CSBD Math Seminar

In this talk, I will introduce cross-ratio puzzling, a new quartet-based algorithm for phylogenetic reconstruction that combines ideas from quartet puzzling with recent work on cross-ratio degrees. I will present preliminary results from applying the algorithm to sequence-alignment data sets. I will also highlight the interplay between the algebraic structure of cross-ratio degrees and the discrete problem of reconstructing phylogenetic trees from quartets.

In this overview talk, I will discuss different techniques that are required for the study of real algebraic and semialgebraic sets. Tarki's transfer principle is the key algebraic ingredient: a semialgebraic formula defines a nonempty semialgebraic set over the real numbers if and only if the same formula defines a nonempty semialgebraic set over an arbitrary real closed field extension. For statements which are not semialgebraic in nature, the tranfer principle fails. One is then forced to introduce metric considerations, which typically take the form of a "distance from the discriminant". I will present this dual perspective by concrete examples, arising from the study of positive polynomials and (if time permits) of real algebraic curves.

The talk is aimed at sharing my own experiences of understanding new mathematics from what I have been working on.

The flows of tissues of epithelial cells often involve neighbour exchanges called T1 transitions. Mechanically, they are irreversible rearrangements crossing an energy barrier. In this talk, I will deploy geometric constructions of classical Euclidean geometry to calculate this energy barrier for general, isolated T1 transitions dominated by line tensions. I will show how regularity of cell packings, tension fluctuations, and nonlinear tensions increase this energy barrier, providing the basis for coarse-grained understanding of cell neighbour exchanges in continuum models of epithelia.

I will describe the synchronization behavior of discrete-time Markov chains on countable state spaces. Representing a Markov chain in terms of a random dynamical system, which describes the collective dynamics of trajectories driven by the same noise, allows for the characterization of synchronization via random attractors. We establish the existence and uniqueness of a random attractor under mild conditions and show that forward and pullback attraction are equivalent in our setting. Additionally, we provide a sufficient condition for reaching the random attractor, or synchronization respectively, in a time of finite mean. This is joint work with Robin Chemnitz and Guillermo Olicón-Méndez

Communication between single cells or higher organisms by means of diffusive compounds is an important phenomenon in biological systems. A straightforward model is by a diffusion equation with suitable flux boundary conditions at the cell boundaries. Such a model will become computationally inefficient and analytically complex when there are many cells, even more so when they are moving. We propose to consider also a point source model. Each cell is virtually reduced to a point and appears in the diffusion equation for the compound on the full spatial domain as a singular reaction term in the form of a Dirac delta measure located at the cell’s centre. The amplitude of the Dirac delta measure is a nonlocal term of the compound’s concentration near the virtual cell boundary so as to preserve the essential biological features. To investigate the positivity of the solution and the structure of steady states, we employ the Laplace transform. In addition, the theory of elliptic curve is also involved.

Since the invention of the compound microscope in the early seventeenth century, scientists have marvelled over red blood cells and their surprising shape. An influential model of Canham predicts the shapes of blood cells and similar biomembranes come from a variational problem minimizing the “bending energy” of these surfaces. Because observed cells have the same shape in humans, it is natural to ask whether the model admits a unique solution. Yu and Chen reduced solution uniqueness for the genus one Canham problem (for a range of isoperimetric ratios) to proving positivity of a P-recursive sequence defined by an explicit linear recurrence relation with polynomial coefficients. In this talk we discuss a method of proving this positivity property, joint with Marc Mezzarobba, and its generalization, with Mezzarobba and Ruiwen Dong, to a wide range of P-recursive sequences. We combine rigorous numeric analytic continuation of D-finite functions with classic bounds from singularity analysis to derive an effective index where the asymptotic behaviour of the sequence, which is positive, dominates the sequence behaviour. Positivity of the finite number of remaining terms can then be checked computationally. Our work has been incorporated into the SageMath ore_algebra package, and can be used by researchers to automatically prove positivity for “generic” positive P-recursive sequences.

Hill functions, $H_{K,n}(x) = \frac{x^n}{K^n + x^n}$, are widely used to model biological input-output responses. While long treated as empirical fits, Martinez-Corral, Nam, DePace, and Gunawardena proposed Hill functions as the universal Hopfield barrier for sharpness of systems at thermodynamic equilibrium. Their case rested on numerical evidence for $n=4$ and $6$. I will introduce a new measure of sharpness and use it to prove that this barrier holds for all $n$.

I will discuss stochastic systems of interacting particles with non-overlapping constraints, which give rise to so-called excluded-volume interactions. The aim is to derive effective macroscopic equations governing the evolution of particle densities from the underlying microscopic dynamics. When particles possess nontrivial size or shape, geometric constraints become essential: they complicate the coarse-graining process and strongly influence the emergent behaviour of the system. I will present two representative examples, hard spheres and infinitely thin needles, highlighting how geometry enters the macroscopic description.

Topological Data Analysis via persistent homology is a new emerging area of data analysis that uses methods from simplicial topology. The persistent homology of a data set can be calculated using a simple algorithm called the reduction algorithm. In this talk, I will present a new construction of worst-case examples for this algorithm. Our constructions are similar to the worst-case examples introduced by Morozov, but replace the single-triangle arrangement with a strip formed by base and fin triangles. This structure allows us to give an explicit algorithm for their construction and to perform experiments comparing the runtime of different versions of the reduction algorithm. We further show that, after suitable edge and triangle subdivisions, these strip examples remain worst-case and can be realized as clique complexes of filtered graphs, and hence as Vietoris–Rips complexes of finite point clouds for a sequence of scale parameters.

Hematopoietic stem cells (HSCs) sustain lifelong blood production by balancing self-renewal and differentiation, with stemness emerging as a reversible property shaped by cell-environment interactions. These dynamic interactions can be formally captured through mechanistic models that allow for reversible transitions between different cellular states, introducing intrinsic heterogeneity into the system. Mathematical frameworks range from ordinary differential equations to stochastic compartmental models and have been developed and validated against diverse biological phenomena, including transplantation assays, label dilution experiments, phenomena of stem cell aging and the pathogenesis and treatment of acute and chronic leukemias. I will highlight a few of applications of such integrative approaches with a focus towards clinical relevance. As one example, longitudinal clonal tracking demonstrates that stochastic two-compartment models with reversible quiescence more accurately reproduce observed clone size distributions than homogeneous models. Furthermore, scale-based analyses reveal that long-term response to tyrosine kinase inhibitor treatment in chronic myeloid leukemia is governed by rare leukemic stem cell activation, thereby supporting dose de-escalation strategies in maintenance therapy. The examples illustrate the potential of mechanistic models to connect experimental data with clinical practice, providing a formal framework for risk stratification, optimizing therapies and advancing personalized medicine in hematological diseases.

We introduce a new homology theory for graphs and digraphs called avalanche homology, which is based on the Abelian sandpile model. This homology theory captures not just the structure of the graph but the dynamics of the graph. We will prove some combinatorial results on avalanche homology for simple digraphs, and present some links and comparisons to other theories such as nerve complexes of circular arcs and burning homology. Finally we introduce a persistent variant of avalanche homology which results in a new measure between network dynamics.

Two fundamental optimization problems in linear time-invariant system theory are the System Identification (SI) problem and the Model Order Reduction (MOR) problem: Given a finite discrete time signal in $\mathbb{R}^N$ (SI) or a higher-order LTI system described by a transfer function of degree $N$ (MOR), the goal is to approximate it by a model of degree $n<N$ such that the approximation error is minimized. In this talk we will present critical equations for these optimization problems and study their algebraic degree. While System Identification corresponds to Euclidean distance minimization to the variety of low-rank Hankel matrices $X_{N,n}$, MOR can be seen as an infinite-dimensional limiting analogue $X_{\infty,n}$ of the former. We also take a look at different weighted norms and present new conjectural values for specific Euclidean Distance degrees of $X_{N,n}$. Our results rely on transforming the critical equations into degeneracy loci of vector bundle maps on projective space and applying Thom-Porteous formula. This is based on joint work with Sibren Lagauw from KU Leuven.

In 2010 Williams and Beer proposed a new framework to quantify multivariate dependencies including redundancies and synergies. Being a small community first, the framework is meanwhile used and applied in many fields including neuroscience and machine learning. In my talk I will provide my perspective on its history and present state, the contribution of our group in Leipzig, the relationship to higher order interactions and some ideas about using the framework in the context of causal inference.

By Alexander's theorem, every link in the 3-sphere can be represented as the closure of a braid. Lorenz links and twisted torus links are two well-studied families that admit explicit braid descriptions. In this talk, we introduce a natural generalization of these families that encompasses all links in the 3-sphere, yielding a simpler braid description for all links in the 3-sphere. This introduces a new perspective in knot theory.

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.

Tensor train varieties are parametrized projective varieties used to approximate the solutions to the electronic Schroedinger equation in second quantization. In particular, Rayleigh-Ritz optimization on these varieties plays a prominent role. In the talk, I will use Rayleigh-Ritz optimization as a motivation to study tensor trains based on work with Borovik, Friedman, and Pfeffer. Then I will focus on the equations of the defining ideal of any tensor train variety. They consist of minors of particular flattenings of the underlying tensors and these equations were conjectured by Sturmfels in an unpublished note. The proof uses the description of the ideal of the general Markov model on trees by Draisma and Kuttler. Time permitting I will touch on Groebner bases, at least for the case of tensor trains in the space of tensors of order 3. The ongoing work on equations is with Skurt.

Knot theory aims to classify embeddings of the circle to 3-space up to isotopies. A classical way to distinguish non-equivalent knots is to find a suitable invariant that takes different values on the two given knots. In the same spirit, given a smooth projective curve over a field, we want to classify its embeddings to projective 3-space up to a suitable notion of isotopy. We will explain how determinantal representations of secant varieties give rise to invariants and discuss to which extent they completely classify embeddings up to our notion of isotopy. A prominent role will be played by various variants of the writhe. This is joint work with Daniele Agostini.

The Kadomtsev–Petviashvili (KP) equation is a central example of an integrable nonlinear PDE with deep connections to algebraic geometry. A classical construction of Krichever produces quasi-periodic solutions from algebraic curves together with divisor data via Riemann theta functions. At the same time, soliton solutions have a rich combinatorial structure: work of Kodama-Williams and others relates them to the geometry and combinatorics of the positive Grassmannian. In this talk I will describe recent and ongoing work with several collaborators that develops a “tropical KP theory’’ connecting these two viewpoints. When an algebraic curve degenerates to a tropical curve, the associated theta-function solutions collapse to soliton solutions. We show that the algebro-geometric data in the Krichever construction admits a direct tropical/combinatorial description that determines the resulting soliton solution. In particular, one can translate the geometric data of the degeneration into purely combinatorial objects that encode the soliton structure. This perspective provides a concrete way to pass from algebraic curves to soliton solutions and reveals a new combinatorial layer underlying the classical algebro-geometric theory of KP.

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.

Using discrete derivatives, one can define a notion of polynomials between arbitrary groups. Such polynomials arise naturally in inverse Gowers theory through a fundamental (and still only partially established) dichotomy: a bounded function $f: G \to S^1$ either behaves pseudorandomly, or it correlates with a polynomial phase. This principle is crucial in establishing the existence of arithmetic progressions in subsets $A \subset G$. Despite their importance, polynomial maps are only partially understood yet. To remedy this, it is valuable to develop algebraic characterizations of such functions. In this talk, we describe the construction of a universal group $\mathrm{Pol}_k^{ab}(G)$ which classifies all unital polynomials of degree at most $k$ from $G$ into an abelian group, building on work of Jamneshan and Thom. We then present a classification of $\mathrm{Pol}_k^{ab}(G)$ for any abelian torsion group $G$ whose $p$-primary components are trivial for all primes $p \leq k$, making use of group rings and symmetric tensors.

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!).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.