Past Talks

In this workshop the goal is to introduce a class of tangles of ropes called rational tangles. We start by introducing them by actually tangling ropes, using two simple actions on them. Showing, with examples, that we can untangle any tangle with the same two moves. I'll let you work on optimizing solutions to tangles (with real ropes!). For the second half I will connect them back to math, showing that there is a bijection between these tangles and the rational numbers that maps these 2 actions to functions, in a commutating way. We further explore this bijection by introducing addition and multiplication of tangles. This is directly based on a workshop I do with gymnasium kids at Vetenskapens Hus.

Homotopy Type Theory (HoTT) is a field of mathematics that unifies logic, computer science, and homotopy theory. It builds upon Martin-Löf Type Theory (MLTT), reinterpreting its types as spaces and its proofs of equality as paths between points, including higher structure that arises from this interpretation. In this talk, I will first introduce the basics of MLTT, including its identity types and the principle of path induction (aka the J-rule). We will then explore the homotopical interpretation of types as ∞-groupoids, leading to the Univalence Axiom. While conceptually powerful, Univalence presents computational challenges in traditional MLTT. I will explain how modern approaches, such as Cubical Type Theory (CTT), resolve these issues. The session will conclude with a short demonstration of formalizing mathematics in Agda within a HoTT-based metatheory.

In this workshop/talk my goal is to introduce you to a class of tangles of ropes called rational tangles. We start by introducing them by actually tangling ropes, using two simple actions on them. We will show, by examples, that we can untangle any tangle with the same two moves. I'll let you work on optimizing solutions to tangles. For the second half I will connect them back to math, showing that there is a bijection between these tangles and the rational numbers that maps these 2 actions to functions, in a commuting way. We further explore this bijection by introducing addition and multiplication of tangles. This is directly based on a workshop I do with high-school kids at Vetenskapens Hus.

I will talk about Eilenberg's axioms on homology theories, go through the definition of homology for CW-Complexes and do a few concrete computations such as computing the homology of quotients and the sphere. In the end, I want to finish off by going through a slick proof of the Jordan Curve theorem using homology.

Locales are mathematical objects that both generalize topological spaces (and are the foundation of "pointless topology"), and also classify certain logical theories, the so called "geometric theories". This talk will be a gentle introduction to locales from both these perspectives, and perhaps at the same time a fresh insight into topology and logic from the perspective of locales. A listener familiar with category theory will find a third connection. We will sketch a proof of the classification theorem connecting the two areas, and doing so get a glimpse into an idea behind "topos theory". It is interesting but not necessary to have some previous knowledge about topology, logic or category when following the talk.

General relativity describes our space time as being not flat, but curved. This means that the theory of special relativity (flat/Minkowski space) is not really complete. General relativity is a theory of gravity, showing that gravity is not a force in itself, but that it is just a consequence of the fact that space is curved. One of the consequences is that gravity is not only acting on bodies, but also on objects with no mass, such as light! This way, we can guess that black hole must exist! In this seminar, I will introduce the mathematical tools from differential geometry that are used in physics and give some "pictures" from physics to help mathematicians to have a better feeling about what differential geometry is and what it implies.

My goal is to introduce the necessary preliminaries in homotopy theory to understand the Brown Representability theorem. The theorem, named after Edgar Brown, gives sufficient conditions for when a contravariant functor from the homotopy category of connected CW complexes to pointed sets is represented by a CW complex. Since the theorem is beautiful and has interesting applications in cohomology theory, I wish to introduce it to anyone that has just started reading topology and talk about its connection to general cohomology.

The study of groups is arguably one of the most important in mathematics, involving many of the greatest mathematicians, like Hanna Neumann, Jacques Tits and (earlier) Alfred Tarksi. I will give a short introduction to representation theory in order to study groups and their amenability. We will study some basic examples of groups like the integers and the free group in two generators. In the end, we might dip into the so-called Property (T) and its connection with amenability.

I will give an introduction to the area of symmetric functions. One big open problem in this area is the Stanley-Stembridge conjecture, which involves certain symmetric functions arising from graph colorings. Briefly, the conjecture states, that one should be able count colorings of certain graphs with a formula of a particular format. I will give an overview of this problem, and discuss some of my research efforts in this area.

Spectral geometry deals with relations between eigenvalues of a differential operator and the geometry of the underlying space, e.g. a domain in Euclidean space, a manifold or a graph. In this talk I will give a gentle introduction to the topic and will discuss questions related to, amongst others, isospectrality ("Can one hear the shape of a drum?") and shape optimization problems (the famous Faber-Krahn inequality). I plan to proceed from more classical results to recent research and some open problems.

The theory of ordered groups studies the structure of groups whose underlying set is ordered in such a way that the binary operation is compatible with the order. One fundamental result in this area is Hahn’s embedding theorem, which provides a characterisation of linearly ordered abelian groups by means of an isomorphism with an additive subgroup of some product of the real numbers. In this talk, I will develop the theory of ordered groups necessary to obtain a self-contained proof of this theorem, namely stating Hölder’s theorem on Archimedean groups first. In addition, the introduced concepts will allow us to discuss central results on orderability, showing examples of families of groups that admit a compatible order. Finally, I will comment some direct applications of Hahn’s theorem, especially its relation to the search for power series solutions of differential equations.

I will give an overview of what smooth projective and quasi-projective varieties are and introduce the problem of classifying them. I will state celebrated classification theorems of Castelnuovo, Enriques, Chen–Hacon, as well as a very recent extension by myself (joint with M. Mendes Lopes and R. Pardini).

Tumbleweed is a two-player abstract strategy board game invented in 2020 by Mike Zapawa. In 2022, Lear Bahack proved that determining which player has a winning strategy from a given position is PSPACE-complete. In this presentation we instead consider the problem of determining whether a given position can be reached, by some sequence of legal moves, from another given position. We prove that this problem is NP-complete using a reduction from a Hamiltonian cycle problem on directed graphs.

Code analysis is an important aspect for code safety and security. It allows electronics devices to run smoothly and without error. During an analysis choosing the right strategy to analyze a code is crucial to enhance precision and reduce the time taken by analysis. I will present to you the method that I have implemented to choose the right strategies in function of the target code. This method relies on Graph Neural Network, a variation of classical Neural Network. I will present what Graph Neural Network is, along with a short overview of what Machine Learning is and common techniques.

The aim of the talk is to introduce a height function on domino tilings of simply connected boards. The height function induces a partial ordering of tilings of a region, in which the covering relation encodes a flip. A flip is taking two adjacent dominos and rotating them 90 degrees, it turns out that the set of tilings under this ordering is a distributive lattice and thus that you can find a sequence of flips which connects any two tilings. Lastly, we apply Birkhoff's representation theorem on the sets of tilings of some rectangular boards and discuss the unimodality of their flip-graphs.

Applying the definition of matrix multiplication directly, the time complexity of computing the product of two $n \times n$ matrices is $\mathcal{O}(n^3)$, assuming arithmetic operations of numbers taking constant time. In the 1960s, faster methods were discovered for the first time, leading to further research in how low the time complexity could go. It is popularly conjectured to be $\mathcal{O}(n^2)$ - as low as addition of matrices. The general method of the developments has been viewing the bilinear mapping that matrix multiplication is as a three-dimensional tensor, where there is a natural correspondence between time complexity of the multiplication algorithm and tensor rank. This talk will act as an introduction to the history of achieving faster matrix multiplication.

The main problem of knot theory is to tell the knot type of a given knot and especially if a given knot is equivalent to a circle. One idea inspired by physics is to define a knot energy which increases with the "complexity" of the knot. I will talk about one such energy called the möbius energy as well as its corresponding heat equation.

We discuss connectivity properties of Julia sets of the parameterized Dixon elliptic functions. Our main result is that the connectivity locus of the parameterized Dixon sine function is the exterior of the open unit disk, and the Julia set is Cantor in the open unit disk minus the origin. We prove the parameterized Dixon cosine function also exhibits a fundamental dichotomy in the connectivity of the Julia set. The Julia and Fatou sets exhibit a variety of rotational symmetries, and no Herman rings exist for any function in either family.

Many things change when we turn to normed spaces of infinite dimension, and some of them can defy our intuition that has been shaped by the finite dimensional case. For example, the closed unit ball is not compact with respect to the norm topology in a space of infinite dimension. In this talk we will introduce the weak topology on a normed space X and the weak* topology on its dual X* and see how the problem of the compactedness of the unit ball can be addressed with these tools. We will state and give an outline of the proof of the Banach-Alaoglu theorem, which has been said to be a result that "echoes through functional analysis"

Hilbert's 14th problem asks if the ring of invariants of an algebraic group acting on a polynomial ring is always finitely generated. While this was disproved by a counterexample (Nagata 1959), there are still interesting questions to ask in the case that the ring of invariants is finitely generated. For example, "how large is the invariant ring with respect to the ambient ring?" In this talk, we will restrict to finite groups and finite-dimensional vector spaces. We will prove that invariant rings are finitely generated, then build up to two combinatorially motivated measures of the relative size of invariant rings as well as results: the size of minimal generating sets (Noether's bound) and Poincaré series (Molien's theorem). If time permits, we will further restrict to finite abelian groups and discuss an elementary means of constructing minimal generating sets. 

Lattices are everywhere: from logic to collections of substructures to geometry. An important result in lattice theory is Birkhoff’s duality between finite distributive lattices and posets from the 1930’s. Later Urquhart gave a representation for arbitrary finite lattice, generalising Brikhoff’s result. Recently, it has been found that there is a one-to-one correspondence between finite lattices and a special class of digraphs called TiRS graphs. In this talk, I will present Birkhoff’s representation and this correspondence between lattice and TiRS. I will also include some results characterizing the TiRS graphs of semidistributive lattices which is joint work with Andrew Craig (University of Johannesburg, South Africa) and Miroslav Haviar (Matej Bel University, Slovakia).

Schemes, as we know them, were introduced by Grothendieck in 1960. 62 years later, it is safe to say that Schemes have revolutionized algebraic geometry and with it, much of pure mathematics. There is no real easy introduction to schemes, but in this talk I will try to define and motivate what it's all about.

In this workshop/talk my goal is to introduce you to a class of tangles of ropes called rational tangles. We start by introducing them by actually tangling ropes, using two simple actions on them. We will show, by examples, that we can untangle any tangle with the same two moves. I'll let you work on optimizing solutions to tangles. For the second half I will connect them back to math, showing that there is a bijection between these tangles and the rational numbers that maps these 2 actions to functions, in a commuting way. We further explore this bijection by introducing addition and multiplication of tangles. This is directly based on a workshop I do with high-school kids at Vetenskapens Hus.