Summer School on Geometry19-30 August 2019

Course leader: Dr. Johan van de Leur Email: j.w.vandeleur@uu.nl

[More information and apply: www.utrechtsummerschool.nl/courses/science/geometry]

The summer school will take place in room 611 at the 6th floor of the Hans Freudenthal Building, Budapestlaan 6, Utrecht. Find information about travel directions at the end of the programme.

The Mathematical Institute will provide lunch on Monday August 19 and Friday August 30 and dinner on Monday August 19 and Tuesday August 27. At all other times you will be expected to bring/buy

your own lunch.

Week 1

Saturday and Sunday, August 17 and 18

Key pick-up You will find the exact key pick-up location in the pre-departure information, which becomes available after you have paid the course fee.

Monday, August 19 Dr. Johan van de Leur: Grassmannians and Representation Theory of GLn

We introduce the notion of Grassmannians, relate this to the exterior tensor algebra and show that there is a correspondence with a certain GLn orbit.

10.00 – Registration and tea/coffee 10.30 – Lecture 11.45 – Exercise class 12.30 – Lunch at the Mathematics Library 13.30 – Lecture 15.30 – Exercise class 17.00 – Walk along the river Kromme Rijn to De Moestuin 18.00 – Dinner at De Moestuin, Laan van Maarschalkerweerd 2, Utrecht

Last updated: July 5.

Tuesday, August 20 Dr. Jaap van Oosten: Model Theory of Fields

This mini-course will give you a first glimpse of a body of mathematical research where logic and algebraic geometry meet. After rehearsing the basic tools of Logic (the notions of a language, a sentence and a theory), including the Compactness Theorem, we start by mentioning Quantifier Elimination for algebraically closed fields (Tarski, 1948) and have a look at some surprising applications it has. Then, we have a look at so-called "types". Types are a bit like ideals: they represent elements of specific extensions of the structure. For any structure, its set of types is naturally a topological space. An important property that structures can satisfy is ω-stability: this property guarantees that the cardinality of the set of types is under control. Finally, we shall look at real closed fields (fields in which every odd-degree nonzero polynomial has a root) and discuss the important notion of o-minimality. In the exercise classes, there will be ample opportunity to play with all abstract notions presented.

09.00 – Lecture 11.00 – Exercise class 12.30 – Lunch break 13.30 – Lecture 15.30 till 17.00 – Exercise class

Wednesday, August 21 Dr. Martijn Kool: Projective Duality

Projective duality is a principle which translates results in projective space to (different looking) results in the dual projective space. In these lectures, I will focus on projective duality for the projective plane and the dual of a plane algebraic curve C, which is defined as the (closure of) the collection of its tangent lines. We will see that (over the complex numbers) the dual of the dual is the original curve (reflexivity theorem). After discussing connections to the Legendre transform and caustic curves, I will focus on an interesting set of formulae due to Clebsch-Plücker, which relate topological, algebraic, and enumerative information of C and its dual C*. As an application, we will see an interesting relation between the 28 bitangents of a general quartic curve and the 27 lines on a general cubic surface. Instead of focusing on general proofs, I will stress special cases and calculation. I will work over the complex numbers. No prior knowledge is assumed (other than basic algebra), e.g. projective spaces, algebraic curves, singularities, etc. are all introduced.

09.00 – Lecture 11.00 – Exercise class 12.30 – Lunch break 13.30 – Lecture 15.30 till 17.00 – Exercise class

Thursday, August 22

Friday, August 23 Dr. Adrien Sauvaget: Around Modular Forms

Modular form are analytic functions that admits certain good behavior under the action of the SL(2, Z). There are deeply connected to the geometry of the moduli space of elliptic curves and one can encounter them in many areas of modern mathematics: theoretical physics, representation theory, number theory. The purpose is to explain the definition of modular form and to construct the first two of them (the Eisenstein series) from different point of views.

09.00 – Lecture 11.00 – Exercise class 12.30 – Lunch break 13.30 – Lecture 15.30 till 17.00 – Exercise class

Prof. Frans Oort: Elliptic Curves

In this course we discuss three applications, in geometry and in number theory, of the theory of elliptic curves. We will give definitions and state properties of elliptic curves, some without proofs (this is an introduction to the theory of elliptic curves, but not a complete course in this topic). We give examples and exercises. In the first application we need the dual of a plane conic, a connection with the course by Martijn Kool. In some cases we need reduction modulo p, a connection with the course by Stefano Marseglia. We expect active participation of the audience in exercises, examples and questions.

09.00 – Lecture 11.00 – Exercise class 12.30 – Lunch break 13.30 – Lecture 15.30 till 17.00 – Exercise class

Week 2Saturday and Sunday, August 24 and 25

For the social programme organised by the university for all the summer school students, see:https://www.utrechtsummerschool.nl/social-programme

Monday, August 26 Prof. Frits Beukers: Dessins d'enfant

The field of rational functions (quotients of polynomials) with complex coefficients is uncountable. However, the subset of those rational functions F such that F(z0) ∈{0,1,∞ } for all critical points z0 (i.e. F'(z0) = 0) turns out to be countable, up to some isomorphisms. They form the set of so-called Belyi maps, which can be enumerated by graphs in the plane that are reminiscent of children's drawings. Hence the title. The original idea was conceived by A. Grothendieck in the 1980s. In full generality he proved that the set of algebraic curves defined over a number field can be enumerated using such 'dessins d'enfant'. During this day we will compute examples of some Belyi maps, learn some applications and formulate Grothendieck's general result.

09.00 – Lecture 11.00 – Exercise class 12.30 – Lunch break 13.30 – Lecture 15.30 till 17.00 – Exercise class

Tuesday, August 27 Dr. Gijs Heuts/Dr. Lennart Meier: Fixed Point Theorems and Their Applications

A fixed point theorem states that a certain kind of continuous map will automatically have a fixed point, i.e. a point mapped to itself. The star of the lectures will be Brouwer's fixed point theorem that says that every continuous self-map of an n-dimensional ball will have a fixed point. We will prove this theorem and provide several applications. In particular, we will show how John Nash applied this theorem to show the existence of equilibria in game theory.

09.00 – Lecture 11.00 – Exercise class 12.30 – Lunch break 13.30 – Lecture 15.30 till 17.00 – Exercise class18.00 – Dinner in the city centre of Utrecht

Wednesday, August 28 Dr. Stefano Marseglia: Elliptic Curves over Finite Fields

In this course we will build on the material presented by Frans Oort, with focus on elliptic curves defined over finite fields and some of their applications to public key cryptography. More precisely, we will discuss what is the mathematics behind the Elliptic Curve Discrete Logarithm Problem (ECDLP) and what are the advantages with respect to the classical DLP.

09.30 – Lecture 11.00 – Exercise class 12.30 – Lunch break 13.30 – Lecture 15.30 till 17.00 – Exercise class

Thursday, August 29 Dr. Jens Forsgaard: Hilbert's 16th Problem

We introduce Hilbert's 16th problem and learn how to construct real algebraic curves with controlled topology using triangulations and patchworking.

09.00 – Lecture 11.00 – Exercise class 12.30 – Lunch break 13.30 – Lecture 15.30 till 17.00 – Exercise class

Friday, August 30 Dr. Shuntaro Yamagishi: Twin Prime Conjecture

Twin prime conjecture states that there are infinitely many pairs of prime numbers which differ by 2. This conjecture has not been proved yet, but recently there has been significant progress towards it. In this course, I will give an overview of James Maynard's work, where he proved that there are infinitely many pairs of primes which differ by h for some h ≤ 600.

09.00 – Lecture 11.00 – Exercise class 12.00 – Lunch at the Mathematics Library

The Summer School on Geometry will take place in the Hans Freudenthal Building, Budapestlaan 6 in Utrecht. You can easily bike from the centre of the city to the university. On the website of Utrecht Summer School you can find information about renting bikes or using the public transport in Utrecht:https://www.utrechtsummerschool.nl/utrecht/getting-around A useful website for public transport is: http://9292.nl/en/