Seminar: Basic Notions, Winter Term 2016/17
Organisers: Regula Krapf, Néstor León Delgado, Joris Roos, Catharina Stroppel
This seminar is organised by students as a BIGS event. The goal is to present topics from all areas of mathematics in an elementary and informal way. The talks should be accessible to a general mathematical audience.
Everybody (students, postdocs, faculty, guests) is welcome to attend.
Coffee, tea and cookies will be served afterwards.
If you would like to give a talk please contact us. Our e-mail address is basicnotions(at)hcm.uni-bonn.de.
For receiving updates and information on upcoming talks, you can also subscribe to our Facebook page.
The seminar will take place Tuesdays 14-16 in room N0.008.
Previous instances of this seminar:
Summer Term 2016, Winter Term 2015/16,
Summer Term 2015, Winter Term 2014/15,
Summer Term 2014, Winter Term 2013/14.
|25.10.2016||Peter Holy||What is a surreal number? (Abstract)|
|08.11.2016||Federico Zerbini||Elliptic curves and modular forms (Abstract)|
|15.11.2016||Elba García-Failde||An overview of the topological recursion (Abstract)|
|22.11.2016||Erick Knight||Mathematical Origami: Foldable Numbers and Rigid Origami (Abstract)|
|29.11.2016||Achim Krause||Spectral Sequences (Abstract)|
|13.12.2016||Jack Davies||An Application of Algebraic Topology to algebra and topology (Abstract)|
|10.01.2017||Prof. Joel Hamkins (CUNY)||Transfinite game values in infinite chess, including new progress (Abstract) (Speaker's webpage)|
|17.01.2017||Lothar Sebastian Krapp (Konstanz)||Non-archimedean Exponential Fields (Abstract)|
|24.01.2017||Christian Berghoff||Cryptography and elliptic curves (Abstract)|
What is a surreal number?
Abstract. I will introduce the number field of the surreal numbers, that contains
both the real numbers and the transfinite numbers, and also contains
infinitesimals, with the operations of addition and multiplication. While
mostly omitting formal definitions, I will present two approaches towards
a presentation of this field that should provide you with a good picture
of this curious and interesting joint generalization of two of the most
basic objects in mathematics.
Elliptic curves and modular forms
Abstract. Elliptic curves are smooth projective algebraic curves of genus 1 (with a marked point). In this talk we mainly want to consider their complex points, in which case we can think of them as complex tori of dimension 1.
The study of their moduli space leads to the definition of a very important class of functions on the complex upper half plane, called modular forms.
These functions, or suitable generalizations, find beautiful applications in a huge range of math and physics subjects, from algebraic number theory to quantum black holes, from representation theory to diophantine approximation, from enumerative geometry and mirror symmetry to combinatorics and then back to the deepest arithmetic properties of elliptic curves, just to cite a few.
In this talk I want to define (complex) elliptic curves, consider their moduli space, define modular forms and give (or most probably sketch) as many applications as I can.
- An overview of the topological recursion
- Abstract. In this talk I will introduce an exciting topic which touches and relates many different fields such as combinatorics, hyperbolic geometry, Gromov-Witten theory, integrable systems, matrix models, knot theory and string theory. The common tool keeping together all these pieces is a recursive procedure which associates a doubly indexed family of differential forms to some initial data called spectral curve using basically residue computations on a Riemann surface. This was discovered in 2007 by B. Eynard and N. Oration while working on statistical mechanics and random matrix theory. We will see why this recursion is called “topological” and some of its many interesting properties. I intend to combine two styles: on the one hand, introducing all the necessary background and studying an elementary example in detail, and on the other hand, giving some informal ideas about why this is worth considering and is the starting or gluing point of an active field of research.
- Mathematical Origami: Foldable Numbers and Rigid Origami
- Abstract. This talk will be about two different aspects of mathematical origami. The first half of the the talk will be about the foldable numbers, which are the numbers that can be constructed by ``doing origami.'' Origami is more powerful than just compass and straightedge, as one can e.g. trisect an angle using origami. The second half of the talk will be about one model of what is happening when someone ``does origami.'' Rigid origami assumes that the paper remains perfectly flat throughout the entire folding process. This has applications to various aspects in engineering, but the main focus will be on what can not be done with this model.
- Spectral Sequences
- Abstract. Spectral sequences are one of the main computational tools in Algebraic Topology, Homological Algebra, Algebraic Geometry, basically any field where long exact sequences appear. Despite their usefulness, they are somewhat feared due to their technical nature.
In this talk, we first want to introduce them and take a closer look at spectral sequences in an algebraic context, where we can really "see what's going on". In the second half, we will discuss a number of examples that appear e.g. in Topology.
- An Application of Algebraic Topology to algebra and topology
- Abstract. In 1931 Heinz Hopf constructed the first interesting (because it's not null-homotopic) map between spheres of different dimensions. As mathematicians tried to generalise this map they came across an obstruction
called the Hopf Invariant one problem. This problem is famous because it is equivalent to the existence of real division algebras, the triviality of tangent bundles on spheres, and the survival of elements in the Adams spectral sequence. Frank Adams originally provided an answer to the Hopf Invariant one problem in 1960 using some complicated algebraic topology, but 6 years later, a much shorter proof was discovered by Adams and Michael Atiyah. This is the proof I will deliver, since it is only about the size of a blackboard, once we have a few definitions and theorems about vector bundles and K-theory.
Abstract. I shall give a general introduction to the theory of infinite games, using infinite chess - chess played on an infinite edgeless chessboard as a central example. Since chess, when won, is won at a finite stage of play, infinite chess is an example of what is known technically as an open game, and such games admit the theory of transfinite ordinal game values. I shall exhibit several interesting positions in infinite chess with very high transfinite game values. The precise value of the omega one of chess is an open mathematical question. This talk will include some of the latest progress, which includes a position with game value ω4.
Non-archimedean Exponential Fields
Abstract. In the first part of my talk, I will give an introduction to ordered exponential fields. The most prominent example of an exponential field is the ordered field of real numbers with its standard exponential function exp. I will explain how the notion of an exponential function can be generalised for ordered fields other than the real numbers. In the second part, I will focus on the class of non-archimedean exponential fields. These fields also contain elements which are infinitely large or infinitesimal.
Cryptography and elliptic curves
Abstract. This talk will first provide a brief overview over various cryptographic algorithms. Starting from historical permutation-based algorithms we will turn our attention to methods stemming from number-theoretical results, which are widely used today, e. g. RSA, ElGamal. We will especially consider elliptic curve cryptography, presenting some of the attacks known so far. In order to find curves resisting these attacks, it is necessary to have an efficient algorithm for counting the number of points on an elliptic curve over a finite field. We will treat this problem in detail and see how number theory comes into play.