# 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.

For receiving updates and information on upcoming talks, you can also subscribe to our Facebook page.

The seminar will take place Wednesdays 14-16 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 DaboulThe five color theorem (Abstract)
10.05.2017 Andreas GerhardusCounting rational curves with quantum gauge theories (Abstract)
17.05.2017
24.05.2017
No talk.
31.05.2017 Thomas PoguntkeEuclidean lattices and ideal class groups (Abstract)
07.06.2017
Pentecost Holiday
14.06.2017 Dimitrije CicmilovićSymplectic geometry and Hamiltonian dynamics (Abstract)
21.06.2017 Lory AintablianGeometrical constructions using only a ruler (Abstract)
28.06.2017 Philipp SchlichtLarge infinities in mathematics (Abstract)
05.07.2017 Tashi WaldeInfinity-categories (Abstract)
12.07.2017 James MoodyReally random reals (Abstract)

• April 26, 2017: Deniz Kus

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.

• May 3, 2017: Siad Daboul

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.

• May 10, 2017: Andreas Gerhardus

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 Calabi-Yau threefold, by using modern methods of mathematical physics. Here, the enumerative problem amounts to counting instantons in a suitable two-dimensional quantum gauge theory. I will demonstrate calculations for the quintic and comment on the generalization to more complicated cases.

• May 31, 2017: Thomas Poguntke

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.

• June 14, 2017: Dimitrije Cicmilović

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 non-degenerate 2-form, 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 non-squeezing 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 non-squeezing theorem. Lastly, I will come back to the beginning and discuss some open problems with non-squeezing theorem in infinite dimensional Hilbert space and how that relates to Hamiltonian partial differential equations.

• June 21, 2017: Lory Aintablian

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.

• June 28, 2017: Philipp Schlicht

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.

• July 5, 2017: Tashi Walde

Infinity-categories

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 well-defined and associative up to homotopy (informally: deforming the path over time). Infinity-categories 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.

• July 12, 2017: James Moody (UC Berkeley)

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.