Abstract: We consider in this talk the quaternionic contact (qc) version of the almost Schur lemma. Namely, we show that a compact quaternionic contact Einstein manifold of dimension bigger than seven with positive qc-Ricci tensor must be with constant qc scalar curvature. The latter is a consequence of a more general result, namely an integral inequality, that informally states that if the traceless qc Ricci tensor and some traces of the covariant derivatives of the qc torsion tensor are “close” to zero, then the qc scalar curvature is “close” to a constant.

Rigidity in Dimension 2

Abstract: I will discuss an approach to proving the conjecture that a normal rigid surface is smooth. The approach is based on the notion of deficient conormal singularities introduced by the speaker.

Hidden Variables in Linear Causal Models

Abstract: Identifying causal relationships between random variables from observational data is an important hard problem in many areas of data science. The presence of hidden variables, though quite realistic, brings up a variety of further problems. Linear structural equation models, which express each random variable as a linear combination of all of its parent variables, have long been used for learning causal structure from observational data.

Surprisingly, when the random variables in a linear structural equation model are non-Gaussian the full causal structure can be learned without interventions, while in the Gaussian case one can only learn the underlying graph up to a Markov equivalence class. In this talk, we first discuss how one can use high-order cumulant information to learn the structure of a linear non- Gaussian structural equation model with hidden variables. While prior work posits that each hidden variable is the common cause of two observed variables, we allow each hidden variable to be the common cause of multiple observed variables. Next, we discuss hidden variable Gaussian causal models and the difficulties that arise with learning those. We show it is hard to even describe the Markov equivalence classes in this case, and we give a semialgebraic description of a large class of these models.

Pólya Sequences with Dominant Colors

Abstract:

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Abstract: A classical result of Gauss states that among all monic polynomials of degree d over a finite field, approximately 1/d are irreducible. Extending previous results in the literature, we prove that under a mild assumption, the proportion of irreducible polynomials does not change even if only the last two coefficients are allowed to vary. Our approach is geometric. The talk will be nontechnical and accessible to a broad audience.

Abstract: We investigate an optimal control problem with a large number of agents (possibly infinitely many). Although the dynamical system (a controlled ordinary differential equation) is of the same type for every agent, each agent may have a different control. So, the multi-agent dynamical system has two levels: a microscopic one, which concerns the control system of each agent, and a macroscopic level, which describes the evolution of the crowd of all agents. The
state variable of the macroscopic system is the set of positions of the agents. We define and study the evolution of such a global dynamical system whose solutions are called solution tubes. We also consider a minimization problem associated with the multi-agent system and we give a new characterization of the corresponding value function as the unique solution of a Hamilton- Jacobi-Bellman equation stated on the space of compact subsets of $R^d$.

The talk is based on joint work with Marc Quincampoix.

What is the smallest algebraic integer?

Abstract:  We survey Lehmer’s problem on the smallest Mahler measure of an integer non-cyclotomic polynomial, synthesizing an introduction to this subject along with the state-of-art results known today. Then we will present the solution of a closely related conjecture of Schinzel and Zassenhaus, and propose some answers towards the question of our title. Time permitting, we discuss also some related recent results on the geometry of integer polynomials.