*-dimensional Riesz kernel and the capacity Ċ_{2/3(d−s),3/2} from non-linear potential theory associated to the Wolff potential.

A set *B* is an additive (resp. multiplicative) basis for a set *A* if any element *a* of *A* can be represented as a sum *b*_{1}+*b*_{2} (resp. product *b*_{1}*b*_{2}) with *b*_{1}, *b*_{2}∈*B*. An important question in arithmetic combinatorics is to determine the minimal possible size of a basis for a given set *A*.

One of the most famous conjectures in number theory is the Hardy-Littlewood conjecture, which gives an asymptotic for the number of integers *n* up to *X* such that for a given tuple of integers *a _{1}*,..,

I will describe a diffusion model for particles carried by a turbulent flow. When different particles are brought together by the flow they form a cluster that lives for a short time before breaking up. The question of interest is to describe the sizes of the pieces into which a cluster breaks. This is achieved by studying the behaviour of a certain multidimensional diffusion process and approximating its behaviour on two different scales.

Large scale fluctuations of a wide class of one dimensional microscopic interacting particle models with weak asymmetry and in non-equilibrium stationary states are expected to be described by the Kardar-Parisi-Zhang equation. This is the content of the weak universality conjecture for KPZ. I will explain how the notion of energy solution introduced by Jara and Goncalves and later improved by Jara and myself can be used to tackle a partial resolution of this conjecture. These results rely on an uniqueness result for energy solution recently established by Perkowski and myself.

We describe computations of two new "first" number fields in degree 8: those with smallest absolute discriminant and Galois groups PSL(2,7) and 2^3:GL(3,2). After presenting the results and what is novel in the technique, we look at the data from the perspective of arithmetic statistics.

Given some class of structures, and some monotone decreasing property P, the typical extremal problem asks for the largest structure with property P. The Erdös-Rothschild problem is an extension of this extremal problem, asking for the structure with the maximum number of r-colourings where every colour class has property P. While it first was asked in the context of graph theory, there has been much recent investigation of the Erdös-Rothschild problem for intersecting families of various kinds. In this talk we will see a unified proof for the three-colour case that covers the previous results and introduces some new ones. This method also provides far better (and sometimes optimal) bounds on the parameters of the families in question. Time permitting, we shall briefly discuss the multi-coloured setting as well. This is joint work with Dennis Clemens and Tuan Tran.

Starting from a skew polynomial ring, we define a class of unital nonassociative algebras introduced by Petit in 1966 (but largely ignored so far), and discuss their structure and applications to coding theory. For instance, these algebras can be employed to systematically build fast-decodable space-time block codes used for wireless digital data transmission, e.g. in mobile phones, laptops or portable TVs.

The Bernoulli convolution with parameter λ is the law of the random variable: ∑*X _{i}λ^{i}*, where

The ring of profinite integers, commonly known as 'Zet-hat', is a familiar acquaintance of researchers in various branches of algebra and number theory. The lecture provides an introduction to profinite numbers that can be enjoyed at all levels, and it will highlight their unconventional algebraic and analytic properties.

The topic of "Quantum ergodicity" deals with the (de)localization of eigenfunctions of Schrödinger operators, for classically chaotic systems. In this talk I will review the rigourous results about the (de)localization of laplacian eigenfunctions, on compact riemannian manifolds. One of the main tool is the relation between the ergodic properties of the geodesic flow and the large time behaviour of solutions of the Schrödinger equation. I will also talk about my work with Etienne Le Masson, where we studied the delocalization of eigenfunctions on large regular (discrete) graphs.

The Sine_{β} process is the bulk limit process of the Gaussian beta-ensembles. We show that this process can be obtained as the spectrum of a self-adjoint random differential operator. The result connects the Montgomery-Dyson conjecture about the Sine_{2} process and the non-trivial zeros of the Riemann zeta function, the Hilbert-Polya conjecture, and de Brange’s approach of possibly proving the Riemann hypothesis. Our proof relies on the Brownian carousel representation of the Sine_{β} process and a connection between hyperbolic carousels and first order differential operators acting on **R**^{2} valued functions.

Let *G=G(n,d)* denote the random *d*-regular graph on *n* vertices. A celebrated result by J. Friedman solves Alon's second eigenvalue conjecture saying that if *d* is fixed and *n* is large then *G* is close to be Ramanujan. Despite of significant effort, much less was known about the structure of the eigenvectors of *G*. We use a combination of graph limit theory and information theory to prove that every eigenvector of *G* (when normalized to have length equal to square root of *n*) has an entry distribution that is close to some Gaussian distribution in the weak topology. Our results also work in the more general setting of almost-eigenvectors. Joint work with A. Backhausz.

We consider the problem of finding metrics with constant Gaussian curvature on a compact, closed surface imposing a conical behaviour at a finite number of points. Differently from the classical uniformization problem, solutions may not exist depending on the topology of the surface and on the conical angles. We discuss methods, variational in nature, to derive general existence results from suitable improvements of the Moser-Trudinger inequality combined with Morse-theoretical arguments. These improvements are derived by carefully analysing the distribution of volume over the surface for a given conformal metric.

My lecture is meant as a survey on some of the most recent developments of the synthetic theory of Ricci lower bounds for metric measure spaces. After the illustration of some key recent results, I will focus in particular on the connections between the different points of view, namely the Lagrangian point of view based on optimal transport (after Lott-Villani and Sturm) and the Eulerian point of view (based on Gamma-calculus, the theory of semigroups and the Bakry-Emery contractivity estimates). This Eulerian-Lagrangian duality is not only conceptually important, but also fundamental for the proof of some results, as already the classical calculus shows. In particular I will illustrate the foundations of first and second order differentiable calculus in metric measure spaces, underlying most of these recent results.