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 online. To get access to the online talks, please fill the form on the registration page.
All talks will be recorded in video, but the video is released publicly only upon permission of the speakers; 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.
Pattern formation in materials can often be explained in the context of the calculus of variations. The resulting minimization problems are often analytically and numerically challenging due to their nonlocal and/or nonconvex structure. In many cases, explicit minimizers cannot be computed explicitly. As a first step towards the understanding of low-energy states and qualitative properties of minimizers (such as periodicity or self-similarity) one often focuses on the scaling of the minimal energy in terms of the problem parameters. This requires the explicit construction of good competitors and the proof of a matching ansatz-free lower bound.
In this talk, I will illustrate some of the techniques in the context of a model for helimagnets.
This talk is partly based on joint work with Janusz Ginster.
More information about Barbara Zwicknagl can be found here.
Three-dimensional Riemannian manifolds are called asymptotically Euclidean if, outside a compact set, they are diffeomorphic to the exterior region of a ball in Euclidean space, and if the Riemannian metric converges to the Euclidean metric as the Euclidean radial coordinate r tends to infinity. In 1996, Huisken and Yau proved existence of a foliation by constant mean curvature (CMC) surfaces in the asymptotic end of an asymptotically Euclidean Riemannian three-manifold. Their work has inspired the study of various other foliations in asymptotic ends, most notably the foliations by constrained Willmore surfaces (Lamm—Metzger—Schulze) and by constant expansion/null mean curvature surfaces in the context of asymptotically Euclidean initial data sets in General Relativity (Metzger, Nerz).
After a rather extensive introduction of the central concepts and ideas, I will present a new foliation by constant spacetime mean curvature surfaces (STCMC), also in the context of asymptotically Euclidean initial data sets in General Relativity (joint work with Sakovich). This STCMC-foliation is well-suited to define the center of mass of an isolated system in General Relativity and thereby answers some previously open questions of relevance in General Relativity. Previous knowledge of General Relativity and Riemannian Geometry will not be assumed.
The video of the talk can be accessed in this link.
More information about Carla Cederbaum can be found here.
Toric geometry allows us to translate algebro-geometric problems to questions in discrete geometry, in particular the geometry of lattice polyhedra. This provides accessible tools for computing interesting invariants of toric varieties. For instance, the cohomology of sheaves associated to divisors on toric varieties can be calculated from the singular cohomology of certain discretely defined sets. One method of calculating this has been know for several decades, another description was given by Klaus Altmann and others in two papers in 2018 and 2019.
In my master’s thesis, I used this new description of the sheaf cohomology of divisors to give a discrete-geometric interpretation of the first extension group \(Ext^1(\mathcal{L^-},\mathcal{L^+})\) of two line bundles \(\mathcal{L^-}\) and \(\mathcal{L^+}\) on a toric variety. I showed how an inclusion/exclusion sequence of lattice polyhedra gives rise to a universal extension sequence for \(Ext^1(\mathcal{L^-},\mathcal{L^+})\).
In this talk I will give a brief introduction to toric geometry and show how discrete geometry comes into play. I will show examples of how invariants of toric varieties can be calculated from discrete-geometric data. Finally, I will sketch the result of my master’s thesis connecting a "polyhedral extension sequence'' to the algebro-geometric \(Ext^1(\mathcal{L^-},\mathcal{L^+})\).
The video of the talk can be accessed in this link.
A cycle in a graph is said to be Hamilton if it spans all the vertices of the graph. Similarly, a graph will be Hamiltonian if it contains a Hamilton cycle. In 1969, Lovász conjectured that the symmetry of a graph heavily influences its Hamiltonicity. This idea gave birth to Lovász conjecture: Every connected symmetric graph is Hamiltonian except for five known counterexamples. There is vast literature showing that Lovász conjecture holds for natural classes of graphs; e.g. Hypercubes, Permutahedra, dense Kneser Graphs, etc. Very recently, Lovász conjecture was shown to hold for the middle levels graph; which is a particularly nice symmetric subgraph of the Hypercube. The known constructions of Hamilton cycles in the middle levels are not particularly symmetric, not giving clues towards the conjectured relation between symmetry and Hamiltonicity. Hence, in 2011 Knuth conjectured the existence of a Hamilton cycle which respects the natural symmetry of the middle levels (i.e. that is automorphism invariant).
In this talk we present:
The video of the talk can be accessed in this link.
In this short talk we will discuss a very famous Fermat’s theorem on sum of two squares which states that a prime number can be written as a sum of two squares iff it is not of the form \(4k+3\). This theorem, although very elementary, has several different proofs and perspectives which can be used as introductions to major areas of number theory such as algebraic number theory, analytic number theory, theory of Diophantine approximations etc. We will let the audience choose in which direction this talk will flow.
The video of the talk can be accessed in this link.
The modeling framework of port-Hamiltonian (pH) systems is a universal model class, associated with a Dirac structure that encloses its energy balance properties. In this talk, pH systems will be introduced, then the possible structures of pH differential-algebraic equations (DAE) and the linear systems associated with them will be discussed. Lastly, we talk about solution of the associated linear systems which have a positive (semi-)definite symmetric part that can be exploited in the numerical solution. Lanczos-like methods for the solution of linear systems coming from pHDAEs are presented.
The video of the talk can be accessed in this link.
First, we give an introduction to (stochastic) optimal control in finite dimensions and present the two classical approaches to optimal control problems: The dynamic programming approach and Pontryagin’s maximum principle. In the second part of the talk, we focus on the optimal control of semilinear stochastic partial differential equations and emphasize the challenges in current research on stochastic optimal control in infinite dimensions.
Arnold’s question about fixed points of Hamiltonian diffeomorphisms and Lagrangian intersections led to Moser's study of coisotropic leaf-wise intersections. The result showed that leaf-wise intersections exist for Hamiltonian diffeomorphisms \(C^1\)-close to identity. Ekeland, Hofer proved a generalization of this result for contact-type hypersurfaces and Hamiltonian diffeomorphisms with energy bounded by certain symplectic capacity, turning this into a question of displacement energy. We will explain results of a variational approach for contact coisotropic submanifolds using generalized perturbed Rabinowitz action functional introduced by Albers, Frauenfelder, Kang.
Siegel modular forms are higher dimensional analogues of classical modular forms: while the latter are sections of a line bundle in the moduli of elliptic curves, the former are such in the moduli of principally polarized abelian varieties. A phenomenon that appears with this generalization is the Fourier- Jacobi series of a Siegel modular form. One might wonder if any formal series of that shape comes from a Siegel modular form. Bruinier and Raum answered the question affirmatively, over the complex numbers, in 2014.
In this talk we study the problem over the integers, and we reformulate it in a way that will lead us to investigate the cohomology of certain sheaves in toric varieties.
The video of the talk can be accessed in this link.
The Multiplicative Ergodic Theorem (MET) is a powerful tool with various applications in different fields of mathematics, including analysis, probability theory, and geometry, and a cornerstone in smooth ergodic theory. It was first proved by Oseledets for matrix cocycles, since then, the theorem attracted many researchers to provide new proofs and formulations with increasing generality.
In this talk motivated by our models in stochastic delay equations and stochastic partial differential equations (SPDE), we will present a version of MET for stationary compositions on a (possibly random) field of (potentially distinct) Banach spaces, depending on the random sample. (Joint work with Sebastian Riedel)
Algebraic geometry studies algebraic varieties which are locally defined by polynomials and Analytic geometry deals with complex manifolds or more generally analytic manifolds which are locally defined by holomorphic functions. During 1950s and 1960s, Jean Pierre Serre and Alexander Grothendieck laid the foundational work for the modern algebraic geometry. Two major papers by Serre were Faisceaux Algébriques Cohérents (FAC, 1955) and Géometrie Algébrique et Géométrie Analytique (GAGA, 1956). The latter describes a beautiful and deep underlying connection between algebraic and analytic geometry.
In this talk, I will give a brief overview of this connection using Riemann surfaces as the prime example.
The video of the talk can be accessed in this link.
For a Banach space \(X\) a multi-function with values in \(X\) is a map \(f:[0,1]\to 2^{X}\setminus \{\emptyset\}\). Using Minkowski addition and convergence in the sense of Hausdorff distance, one can define Riemann integral of a multi-function similarly to that of a real-valued function.
The convex hull of a multi-function \(F\) is the multi-function \(convF : t \to conv(F(t))\). It is known that the convex hull of an integrable multi-function is also integrable. The inverse is known for Hilbert spaces. That is, every multi-function (with values in a Hilbert space) with an integrable convex hull is itself integrable. However, it is an open question for what Banach spaces the same result holds true. We present a complete solution for this open problem.
This is a joint work with Vladimir Kadets and Artur Kulykov.
The research was supported by the National Research Foundation of Ukraine funded by Ukrainian State budget as part of the project 2020.02/0096 "Operators in infinite-dimensional spaces: the interplay between geometry, algebra and topology"
The video of the talk can be accessed in this link.
We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process \(X\). We consider classic and randomized stopping times represented by linear functionals of the rough path signature \(\mathbb{X}^{<\infty}\) associated to \(X\), and prove that maximizing over the class of signature stopping times, in fact, solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a deterministic optimization problem depending only on the truncated expected signature \(E[ X^{\le N} ]\). The only assumption on the process \(X\) is that it admits a lift to a continuous (geometric) random rough path. Hence, the theory encompasses processes such as fractional Brownian motion which fail to be either semi-martingales or Markov processes.
The study of random graphs is a central topic that lies at the intersection of Probability Theory and Combinatorics. In the past couple of decades, there has been great interest in so-called ‘pseudorandom’ graphs: graphs which resemble purely random graphs. This study is motivated by applications in Computer Science where true randomness is hard to come by. In this talk we will introduce the concept of pseudorandom graphs and discuss the problem of determining when a pseudorandom graph contains a certain subgraph. We will focus on one of the few cases where we actually have a satisfactory answer to this question; a recent breakthrough giving a tight condition on the level of pseudorandomness which guarantees the existence of a triangle factor in a graph.
\(J\)-holomorphic curves are very powerful in studying the topology of symplectic manifolds. The existence of certain \(J\)-holomorphic curves in a symplectic manifold obstruct symplectic embeddings into the underlying symplectic manifold. In fact, there is this Eliashberg’s Principle that says that any obstruction to a symplectic embedding (beyond the volume condition) can be described by a \(J\)-holomorphic curve. In this talk, we give an example of this phenomenon, namely, the non-squeezing theorem of Gromov. We will demonstrate how the presence of a \(J\)-holomorphic sphere forbids the symplectic embedding of some Euclidean balls into some symplectic manifolds.
The video of the talk can be accessed in this link.
Generalized permutahedra are a class of polytopes with many interesting combinatorial subclasses. We introduce pruned inside-out polytopes, a generalization of concepts introduced by Beck–Zaslavsky (2006),which have many applications such as recovering the famous reciprocity result for graph colorings by Stanley. We study the integer point count of pruned inside-out polytopes by applying classical Ehrhart polynomials and Ehrhart–Macdonald reciprocity. This yields a geometric perspective on and a generalization of a combinatorial reciprocity theorem for generalized permutahedra by Aguiar–Ardila (2017) and Billera–Jia–Reiner (2009). Applying this reciprocity theorem to hypergraphic polytopes allows us to give an arguably simpler proof of a recent combinatorial reciprocity theorem for hypergraph colorings by Aval–Karaboghossian–Tanasa (2020). Our proof relies, aside from the reciprocity for generalized permutahedra, only on elementary geometric and combinatorial properties of hypergraphs and their associated polytopes.
We review classical and modern results in approximation theory of neural networks. First, the density of neural networks within different function spaces under various assumptions on the activation function is considered. Next, lower and upper bounds on the order of approximation with neural networks are given based on the input dimension, the number of neurons and a parameter quantifying the smoothness of the target function. Lastly, a family of compositional target functions for which the curse of dimensionality can be overcome using deep neural networks is examined.
The video of the talk can be accessed in this link.
A model for the description of dynamic crack growth in (visco-)elastic materials will be introduced. Using a phase-field approach where the sharp crack interface is regularized with a volumetric approximation, certain challenges regarding the analysis of the model are discussed and an application oriented solution strategy is presented.
Löwner chains provide a way to encode certain planar domains and curves by real-valued “driving” functions. Originally a purely complex analytic tool to study conformal maps, it has turned out to be very useful in constructing SLE, a “uniformly random” curve in a domain. Not any curve can be described as a Löwner chain, but the ones that can (we call them traces) may look very wild and even be space-filling. Intuitively, traces are characterised by the property that whenever they self-intersect, they need to “bounce off” instead of “crossing over”.
In this talk, I will introduce Löwner chains and present three equivalent ways of describing the property that characterise traces.