Math Seminar at CSBD

This is the website for the Math Seminar at CSBD. Upcoming talks and abstracts are posted here. The seminar, if not announced differently, takes place on Thursdays at 3pm. Currently, the organizers are Nikola Sadovek, Maximilian Wiesmann and Giulio Zucal. Feel free to reach out to them if you would like to suggest a speaker or for any organizational questions.

Each talk is scheduled for 50 minutes, with an additional 10 minutes for questions.

Location: CSBD (location on Google maps), top floor seminar room.

Upcoming Talks

Date	Speaker	Affiliation	Title	Abstract
Mar 05	Bernat Jordà Carbonell	VU Amsterdam	Hypergraph homologies (BONUS: ECT for paleontology and the origin of life)	Hypergraphs extend graphs and simplicial complexes by representing higher-order interactions with more degrees of freedom. However, the absence of downward closure makes the usual chain-level boundary operators fail. How to get around this is not trivial and different authors have proposed different approaches that get to completely different homologies. While trying to better understand what information is relevant from the hypergraphs, we notice that one of the most promising homology theories (relative barycentric homology) is equivalent to the compact support homology of our geometric realisation of a hypergraph. We prove a natural isomorphism between these homologies and determine the functoriality of the former. BONUS: How can we adapt TDA tools generally used for biology to paleontology? In Paleo we don’t have access to 3d structures, only images of sections of rock that might be degraded and transformed after such a long time. More specifically, if we want to bring life to the scientific discussion on origins of life, we need to look at fossils of bacteria. The small scale and degradation makes this task quite complicated and the use of morphometry has not been applied successfully (yet!).
Mar 12	Peter Elias Schuster	University of Bonn	tba	tba
Mar 19	Andreas Thom	TU Dresden	On planar sections of the dodecahedron	In the analysis of three-dimensional biological microstructures such as organoids, microscopy frequently yields two-dimensional optical sections without access to their orientation. Motivated by the question of whether such random planar sections determine the underlying three-dimensional structure, we investigate a discrete analogue in which the ambient structure is the vertex set of a Platonic solid and the observed data are congruence classes of planar intersections. For the regular dodecahedron with vertex set $V$, we define the planar statistic of a subset $X \subseteq V$ of vertices as the distribution of isometry types of inclusions $\Pi \cap X \subseteq \Pi\cap V\subseteq V$, and ask whether this statistic determines $X$ up to isometry. We show that this is not the case: there exist two non-isometric 7-element subsets with identical planar statistics. As a consequence, there exist two polytopes in $\mathbb{R}^3$, whose distribution of isometry classes of two-dimensional intersections is identical, while the polytopes are not themselves isometric. This result is an analogue of classical non-uniqueness phenomena in geometric tomography.
Mar 26	Yelena Mandelshtam	University of Michigan	tba	tba
Apr 02	Mario Kummer	TU Dresden	tba	tba
Apr 09	Joscha Diehl	University of Greifswald	tba	tba
Apr 16	Selvi Kara	Bryn Mawr College	A Three-Regime Theorem for Flow-Firing	Flow-firing, introduced by Felzenszwalb-Klivans, is a 2D analogue of chip-firing: an integer flow on the edges of a cell complex evolves by repeatedly applying local rerouting moves around faces. In their original work, the only proven confluent family on the grid stabilized (independent of firing choices) into an Aztec diamond, a centered diamond-shaped patch of unit squares. In this talk I will explain how far this phenomenon extends. For a natural family of conservative “pulse” initial conditions, we prove a three-regime theorem: there is a small-support regime with unique stabilization to the Aztec diamond, an intermediate regime where stabilization occurs but the terminal state is not unique (though the Aztec diamond can still occur), and a large-support regime where confluence fails, including a range where the Aztec-diamond outcome is impossible.
Apr 23	Eckehard Olbrich	MPI MiS	tba	tba
Apr 30	Leonie Kayser	MPI MiS	tba	tba
May 07	Jason P. Smith	Nottingham Trent University	tba	tba
May 14	public holiday	no talk

Past Talks

Date	Speaker	Affiliation	Title	Abstract
Feb 26	Hannah Friedman	UC Berkeley	Counting Homogeneous Einstein Metrics	The problem of finding homogeneous Einstein metrics on a compact homogeneous space reduces to solving a system of Laurent polynomial equations. This is an example of a vertically parametrized system; such systems arise in algebraic statistics and chemical reaction networks. We prove that the number of isolated solutions of this system is bounded above by the central Delannoy numbers and we describe the discriminant locus where the number of isolated solutions drops in terms of the principal A-determinant.
Feb 19	Dominik Sturm	MPI CBG	Robust variable selection for spatial point processes observed with noise	We propose a method for variable selection in the intensity function of spatial point processes that combines sparsity-promoting estimation with noise-robust model selection. As high-resolution spatial data becomes increasingly available through remote sensing and automated image analysis, identifying spatial covariates that influence the localization of events is crucial to understand the underlying mechanism. However, results from automated acquisition techniques are often noisy, for example due to measurement uncertainties or detection errors, which leads to spurious displacements and missed events. We study the impact of such noise on sparse point-process estimation across different models, including Poisson and Thomas processes. To improve noise robustness, we propose to use stability selection based on point-process subsampling and to incorporate a non-convex best-subset penalty to enhance model-selection performance. In extensive simulations, we demonstrate that such an approach reliably recovers true covariates under diverse noise scenarios and improves both selection accuracy and stability. We then apply the proposed method to a forestry data set, analyzing the distribution of trees in relation to elevation and soil nutrients in a tropical rain forest. This shows the practical utility of the method, which provides a systematic framework for robust variable selection in spatial point-process models under noise, without requiring additional knowledge of the process.
Feb 12	Pardis Semnani	University of British Columbia	Path-Dependent SDEs: Solutions and Parameter Estimation	In this talk, we discuss how temporal causal structures can be modelled using a path-dependent stochastic differential equation (SDE). We then consider a rich class of path-dependent SDEs, called signature SDEs, which can model general path-dependent phenomena. We provide conditions that ensure the existence and uniqueness of solutions to a general signature SDE. Path signatures are iterated integrals of a given path with the property that any sufficiently nice function of the path can be approximated by a linear functional of its signatures. This is why we model the drift and diffusion of our signature SDE as linear functions of path signatures, and then introduce the Expected Signature Matching Method (ESMM) for linear signature SDEs, which enables inference of the signature-dependent drift and diffusion coefficients from observed trajectories. Furthermore, we show that the ESMM is consistent: given sufficiently many samples and Picard iterations used by the method, the parameters estimated by the ESMM approach the true parameter with arbitrary precision. We discuss the asymptotic distribution of the estimator obtained from the ESMM, and finally, demonstrate on a variety of empirical simulations that our ESMM accurately infers the drift and diffusion parameters from observed trajectories. While parameter estimation is often restricted by the need for a suitable parametric model, this study makes progress toward a completely general framework for SDE parameter estimation, using signature terms to model arbitrary path-independent and path-dependent processes.This talk is based on joint work with Vincent Guan, Elina Robeva, and Darrick Lee.
Feb 05	Martin Keller-Ressel	TU Dresden	Polynomial-preserving Stochastic Processes and Applications	Polynomial processes are a large class of stochastic processes, for which moments and related quantities, such as their auto-covariance, can be efficiently computed. Mathematically, they can be characterised as Markov processes, whose transition semigroup leaves the vector spaces of polynomials of given degree invariant. In my talk I give an overview of their general theory and discuss an application to a model of RNA transcription by Gorin, Vastola, Feng and Pachter. I also present some recent non-Markovian generalisations of polynomial processes and their connection to the algebraic theory of K-positivity preservers.
Jan 29	Alvaro Ribot	Harvard University	Beyond independent component analysis: identifiability and algorithms	Independent Component Analysis (ICA) is a classical method for recovering latent variables with useful identifiability properties. However, full independence is a strong assumption that may not hold in many real-world settings. In this talk, I will discuss how much we can relax the independence assumption without losing identifiability of the model. We show that the weakest such assumption is pairwise mean independence. Our identifiability result is based on a generalization of the spectral theorem from matrices to higher-order tensors, which implies a unique tensor decomposition of the cumulant tensors arising in the model. This is joint work with Anna Seigal and Piotr Zwiernik.
Jan 26	Michael Hecht	University Wrocław, Helmholtz-Zentrum Dresden-Rossendorf	The Fast Newton Transform (FNT)	The FNT is a novel algorithm for multivariate polynomial interpolation with a runtime of nearly Nlog(N), where N scales only sub-exponentially with spatial dimension, surpassing the runtime of the tensorial Fast Fourier Transform (FFT). We have proven and demonstrated the optimal geometric approximation rates for a class of analytic functions—termed Bos–Levenberg–Trefethen functions—to be reached by the FNT and to be maintained for the derivatives of the interpolants. This establishes the FNT as a new standard in spectral methods, particularly suitable for high-dimensional, non-periodic PDE problems, interpolation tasks, arising as the computational bottleneck in solving, e.g. 6D Boltzmann, Fokker-Planck, or Vlasov equations, multi-body Hamiltonian systems, and the inference of governing equations in complex self-organizing systems.
Jan 22	Ivan Spirandelli	University of Potsdam	Geometric and topological potentials driving self-assembly	The assembly of molecular building blocks into functional complexes is central to biology and materials science. We investigate the generative and predictive capabilities of a geometric model, the morphometric approach to solvation free energy, in a simulation setting. We show that biologically relevant structural motifs appear for generic building blocks under geometric optimization. Applying the same method to the self-assembly of protein subunits, we show that geometric fit alone predicts the native nucleation states of various systems. To overcome limitations in efficiency of the geometric model caused by its short-range nature, we introduce a novel energetic bias based on persistent homology. By combining these shape-based potentials we obtain an efficient simulation strategy increasing success rates by an order of magnitude, or enabling assembly in the first place, when compared to the geometric model alone. Integrating topological descriptions into energy functions offers a general strategy for overcoming kinetic barriers in molecular simulations, with potential applications in drug design, material development, and the study of complex self-assembly processes.
Jan 15	Jannis Kremer	University of Bonn	VC Dimension of DTW-Based Classifiers	This thesis investigates the VC dimension of classifiers defined by Dynamic Time Warping (DTW) metric balls. We analyze the geometric structure of these DTW balls and demonstrate that they can be characterized as specific types of ellipsoids. By applying classical results, such as Warren’s theorems, we derive upper bounds on the VC dimension. Furthermore, we employ techniques from discrete geometry and convex optimization to computationally explore lower bounds.
Jan 09	Bernd Sturmfels	MPI MiS	Lines in 3-Space	We consider configurations of lines in 3-space with incidences prescribed by a graph. This defines a subvariety in a product of Grassmannians. Leveraging a connection with rigidity theory in the plane, for any graph, we determine the dimension of the incidence variety and characterize when it is irreducible or a complete intersection. We study its multidegree and the family of Schubert problems it encodes. Our spanning-tree coordinates enable efficient symbolic computations. We also provide numerical irreducible decompositions for incidence varieties with up to eight lines. These constructions with lines play a key role in the Landau analysis of scattering amplitudes in particle physics.
Dec 11	Sjoerd Van der Niet	Renyi Institute Budapest, TU Munich	Spectral convergence of Laplacians on dense hypergraph sequences	Higher-order networks have become a popular tool in the network science community to model dynamics such as synchronization and diffusion. The linearized system often depends on a Laplacian operator and its spectral properties. We introduce a Laplacian operator for uniform hypergraphs and study the limiting operator for an increasing sequence of dense uniform hypergraphs using the theory of graph limits. Although a theory of dense hypergraph limits has been developed by Elek and Szegedy, and independently Zhao, not much of its implications to spectral properties is known. We show that a weaker notion of convergence for the sequence of hypergraphs is sufficient to obtain pointwise convergence of the spectrum of the Laplacians.
Dec 04	Timo de Wolff	TU Braunschweig	An Introduction to Nonnegativity and Polynomial Optimization	In science and engineering, we regularly face (constrained) polynomial optimization problems (CPOP). That is the task to minimize a real, multivariate polynomial under polynomial constraints. Solving these problems is essentially equivalent to certifying nonnegativity of real polynomials - a key problem in real algebraic geometry since the 19th century. Since this is a notoriously hard to solve problem (e.g., various NP-complete problems admit a CPOP formulation), one is interested in certificates that imply nonnegativity and are easier to check than nonnegativity itself. In particular, a polynomial is nonnegative if it is a sums of squares (SOS) of other polynomials. Being an SOS can be detected effectively via semidefinite programming (SDP) in practice. In 2014, Iliman and I introduced a new certificate of nonnegativity based on sums of nonnegative circuit polynomials (SONC), which I have developed further since then both in theory and practice joint with different coauthors. Circuit polynomials are a particular type of very sparse polynomial, which allow to decide nonnegativity easily. SONC certificates are interesting both from a theoretical and practical viewpoint, as they are independent of sums of squares and can be computed effectively via relative entropy programs. In this talk, I will give an introduction to polynomial optimization, nonnegativity, and the role of sparsity within these problem ensembles. I will moreover introduce SOS and SONC, and, give some examples of applications of nonnegativity certificatess.
Nov 27	Daniel McGinnis	Princeton University	A necessary and sufficient condition for $k$-transversals	We solve a long-standing open problem posed by Goodman and Pollack in 1988 by establishing a necessary and sufficient condition for a family of convex sets in $\mathbb{R}^d$ to admit a $k$-transversal (a $k$-dimensional affine subspace that intersects each set in the family) for any $0 \le k \le d-1$. This result is a common generalization of Helly’s theorem ($k=0$) and the Goodman-Pollack-Wenger theorem ($k=d-1$). Our approach is topological and employs a Borsuk-Ulam-type theorem on Stiefel manifolds. This is joint work with Nikola Sadovek.
Nov 20	Clemens Brüser	TU Dresden	Smoothness and Determinantal Representations of Adjoint Hypersurfaces	Adjoint polynomials of convex polytopes have recently received attention from the field of particle physics, and the question has been raised whether they admit determinantal representations. In this talk we define the notion of adjoint polynomials/hypersurfaces and characterize them through their degree and a simple vanishing condition. Through this vanishing condition we derive a certificate for the existence of singularities on the adjoint hypersurface. We then survey the classical theory on determinantal representations. We prove that the adjoint curve of a polygon always has a natural symmetric determinantal representation that certifies hyperbolicity. For three-dimensional polytopes we show that if the adjoint is smooth, then a determinantal representation exists. The methods to find these representations are computationally viable. There are also some negative results for higher dimensions. The presented results are based on joint work with Mario Kummer and Dmitrii Pavlov (both TU Dresden) and with Julian Weigert (MPI-MIS Leipzig).
Nov 13	Thomas Bouchet	MPI CBG	An invariant theoretic approach to algebraic curves	In this talk, I will present key tools from invariant theory and show how they can be used for explicit computations with algebraic curves. I will begin by introducing invariants that classify curves of a given genus up to geometric isomorphism. Beyond providing explicit equations for moduli spaces of curves, these invariants play a major role in constructing explicit examples of curves. Then, I will introduce the notion of covariants, and explain how one can reconstruct a curve/hypersurface from its invariants. I will illustrate this process through examples of curves with “interesting properties” obtained in this way. Finally, I will show how covariants can provide an efficient way to compute linear changes of variables between homogeneous polynomials, largely outperforming existing implementations.
Nov 06	Iolo Jones	Durham University	Computing Diffusion Geometry	Calculus and geometry are ubiquitous in the theoretical modelling of scientific phenomena, but have historically been very challenging to apply directly to real data as statistics. Diffusion geometry is a new theory that reformulates classical calculus and geometry in terms of a diffusion process, allowing these theories to generalise beyond manifolds and be computed from data. In this talk, I will describe a new, simple computational framework for diffusion geometry that substantially broadens its practical scope and improves its precision, robustness to noise, and computational complexity. We introduce a range of new computational methods, including all the standard objects from vector calculus and Riemannian geometry, spatial PDEs and vector field flows, and topological features like cohomology, circular coordinates, and Morse theory. These methods are fully data-driven, parameter-free, scalable, and can be computed in near-linear time and space.
Oct 30	Lin Wan	Academy of Mathematics and Systems Science, Chinese Academy of Sciences; ELBE Visiting Faculty of CSBD	Learning Collective Multicellular Dynamics with an Interacting Mean-Field Neural SDE Model	The advent of temporal single-cell RNA sequencing (scRNA-seq) data has enabled in-depth investigation of dynamic processes in heterogeneous multicellular systems. Despite remarkable advancements in computational methods for modeling cellular dynamics, integrating cell-cell interactions (CCIs) into these models remains a major challenge. This is particularly true when dealing with high-dimensional gene expression profiles from large populations of interacting cells, where the intricate interplay between cells can be obscured by data complexity. In this talk, I will present our recent work on a neural interacting mean-field stochastic differential equation (SDE) framework for temporal scRNA-seq data. Our approach combines mean-field modeling with neural networks to learn the dynamics of large, interacting cell populations directly from data. It enables the reconstruction of intrinsic cell population trajectories and the systematic characterization of CCIs. Notably, the model uncovers biologically interpretable, non-reciprocal interaction patterns and offers a principled way to study complex, non-equilibrium multicellular systems.
Oct 23	Oskar Henriksson	MPI CBG	Exploring the parameter space of polynomial systems using pseudo-witness sets	Polynomial systems that arise in applications are often parametric (in the sense that the coefficients depend on some model parameters), and many problems in applied math boils down to understanding what possible geometries the solution sets to such systems can take as the parameters vary. A key tool for studying such questions is the discriminant variety, which partitions the parameter space into regions of constant qualitative and quantitative properties of the solutions. A common challenge, however, is that many methods rely on having access to an explicit equation for the discriminant, which generally requires solving a costly elimination problem. In this talk, I will present a new approach for finding sample points in all connected components of the complement of a discriminant variety, which uses pseudo-witness sets to circumvent the need for symbolic elimination.
Oct 16	Renee Hoekzema	Free University of Amsterdam	Spectral gene selection methods and models for host/parasite co-phylogeny	I will talk about two disjoint projects. Firstly, I will talk about an application of spectral graph theory to the study of single cell transcriptomics, in particular the problem of feature selection of relevant genes in such experiments. Single cell transcriptomics is a powerful technique in biology that allows for the measurement of gene expression levels in many individual cells simultaneously. Current methods for analysis assume that cell types are discrete. However, in practice there is also continuous variation between cells: subtypes of subtypes, differentiation pathways, responses to environment or treatment, et cetera. We propose topologically-inspired data analysis methods that identify coherent gene expression patterns considering discrete and continuous patterns on equal footing. This is joint work with Lewis Marsh, Otto Sumray, Thomas Carroll, Xin Lu, Helen Byrne and Heather Harrington. Secondly, I will talk about ongoing work with Gillian Grindstaff on models for co-evolution of “nested’’ systems, such as parasite/host systems, individuals within a species, or “phylosymbiosis” – the coupled evolution of the microbiome and its hosts. We create a space of nested phylogenetic trees and study its intricate geometry. In particular we show that this space is CAT(0) – in analogy with the influential work of Billera, Holmes and Vogtmann (2001) – implying the existence of unique averages over nested trees.
Oct 09	Edmilson Roque	MPI PKS	Ergodic basis pursuit leads to robust reconstruction of sparse network dynamics	Networks of coupled dynamical systems are successful models in diverse fields of science, ranging from physics to neuroscience. The network interaction structure impacts the dynamics; in fact, many malfunctions are associated with disorders in the network structure. Yet, typically, we cannot measure the interaction structure; we only have access to multivariate time series of nodes’ states. This led to considerable effort in reconstructing the network from multivariate data. This reconstruction problem is ill-posed for large networks, leading to the reconstruction of false network structures. In this talk, I will present an approach that uses the network dynamics’ statistical properties to ensure the exact reconstruction of weakly coupled sparse networks. Moreover, this approach exhibits robustness against noise. I will illustrate its reconstruction power using experimental multivariate time series data obtained from optoelectronic networks.
Oct 02	Sabina Haque	University of Michigan Ann Arbor	Graph-theoretic and algebraic geometric approaches to biochemical reaction networks	Under mass-action kinetics, systems of biochemical reactions are modeled by chemical reaction networks (CRNs), a class of graphs that gives rise to polynomial dynamical systems. Approaches in this field include chemical reaction network theory and the more recent linear framework. In this talk, I will focus primarily on the linear framework, a graph-theoretic approach to timescale separation in biochemical systems. I will discuss a graph-theoretic construction within the framework that mimics what would happen if a single parameter in a graph is taken to infinity, producing what we call an asymptotic graph. I consider how properties of the asymptotic graph, such as its steady states, serve as an appropriate representation for a linear framework graph in this limit. I also speculate on some extensions of this construction beyond the scope of the linear framework to parameter identifiability and the steady state varieties of CRNs, suggesting areas for future work at the intersection of graph theory, algebraic geometry, and dynamical systems.
Sep 25	Cerene Rathilal	University of Kwa-Zulu Natal	On frames and the Peano compactification	This talk will provide an introduction to pointfree topology and have a focus on some recent work on compactifications of frames. In [Curtis (1980): Hyperspaces of Noncompact Metric Spaces], Curtis introduced the concept of a locally non-separating remainder in order to study the hyperspace of a non-compact space $X$. Using the property of a locally non-separating remainder, Curtis established the conditions under which a Peano compactification of a connected space $X$ would exist. In this talk, we will present the analog of the concept of locally non-separating sets, in frames. We will discuss properties of sublocales, after which we define a locally non-separating sublocale and conclude by providing a generalisation for a special case of Curtis’s result.