Numerical (Nonlinear) Algebra in the Real World

Contribution List

38. Collaboration rooms available, CSBD 121, 205, 306

2/3/25, 1:00 PM

32. Registration and greeting

2/3/25, 2:00 PM

5. Welcome

2/3/25, 2:30 PM

1. Jon Hauenstein - Four computations in real numerical algebraic geometry

2/3/25, 2:45 PM

Many problems in science and engineering can be formulated as computing information about the set of real points satisfying polynomial equations and inequalities. Some examples include synthesizing a linkage in kinematics, computing and analyzing the steady-state solutions to a polynomial dynamical system, and reconstructing a scene in computer vision. Since different problems could require...

3. Andreas Münch - Continuum models of solutions and gels of charge regulating polyelectrolytes such as RNA and DNA and their phase behaviour under varying environmental conditions

2/3/25, 4:15 PM

We develop continuum models for solutions of polyelectrolytes as a minimal system for biological molecules. We investigate their propensity to undergo phase separation by a combination of bifurcation analysis and time-dependent numerical solutions. In the case of polyelectrolyte gels, we describe the impact of salt concentration in the environment on the pattern forming behaviour under the...

33. Evening collaboration

2/3/25, 5:00 PM

39. Collaboration rooms available, CSBD 121, 205, 306

2/4/25, 9:00 AM

6. Frank Jülicher - Active Matter Hydrodynamics: From Droplets to Tissues

2/4/25, 9:00 AM

One of the most fascinating features of living matter is its extraordinary dynamics. This is revealed when observing cells under the microscope. They move, they divide and the undergo shape changes. Cells can generate forces and movements. From a materials perspective, cellular matter is soft and liquid-like, but exhibits complex spatio-temporal organization. To understand cells and tissue as...

7. Pierre Haas - Stories of ecological structures

2/4/25, 9:45 AM

The dynamics of ecological communities are intrinsically linked to ecological structures such as the network of species interactions or the phenotypic substructure of different species. In this talk, I will give two examples of minimal models that elucidate the interplay of these structures and ecological dynamics. In both cases, I will highlight open questions that (new) (numerical) algebraic...

27. Ada Wang - Computing Arrangements of Hypersurfaces

2/4/25, 11:15 AM

In this talk, I will present a Julia package, HypersurfaceRegions.jl, for computing all connected components in the complement of an arrangement of real algebraic hypersurfaces in $R^n$. The package is based on a modified implementation of the algorithm from the paper "Smooth Connectivity in Real Algebraic Varieties" by Cummings et al. I will outline the theory behind the algorithm and our...

10. Ivo Sbalzarini - Particle Methods: Numerical Computation on Point Clouds

2/4/25, 12:00 PM

Point clouds provide an expressive abstraction for tasks across computational mathematics. They can be used as collocation point sets in numerical analysis, but also as a data structure for machine learning and statistical inference. I provide an introduction to how differential operators can be consistently approximated on point clouds by solving polynomial systems. This includes a meshfree...

11. Announcements

2/4/25, 12:45 PM

35. Collaboration

2/4/25, 2:00 PM

13. Torkel Loman - Catalyst: Fast and flexible modelling of reaction networks

2/4/25, 2:45 PM

Chemical reaction networks (CRNs) are a type of model commonly used in biology and chemistry. Their applications include the investigation of cellular system functions, designing drugs (pharmacology), forecasting epidemic progression (epidemiology), and optimisation of chemical synthesis pipelines. The Catalyst.jl modelling tool provides an interface for creating such CRN models in the Julia...

15. Aida Maraj - Toric Geometry in Brownian motion tree models and their generalizations

2/4/25, 4:15 PM

Algebraic geometry has recently provided a new approach to advancing problems in multivariate Gaussian models. This is achieved by identifying Gaussian distributions with symmetric matrices and analyzing the polynomials that vanish on these matrices, known as ideals. The talk will focus on Brownian motion tree (BMT) models, a type of Gaussian model used in phylogenetics, and their...

34. Evening collaboration

2/4/25, 5:00 PM

40. Collaboration rooms available, CSBD 121, 205, 306

2/5/25, 9:00 AM

16. Oskar Henriksson - The numerical algebraic geometry of steady state equations

2/5/25, 9:00 AM

The steady states of a chemical reaction network with power-law kinetics can be described by a polynomial system with fixed support and coefficients that depend on parameters called rate constants and total amounts. This so-called vertical parametrization of the coefficients typically introduces dependencies among them, such that the generic root count over the complex numbers drops below the...

17. Viktoriia Borovik - Numerical homotopies from SAGBI bases

2/5/25, 9:45 AM

I will present homotopy continuation methods for solving 0-dimensional polynomial systems, where each polynomial is expressed as a general linear combination of prescribed, fixed polynomials. This approach involves selecting a specific starting system for homotopy continuation, leveraging the theory of SAGBI (Khovanskii bases) and toric geometry. For square systems, SAGBI homotopies can...

18. Fabian Faulstich - Numerical Algebraic Geometry and Correlated Electrons

2/5/25, 11:15 AM

We discuss the algebra and combinatorics underpinning coupled cluster (CC) theory for quantum many-body systems. The high-dimensional eigenvalue problems that encode the electronic Schrödinger equation are approximated by polynomial systems at various levels of truncation. The exponential parametrization of the eigenstates gives rise to truncation varieties. These generalize Grassmannians in...

19. Irem Portakal - Computing equilibra in game theory

2/5/25, 12:00 PM

In 1950, Nash published a very influential two-page paper proving the existence of Nash equilibria for any finite game. The proof uses an elegant application of the Kakutani fixed-point theorem from the field of topology. It has, however, been noted that in some cases the Nash equilibrium fails to predict the most beneficial outcome for all players. To address this, generalizations of Nash...

20. Announcements

2/5/25, 12:45 PM

36. Collaboration

2/5/25, 2:00 PM

21. Georgy Scholten - Globtim: An attempt at global optimization over compact domain via polynomial approximations

2/5/25, 2:45 PM

Let $f$ be a $L^2$ measurable function defined over the $n$-dimensional unit-cube and enjoying a quadratic growth property around all of its local minimizers on that domain. Our objective is to design an algorithm that can compute all local minimizers of $f$.
We work in a computational framework where the function $f$ is given by an evaluation program $\Gamma$. This program takes as input...

22. Türkü Özlüm Çelik - Wave Parameters from Discrete Data

2/5/25, 4:15 PM

We study approximating underlying mathematical structures via point cloud data within the framework of integrable systems, with a particular emphasis on the Kadomtsev-Petviashvili (KP) equation. This is a pivotal element in the theory of integrable systems, and models nonlinear wave interactions. Our objective is to determine the finite-genus KP solution parameters that optimally correspond to...

37. Evening collaboration

2/5/25, 5:00 PM

24. Conference meal at MPI-CBG Kantine

2/5/25, 6:00 PM

We will have a delightful meal together at MPI-CBG in our Kantine, with food provided for participants by our expert internal catering company ISSMA

41. Collaboration rooms available, CSBD 121, 205, 306

2/6/25, 9:00 AM

25. Simon Telen - Chebyshev varieties

2/6/25, 9:00 AM

Chebyshev varieties are algebraic varieties parametrized by Chebyshev polynomials or their multivariate generalizations. We determine the dimension, degree, singular locus and defining equations of these varieties. We explain how they play the role of toric varieties in sparse polynomial root finding, when monomials are replaced by Chebyshev polynomials. We present numerical root finding...

26. Dmitrii Pavlov - Gibbs manifolds

2/6/25, 9:45 AM

Gibbs manifolds are images of affine spaces of symmetric matrices under the exponential map. They arise in applications such as optimization, statistics and quantum physics, where they extend the ubiquitous role of toric geometry. The Gibbs variety is the zero locus of all polynomials that vanish on the Gibbs manifold. In this talk, I will describe some properties of this variety, and present...

9. Jiayi Li - Taming Non-convexity in Shallow Neural Networks with Algebraic Activations

2/6/25, 11:15 AM

One of the key challenges in optimizing neural networks is the inherent high-dimensionality and non-convexity of the objective function. A single neuron with Sigmoid activation is known to have the number of local minima grow exponentially in the dimension based on square loss. Properly tuned gradient-based methods converge to a stationary point, prompting the question: which stationary point...

28. Lightning presentations

2/6/25, 12:00 PM

29. Announcements

2/6/25, 12:45 PM

30. Open collaboration time

2/6/25, 2:00 PM

31. Open collaboration time

2/6/25, 4:00 PM

14. Coffee and collaboration

8. Coffee and colloboration

2. coffee break

23. Conference meal?

4. Lunch

12. Lunch