Lectures

Number systems and Tilings (S. Akiyama)

Number systems. Number system is the way to express numbers by finite alphabets. We start with basics of abstract number system and their classification. Then we discuss associated fractal tilings. One can view number system as a pair of an expanding matrix and a representatives in a lattice. The goal of the 1-st lecture is to obtain a dynamical undestanding of number systems. Boundary of tiles, Duality of tilings and point sets. We then introduce an adding machine of number system to investigate the fractal boundary of the tile. Tiling and Delone point sets are dual objects. We discuss self-affine tilings and its representative point sets. This gives a natural and easy understanding of inflation rules. The 2nd lecture makes you familiar to techniques used in tiling theory. Almost periodicity of tilings.Tiling is almost periodic if it is extremely close to its translations. Not all tiling are almost periodic and it is a subtle question to determine this property. In the 3rd lecture, we discuss how to check whether a given tiling is almost periodic.

Optimization and aperiodicity (Th. Fernique)

Local rules and aperiodicity. This first lecture is an introduction to the theory of local rules and their connections with various domains : computability theory, combinatorics, symbolic dynamics and condensed material theory. Basic notions (aperiodicity, calculability, tilings, subshifts of finite type) shall be defined and explained with simple examples. Beenker tilings. Beenker tilings are tilings by two tiles, a square and a 45° rhombus, with simple restrictions on the way two neighboor tiles can be adjacent (local rules). They can also be defined as digitization of a one-parameter family of two-dimensional planes in the four-dimensional real space. Among them, those which maximize the proportion of rhombus tiles turn out to be aperiodic. This is one of the few and most elegant example of connection between aperiodicity, tilings and optimization. The aim of this lecture is to present in details this example. Packing by discs with different sizes.It is well known (although not straighforward) that the most efficient way to pack identical discs on the plane is to put their center on the triangular grid. What if we have two different sizes of circles? Or more? Or spheres instead of discs? We shall review existent results on this topics, provide a connection with tilings by triangles (or tetrahedra) satisfying some optimization constraint and examine whether aperiodicity can be achieved in this way. This lectures will be an introduction to numerous open questions in this field, the answer of which should be extremely interesting in condensed matter theory (packing of different atoms in alloys).

Variational inequalities, complementary problems, equilibrium (M. Fukushima)

Variational Inequalities and Related Problems: Basic Theory: definition of variational inequalities and complementarity problems; existence and uniqueness of solution. Algorithms for Variational Inequalities and Complementarity Problems: iterative methods such as projection methods and Newton's method; reformulation methods such as semismooth and smoothing Newton methods; splitting methods including alternating direction method of multipliers. Applications of Variational Inequalities: traffic equilibrium problem, Nash equilibrium problem (NEP); mathematical programs with equilibrium constraints (MPEC); equilibrium problems with equilibrium constraints (EPEC); multi-leader-follower games.Extensions of variational inequalities: generalized Nash equilibrium problem (GNEP) and quasi variational inequalities (QVI); stochastic variational inequalities and stochastic complementarity problem.

Extremal repetitions and no-repetitions in words (D. Jamet)

Combinatorics on words, i.e., the study of finite sequences (called words) taking their values in a finite set (called alphabet) is dated back to 1906, when Axel Thue published his now famous papers. A. Thue provided a method to construct an infinite sequence on three symbols with no squares (i.e., without two consecutive identical blocks) and an infinite sequence on two symbols with no cubes (i.e., without three consecutive identical blocks) in it. Thue’s sequence on two symbols is also described in a number-theoretic paper by Prouhet addressing what is now known as the Prouhet-Tarry-Escott problem. Recently, several authors have investigated different kind of repetitions such as abelian k-powers (k consecutive blocks, identical up to permutations) and additive k-powers (k consecutive blocks, with same size and same sum). One knows that 3 letters are necessary and sufficient to avoid abelian cubes in a infinite sequences while one needs 4 letters to construct an infinite sequence without abelian squares. M. Rao showed how to construct an infinite sequence without additive cubes. The question of avoiding additive squares over a finite alphabet is still open.Given a finite alphabet and an integer k, two natural questions are :

1. What is the smallest number of k-(pure/abelian/additive) powers that an infinite sequence must contain ?
2. What is the highest number of k-(pure/abelian/additive) powers that a string (finite word) can contain ?

The main goal of the present lecture is to give an overview of the most recent results concerning these two last questions which give a prominent position to algorithmic proofs.

Linear and Nonlinear Optimization (E. Nurminskiy and N. Shamray)

Introduction to linear optimization : examples of linear problems (continuous and discrete) and modeling techniques; geometry of linear programming; duality theory; sensitivity analysis. Methods and Complexity: simplex method; ellipsoid method; interior point method; projection method; large scale optimization and decomposition.The art in linear optimization: modeling languages and software.

Quadratic programming: problem formulation; Lagrangian duality; active set and others finite methods; large-scale quadratic optimization; projection problems. Nonlinear programming: nonlinear optimality conditions; regularity and sensitivity theory; gradient methods (classic and modern); Newton and Newton-like methods. Modern methods: conjugate gradient family; interior point methods; convex analysis; subgradient algorithms; bundle and epi-projection methods.Sharp optimization.

Some current challenges in packing theory (C. Zong)

Packing regular tetrahedra. The regular tetrahedron is the simplest Platonic solid. Nevertheless, in studying its packing properties several renowned scholars made mistakes, and many questions about it remain unsolved. Currently no one knows the density of its densest packings, the density of its densest translative packings, or the exact value of its congruent kissing number. In this lecture, we recount historical developments on packing regular tetrahedra, report new results on its translative packing density and congruent kissing number, and formulate several unsolved problems. Random packing of spheres. Packings of identical objects have fascinated both scientists and laymen alike for centuries, in particular the sphere packings and the packings of identical regular tetrahedra. Mathematicians have tried for centuries to determine the densest packings; Crystallographers and chemists have been fascinated by the lattice packings for centuries as well. On the other hand, physicists, geologists, material scientists and engineers have been challenged by the mysterious random packings for decades. Experiments have shown the existence of the dense random sphere packings and the loose random sphere packings for more than half a century. However, rigorous definitions for them are still missing. This lecture will review the random solid packings and try to introduce a mathematical treatment. Tilings of the plane by pentagons and other convex sets. Everybody knows that identical regular triangles or squares can tile the whole plane. Many people know that identical regular hexagons can tile the plane properly as well. In fact, even the bees know and use this fact! Is there any other convex domain which can tile the Euclidean plane? Of course, there is a long list of them! To find the list and to show the completeness of the list is a unique drama in mathematics, which has lasted for more than one century and the completeness of the list has been mistakenly announced not only once! Up to now, the list consists of triangles, quadrilaterals, fifteen types of pentagons, and three types of hexagons. In 2017, Michaël Rao announced a computer proof for the completeness of the list. Meanwhile, Qi Yang and Chuanming Zong made a series of unexpected discoveries in multiple tilings in the Euclidean plane. For examples, besides parallelograms and centrally symmetric hexagons, there is no other convex domain which can form any two-, three- or four-fold translative tiling in the plane. However, there are two types of octagons and one type of decagons which can form nontrivial five-fold translative tilings. Furthermore, a convex domain can form a five-fold translative tiling of the plane if and only if it can form a five-fold lattice tiling. This lecture tells the stories of these discoveries.