Abstract: We often encounter proofs by induction. But how can we prove induction itself? How can we prove without induction? Can we induce over real numbers? In this lecture the idea of induction on natural numbers will be generalised to tuples, graphs, and even real numbers, and even proofs which usually seem completely different. We will see how these tricks can help us with calculus, discrete math, graph theory, and beyond.

Abstract: We mathematicians like to work with numbers. One could even say that numbers are our native language. While quantum mechanics often deals with incredibly small numbers, astronomers work with extraordinarily large ones. In this talk, however, we will dive into the world of absurdly HUGE numbers.

We will humbly begin by introducing the well-known Graham’s number before moving on to the TREE sequence. Along the way, we will make a few remarks on ordinal numbers, Kruskal’s tree theorem, and the construction of the fast-growing hierarchy of functions. We will then attempt to place Graham’s number and the TREE sequence within this hierarchy.

Finally, we will ask whether it is even possible for the human mind to comprehend numbers of such unimaginable magnitude.

Disclaimer: This talk belongs exclusively to the special section of mathematical obscurities.

Abstract: Why does the Kepler problem have more conserved quantities than it seems to need? A forgotten 19th-century picture, the hodograph, leads to the Laplace–Runge–Lenz vector, an unexpected four-dimensional symmetry, and Pauli's derivation of the hydrogen spectrum without solving any differential equation. Closed orbits, degenerate energies, and a sphere in momentum space all turn out to be the same fact.

Abstract: It is a jokingly said that linear algebra is the only effective tool for calculations in mathematics. I will present some very basic but useful concepts which are usually omitted from first courses: characterizations of subspaces, annihilators and linear relations, and explain how they can be applied.

Abstract: The properties of Manin triples and Drinfeld doubles will be discussed, including their definitions, structures, equivalence, and classifications in low dimensions. Moreover, field-theoretic sigma models will be introduced, along with their symmetries and their significance for string theory. Finally, the connection between Manin triples and the duality and plurality of string theories will be outlined.

Abstract: The aim of this seminar is to convince the dear participant that Hopf algebras are crucial in our day-to-day lives. This will be achieved by discussing examples, reaching from the forest itself to Feynman diagrams and quantum field theory.

Abstract: We all have some preferences, therefore it would make sense for us to find something that we like the most. Is that always the case though? How can we order things in mathematics? Can we order complex numbers? Vectors? Sets? What about functions?

Order theory gives us practical tools, how to navigate such situations. We will demonstrate that even if it's rarely mentioned, order structures can be found in perhaps every subject of the math department and even in real life.

Abstract: How can causality be described mathematically, or into which mathematical structures can we encode the fact that a cause must precede its effect? Using the examples of the spacetimes of Newtonian mechanics, special relativity, and general relativity, we will see how causal structure becomes progressively more intricate.

Abstract: Poisson geometry is a well-established branch of mathematics that originates in the mathematical formulation of classical mechanics introduced by Sir William Rowan Hamilton. Over the past few decades, the field has expanded significantly and become a rich theory of its own. In this lecture, I will provide an introductory overview of Poisson geometry. In particular, I will show how we can use the theory to “draw” finite-dimensional Lie algebras or to describe the dynamics of populations of n interacting biological species within an ecosystem.

Abstract: Linear algebra together with statistics give us a simple manual how to get rich on the stock market. So why do I spend time lecturing on CoMa instead of swimming in a pool full of money and what does this manual tell about your financial advisor?

Abstract: Imagine being a particle living in a 2D plane. You can fly up and down, left and right. Life is good. But then a maleficent physicist appears and constraints you to the end of a massless rod, and starts calling you a mathematical pendulum!

If this story sounds familiar, or if you are a current/past/future student of TEF1, then this talk is for you! We will view constraints in physics from the mathematical perspective of differential geometry, using terms like vector fields, integrable and non-integrable distributions, Frobenius Theorem, integral submanifolds and Constant Rank Level-Set Theorem.

Abstract: The mathematical theory of knots is an area of topology that studies embeddings of one or more circles into three-dimensional ambient space. In this lecture I will introduce some basic concepts of knot theory (Reidemeister moves, linking numbers) as well as more advanced ones (the Jones polynomial). I will also present the braid group, which provides an efficient algebraic description of knots, and carry out a number of experiments to illustrate the theory.

Abstract: Statistics tell us that statistical literacy in society is very low. And since probability lies at the foundation of statistics, this fact probably also implies a low level of probability literacy. In this lecture, we will therefore take a brisk journey through both fields and introduce concepts such as random variables, characteristics of random variables, parameter estimation, and statistical hypothesis testing. 

Abstract: What precious secret did the mathematician del Ferro share with his student just before his death in 1526? When did mathemacians first discovered complex numbers? We all know how to solve linear and quadratic equations (in one variable) - but what about equations like x³+3x+2=0? After discussing the turbulent history of these discoveries, we will have a look at how to solve general equations of third and fourth order.

Abstract: They lurk behind every bush, they hide under every stone. Commutative, associative, even Lie if you’re lucky. From tame to wild, nature is teeming with algebras. Come visit our seminar and see for yourself! 

Abstract: In today's hectic times, too much information often throws us into decision paralysis. In mathematics, there is a way to get rid of unwanted information by identifying certain elements of a set according to an equivalence relation, producing a so-called quotient set. This phenomenon occurs surprisingly often within various mathematical structures. The aim of this talk is to show examples from the world of vector spaces, groups, algebras and geometry.

Abstract: In today's hectic times, too much information often throws us into decision paralysis. In mathematics, there is a way to get rid of unwanted information by identifying certain elements of a set according to an equivalence relation, producing a so-called quotient set. This phenomenon occurs surprisingly often within various mathematical structures. The aim of this talk is to show examples from the world of vector spaces, groups, algebras and geometry.

Abstract: Students encounter trigonometry for the first time in the context of geometry and right-angle triangles. Later they encounter the algebraic properties of these functions, and later still properties from calculus. But how are they linked? What's an angle and a radian? How do we measure a circle? Why do harmonic oscillations follow a triangle function? And where did Eulers formula come from? Are complex number matrices? During this session we will try to discuss and answer these questions, and during that time we will hopefully learn something about constructing mathematics for objects around us.

Abstract: The exponential function agrees with its Taylor series for all real (or complex) arguments, which allows us to quickly and efficiently extend the definition of an exponential to real (or complex) square matrices. In this talk, we shall hint on some possible uses for this matrix exponential, as well as examine what properties it shares with the ordinary (number valued) exponential. Furthermore, we will take a look at the relationship between the properties of the exponent and the exponential. All will be illustrated on pleasant low dimensional examples.

Abstract: Differential calculus of real functions of one real variable can be seen as a special case of the study of differential geometry of so-called smooth manifolds.  In this talk we will focus on the manifolds of dimension 1 – the line and the circle. We show what is the difference between the differential calculus on the circle and the classical theory known from Calculus 1.