Schedule of the lectures (Paris time, CET, UTC+2).
|Monday||Welcome address||Kohji Matsumuto||Catherine Goldstein||Jehanne Dousse||Yvan Saint-Aubin||James Parks||Yuri Lima|
|Tuesday||Amin Aminzadeh Gohari||Serge Cohen||Rachel Bawden||François Bergeron||Thomas Brüstle|
|Wednesday||Gaël Raoul||Antoine Chambert-Loir||Édouard Rousseau|
Natural Language Processing: Applications and Neural Modelling
Natural Language Processing (NLP) applications are a part of our daily modern lives: spam email detection, spelling and grammar correction, chatbots, machine translation, fake news detection, automatic journalism to name just a few. All these applications are designed to process natural language (i.e. produced by humans) either by analysing them, transforming them or by generating new text. In this introduction to NLP, I will take you through different NLP applications to give you an overview of the field. I will then present how neural networks can be applied to NLP, including applications to machine translation and language modelling. In this presentation I hope to give you a good overview of the domain and inspire you to learn more about NLP and its applications.
The study of the interplay between Catalan combinatorics and abstract algebra has interesting ties with many subjects including: Computer Science, Algebraic Geometry, Knot Theory, and Quantum Mechanics. On the combinatorial side, the story involves many classical structures including triangulation of polygons, binary trees, Dyck paths, partitions contained in a staircase, well-parenthesized expressions, two-row standard Young tableaux, and pattern avoiding permutations; to mention but a few. The enumeration of all of these objects involves Catalan numbers. Several interesting families of polynomials arise when one considers weighted enumerations of the combinatorial objects considered in this broad subject. As a matter of fact, since they arise in so many areas, several generalizations of these families of object have been introduced over the years. On top of this rich combinatorics, ties with abstract algebra also abound. Typically, one sees the Catalan numbers appear as the dimensions of interesting vector spaces. The purpose of this lecture is to give a small first taste of this vast subject, as well as some indications on where to find further material.
Persistence theory and quiver representations
Topological data analysis is using algebraic methods to identify which features of the data persist the longest (and thus are relevant). Algebraic persistence theory provides tools and algorithms to accomplish this goal using representation theory of quivers. We present the most successful instance of that theory, the representation theory of a linearly ordered quiver, which is completely understood.
The course and the exercises will present the underlying notions of quiver representation theory and homology necessary to understand and apply persistence theory.
Diophantine equations : from geometry to algebra… to geometry
I will retrace 4000 years of diophantine equations, focusing on a few landmarks of this ongoing mathematical quest that started from apparently innocuous geometric puzzles of Antiquity, was revolutionized by the invention of algebra during Renaissance, to be re-geometrized in the 20th century.
The exercise session will propose a (hopefully delicious) feast of number theoretical puzzles of all kinds.
Vincent Viallat Cohen-Addad
Approximation Algorithms for Data Mining and Machine Learning
Extracting and Learning information from datasets is nowadays central for most research areas, or for large companies infrastructures. Thus, designing algorithms that can do that automatically and efficiently is an important research areas. In this context, one of the main algorithmic goal is to compute clusters of similar data elements, or in other words, find a procedure that identifies groups of points that are much more similar to each other than to the rest of the data elements.
The goal of this talk is to illustrate the algorithm researchers journey in the above context:
(1) formalize a real-world problem into a model problem.
(2) design an algorithm and prove that it solves the model problem defined at (1), and
(3) check that the algorithm design at (2) indeed produces good solution for the real-world problem.
We will review the classic formalizations of some clustering problems, explain a state-of-the-art
algorithm and show how it performs on some machine learning datasets.
Counting integer partitions
A partition of a positive integer n is a non-increasing sequence of positive integers (called parts) whose sum is n. Despite being simple objects to define, partitions are surprisingly difficult to count. Indeed, no simple formula is known for the number of partitions of n. We will introduce two tools that allow us to work around this issue and prove interesting results on partitions: generating functions and bijections. Among other results, we will show Euler’s partition identity, which states that for all n, the number of partitions of n into odd parts equals the number of partitions of n into distinct parts.
Amin Aminzadeh Gohari
Measuring and communicating information
What is information and how can we measure it (from a mathematical perspective)? An answer to this question was given by Claude Shannon, a mathematician and communication engineer back in 1948. Shannon’s answer to this question laid the foundation of modern communication systems and started the digital revolution. I will discuss Shannon’s notions of information and uncertainty and cover some of their applications.
A Book with Seven Seals : C.F. Gauss’s Disquisitiones arithmeticae in the development of mathematics
What has been called ‘that great book with seven seals,’ C.F. Gauss’s Disquisitiones arithmeticae [Arithmetical Investigations] has served as a source of mathematical inspiration since its publication in 1801. Gauss (in his twenties at the time) introduced in it new concepts, new notations and new ways of working in mathematics, all of which have been widely used since then, from primality tests to the theory of algebraic equations, from rational approximations to geometrical problems. Even in our century, Manjul Bhargava’s thesis (Fields Medal 2014) was based on an extension of the so-called composition laws of the Disquisitiones. In the course, I shall survey this book and trace some of its effects on the development of mathematics.
An invitation to dynamical systems
Describing the mechanics of the universe is yet an open problem. Many attempts were made, and each of them contributed in its own way to a more precise description. One of them was proposed 125 years ago by the French mathematician Henri Poincaré, who suggested an approach less focused on the explicit solution (quantitative) and more focused on the geometric description (qualitative). This idea started a new area in Mathematics, called dynamical systems. In this talk, we will discuss the ideas that led to the creation of this new area, from the geocentric models to Poincaré’s proposal. And, to give a taste of the area, we will present the proof of a beautiful theorem about periodic points called Sharkovsky’s theorem.
An introduction to prime number theory
In this lecture I will give a short introduction to number theory, mainly on the theory of distribution of prime numbers. We begin with very elementary matters, then we introduce the notion of the zeta-function, discuss its basic properties, and finally we sketch how to prove the celebrated “prime number theorem’’ by using the zeta-function. On the first part of the lecture no prerequisite knowledge is necessary. To understand the zeta-function we will develop some infinite-series argument. In the final part of the lecture I will use differential and integral calculus.
Renormalize! Renormalize! Renormalize!
Every dynamical system describes a change of (some) space through time. Modern theory of dynamical systems doesn’t study such systems one by one (time consuming…) but rather by families.
Renormalization is a new dynamical system defined (and here comes a conceptual jump!) in the space of dynamical systems themselves, in other words, in the space of parameters (which is oftentimes of infinite dimension).
In certain situations, two lucky observations coexist. First, renormalization is simpler to understand than the initial dynamics. Second, some of the features of renormalization impact the behaviour of the underlying dynamics. Renormalization is a wonderful tool and a wonder in itself — finding it and working out its properties is hard work. This work is in the heart of research in dynamics today.
My talk and exercise session give an introduction to renormalization via two examples — rotations of the circle and unimodal maps of the interval.
An Introduction to Data Science: Song Popularity Prediction
As more and more data is produced every day from our interactions with what we watch, what we listen to and where we shop, the demand for people with skills in mathematics and statistics to analyze and explain this data increases. This is the role of a data scientist and in this talk we give an introduction to the discipline from a practical perspective. In the first part we give an overview of what it is like to work as a data scientist. In the second part, we’ll walk through how to build a machine learning model to predict a song’s popularity on Spotify using publicly available data sets.
Mathematical models in ecology
Mathematical models have always played an important role in ecology and in evolutionary biology, as illustrated by the work of Mendel or Fisher. In the last decades, these biological fields have grown considerably, driven by societal problems (invasive species, desertification, resistance to antibiotics, etc.). To tackle these challenges, new mathematical models have to be developed and analyzed. In this lecture we will discuss some recent progress made on deterministic and stochastic models for spatially structured populations. We will describe how these models can provide some insight on the interplay between the local adaptation of species and the effect of climate change.
The mathematics of secrets
In this talk, we will introduce cryptography, some of its uses in our world nowadays, and its link with mathematics. Cryptography is the study of the techniques used to secure communications, it is also called the “science of secret writing”. The basic question is “How can I send a message to someone so that only this person can read the message?”. It is a very old field of study, and it is still widely used everyday. In fact, these lines were probably brought to you through a secured communication channel, using cryptography. A lot of cryptographic protocols are based on mathematical problems, and we will explain how this works.
In the exercise session, we will introduce and analyze some famous cryptographic protocols in order to manipulate and understand them.
On a woodcut by M.C. Escher
Certain mathematical concepts or objects were discovered or used intuitively a long time before they were given rigorous mathematical definitions. Various reasons explain such a long genesis: sometimes the concept did not have any obvious mathematical use, sometimes mathematicians were stubborn in accepting the existence of the given object. A woodcut by Escher is tied to two mathematical objects whose emergence is spread over a few millennia and it provides an intriguing introduction to them.