# Category Archives: Math

Today I gave my talk in the Seminar on Irrationality and Transcendence (Number Theory) offered by Prof. Sander Zwegers (University of Cologne). In my talk I defined E-Functions, proved some properties, and derived the Lindemann-Weierstraß Theorem from Shidlovskii's Theorem. You are welcome to download the German presentation slides.

# Jacobi Triple Product Identity

Today I gave my talk in the Proseminar on Partitions (Number Theory) offered by Prof. Kathrin Bringmann (University of Cologne). In my talk I gave a proof of the Jacobi Triple Product Identity and also proved some beautiful corollaries, e.g. Euler's Pentagonal Number Theorem.

Using the following definition of the Pochhammer Symbol

$(a;q)_n := \prod_{j=0}^{n-1}(1-aq^j)$

the Jacobi Triple Product Identity becomes

$\sum_{n\in \mathbb{Z}}z^nq^{n^2}=(q^2,q^2)_{\infty}(-zq,q^2)_{\infty}(-z^{-1}q,q^2)_{\infty}$ for $z\neq 0$

Using this equation, the Pentagonal Number Theorem

$(q;q)_{\infty}=\sum_{n\in \mathbb{Z}}(-1)^nq^{n(3n-1)/2}$

can be obtained by replacing $q$ by $q^{3/2}$.

# TopMath Spring School 2014

Last week I have been at TopMath Spring School in Oberschönenfeld offered by Prof. Bernd Schmidt (University of Augsburg). The topic was "Convex Sets and their Applications" and my talk was on Polytopes. One can download the handout right here: Polytopes Handout.