The student talks displayed on this page are ordered alphabetically by the speakers' first names. See the schedule and venue on the linked pages.
Our conference features plenary talks from three invited speakers and student talks:
Projective spaces or, more general, so-called toric varieties X can be encoded by convex polyhedra. Actually, these gadgets carry more information than just that of the variety—namely also that of special projective embeddings of X.
The latter is linked to so-called invertible sheaves on X. We will explain both this notion and relationship. Moreover, we show that even the cohomology of these sheaves can be read off the polyhedra.
More information about Prof. Altmann can be found here.
Abstract coming soon.
Abstract coming soon.
Stochastic partial differential equations (SPDEs) and PDEs are often used to model the evolution of particle densities in systems with many interacting particles. In this talk, I will introduce the Dean–Kawasaki SPDE and discuss some of the difficulties that arise when studying its solutions. We are particularly interested in how solutions to such equations can be approximated numerically.
"What does discrete geometry have to do with neural networks? An important class of neural networks (namely, those with ReLU activations) represents exactly piecewise-linear functions. These are, among others, application areas of combinatorial and polyhedral techniques. In the talk, I will give a short introduction to the developing research field of polyhedral geometry in neural networks and present an example to illustrate how polytopes can be used to study networks."
Abstract coming soon.
We give a general overview of moduli theory and it's place in mathematics. The main example of our talk will be the moduli space of curves with an emphasis on its birational geometry.
Every fibered link in the 3-sphere arises as the binding of an open book. It remains unknown whether this can be strengthened to say that every such link arises as the binding of a braided open book. I will introduce braided open books and discuss recent progress on this question, based on joint work with B. Bode.
Abstract coming soon.
Abstract coming soon.
Abstract coming soon.
Interface Problems is a broader topic in the field of Numerical Methods for Partial Differential Equations. It can be seen as an intermediate step between the continuous and the discrete problem that gives access to both the design and the analysis (stability, error control) of the resulting numerical method. Of particular interest and recent development is the paradigm of discontinuous Petrov-Galerkin methods.
"A complete graph consists of a finite set of vertices, with an edge joining every two distinct vertices. When the edges of such a graph are colored, certain monochromatic structures turn out to be unavoidable, regardless of how the coloring is chosen. An example of this phenomenon is a result conjectured by Lehel: every complete graph whose edges are colored red and blue admits a partition of its vertices into one red cycle and one blue cycle. In this talk, we present the main ideas behind the proof by Bessy and Thomassé. No prior knowledge of graph theory is assumed, as the proof relies only on elementary combinatorial arguments."
"Homotopy type theory (HoTT) is a formal system for mathematics which rather than sets and propositions, takes 'types' and functions as its building blocks. One can think of types as collections of mathematical objects with a notion of isomorphism between them built-in. Every construction in HoTT is *automatically* invariant under isomorphism, which mimics patterns observed in mathematical practice, and has concrete advantages. For example, proving an object satisfying a certain universal property exists, is the same as giving a construction of it; allowing one to avoid the use of the axiom of choice in category theory. Alternatively one can interpret types as 'homotopy types', with terms corresponding to points, isomorphisms corresponding to paths, isomorphisms between isomorphisms corresponding to homotopies of paths, etc. This allows one to use the very same theory to prove statements in homotopy theory, completely computer verifiably, without ever mentioning topological spaces. Even more generally, HoTT can be interpreted in any ∞-topos. Choosing this topos carefully allows one to reason about higher categories (notoriously hard to define objects) in a very efficient way. In this talk I'll give an introduction to homotopy type theory, show some of the advantages of taking it as your mathematical foundation and sketch some of these applications beyond just foundational concerns."
Abstract coming soon.
When pouring some milk into hot tea, the milk forms increasingly filamented patterns before eventually becoming a uniform blend of milky tea. Mathematically, this process is modelled by the advection-diffusion equation. It is conjectured that the filementations of a passive scalar (the milk) get smaller and smaller until they reach the Batchelor lenghscale. In this talk, passive scalar advection and the Batchelor scale are introduced and we provide a simple stochastic fluid model for which the Batchelor scale conjecture can be verified.
"Gradient-based optimization algorithms are indispensable for most modern machine learning applications, since they are used to train convolutional neural networks (CNNs) for image classification or transformers for natural language processing. Often, the weights are optimized using the Adam algorithm, a momentum-based modification of gradient descent. In the vanishing step-size limit, these momentum-based algorithms can be described as second-order damped dynamics with a time-dependent Hamiltonian interpretation. In this talk, I will gently introduce the cocontact / conformally symplectic geometry behind time-dependent Hamiltonian systems and show how viewing momentum-based methods in machine learning through this geometric lens can help elucidate their properties and aid in designing new momentum-based optimization algorithms on infinite-dimensional manifolds of probability measures."