Talks displayed in this page are ordered by alphabetical order of the authors, with respect to their surnames. The schedule can be consulted on the schedule page.
The talks will take place at the lecture hall MA 042 in the mathematics building of the Technische Universität Berlin, at Straße des 17. Juni 136. More information about the location can be found on the venue page.
All talks will be recorded in video, but the video is released publicly only upon permission of the speaker; you can find the link to each video below the corresponding abstract. Some videos are only accessible with a password; if you are interested in one of these videos, please contact the speaker directly.
Our conference will feature talks from two invited speakers and seventeen student speakers:
I will give a gentle introduction to recent research for finding efficient algorithms for natural problems in geometric invariant theory. While the original motivation for this research came from algebraic complexity theory, surprising connections to a diverse set of problems in different areas of mathematics, computer science, and physics popped up.
The video of the talk can be accessed in this link.
More information about Peter Bürgisser can be found here.
Since the 1970’s it has become well understood that many apparently simple deterministic dynamical systems have unpredictable and chaotic behaviour. This has led to a huge amount of research and very deep results about the structure of dynamical systems. Amongst the many and varied approaches to the topic, a very effective one has been the application of ideas from ergodic theory and statistical mechanics to study deterministic systems with the language and tools of probability theory.
In this talk I will give an elementary introduction to some of these ideas and outline some of the main open problems and conjectures in the field. In particular I will introduce and motivate the notion of a “physical measure” as a very powerful way to study complicated dynamics, and formulate some conjectures and results on the existence of physical measures.
The video of the talk can be accessed in this link.
More information about Stefano Luzzatto can be found here.
A graph \(G\) is Ramsey for another graph \(H\) if any red/blue-coloring of the edges of \(G\) contains a monochromatic copy of \(H\). A classical result due to Ramsey shows that Ramsey graphs exist for any choice of \(H\). Once we know that Ramsey graphs for \(H\) exist, it is natural to ask what such graphs look like and what properties they must satisfy. For instance, if you have taken a basic course in combinatorics, you might be familiar with the problem of determining the Ramsey number of a graph \(H\), which in this language can be defined as the minimum number of vertices in a graph that is Ramsey for \(H\).
In this talk, we will give a short introduction to the study of (minimal) Ramsey graphs, focusing on questions related to minimum degrees. We will mainly discuss classical results, aiming to illustrate some of the ideas and methods used in this field.
The Poincaré-Bendixson theorem for ordinary differential equations on the plane is one of the most iconic results in dynamical systems theory. The core idea is that the long term behavior of the solution curves is heavily constrained by the \(2\)-dimensional phase space.
Surprisingly, Mallet-Paret and Sell (1) have drawn analogous conclusions for delay differential equations with monotone feedback, for which the phase space is infinite-dimensional.
In the talk I will comment on the results in (1). As an application, we characterize the, genuinely infinite-dimensional, periodic solutions of a particular delay differential equation in terms of a planar ordinary differential equation.
QRT map appears in various contexts in mathematical physics and discretizations. Geometrically, one obtains a QRT map by composing two involutions defined on a pencil of biquadratic curves on the plane. The QRT map is then a planar birational map with singularities at the base points of the pencil. One can resolve these singularities by blowing up and lifting the QRT map to a surface automorphism. Some dynamical properties can be derived using the help of elliptic surface theory.
The video of the talk can be accessed in this link.
At the end of the nineteenth century, Gabriel Koenigs (1858-1931) proved some interesting theorems about conjugate nets with equal Laplace invariants. Nowadays, discrete conjugate nets with equal Laplace invariants are pervasive in discrete differential geometry. In recognition of Gabriel Koenigs' work, they are usually called Koenigs nets. Via the projective geometry of inscribed conics, I will introduce Koenigs nets and I will explain why they can be regarded as a discretisation of conjugate nets with equal Laplace invariants.
Energy system models are a crucial component of energy system design and operations, as well as energy policy consulting. If detailed enough, such models lead to large-scale linear optimization problems (linear programs, or short LPs) that are intractable even for the best state-of-the-art solvers. This talk describes a new interior-point solver that exploits common structures of energy system LPs to efficiently solve them in parallel on distributed-memory systems.
In particular, we will describe a preconditioned Schur complement based decomposition approach for solving the linear systems arising within the interior-point algorithm. As a result, energy system LPs with more than one billion variables and constraints can now be solved within two hours.
The video of the talk can be accessed in this link.
Characteristic classes is a name for a group of cohomological invariants which satisfy certain axioms. In a sense, they measure how much twisted a vector (or, more generally, a principle) bundle is. This is a very powerful tool of Algebraic Topology. For instance, you can use them to prove Hairly ball theorem, which states that you can't draw a continuous non vanishing vector field on a 2-dimensional sphere. You will find characteristic classes in Atiyah–Singer index theorem, one of the most powerful theorems I know, which has application all over the math, from Algebraic Geometry to Mathematical Physics. They are also one of the key tools in modern condensed matter physics.
The goal of my talk is to give you a feeling of what they are and why we care. Don't be scared if you don't understand words in the first two sentences of this abstract - I will give an intuitive explanation, as well as formal definition.
The video of the talk can be accessed in this link.
Compressible fluid models describe a large spectrum of possible models in meteorology, geophysics and astrophysics. The models are given by systems of partial differential equations. Writing these systems in dimensionless form involve characteristic numbers (i.e Mach number, Rossby number, Froude number, Reynolds number etc.). If these numbers are small or large we observe some interesting phenomena. In this talk my aim is to give an idea about this limit behavior.
The video of the talk can be accessed in this link.
I will explain what pouring e.g. honey on a lifebuoy tells you about its topology. It turns out that similar ideas can be applied in a rather different setting, which provides insight into planetary motion. Along the way, we will learn about Morse and Hamiltonian Floer homology and catch a glimpse at how working in symplectic topology can be like.
Rising air bubbles can roll up into bubble rings with mesmerizing dynamics. A drop of ink in water can create a beautiful chandelier like patterns. What causes these fluid phenomena and how can you simulate their behavior? In this talk a novel model using viscous vortex filaments with buoyancy is presented that accurately recreates real life footage of bubble rings and ink chandeliers.
See videos and more on the speaker's webpage.
The video of the talk can be accessed in this link.
In this talk, we present a classical result on algebraic geometry namely, that every smooth cubic surface has exactly 27 lines. Our aim is to introduce the terminologies behind this result to non-algebraic geometers and to sketch its proof. The proof of this theorem is interesting not only because it presents a justification for a nice result but also because it combines important ideas in algebraic geometry like parameter spaces, which will appear naturally in the course of the proof. Moreover, for fun, we will give evidence for this fact computationally by running a code enumerating them for various “randomly chosen” surfaces.
We use differential geometric methods to deal with the following problem coming from continuum mechanics: Consider an elastic membrane, fixed to a boundary. By deforming, bending and stretching that boundary, the elastic membrane will take shape such a way that the resulting elastic strain energy is minimal and the external and internal elastic forces balance out. What do these elastic surfaces look like? We look at numerical simulations of rotatio- nally symmetric equilibrium shapes as well as compare these to experiments. To model different elastic materials, stiff materials or stretchy materials, we employ different material laws. Like for minimal surfaces, questions about existence, regularity and stability of the elastic surfaces pop up.
It is well-known that in a battlefield, soldiers are placed on \(\mathbb{Z}^2\) and advance by jumping over one another, causing the jumped-over soldier to perish. We introduce the technique of potential functions, and use it to show *many* soldiers are needed to advance more than a couple of coordinates in a battlefield. This shows that wars should be best left as a thought-exercise.
The video of the talk can be accessed in this link.
The relation between sums of squares and non-negativity of polynomials was firstly studied by Hilbert at the end of the \(19^\text{th}\) century and led to the formulation of the \(17^\text{th}\) problem from his celebrated list of problems. This drove to significant developments in real algebraic geometry and, more recently, sums of squares were found to be useful in optimization. In this talk, I will present general facts about sums of squares and how they are related to semidefinite programming. In the end, we will see a few applications.
The video of the talk can be accessed in this link.
A \(p\)-centered coloring is a vertex-coloring of a graph \(G\) such that for every connected subgraph \(H\) of \(G\) either \(H\) receives more than \(p\) colors or there is a color that appears exactly once in \(H\). The concept was introduced by Nšetřil and Ossona de Mendez to provide a local condition suitable to measure sparsity of graphs.
We develop first non-trivial lower bounds on the \(p\)-centered coloring numbers. For outerplanar graphs, we prove that their maximum \(p\)-centered coloring number is in \(\Theta(p\log p)\). For planar graphs, we show that some require \(\Omega(p^2 \log(p))\) colors, while all of them admit a \(p\)-centered coloring with \(\mathcal{O}(p^3 \log(p))\) colors. This improves an \(\mathcal{O}(p^{19})\) bound by Pilipczuk and Siebertz. For graphs of degree at most \(\Delta\), we give a coloring with \(\mathcal{O}(\Delta^2 p)\) colors which is in strong contrast to the result for the related weak coloring numbers.
This is joint work with Michał Dębski, Stefan Felsner and Piotr Micek.
The video of the talk can be accessed in this link.
In my talk, we consider linear reaction systems (which can also be interpreted as master equations or Kolmogorov forward equations for Markov processes on a finite state space). Modeling real-world processes, one often assumes that some reactions are slow and others are fast.
We investigate the limit behavior of the reaction system if the fast reaction rates tend to infinity. As we will see, this leads to a coarse-grained model on a smaller state space, where the fast reactions create microscopically equilibrated clusters, while the exchange mass between the clusters occurs on the slow time scale. Interestingly, the whole evolution equation can be reconstructed from the coarse-grained system without loosing any information.
Moreover, one can also show that the coarse-graining and reconstruction procedure does even work on the level of the underlying physical principle (namely the energy-dissipation-principle) and not only for the evolution equation itself. In particular, this justifies the coarse-graining procedure from the physical point of view.
When trying to fly on an airplane one is given the daunting task of selecting which items to bring along. To make this task easier, for each object \(i\) in your possession, you do the following:
Your plan is to select a subset of items that satisfy the weight restriction of the airline while maximizing happiness. Sadly, even though the problem seems very simple, it is not possible to solve it optimally and efficiently (unless \(\texttt{P}=\texttt{NP}\)). This last observation motivates the search for “good-enough” or approximate solutions that can be found efficiently.
In this talk we will first discuss the traditional Knapsack problem and the methods used to deal with it. Afterwards, we will consider a two-dimensional generalization of the problem. This time the Knapsack is a square of fixed size, and the objects to pack are now polygons. Our understanding of this problem highly depends on the input polygons. For example, if all polygons are axis-parallel squares you can find arbitrarily-good solutions efficiently.
Our main result is an approximation algorithm that is quasi-efficient (i.e. that runs in quasi-polynomial time) and allows for convex polygons. To the best of our knowledge, these are the first results for two-dimensional Knapsack in which the input objects are not limited to axis-parallel rectangles or circles and in which you can rotate by arbitrary angles.
This is joint work with Andreas Wiese.
A limit order book (LOB) is an electronical tool used in economics to store and display unexecuted orders. One research objective is to describe realistic discrete dynamics of a LOB which can be approximated by a continuous time model. In order to apply such a model for intraday electricity markets, we aim to extend existing mathematical analysis of LOB models for purely financial markets. In contrast to traditional stock markets, electricity markets are not as liquid, exhibit large spreads and unforeseen power outages can lead to extreme spikes in the prices, including possibly negative prices. Hence, one starting point in extending existing convergence results for financial markets is to allow larger sizes of price changes which do not become small in the limit. Ensuring that these large price changes only appear with a small probability that scales in the right way, a convergence to a diffusion approximation can still be proven. How can we extend these results to an approximation by a diffusion with jumps?