Seminar: Basic Notions, Summer Term 2017
Organisers: João Pedro Ramos, Joris Roos, Tim Seynnaeve, Arik Wilbert, 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 email address is basicnotions(at)hcm.unibonn.de.
For receiving updates and information on upcoming talks, you can also subscribe to our Facebook page.
The seminar will take place Wednesdays 1416 in room 1.007.
Previous instances of this seminar:
Winter Term 2016/17, Summer Term 2016,
Winter Term 2015/16, Summer Term 2015,
Winter Term 2014/15, Summer Term 2014,
Winter Term 2013/14.
Date  Speaker  Topic 

26.04.2017  Deniz Kus  What is representation theory? An introduction via the symmetric group (Abstract) 
03.05.2017  Siad Daboul  The five color theorem (Abstract) 
10.05.2017  Andreas Gerhardus  Counting rational curves with quantum gauge theories (Abstract) 
17.05.2017  
24.05.2017  
31.05.2017  Thomas Poguntke  Euclidean lattices and ideal class groups (Abstract) 
07.06.2017  
14.06.2017  Dimitrije Cicmilović  Symplectic geometry and Hamiltonian dynamics (Abstract) 
21.06.2017  Lory Aintablian  Geometrical constructions using only a ruler (Abstract) 
28.06.2017  Philipp Schlicht  Large infinities in mathematics (Abstract) 
05.07.2017  Tashi Walde  Infinitycategories (Abstract) 
12.07.2017  James Moody  Really random reals (Abstract) 

What is representation theory? An introduction via the symmetric
group

Abstract. Representation theory is a branch of mathematics that reduces
problems in abstract algebra to problems in linear algebra, a subject
that is well understood. In the first part of the talk we will present
the basic concepts about the representation theory of finite groups:
indecomposable objects, simple objects, module homomorphisms, Schur's
Lemma and Maschke's theorem. The running example will be the symmetric
group. In the second part of the talk we discuss several applications
and answer the question why representation theory is important.

The five color theorem

Abstract. The four color theorem states that the regions of a planar map can be colored with at most four colors such that all neighboring regions are colored differently. In the first part of the talk I will give a brief introduction to graph theory and finally prove the weaker statement where we allow five colors instead of four. I will then discuss some extensions and provide examples for real world applications of coloring problems that arise in the design of computer chips.

Counting rational curves with quantum gauge theories

Abstract.
Rational curves in \(\mathbb{C}P^n\) are the images of homogeneous polynomial maps \(\phi: \mathbb{C}P \to \mathbb{C}P^n\). It is a classical problem of enumerative geometry to count the number of rational curves that are contained in a given hypersurface \(X \subset \mathbb{C}P^n\). In this talk I will address this problem for \(X\) being the vanishing locus of a degree \(5\) polynomial in \(\mathbb{C}P^4\), the well studied quintic CalabiYau threefold, by using modern methods of mathematical physics. Here, the enumerative problem amounts to counting instantons in a suitable twodimensional quantum gauge theory. I will demonstrate calculations for the quintic and comment on the generalization to more complicated cases.

Euclidean lattices and ideal class groups

Abstract.
We introduce some basic concepts of algebraic number theory, with an emphasis on results surrounding the ideal class group of the ring of integers in a number field. In particular, we indicate how Minkowski's theorem on lattice points in compact convex symmetric regions in Euclidean space can be used to deduce its finiteness, and lends itself to compute it in examples.

Symplectic geometry and Hamiltonian dynamics

Abstract.
In this lecture, I will try to present the symplectic geometry that
initially rose in classical mechanics. Symplectic structure on a manifold
is given by a closed nondegenerate 2form, which gives some interesting
obstructions for what such manifold can be. For example, every symplectic
manifold is locally isomorphic to \(\mathbb{C}^n\), thus locally we cannot distinguish
it from Euclidean space. This is the reason the term symplectic topology
is
used instead of geometry, as the global topological obstructions play a
significant role. In the first part of the lecture I will provide some
insight into the development of symplectic geometry as a separate field,
to present some differences from Riemannian geometry and will discuss the
nonsqueezing theorem which was a cornerstone in research in symplectic
geometry. Subsequently, I will discuss some invariants which are called
symplectic capacities and whose existence is equivalent to the
nonsqueezing theorem.
Lastly, I will come back to the beginning and discuss some open problems
with nonsqueezing theorem in infinite dimensional Hilbert space and how
that relates to Hamiltonian partial differential equations.

Geometrical constructions using only a ruler

Abstract. From the times of ancient Greece, mathematicians attempted geometrical constructions in the
Euclidean plane using a compass and straight edge only. In 1822 Jean Victor Poncelet
conjectured and in 1833 Jakob Steiner proved that every point constructible with a compass
and a straight edge can be constructed using a straight edge alone given a fixed circle and its
center in the plane. Using geometrical tools and mainly following Steiner's method, we will
prove this latter statement. The talk will have an interactive and visual nature.

Large infinities in mathematics

Abstract. Beyond the sets of natural numbers and real numbers, there are many much larger sets that appear in mathematics and in set theory. Which possible infinities are there? I will introduce some large infinities such as the inaccessible cardinals, which cannot be approached from below. I will show that the existence of very large infinities cannot be proved; this is closely related to Gödel's incompleteness theorems.

Infinitycategories

Abstract. There seems to be an interesting comparison between paths in a topological space and arrows in a category. In a category we can (uniquely!) compose arrows, composition is associative, and not every arrow is required to be invertible. In a topological space each path has an inverse (just walk the same path backwards) but the composition of paths is only welldefined and associative up to homotopy (informally: deforming the path over time).
Infinitycategories can be seen as the least common generalization of categories and topological spaces: composition is as flexible as in spaces but we drop the requirement that every path/arrow be invertible. This talk will be rather informal; no particular knowledge about topology or category theory is needed beyond the very basic definitions.

Really random reals

Abstract.
There are a few intuitive ideas about what makes an infinite binary sequence
"random". If we treat the digits of a sequence as coin flips, we expect that no
computable way of placing fair bets on these coin flips could yield unbounded
profit. We also think that the information in a random binary sequence should
be incompressible (i.e., a random sequence is its own shortest description). We'll
explore some putative definitions of randomness, and show that some (but not
all) of them are equivalent. Then we'll explore what happens to the notion of
randomness when we use different measures on the underlying space.
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