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 took place online. To get access to the online talks, please fill the form on the registration page.
All talks were 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.
Our conference featured talks from two invited speakers and fifteen student speakers:
Smooth curves and surfaces can be characterized as minimizers of squared curvature bending energies subject to constraints. In the univariate case with an isometry (length) constraint this leads to classic non-linear splines. For surfaces, isometry is too rigid a constraint and instead one asks for minimizers of the Willmore (squared mean curvature) energy subject to a conformality constraint. We present an efficient algorithm for (conformally) constrained Willmore surfaces using triangle meshes of arbitrary topology with or without boundary. Our conformal class constraint is based on the discrete notion of conformal equivalence of triangle meshes. The resulting non-linear constrained optimization problem can be solved efficiently using the competitive gradient descent method together with appropriate Sobolev metrics. The surfaces can be represented either through point positions or differential coordinates. The latter enable the realization of abstract metric surfaces without an initial immersion. A versatile toolkit for extrinsic conformal geometry processing, suitable for the construction and manipulation of smooth surfaces, results through the inclusion of additional point, area, and volume constraints.
More information about Peter Schröder can be found here.
Somewhere in the convex hull of symplectic topology, algebraic geometry and mathematical physics, ideas from all three subjects can be applied toward the problem of "counting curves", leading to the subject known as Gromov-Witten theory. I'll start by sketching the main ideas of this theory from the symplectic perspective, including the important role played by pseudoholomorphic curves in symplectic topology, and then discuss a conjecture that was formulated by algebraic geometers about 20 years ago: for generic choices of geometric data on a symplectic Calabi-Yau 6-manifold, all holomorphic curves are super-rigid. The difficulty behind this conjecture involves the fundamental incompatibility between transversality and symmetry, and its solution suggests that there is much more room for harmony between these two concepts than one might expect. However, the solution did not come easily: I am aware of at least five false claims of proofs that were put forward between 2000 and 2016, the last two of which (preceding an actual correct proof in 2019) were by me. I will recount the history of this drama as best I can, from my own biased perspectve; maybe you can learn from my mistakes.
More information about Chris Wendl can be found here.
Epitaxy is a special form of crystal growth and of great importance in modern technology, especially in the making of semi-conducting devices. We consider a crystalline film that is deposited on a (rigid) substrate. The misfit between the crystal structure of the film and substrate can lead to dislocations and has an influence on the morphology of the film. Dislocations are topological defects of the crystallographic lattice.
In this talk we will introduce epitaxy and dislocations. Then we will consider various mathematical models of epitaxially strained films and discuss their features.
In this talk we discuss classic theorems from Convex Geometry in the context of topological drawings and beyond. In a simple topological drawing of the complete graph \(K_n\), any two edges share at most one point: either a common vertex or a point where they cross. Triangles of simple topological drawings can be viewed as convex sets. This gives a link to convex geometry. Among the classic theorems from convex geometry, we discuss Kirchberger's theorem, Carathéodory's theorem and Helly's theorem.
How many affine hyperplanes do we need to cover all points of the \(n\)-dimensional hypercube \(\{0,1\}^n\subseteq \mathbb{F}^n\)? It is not difficult to convince ourselves that the answer is two. What happens, however, if we are not allowed to cover the point \(\mathbf{0}\)? As shown in a seminal paper by Alon and Füredi, this simple restriction makes the answer much larger. In this talk, we will discuss a natural generalization of this problem, in which the points have to be covered with certain multiplicities, and present some new results for the binary field \(\mathbb{F}_2\).
This talk is based on joint work with Anurag Bishnoi, Shagnik Das, and Tamás Mészáros.
The sensitivity of a dynamic on a network describes how an equilibrium changes when slightly perturbing the network parameters. In general, network processes can be highly complex, including non-linearities, which often makes direct calculations infeasible. However, the sensitivity only depends on the linearization in an equilibrium, which is in structure very close to a Markov chain. In this talk, I will present how this connection can be used to compute the sensitivity of chemical reaction networks only using results on the sensitivity of Markov chains.
The Ensemble Kalman-Bucy filter (EnKBF) is an important tool in stochastic filtering, that aims to approximate the law of a diffusion process \(X\), called the signal, conditioned on noisy observations \(Y\), by employing a system of diffusion processes interacting through their ensemble mean and covariance.
In this talk we first derive an EnKBF in the case that the observation noise and the signal process are correlated, using a consistent representation of the conditional distribution of \(X\) given \(Y\) by a McKean-Vlasov equation.
Then we discuss the mean-field limit and its governing McKean-Vlasov equation, proving existence and uniqueness of solutions even in the case of a nonlinear, non-Gaussian signal.
Finally we identify the stochastic partial differential equation driving the distribution of the mean-field limit and discuss its (in)consistency with respect to the conditional distribution of \(X\) given \(Y\).
The talk will begin with an introduction to circular nets. In discrete differential geometry, circular nets provide a discretisation of surfaces that are parametrised along curvature lines. I will present a sequence of incidence theorems that are essential to construct a class of circular nets that provide a discretisation of surfaces with linear, circular, planar or spherical curvature lines. The foregoing surfaces have remarkable geometric properties. For instance, surfaces with circular curvature lines are quartic surfaces that are classically known as Dupin cyclides.
The article considers minimization of the expectation of convex function. Problems of such type often arise in machine learning and a number of other applications. In practice, stochastic gradient descent (SGD) and similar procedures are often used to solve such problems. We propose to use Vaidya's cutting plane method with minibatching, which converges linearly and requires significantly less iterations than SGD on a class of problems. This is verified in our experiment, where the algorithm is applied to train logistic regression model for the binary classification problem. The algorithm does not require neither smoothness nor strong convexity of target function to achieve linear convergence. We prove that the method arrives at approximate solution with given probability when using minibatches of size proportional to \(\varepsilon^{-2}\), where \(\varepsilon\) is the desired precision. This enables efficient parallel execution of the algorithm, whereas possibilities for batch parallelization of SGD are rather limited. Despite fast convergence, Vaidya's cutting plane method can result in a greater total number of calls to oracle than SGD, which works decently with small batches. Complexity is proportional to \(n \ln n\), where \(n\) is the dimension of the problem, hence the method is suitable for relatively small dimensionalities.
If a previously deformed wire made out of special Nickel-Titanium is heated, the wire returns to its original form. This phenomenon is called the shape-memory effect. The cooling (or heating) of a material leads to a first order solid-to-solid phase transformation. During the cooling the lattice structure of a material changes abruptly at a critical temperature. The high temperature phase is called austenite and the low temperature phase is called martensite. Cooling of the austenite leads to the abrupt formation of martensitic nuclei. Heating reverts the martensitic transformation, i.e. the lattice structure changes at a high temperature back to the lattice structure of the austenite.
We are interested in the martensitic phase transformation with two martensitic variants. In particular, we aim at a better understanding of the appearing microstructure, i.e. the geometry at the very small scale-structure, emerging during a martensitic phase transformation. Therfore, we study a mathematical model postulated by Kohn and Müller (1992 & 1994), which leads us to a nonconvex singularly perturbed variational problem. We prove asymptotic self-similarity of minimizers. Our results generalize the result by Conti (2000) to various physically relevant boundary conditions, more general domains, and arbitrary volume fractions.
This is joint work with Sergio Conti, Johannes Diermeier, and Barbara Zwicknagl.
We study valuated matroids, their tropical incidence relations, flag matroids and total positivity. Our techniques employ the polyhedral geometry of the hypersimplices, the regular permutahedra and their subdivisions.
The progressive technical and scientific development within human society requires us to work on increasingly complex systems. For example, the transition to green energy introduces new kinds of energy sources in the power network, like solar panels and wind turbines, that are inherently more unstable and erratic than the ones based on fossil fuel combustion. The models and methods currently used to regulate power networks are unfortunately insufficient to deal with these much more dynamical systems. These challenges have been driving the development of modern and flexible modeling paradigms. One of these is the one of port-Hamiltonian systems.
Port-Hamiltonian (pH) systems are ubiquitous, covering diverse physical domains and allowing easy modularization. This is achieved by recognizing energy as the "lingua franca" between physical domains, and using it both to drive the dynamics of the system and to interconnect different systems in a conservative way. Mathematically, pH systems bring together port-based modeling, geometric mechanics, and systems and control theory. PH systems present a plethora of good qualities, which make them particularly suitable for modeling, analysis, numerics, and control.
In this talk, we will first introduce the modeling paradigm and the geometric interpretation of port-Hamiltonian systems. This will lead us to an alternative formulation as pH differential-algebraic equations (pHDAE) with input and output variables, also known as port-Hamiltonian descriptor systems. After listing some of the good properties of pHDAEs, we will focus on a specific but simple numerical scheme: collocation methods for time integration. In particular, we will show that the Gauss-Legendre collocation scheme always preserves the structure of a certain class of pHDAEs.
The study of the moduli stack Coh(X) of coherent sheaves on a projective scheme X, has far-reaching results in the theory of moduli spaces, non-abelian Hodge correspondence, minimal model program, hyperkähler geometry and beyond. in this talk we present a construction of such a classifying space through the methods of Geometric Invariant Theory.
Coh as a moduli problem: In the first part, we introduce a proper framework for the development of moduli problems. We first define presheaves of sets over the site of schemes. This approach will lead us to the notion of a moduli space (i.e a moduli functor being represented by a scheme). Then we review two of the most well known phenomena that arise whenever this representability is not possible, namely the jump phenomena and the unboundedness of a moduli problem. We give a detailed example that shows that the moduli problem of coherent sheaves is not bounded. Even though we cannot build the desired moduli space, Geometric Invariant Theory tells us how to further restrict the moduli problem in order to salvage a relevant smaller moduli space.
GIT and corepresentability: In the second part, we present the GIT solution to the corepresentability of a moduli problem. First we construct the Quot moduli functor and discuss its representability by a projective scheme, which is suitably embedded into a Grassmannian. Then we define G-linearisations for the action of a reductive algebraic group on a k-scheme over an algebraically closed field k. After that we relate corepresentability of a moduli problem of interest to the problem of finding a quotient for the group action. And finally we present the main theorem of GIT which produces the moduli space of coherent semistable sheaves over X.
In-depth understanding of the bending behaviour of a membrane is often considered essential for explaining cellular properties, as well as some material properties and can lead to advancements in fields of biology and material science. In this talk we present two classical approaches to model a membrane - a deterministic and a stochastic one. We first consider a variational model for a membrane that is strongly attracted to two walls, and characterize optimal membrane shapes by means of Γ-convergence. We show how this approach is linked to the variational model considered above via Large Deviations Principles. The incorporation of various cons- traints is also discussed.
This talk is based on a Master Thesis done under the supervision of B. Zwicknagl and J. Deuschel.
The topic of the present paper has been motivated by a recent computational framework that provides highly scalable identification of reduced Bayesian and Markov relation models and the inclusion of a priori physical information. An efficient approach to date for the identification of finite-time coherent sets in time-dependent dynamical systems on the other hand are transfer operators. Numerical techniques for this purpose rely on the estimation and further numerical processing of a transition matrix. In this article, we reinterpret a directly low-rank transfer operator as a product of the Ulam matrix and a stochastic projection which depends on affiliations to the so-called latent states. Our method can be seen as an extension of Bayesian model reduction directly from trajectory data, and it allows us to identify a low rank projected transfer operator together with the associated transition probabilities. We illustrate the potential of the method on several examples.
We study the Nash system of a (non-necessarily homogeneous) \(n\)-player game of optimal stopping in the presence of common strongly Markovian noise. For linear diffusions, we derive a one-to-one characterisation of subgame-perfect feedback Nash equilibria in terms of this coupled free-boundary type system of differential equations, whence a method of constructing these equilibria and devising conditions on both their existence and number. If the game's rules can be controlled by a principal through a decision at its beginning, the decision problem of choosing between different regimes of equilibria can be mathematically addressed.
A major motivation for this work stems from the challenge of technological transitions in second-best, strategic market interactions under a commonly faced uncertainty, an example being the transition form internal combustion engine to electric vehicles in the transport sector. This project is ongoing doctoral research supervised by Christoph Belak.
Knot theory - the study of embedded 1-spheres in 3-dimensional space - lies at the intersection of many mathematical disciplines: topology, geometry, algebra and combinatorics. A particularly surprising connection, which was first studied in detail by Bill Thurston in the 1970s, is the many knots admit a unique complete hyperbolic structure on their complement, thus allowing us to use geometric tools to define knot invariants.
In this talk, I will give a brief introduction to the relevant knot-theoretical and geometric concepts, and show how one can triangulate the complement of a knot in order to find the hyperbolic structure algorithmically.