The students talks displayed in this page are ordered by alphabetical order of the speakers, with respect to their first names. The schedule can be consulted on the schedule page.
The talks will take place in the main lecture hall, which is room MA 041 of TU Berlin. More information on the venue and how to find the talks can be found on the venue page.
Many talks were recorded on video, but the video is released publicly only upon permission of the speakers; you can find all the videos using this link.
Our conference is going to feature talks from two invited speakers and fifteen student speakers:
In this talk I will recall motivations and foundational results on the classical problem of finding “the best” metric on a given algebraic manifold, and how analytic, geometrical and algebraic techniques get together to help this long standing (still very much open) problem. After recalling a general framework to prove existence of Einstein and constant scalar curvature metrics on algebraic varieties, we apply a recent breakthrough by Chen-Cheng to study the existence problem of such metrics on Galois covers, connecting the analytic theory to various geometrical constructions in the cyclic, abelian and non-abelian cases.
More information about Claudio Arezzo can be found here.
How I met Convex Polytopes, why books are important, why examples are important, what my students contributed, and whether there are still interesting problems to work on...
More information about Günter Ziegler can be found here.
The trace of a linear map is a fundamental invariant that is both easy to compute and incredibly useful. For example the trace of the identity map of a given vector space computes its dimension, and we can detect fixed points of continuous maps between topological spaces by computing the traces of certain induced linear maps.
This can be generalised to a much more abstract setting. If \(\mathcal C\) is a category that has an appropriate notion of tensor product and \(V\) is an object that has a "dual object"\(V^\ast\) we can define a notion of trace for any map \(V\to V\) and define the dimension of \(V\) to be the trace of its identity. This can yield surprising results depending on the ambient category. For example, we can recover the Euler characteristic of a space as the dimension of a chain complex. Furthermore we could ask questions such as “What is the dimension of a category?”
In this talk I will first go over the minimum amount of background knowledge in category theory needed and then sketch the general theory.
The goal of Hodge theory is to impose restrictions on the topology of algebraic varieties (or more generally compact Kähler manifolds). These can be conditions on the cohomology or on the fundamental group. For example, it is always true that the first Betti number of a compact Kähler manifold is even and that \(\mathrm{SL}_3(\mathbb{Z})\) cannot be the fundamental group of a compact Kähler manifold.
In this talk, I will sketch how these kind of restrictions arise from abelian and non-abelian Hodge theory. I will try to explain as much as possible from scratch. My goal is that after half an hour everyone will be able to recite Simpsons’s non-abelian Hodge correspondence.
Lyapunov exponents are a measure for instability and entropy in both deterministic and random dynamical systems. Finding lower bounds for the top Lyapunov exponents of a system is in general difficult and of interest in, for example, the field of fluid dynamics. In this talk I will explain the basic concepts behind Lyapunov exponents in random dynamical systems and outline my recent work on proving the possibility of positive Lyapunov exponents in the Hopf Normal Form with Additive Noise. This is based on joint work with Maximilian Engel.
The video of the talk can be accessed in this link.
During the last decade the so-called double phase operator has drawn attention from researchers. Originally it was introduced by Zhikov in the context of homogenization and elasticity theory and as an example for the Lavrentiev phenomenon. It regained popularity after some novel regularity results for local minimizers of the corresponding functional.
In this talk we consider quasilinear elliptic equations driven by the variable exponent double phase operator with superlinear right-hand sides. Under very general assumptions on the nonlinearity we prove a multiplicity result for such problems, whereby we show the existence of a positive solution, a negative one and a solution with changing sign. The sign-changing solution is obtained via the Nehari manifold approach and, in addition, we can also give information on its nodal domains. Furthermore, we derive a priori estimates on the solutions in the \(L^\infty\)-norm under the very general setting used above.
This is joint work with P. Winkert.
The video of the talk can be accessed in this link.
The Traveling Salesman Problem is one of the prominent NP-hard problems in combinatorial optimization. Given a complete graph with nonnegative edge weights, the task is to find a minimum-weight Hamiltonian cycle. This talk is about a special family of approximation algorithms, the insertion heuristics, which are widely used in practice. Among them, Farthest Insertion yields especially favorable empirical accuracy, although no approximation ratio better than the general case is known for this algorithm.
We present the currently known worst-case instance admitting an approximation ratio of \(6.5\) due to Hurkens and sketch an improvement to this instance that yields a ratio of \(7\). We discuss the possibility of further improvements on the instance, in particular that of obtaining a nonconstant lower bound.
In an attempt to understand insertion heuristics better, we later restrict ourselves to \( (1,2)\)-TSP, that is, TSP on instances with edge weights in \(\{1,2\}\) and prove a tight bound for this special case.
This talk is based on ongoing work.
Early in my undergraduate studies, I was dreaming for a world in which a series would converge, if its terms tend to zero. As widely known, this is very far from being the case... or so have we been told? Notions of convergence are metric-dependent. In the Non-archimedean world, my dreams come true. A series converges whenever its terms tend to zero! However, this comes at a price, every triangle is isosceles, and given any two disks, either they are disjoint, or one is contained in another.
In this talk, we will discuss the motivations to study Non-archimedean geometry, as well as their importance in the modern study of number theory (as time permits). Our main focus will be to touch on the new (fascinating) phenomena that we observe in the Non-archimedean world. The talk is meant for a wider mathematical audience. No knowledge of number theory, algebraic geometry, etc. is assumed. If you know what it means for a sequence to converge, you already know more than enough to fully follow the talk with ease!
The video of the talk can be accessed in this link.
Interacting particle systems are countable systems of locally interacting Markov processes and are often used as toy models for stochastic phenomena with an underlying spatial structure. Even though the definition of an interacting particle system often looks very simple and the major technical issues of existence and uniqueness have long been settled, it is in general surprisingly difficult to say anything non-trivial about their behavior. Some of the main challenges deal with the long-time behavior of the systems. Because of the possible non-uniqueness of the time-stationary distribution the analysis is very delicate and various techniques have been developed to study limit theorems or attractor properties. One particular technique that will play a major role in this talk is due to Richard Holley and involves using the relative entropy functional with respect to some specification \(\gamma\) as a Lyapunov function for the measure-valued differential equation that describes the time evolution of the system in the space of measures. We first explain the general idea behind the method on the simple example of a continuous time Markov chain on a finite state space and then discuss how one can extend it to irreversible interacting particle systems in infinite volumes.
A natural way to understand an algebraic variety is understanding its possible realizations in projective spaces. I will recall some basic facts about line bundles and hyperplane-sections. Then I will present a way to understand higher codimension linear sections via vector bundles. As an application I will compute the degree of irrationality of low genus K3 surfaces.
Uncertainty is inherent in most real-world systems. It places many disadvantages (and sometimes, advantages) on humankind’s efforts, which are usually associated with the quest for optimal results. The decision makers must select an optimal decision among all possible ones to achieve the best expected result related to their goals. Such optimization problems are called stochastic optimal control problems. We study optimal control of stochastic differential equations (SDEs) driven by semimartingales with jumps, where the jumps of the solution are obtained as small relaxation time limits of fast curvilinear motions along the solution of a non-linear ordinary differential equation, and between the jumps, the solution to the SDE moves along the (coefficient) vector field in the sense of Marcus-type.
I will give an outlook on my ongoing research where such stochastic differential equations are to be applied.
The Ericksen-Leslie equations are used to model liquid crystals in their nematic phase. We define a generalized solution concept based on a relative energy approach for the Ericksen--Leslie equations in three spatial dimensions. This solution concept satisfies the weak-strong uniqueness property. For a certain choice of the regularity weight, we construct an energy-variational solution with the help of an implementable, structure-inheriting space-time discretization. Computational studies are performed in order to provide some evidence of the applicability of the proposed algorithm.
The mathematical treatment of classical mechanics naturally leads to symplectic geometry, the study of symplectic manifolds. Symplectic geometry has rich connections to various branches of mathematics, such as enumerative and algebraic geometry, functional analysis, elliptic partial differential equations and dynamical systems. We present some results illuminating these connections.
A fundamental tool for many of these connections are pseudo-holomorphic curves. We give an overview of their theory, discuss some fundamental questions which are still open and indicate some avenues for addressing them. No prior knowledge of symplectic geometry will be assumed.
In 2004, with her PhD thesis, Maryam Mirzakhani presented a recursive formula computing the Weil-Petersson volume of moduli space of bordered Riemann surfaces, subsequently leading to a new proof of Witten’s conjecture/Kontsevich theorem. In this talk I’ll give a brief introduction to the topic, showing in particular how such a result can be viewed as an example of Eynard-Orantin topological recursion, for which I'll provide the basic notions and most notable applications.
In this talk, we will go on a journey through the fascinating world of Math Finance, ending with state-of-the-art rough volatility models.
Using real financial data, we will speed-run the developments of the last century with planned stops at the pioneering Bachelier model and the celebrated Black-Scholes model. Along the way, we will introduce you to some techniques of stochastic analysis and familiarize you with essential financial notations. But the journey does not end there. We will also present a new result that was a joint effort with Peter K. Friz and William Salkeld.
Don't be intimidated - this journey is for everyone. If you roughly know what a normal distribution and an integral is, or even if you don't, join us for an adventure that might not only inform but also inspire you.
The video of the talk can be accessed in this link.