Seminars Archive: Mathematics and Artificial Intelligence

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.

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.

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of minors of particular flattenings of the underlying tensors and these equations were conjectured by Sturmfels in an unpublished note.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⊆V of vertices as the distribution of isometry types of inclusions Π∩X⊆Π∩V⊆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 R3, 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 periodic surfaces as a scaffold is a convenient route to making periodic entanglements, which are interesting in the context of physics, biomaterials and chemical frameworks. I will present a systematic way of enumerating and characterising new tangled periodic structures, using low-dimensional topology and combinatorics. As a second part, the morphometric approach to solvation free energy is a geometry-based theory that incorporates a weighted combination of geometric measures over the solvent accessible surface for solute configurations in a solvent. I will demonstrate that employing this geometric technique in simulating the self assembly of sphere clusters, viruses and short flexible tubes results in an assortment of interesting geometric structures. This gives insight into the role of shape in the physical process of self assembly.

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→S1 either behaves pseudorandomly, or it correlates with a polynomial phase. This principle is crucial in establishing the existence of arithmetic progressions in subsets A⊂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 Polabk(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 Polabk(G) for any abelian torsion group G whose p-primary components are trivial for all primes p≤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!).

We are witnessing an AI revolution. At the heart of this revolution are generative AI models that, powered by advanced architectures and large datasets, are transforming AI across a variety of disciplines. But how can AI facilitate and eventually enable discoveries in life sciences? How can it bring us closer to understanding biology, the functions of our cells and relationships across different molecular layers? In this talk, I will introduce generative AI methods designed to uncover relationships across different omics layers. I will demonstrate how these approaches enable the reassembly of tissues from dissociated single cells and how they can help us to understand the relationship between nuclear morphology and transcriptomic profiles. I will further demonstrate how biomedical challenges can help to drive new fundamental AI advances. Finally, I will discuss challenges and limitations of existing methods for accurately predicting cellular response to perturbations.

Modern molecular and multicellular biology is confronting the same conceptual bottlenecks that defined the great unsolved problems in theoretical physics: vast structured spaces of possibilities, dynamical principles underlying emergent behavior, and the need for interpretable frameworks rather than purely phenomenological fits. In physics, the string landscape demanded methods to navigate exponentially large solution spaces, effective field theory provided tools to extract governing equations at the right level of description, and dualities revealed that apparently distinct theories can encode the same physics. In biology, similarly enormous combinatorial spaces arise in perturbation screens, gene regulatory networks, tissue morphogenesis, and the design of emergent cellular systems. I will present a research program developed at the interface of mathematics, theoretical physics and machine learning, and argue that its methods are naturally suited to discovering organizing principles in biological data: navigate large landscapes with sparse samples, extract effective, interpretable laws directly from data, recognize equivalence classes of mechanisms rather than unique solutions, and invert design problems to generate candidate protocols for target phenotypes. I will argue that these approaches can be directly applied to key open questions in molecular biology, single-cell dynamics, organoid screens, morphogenesis, and programmable multicellular systems. This framework bridges physics, mathematics, machine learning, and biology, and has immediate opportunities for collaboration with the research groups at MPI-CBG.

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.

Time series data often contain latent cyclic structure, but standard tools to recover it often assume regular sampling and reliable time stamps. Cyclicity analysis was introduced as a reparametrization invariant method to detect linear planes which maximize the signed area of time series, acting as a proxy for cyclic behavior. In this talk, we discuss ongoing work to generalize cyclicity analysis to the nonlinear setting by using kernel methods. We introduce computable method to extract nonlinear cyclic coordinates from time series along with geometric phase variables. We’ll discuss some theoretical results, along with some preliminary experiments.

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.

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⊆V of vertices as the distribution of isometry types of inclusions Π∩X⊆Π∩V⊆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 R3, 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.