Abstract: In this paper I present and assess, firstly independently from one another, and then comparatively, the strengths and the weaknesses of three views on semantics: model-theoretic, truth-conditional, and proof-theoretic. The whole philosophical perspective from which I explore these three approaches is motivated by the investigation of the connection between truth and meaning, as well as by the understanding of the role and place of meaning within a physicalist outlook of the world. I also hint at how one can combine and coordinate the three approaches depending on the philosophical issues which those systems seek to model. In the process, I present the philosophical rich and nuanced positions of Donald Davidson who pioneered the Tarski-type truth-conditional semantics for natural languages, and of Wilfrid Sellars who made an essential contribution to the understanding of both semantic and pragmatic aspects of the relations between truth and meaning from an inferentialist (proof) based perspective. I offer a sketch of how certain problems within the Davidsonian framework can be fixed by using the Sellarsian framework. The paper ends with some critical points and challenges that I raise against the proof-theoretic (inferentialist) approach questioning its power to give a complete account of the issue concerning the relationship between truth, meanings, and rules.

Keywords: model-theoretic semantics, truth-conditional semantics, proof-theoretic semantics, inferentialism, Alfred Tarski, Donald Davidson, Wilfrid Sellars. 

Abstract: A Weyl connection on a conformal manifold is a torsion-free connection preserving the conformal class. In this talk I will report on recent progress (obtained in collaboration with Florin Belgun, Brice Flamencourt, Farid Madani and Mihaela Pilca) towards the classification of compact conformal manifolds carrying two Weyl connections with special holonomy.

Abstract: Compact Vaisman manifolds come endowed with a natural complex one-dimensional Lie group G, which has compact closure H inside the group of biholomorphisms. When G=H, it is well known that the Vaisman manifold has maximal algebraic dimension and the quotient by G is the algebraic reduction. 

I will explain how to obtain the algebraic reduction in the general setting, and in particular how the dimension of H determines the algebraic dimension of the Vaisman manifold.

Abstract: For centuries, mathematics has been viewed as a uniquely human endeavor – a realm of pure logic and abstract creativity, where proofs arise from deep intuition and rigorous deduction. In this world, artificial intelligence has often been met with skepticism by the pure mathematics community, frequently associated with brute-force automated theorem provers that lack genuine understanding.

Drawing from my experience as a mathematician working in deep and reinforcement learning since 2016, I will argue for a fundamental shift in this perspective: AI can be used as an interactive collaborator that augments, rather than replaces, the mathematician in doing what we all love: discovering new mathematics.

This talk will serve both as an introduction to these new methods and as an invitation to our community to consider the exciting future of this human–AI partnership in mathematics!

Abstract: We intend to report on some results about conformal Hermitian geometry of complex surfaces.

Abstract: It will be an excursus through some geometric facts that occur just in one or a few exceptional dimensions. I will focus in particular on aspects of 16-dimensional Riemannian geometry, in relation with octonions, the group Spin(9) and even Clifford structures.

Abstract: We discuss cohomology properties of the known classes of examples of locally conformally Kähler manifolds.

Abstract: A hyperhermitian manifold (M, J_1, J_2, J_3, g)  is said to be strong HKT if the Bismut connections associated to the three Hermitian structures (J_i, g), i = 1,2, 3,  coincide, and the common Bismut torsion 3-form is closed.

 In the  talk  I  will  first discuss  some general properties of strong HKT manifolds, including a non-existence result on solvmanifolds.   Then I  will  focus on  the geometry of compact, simply connected strong HKT manifolds in real dimension eight.  This talk  is based on a  joint work with Beatrice Brienza,  Gueo  Grantcharov, and Misha Verbitsky.

Abstract: We show that the Chern-Simons invariant of a connection on a vector bundle admitting a flat, rank-1 complement inside the trivial bundle, must vanish modulo integers. This extends the classical obstruction of Chern and Simons for isometric immersions in Euclidean 4-space, and also a recent result of Cap, Flood and Mettler about equiaffine immersions preserving a volume form.

Abstract: As for LCK structures, a locally conformally symplectic (lcs) structure is a non-dedgenerate 2-form ω that satisfies dω = θ ∧ ω, where the closed 1-form θ is called the Lee form, and the flow of its symplectic dual, called the s-Lee vector field, acts by lcs transformations. We study compact group actions on compact lcs manifolds. Such an action preserves a compatible almost complex structure J and is called J-holomorphic. We focus on the case where such an action includes the s-lee field and call them action of Lee type, the underlying lcs manifold is called J-holomorphic. This generalizes the holomorphic action of the (anti-) Lee vector field on a Vaisman manifold, which can also be seen as a particular case of an LCK metric with potential, more generally an LCK metric of exact type. We characterize the J-holomorphic actions of Lee type on lcs manifolds of exact type and apply this to the LCK setting, in particular we show that there are LCK metrics with holomorphic Lee vector field on Hopf manifolds (which are not LCK with potential unless they are Vaisman). We also compute the leaf cohomology of the foliaton induced by the s-Lee vector field on a J-holomorphic compact lcs manifold, and characterize the case where union of its closed orbits is two-dimensional (minimal), in particular for Vaisman manifolds.

Abstract: Vaisman manifolds are locally conformally Kahler manifolds with the Lee form parallel under the Levi-Civita connection. The corresponding vector field, called the Lee field, is Killing and holomorphic. Taking a closure of its flow, we obtain a compact abelian Lie group G acting on a Vaisman manifold by holomorphic isometries. Let B be a stable holomorphic vector bundle on a Vaisman manifold of dimension ≥ 3. I will prove that B is G-equivariant. This result leads to some interesting algebraic-geometric applications.

Abstract: We shall present older and new results on Catlin and D'Angelo q-type introduced in d-bar problem.