L4

While there have been substantial successes of actor-critic methods in continuous control, simpler critic-only methods such as Q-learning often remain intractable in the associated high-dimensional action spaces. However, most actor-critic methods come at the cost of added complexity: heuristics for stabilisation, compute requirements as well as wider hyperparameter search spaces. To address this limitation, we demonstrate in two stages how a simple variant of Deep Q Learning matches state-of-the-art continuous actor-critic methods when learning from simpler features or even directly from raw pixels. First, we take inspiration from control theory and shift from continuous control with policy distributions whose support covers the entire action space to pure bang-bang control via Bernoulli distributions. And second, we combine this approach with naive value decomposition, framing single-agent control as cooperative multi-agent reinforcement learning (MARL). We finally add illustrative examples from control theory as well as classical bandit examples from cooperative MARL to provide intuition for 1) when action extrema are sufficient and 2) how decoupled value functions leverage state information to coordinate joint optimization.

According to the Yau-Tian-Donaldson conjecture, the existence of a constant scalar curvature Kähler (cscK) metric in the cohomology class of an ample line bundle $L$ on a compact complex manifold $X$ should be equivalent to an algebro-geometric "stability condition" satisfied by the pair $(X,L)$. The cscK metrics are the critical points of Mabuchi's $K$-energy functional $M$, defined on the space of Kähler potentials, and an important result of Chen-Cheng shows that cscK metrics exist iff $M$ satisfies a standard growth condition (coercivity/properness). Recently the speaker has shown that the $K$-energy is indeed proper if and only if the polarized manifold is stable. The stability condition is closely related to the classical notion of Hilbert-Mumford stability. The speaker will give a non-technical account of the many areas of mathematics that are involved in the proof. In particular, he hopes to discuss the surprising role played by arithmetic geometry ​in the spirit of Arakelov, Faltings, and Bismut-Gillet-Soule.

Given a cover U of a family of smooth complex algebraic varieties, we associate with it a class C of structures locally definable in an o-minimal expansion of the reals, containing the cover U.  We prove that the class is ℵ0-homogeneous over submodels and stable. It follows that C is categorical in cardinality ℵ1. In the one-dimensional case we prove that a slight modification of C is an abstract elementary class categorical in all uncountable cardinals.
 

Topological signals associated not only to nodes but also to links and to the higher dimensional simplices of simplicial complexes are attracting increasing interest in signal processing, machine learning and network science. However, little is known about the collective dynamical phenomena involving topological signals. Typically, topological signals of a given dimension are investigated and filtered using the corresponding Hodge Laplacians. In this talk, I will introduce the topological Dirac operator that can be used to process simultaneously topological signals of different dimensions.  I will discuss the main spectral properties of the Dirac operator defined on networks, simplicial complexes and multiplex networks, and their relation to Hodge Laplacians.   I will show that topological signals treated with the Hodge Laplacians or with the Dirac operator can undergo collective synchronization phenomena displaying different types of critical phenomena. Finally, I will show how the Dirac operator allows to couple the dynamics of topological signals of different dimension leading to the Dirac signal processing of signals defined on nodes, links and triangles of simplicial complexes. 

Mathematics has always been part of the fabric of culture. References to mathematics in literature go back at least as far as Aristophanes, and encompass everyone from Dostoevsky to Oscar Wilde. In this talk I’ll explore some of the ways that literature has engaged with mathematical ideas, from the 17^th and 18^th century obsession with the cycloid (the “Helen of Geometry”) to the 19^th century love of the fourth dimension.

The handlebody group is defined as the mapping class group of a three-dimensional handlebody. We will survey some geometric and algebraic properties of the handlebody groups and compare them to those of two of the most studied (classes of) groups in geometric group theory, namely mapping class groups of surfaces, and ${\rm Out}(F_n)$. We will also introduce the disk graph, the handlebody-analogon of the curve graph of a surface, and discuss some of its properties.

Given a group of type ${\rm FP}_n$, one may ask if this property also holds for its subgroups. The BNS invariant is a subset of the character sphere that fully captures this information for subgroups that are kernels of characters. It also provides an interesting connection of finiteness properties of subgroups and group homology. In this talk I am going to give an introduction to this problem and present an attempt to generalize the BNS invariant to more subgroups than just the kernels of characters.

In recent years, a tend of higher category theory is growing from multiple areas of research throughout mathematics, physics and theoretical computer science. Guided by Cobordism Hypothesis, I would like to introduce some basics of `higher representation theory’, i.e. the part of higher category theory where we focus on the fundamental objects: `finite dimensional’ linear n-categories. If time permits, I will also introduce some recent progress in linear higher categories and motivations from condensed matter physics.

Persistent homology is both a powerful framework for data science and a fruitful source of mathematical questions. Here, we will give an introduction to both single-parameter and multiparameter persistent homology. We will see some examples of how topology has been successfully applied to the real world, and also explore some of the pure-mathematical ideas that arise from this new perspective.

Separability is an algebraic property enjoyed by certain subsets of groups. In the world of non-positively curved groups, it has a not-too-well-understood link to geometric properties such as convexity. We explore this connection in the setting of relatively hyperbolic groups and discuss a recent joint work in this area involving products of quasiconvex subgroups.