Semantic tableaux for combined logical systems are usually constructed ad hoc and the problem of developing and applying more general methodologies for combining tableaux is yet to be systematically explored. In this talk I will address that problem and will outline some methodological approaches for combining tableaux for fibring, fusion, and products of logics. I will focus mainly on the case of fibring of tableaux and will discuss the questions of transfer of soundness, completeness, and termination from the components to the combined tableaux, both in general and in the context of some important special cases.
In this talk, I will discuss recent results regarding the intermediate nonlinear Schrödinger equation (INLS). Analytically, INLS is a one-dimensional completely integrable nonlinear Schrödinger equation with a cubic derivative nonlinearity and is $L^2$-critical. A limiting form of INLS is the continuum Calogero-Moser equation (CCM), which is also completely integrable. Interestingly, CCM keeps the Hardy space $L^2_+$ invariant, and, under this assumption, tools from complete integrability have recently resolved the well-posedness problem for CCM in $L^2_+$. I will discuss progress on the well-posedness for INLS and CCM (not relying on complete integrability), outside of the Hardy space and in low-regularity. Our approach combines a gauge transformation, bilinear Strichartz estimates and a refined decomposition for smooth solutions. This is based on joint work with A. Chapouto (CNRS, Monash) and T. Laurens (UW-Madison).
In this talk, we consider the low regularity well-posedness problem for the Korteweg-de Vries equation (KdV) on the real line. Aiming to bridge the regularity gap between the scaling critical space and the known optimal well-posedness in $L^2$-based Sobolev spaces, we consider rough data in Fourier-Lebesgue spaces. Via infinite normal form reductions and exploiting algebraic cancellations, we introduce a new gauged KdV equation, equivalent to the original one at high regularity, but better behaved for rough solutions below the $H^{-1}$-scale. Surprisingly, our method does not rely on the completely integrable structure of KdV and is easily adapted to other equations with quadratic derivative nonlinearities, such as the dispersion-generalized Benjamin-Ono equations.
This talk is based on joint work with Simão Correia (IST, U. Lisboa) and João Pedro Ramos (IMPA).
We show that, under a division condition, the tangential Cauchy-Riemann cohomology of a compact Lie group with a left-invariant CR structure can be computed on a suitable maximal torus. As a consequence, we conclude that the tangential Cauchy-Riemann cohomology is finite-dimensional. We also show that, for a class of CR structures, this division condition is necessary for the total cohomology to be finite-dimensional. The proof combines Fourier analysis on compact Lie groups, highest-weight representations and Lie algebra cohomology. This not only generalizes but provides completely new proofs for the analogous result due to Pittie and for its extensions to Levi-flat CR structures, obtained by Jacobowitz and Jahnke.
We study the family of Cartan-Schouten connections on Lie groups, parameterized by $\lambda\in[0,1]$, whose geodesics through the identity are one-parameter subgroups. We compute their curvature, torsion, parallel transport, and geodesics, and develop Euler-Poincaré and Lie-Poisson reduction for mechanical systems via these connections, unifying the “minus” and “plus” cases. These inspire us to introduce a connection-dependent variational principle where the Lagrangian is expressed in terms of the parallel-transported velocity, leading to an integro-differential Euler-Lagrange equation that explicitly involves torsion and curvature memory terms. The general framework is illustrated on two concrete examples: the Heisenberg group, where the equations simplify to an ODE system, and the rotation group SO(3), where the integro-differential system is solved numerically via a Magnus expansion.
An important aspect of socially interacting rational agents are the strategic abilities of individual agents and groups (coalitions) of agents to guarantee the achievement of their desired goals, while acting and interacting within an entire society of agents. Several logical systems have been proposed for formalising and capturing such reasoning were introduced in the early 2000s, starting with the Coalition Logic (CL), the Alternating Time Temporal Logic (ATL), and some extensions of these. Coalition Logic provides a natural, but rather restricted perspective: the agents in the proponent coalition are viewed as acting in full cooperation with each other but in complete opposition to all agents outside of the coalition, which are thus treated as adversaries. The Alternating Time Temporal Logic extends Coalition Logic with temporal operators allowing for expressing long-term temporised goals. The strategic interaction in real societies is much more complex, usually involving various patterns combining cooperation and competition. To capture these, more expressive and versatile logical frameworks are needed. In this talk I will give a brief overview of some of these, and will then focus on the Logic of Coalitional Goal Assignments (LCGA), capturing reasoning about strategic abilities of the entire society to cooperate in order to ensure achievement of the societal goals, while simultaneously protecting the abilities of individuals and groups within the society to achieve their individual and group goals.
Tennenbaum's theorem states that PA does not admit any nonstandard computable model. In 2022, Fedor Pakhomov proved that this theorem is fragile in regards to how PA is expressed, by constructing a theory that is definitionally equivalent to PA (roughly: “it’s PA but with a different choice of signature”) for which there is a computable nonstandard model. I will introduce the audience to this result and, time allowing, present the way in which we have been able to improve on Pakhomov's original construction and some remaining open questions.
