Talks

Varieties, sheaves, and polytopes

Klaus Altmann (FU Berlin) - Invited Speaker

Area: Algebraic Geometry / Toric Geometry

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.

Talk (title TBA)

Raghda Abdellatif — Research Talk Abstract pending

Area: Algebraic Geometry

Abstract coming soon.

On Moduli Spaces of Curves

Bogdan Carasca — Expository Talk

Area: Algebraic Geometry

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.

An Introduction to Brill-Noether Theory

Riccardo Redigolo— Expository Talk

Area: Algebraic Geometry

Brill-Noether theory aims to bridge the gap between the classical view of curves as one dimensional solutions to some polynomial equations to the more modern one which sees the same abstract curve map to different projective spaces.

In this expository talk we give a brief overview of the subject and discuss some recent advances.

l^2-Betti Numbers

Elia Immanuel Auer — Expository Talk

Area: Topology

In algebraic topology, one studies algebraic invariants attached to a space X: From the so-called chain complex of the space, one obtains certain abelian groups called the homology groups and finally the Betti numbers, which are the ranks of these homology groups.

In this expository talk, I am going to introduce a variant of the classical Betti numbers called l^2-Betti numbers. These are invariants of a space X with an action of a group G. One starts by taking the Hilbert space completion of the chain complex to obtain the so-called l^2-chain complex, which consists of Hilbert spaces with a G-action. The homology groups of the l^2-chain complex are the l^2-homology groups, and they are again Hilbert spaces with a G-action. These have a natural notion of dimension and the dimensions of the l^2-homology groups will be called the l^2-Betti numbers.

This has applications to group theory: To every group G, one associates a contractible space EG with a free G-action. The l^2-Betti numbers of the group G are defined to be the l^2-Betti numbers of the space EG. These are useful invariants of the group G. In the talk, we will compute multiple examples of the l^2-Betti numbers of spaces and groups.

An introduction to Homotopy Type Theory and its applications

Koen Bresters — Expository Talk

Area: Homotopy Type Theory

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.

Combinatorial Slicing Problems of Polytopes

Marie-Charlotte Brandenburg (Ruhr-Universität Bochum) - Invited Speaker

Area: Discrete Geometry

Given a 3-dimensional cube, its intersection with an affine hyperplane is always a polygon with 3, 4, 5, or 6 vertices. But what are the slices of a general polytope?

In this talk, we examine the combinatorial structure of all possible hyperplane sections of a polytope. Based on this, we develop an algorithm to enumerate all combinatorial types of sections. Finally, we consider slices of the cube in dimensions 4, 5, and 6, and derive conjectures about slices of cubes in any dimension.

More information about Prof. Brandenburg can be found here.

When Combinatorics Meets Commutative Algebra: The g-Theorem

Fatemeh Zeinabadi — Expository Talk

Area: Discrete Geometry

The face numbers of a simplicial polytopes are encoded in its f-vector, a fundamental combinatorial invariant. The celebrated g-theorem gives a complete characterization of which vectors arise as f-vectors of simplicial polytopes. Remarkably, although the statement of the theorem is combinatorial, its proof relies on tools from commutative algebra and algebraic geometry.

In this talk, I will introduce f-vectors and the g-theorem, and briefly discuss generalizations to homology spheres.

Partitioning edge-colored graphs into monochromatic cycles and paths

Johanna Grell — Expository Talk

Area: Graph Theory

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.

More information about Prof. Brandenburg can be found here.

The Hypostasis of Algebra’s Many Avatars

Fawzy Naguib Hegab — Expository Talk

Area: Algebra and Geometry

As an undergraduate student in mathematics, one usually encounters many algebraic structures: groups, rings, and Lie algebras are among the most common. As one learns these topics, it becomes clear that there are many similarities between them: homomorphisms, kernels, images, quotients, isomorphism theorems, and so on.

Motivated by these analogies between various theories*, one might hope that there is a universal notion of algebraic structure of which familiar theories such as groups, rings, and Lie algebras are examples. In this talk, we will present an exposition of various known approaches to formalizing this general notion and highlight some of their power and centrality in modern mathematics.

*"A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories" - Stefan Banach

Was ist und was soll die Geometrie?

Fawzy Naguib Hegab — Expository Talk

Area: Algebra and Geometry

First of all, this is a talk in English, not German. So why is the title in German? Come to the talk to find out!

This philosophically oriented talk discusses the modern viewpoint on geometry that has emerged over the last few decades. Our aim is to be conceptual, broad, and example-oriented rather than technically pedantic. By the end of the talk, you should have an intuitive understanding of what the equation

Geometry=algebraic data+spatial data

is really trying to say.

Homotopy colimits in topological combinatorics

Bálint Zsigri — Research Talk

Area: Topological Combinatorics

I will give a brief introduction to homotopy colimits and their applications in topological combinatorics, sampling the work of Ziegler and Živaljević about the wedge decomposition of the one-point compactification of a union of linear subspaces in R^n. I conclude by saying a few words about my research project in which I give a homological analogue of the Wedge Lemma, the main tool leading to the aforementioned result.

Braided open books

Chun-Sheng Hsueh — Research Talk

Area: Geometric Topology

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.

On Discrete Geometry of Neural Networks

Andrei Balakin — Research Talk

Area: Discrete Geometry

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.

Guaranteed plate eigenvalues with adaptivity

Benedikt Gräßle (University of Zürich) - Invited Speaker

Area: Numerical analysis

Spectral problems sit at the crossroads of spectral geometry, computational mathematics, and mechanics. For the Laplacian, Weyl’s law and the famous question “Can you hear the shape of a drum?” exemplify how eigenvalues encode geometry. In thin-plate models, an analogous role is played by the biharmonic operator: its spectrum governs both vibration frequencies and stability thresholds, for instance in the safety analysis of materials.

This lecture gives a gentle introduction to certified eigenvalue computations for plates. I will highlight the basic asymmetry behind eigenvalue bounds: upper bounds follow directly from variational principles, whereas guaranteed lower bounds require carefully designed discretisations and additional machinery. The focus is on the central ideas behind modern adaptive algorithms that provide reliable spectral enclosures together with accurate and efficient numerical results.

More information about Dr. Gräßle can be found here.

Finite Volume Methods For Regularised Dean-Kawasaki equations

Ana Damnjanović — Research Talk

Area: Stochastics & Numerics

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 Adam have to do with symplectic manifolds? The geometry behind momentum methods in machine learning

Viktor Stein — Expository Talk

Area: Applied Analysis

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.

Interface Problems

Georgi Mitsov — Expository Talk

Area: Numerics

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.

Mixing in Fluids -- The Batchelor Scale

Robin Chemnitz — Research Talk

Area: Stochastics

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.