VU Math Student Lunch Seminar

Abstract

I will speak about my favourite theorem in mathematics : Ropes have an even number of ends. This theorem, although intuitively obvious, has very important consequences in many places of mathematics (and outside of it). I will discuss some applications. It will help us answer the questions: Can you escape a maze? Can I find rectangles on closed curves in the plane? Can I fit a Klein bottle in Euclidean space?

Abstract

In the combinatorial game of Hackenbush, the goal is to be the last to finish shaping your shrubbery. Though the rules of the game are straightforward, they allow for a beautiful mathematical analysis. We will explore the world of combinatorial game theory through this game (and perhaps some others), finding some funky algebraic concepts along the way, as well as some applications to real-world games like Go.

Abstract

in the early 20th century, Henri Lebesgue developed measure theory in order to formalize integration. A set of real-valued points is called "Lebesgue Measurable" if it admits a measure function, which intuitively corresponds to this set having a well-defined notion of “size” or “volume”. While this theory helped to solve many problems in analysis, it was also quickly noticed that not all sets of points are measurable, e.g., the Vitali Set. This, in turn, led to counterintuitive results such as the Banach-Tarski Paradox (the paradoxical decomposition of the sphere to produce two spheres of the same size).

Constructing a non-measurable set is easy with the Axiom of Choice. It took set theorists several decades longer to realize that this is only​ possible because of the Axiom of Choice. A more precise analysis of the situation leads us to the field called ​``Descriptive Set Theory'', in which sets of points in are classified in a complexity hierarchy, with the topologically simple sets (open/closed) at the bottom, followed by the Borel hierarchy, and extending further throughout so-called “projective" sets. 

It turns out that there is a remarkable connection between mathematical properties of sets (such as their Lebesgue measurability) and their position in this complexity hierarchy. The Axiom of Choice is the only axiom allowing us to step entirely outside of this hierarchy, where we find many "non-definable”  or ​"non-classifiable" sets, such as the Vitali set and other paradoxical counterexamples. 

In this talk, I will explain how Descriptive Set Theory interacts with classical questions in analysis and topology, and the role that Axiomatic Set Theory    plays for questions of interest to a broader range of mathematicians. I will also briefly touch upon current, state-of-the-art research, and how it can be viewed as a direct descendent of the classical questions described above.

Abstract

You all know that a smooth function on a compact domain has at least 2 critical points, namely the minimum and maximum. But can you always find a function with precisely two critical points? We will see why we can do this on the sphere but not on the torus. Along the way, we will see some interesting interplay between topology and dynamics. At the end I will relate this problem to some physics problems and tell you about the relation with my own research. 

Note: this talk will be a lot of pictures and almost no formulas, so everyone should be able to follow along! 

Abstract

TBD

Abstract

In the field of (algebraic) topology, we try to map out the complicated galaxy that is the collection of all topological spaces - that is, we try to find all mathematically possible shapes. One very interesting solar system is the collection of manifolds, which are particularly nice shapes that come with a dimension. You would expect that, as you increase the dimension, the shapes will get more complicated. But this does not really turn out to be the case. Actually, it is the four-dimensional manifolds that give us the most trouble. In this talk, I'll give a brief introduction to the study of manifolds. I'll show why the first few dimensions are mostly harmless, while the high dimensions are doable, so long as you carry the right tools.

Abstract

How can we find patterns in complicated data, such as recordings of brain activity? One way is to use tools from topology. Persistent homology is a method that looks for shapes and loops hidden in data and records how long they “survive” as we zoom in and out. These long-lasting features often reveal something essential about the data. In this talk, I will introduce the main ideas of persistent homology without assuming background knowledge, and then show how it can be applied to neuroscience. 

Abstract

Mathematics is not immune to mistakes. With lengthy, complex, and increasingly technical proofs, errors can go unnoticed for years even after a paper has been peer-reviewed. This raises a natural question: could we use computers to check the correctness of a mathematical proof? Formalization is the process of encoding mathematics into a formal language, enabling proofs to be mechanically verified by a computer. In this talk, I will give an introduction to the formalization of mathematics using proof assistants (interactive software suitable for this task), with a particular focus on Lean. I will also discuss some of the ways this technology can contribute to mathematical practice. 

Abstract

Finding integer solutions to equations with integer coefficients, so-called Diophantine equations, is the essence of number theory; it is the predecessor to arithmetic geometry. There is no algorithm that decides whether such equations admit integer solutions, and the process of determining solubility can be difficult.

On the other hand, it is often easier to check if an equation admits solutions over a finite field. Simply put, the Hasse principle postulates that the solutions over finite fields are necessary and sufficient to determine if there are solutions over the integers. We will explore various examples where the Hasse principle is valid and others where it fails. In the latter case, we provide some insight for why it fails to hold, especially in the case of elliptic curves. 

Abstract

I will introduce category theory, and explain using examples how it unifies many of the topics studied during the math bachelor's degree. I will conclude with a fun fact that interprets the Fibonacci sequence as a (product-preserving) functor between categories.

Abstract

I will explain the general framework in which forensic statisticians typically work and apply this to the question of how strong a DNA-match actually is. Some beautiful probability theory is used to come to an answer, keeping in mind that there might be familial relationships in a population. 

Abstract

What do a billiard game, the acoustic in a cathedral, and the light pattern in your cup of coffee have in common? They are all described by a straight line motion (of the ball, of the sound, of the light) that gets reflected at the boundary of a region. In this talk, we present the field of mathematical billiards, which aims at predicting the features of this motion. We will discover unexpected properties and an interesting interplay between order and chaos. 

Slides

Videos

Octagon, Blocking Problem,Tokarsky Billiard, Elliptic Billiard, Sinai billiard, Bunimovich Stadium

Abstract

In this talk, I will introduce the field of Algebraic Number Theory, focusing on the role of prime ideals, which provides a deeper understanding beyond the traditional concept of prime numbers. Through this lens, we will explore one of the Mordell equations: y^3+11 =x^2 and work through the process of finding all integral solutions. This approach will highlight how Algebraic Number Theory can be applied to solve Diophantine equations. 

Abstract

In classical physics the future could be predicted with all detail in case the state of the universe was fully known, leading to the loss of free will and other philosophical consequences. With the onset of quantum physics at the start of the last century, it became clear that chance is immanently built into physics. However, there is still no satisfactory model for it. In my talk I will discuss two very different approaches to the probabilistic content of quantum theory and how I plan to explore both approaches with the help of my new research grant.  

Abstract

Philosophy and Mathematics are two seemingly different disciplines with no apparent connection to one another. However, from antiquity onwards, mathematics has been the subject of significant philosophical inquiry and a source of deep dispute among philosophers. The purpose of this talk is twofold. First, we will focus on how answering standard philosophical questions about mathematics is no easy feat. Second, we will briefly present how the area of the foundations of mathematics, specifically set theory, is ripe for both mathematical and philosophical exploration.  

Abstract

Since the 1990s, a lot of work has gone into developing the hardware and software for quantum computers, which are based on quantum-mechanical effects like superposition, interference, and entanglement. However, quantum techniques have also been useful for proving results in diverse areas of mathematics and computer science, such as coding theory, complexity theory, and polynomial approximations, that have little to do with quantum mechanics. This is somewhat reminiscent of the "probabilistic method" where one uses tools from probability theory to prove theorems that have nothing to do with probability, such as the existence of graphs with certain properties. It has the added advantage that one doesn't need to actually build a physical quantum computer for it; the mathematical rules of the game suffice. In this talk I will introduce this quantum way of thinking and sketch some of the results that came out of it. 

Abstract

A spatial graph is the image of an embedding of a graph in a 3-manifold. A given graph may have many different embeddings: An easy example are differently knotted embeddings of a circle into R3. If we allow for more complicated graphs, we encounter notions of entanglement that are not caused by knotting. We will look at these different notions of entanglements and explore some examples.

Abstract

Einstein’s theory of special relativity completely transformed the way we think about time and space. In this talk, we’ll explore the underlying philosophy through some of Einstein’s famous thought experiments and uncover a few of its more baffling consequences. Along the way, we’ll also meet some of the key mathematical tools involved, like invariant intervals and Lorentz transformations. No prior knowledge of physics is assumed — just a willingness to question your intuition (and maybe your watch).