The seminar will take place in F11 for the first 18 people to arrive. Overflow audience and those who are working from home can participate via Zoom with meeting ID
Tuesday, 19 January, 11:15 Katharina Jochemko Points, polytopes, polynomials Ehrhart theory is concerned with the enumeration of lattice points in polytopes. Given a polytope with vertices in the integer lattice, the function counting the lattice points in its integer dilates agrees with a polynomial, the Ehrhart polynomial of the polytope. In this talk, I will give a brief overview of some of my research interests in Ehrhart theory, in particular, geometric models for enumerative problems, structural results for Ehrhart polynomials as well as ties to valuation theory.
Tuesday, 26 January, 11:15 Kaie Kubjas (Aalto University, Helsinki) Uniqueness of nonnegative matrix factorizations Nonnegative matrix factorizations are often encountered in data mining applications where they are used to explain datasets by a small number of parts. For many of these applications it is desirable that there exists a unique nonnegative matrix factorization up to trivial modifications given by scalings and permutations. This means that model parameters are uniquely identifiable from the data. Different sufficient conditions for the uniqueness of nonnegative matrix factorizations have been well studied, however, a little is known about necessary conditions. We will give so far the strongest necessary condition for the uniqueness of a nonnegative factorization. The talk is based on the joint work with Robert Krone.
Tuesday, 2 February, 11:15 Peter Bubenik (University of Florida)
Tuesday, 9 February, 16:15 Rehka Thomas (University of Washington)
Tuesday, 17 November, 11:15 Kathlén Kohn Nonlinear Algebra of Data Science and AI Nonlinear algebra is the study of systems of multivariate polynomial equations and inequalities. At the heart of this lies algebraic geometry, but the field is driven by many other areas of mathematics such as combinatorics, convex and discrete geometry, multilinear algebra, tropical geometry, or representation theory. Systems of polynomial (in)equalities appear vastly throughout the sciences and engineering. In this talk, I will focus on applications in data science and artificial intelligence. My research focuses on uncovering geometric structures in applied problems and on finding connections between seemingly unrelated areas of mathematics or science in general. I will demonstrate this by describing the underlying (algebraic) geometry of machine learning with neural networks and of 3D reconstruction in computer vision. I will also outline surprising connections between statistics, invariant theory, geometric modeling, intersection theory, and physics.
Slides (pdf 4.3 MB)
Tuesday, 10 November, 11:15 Johan Håstad Some of my interests and (old) results I work in complexity theory and recently I have studied efficient approximability of NP-hard optimization problems. In particular I have been interested in constraint satisfaction problems (CSPs). In a CSP you are given a large number of constraints each only depending on a constant number of variables and the goal is to find an assignment to satisfy as many constraints as possible. Central examples are Max-3Sat and sets of linear equations over a finite field where each equation only depends on three variables. I will also briefly mention some problems in post-quantum cryptography.
slides (pdf 264 kB)
Tuesday, 3 November, 11:15 Lorenzo Venturello Wasserstein distance in algebraic statistics Any metric on the set of states of a discrete random variable induces a metric called Wasserstein distance on the probability simplex. The unit ball of this norm is a polytope, well known in discrete and tropical geometry. Given any data distribution, we seek to minimize its Wasserstein distance to an algebraic model, i.e., a model described by polynomials. The solution to this optimization problem is a piecewise algebraic function of the data. After a gentle introduction on the topic, I will comment on the combinatorial and algebraic complexity of the problem. This talk is based on joint work with Türkü Özlüm Ç̧̧elik, Asgar Jamneshan, Guido Montúfar and Bernd Sturmfels.
slides (pdf 1.2 MB)
Tuesday, 27 October, *10:15* ¡ PhD Day !
Tuesday, 20 October, 11:15 Orlando Marigliano Algebraic statistics and the maximum likelihood degree This talk is about statistical models and algebraic varieties. Algebraic Statistics unites these two concepts, turning algebraic structure into statistical insight. We will discuss the basic ideas of this relatively new field and see some examples of the kind of models algebraic statisticians are interested in. We will focus on an important quantity associated to the likelihood estimation problem on these models: the maximum likelihood degree.
Tuesday, 13 October, 11:15 Ludwig Hedlin Convergence of linear neural networks to global minimizers It is known that gradient flow in linear neural networks using Euclidean loss almost always avoids critical points that have at least one eigendirection with negative curvature. Using algebraic invariants of the gradient flow we try to prove that the set of all critical points with no second-order curvature (zero Hessian) for arbitrary networks is associated to a subset of the invariants of lower dimension. This would mean that these critical points are almost surely avoided. We show that this holds for networks with 3 or less hidden layers and a few other special cases. We show by way of explicit counter-example that it is not true for general deep networks.
slides (pdf 477 kB)
Tuesday, 6 October, 11:15 Maria Dostert Exact semidefinite programming bounds for packing problems In the first part of the talk, I present how semidefinite programming methods can provide upper bounds for various geometric packing problems, such as kissing numbers, spherical codes, or packings of spheres into a larger sphere. When these bounds are sharp, they give additional information on optimal configurations, that may lead to prove the uniqueness of such packings. For example, we show that the lattice E8 is the unique solution for the kissing number problem on the hemisphere in dimension 8. However, semidefinite programming solvers provide approximate solutions, and some additional work is required to turn them into an exact solution, giving a certificate that the bound is sharp. In the second part of the talk, I explain how, via our rounding procedure, we can obtain an exact rational solution of a semidefinite program from an approximate solution in floating point given by the solver. This is a joined work with David de Laat and Philippe Moustrou.
slides (pdf 6.3 MB)
Tuesday, 22 September, 11:15 Liam Solus Combinatorics, Algebra, and Intervention The modern theory of causality is founded in our ability to encode both probabilistic and causal information about a data-generating distribution within the structure of a graph. Doing so in the proper way allows us to derive theorems which can be applied when developing data-driven causal structure learning algorithms, as well as probabilistic and causal inference. Such algorithms work best in the cases where the corresponding theorems point to nice combinatorial structure in the graph. Going deeper, if we also consider a parameterization of the joint distribution, we find that nice properties of the model can be attributed to nice properties of its defining algebraic variety in the ambient parameter space. Classic results of this nature are known for probabilistic graphical models. In this talk, we will see how such results generalize to interventional graphical models, which are now being used to learn causal structure from a mixture of observational and interventional data. Time permitting, we will explore how the proofs of these algebraic results give rise to new methods for modeling causation in context-specific settings.
slides (pdf 277 kB)
Tuesday, 15 September, 11:15 Angélica Torres Dynamics of chemical reaction networks and positivity of polynomials In biochemical processes, different molecules are consumed and produced as they interact with each other. The evolution of their concentration through time can be modelled with a system of differential equations that, under certain assumptions, is polynomial. In this case, the equilibrium points form an algebraic variety. In this talk I will present how these systems arise, and how deciding whether a polynomial can attain positive or negative values is used for detecting multistationarity and stability in the biochemical system.
slides (pdf 307 kB)
Tuesday, 8 September, 11:15 Rene Corbet An overview of the pipeline of multiparameter persistence In the last two decades, persistent homology and its generalizations have grown to a substantial branch of mathematical research. It gives rise to various research problems in algebra, category theory, geometry, topology, and other areas. Moreover, having immediate applications in data science, efficient algorithms, computational and probabilistic methods, and concrete implementations deserve the same amount of attention. Multiparameter persistence is the generalization of persistence to multiple independent parameters taken into account at once. While the theory of multiparameter persistence is provably much more complicated than the theory of ordinary persistence, new computational challenges arise as well. In this talk, I give a rough overview of the computational pipeline of multiparameter persistence, mention some important challenges, and outline the projects of my PhD thesis along these lines.
slides (pdf 9.8 MB)