Seminars and short courses

We'll talk about joint work with Jacob Lurie regarding moduli stacks of geometric objects developing natural breaks. If time allows, I'll end with some speculation regarding a 3-D TFT arising from various G2 manifolds.

The Mobius strip; collapsing the equator; exploding a point in the plane; geometric definition of blowups; the secant construction; pull-backs of curves under blowup.

The algebraic definition of blowups; the blowup in affine charts; birational maps; the strict and total transforms of varieties and vector fields.

Axiomatic definition of blowups; Cartier divisors; base change property; persistence properties.

The famous Burau representation of the braid group is known to be unfaithful for braids with at least five strands. In the early 2000s, two constructions were provided to fix faithfulness: the first being the Lawrence–Krammer–Bigelow linear representation, hence proving linearity of braid groups, and the second being the Khovanov–Seidel categorical representation. In this talk, based on joint work in progress with Licata, Queffelec, and Wagner, I will investigate the interplay between these two representations.

First examples of resolutions; curves and surfaces; effect of blowups on polynomials; singularity invariants; lexicographic orderings.

Choice of centers of blowup; descent in dimension; lexicographic decrease of invariant; transversality; obstructions in positive characteristic; resolution of planar vector fields.

How many different problems can a neural network solve? What makes two machine learning problems different? In this talk, we'll show how Topological Data Analysis (TDA) can be used to partition classification problems into equivalence classes, and how the complexity of decision boundaries can be quantified using persistent homology. Then we will look at a network's learning process from a manifold disentanglement perspective. We'll demonstrate why analyzing decision boundaries from a topological standpoint provides clearer insights than previous approaches. We use the topology of the decision boundaries realized by a neural network as a measure of a neural network's expressive power. We show how such a measure of expressive power depends on the properties of the neural networks' architectures, like depth, width and other related quantities.

References

• Ballester et al, Topological Data Analysis for Neural Network Analysis: A Comprehensive Survey
• Papamrakou et al, Position: Topological Deep Learning is the New Frontier for Relational Learning
• Petri and Leitão, On The Topological Expressive Power of Neural Networks

How many different problems can a neural network solve? What makes two machine learning problems different? In this talk, we'll show how Topological Data Analysis (TDA) can be used to partition classification problems into equivalence classes, and how the complexity of decision boundaries can be quantified using persistent homology. Then we will look at a network's learning process from a manifold disentanglement perspective. We'll demonstrate why analyzing decision boundaries from a topological standpoint provides clearer insights than previous approaches. We use the topology of the decision boundaries realized by a neural network as a measure of a neural network's expressive power. We show how such a measure of expressive power depends on the properties of the neural networks' architectures, like depth, width and other related quantities.

References

• Ballester et al, Topological Data Analysis for Neural Network Analysis: A Comprehensive Survey
• Papamrakou et al, Position: Topological Deep Learning is the New Frontier for Relational Learning
• Petri and Leitão, On The Topological Expressive Power of Neural Networks

In this talk I will discuss some results obtained in collaboration with Filipe C. Mena and former PhD student Vítor Bessa on the global dynamics of a minimally coupled scalar field interacting with a perfect-fluid through a friction-like term in spatially flat homogeneous and isotropic spacetimes. In particular, it is shown that the late time dynamics contain a rich varitey of possible asymptotic states which in some cases are described by partially hyperbolic lines of equilibria, bands of periodic orbits or generalised Liénard systems.