Abstracts

Talks

• Luca Accornero: TBA
  
  coming soon
• Iakovos Androulidakis: TBA
  
  coming soon
• Sara Azzali: TBA
  
  coming soon
• Marco Benini: Higher structures in Lorentzian quantum field theory
  
  One of the mathematical challenges posed by quantum field theory (QFT) is the quest for its axiomatization. This quest becomes even more challenging when the higher structures emerging from gauge theories are taken into account. In Lorentzian signature, two prominent axiomatizations are based on algebraic QFTs and factorization algebras, respectively. In the 1-categorical setting, thus ignoring the higher structures coming from gauge theories, these two candidates are equivalent as a consequence of the so-called time-slice axiom, which captures well-posedness of the underlying Cauchy problem. Unfortunately, the higher categorical analog of this equivalence turns out to be quite elusive. After reviewing the 1-categorical equivalence, I shall propose a new approach towards establishing its higher categorical analog, which crucially relies on a homotopically relaxed version of the time-slice axiom. This higher categorical equivalence entails that Lorentzian QFTs, including the higher structures coming from gauge theories, can be axiomatized equivalently by algebraic QFTs and by (time-orderable pre)factorization algebras.
• Eugenia Boffo: On Kodaira-Spencer theory
  
  Topological strings have moduli that are either Kähler h^1,1 or complex structure h^2,1. The deformation problem of complex structures consists of a Maurer-Cartan equation, involving the Dolbeault differential and the Schouten bracket. Unfortunately, various issues (lack of a suitable bilinear symmetric 2-form, or use of a divergence operator in the working expression for the Schouten bracket) concur to prevent an action functional in MC form (i.e. whose field equations are the MC equations). Therefore the original proposals of an action in the Batalin-Vilkovisky formalism, known as Kodaira-Spencer theory of gravity or BCOV theory, involve a non-local kinetic term and a a constrained variation [Bershadsky-Cecotti-Ooguri-Vafa '93]. We propose a fix to this, borrowing constructions from superstring field theory. Based on 2506.02983 with Hulík and Sachs.
• Giovanni Canepa: Poisson infinity structures of field theories on manifolds with corners
  
  In this talk I will describe a local Poisson structure (up to homotopy) associated with corners in field theories. This is achieved through the use of the BFV formalism. I will also present some examples like BF theory, Chern Simons and four dimensional gravity. This talk is based on arXiv:2202.08684, a joint work with A.S. Cattaneo.
• Alberto Canonaco: Uniqueness and strong uniqueness of enhancements for triangulated categories
  
  It is well known that it is often useful to replace triangulated categories with suitable higher categorical enhancements. While for most triangulated categories of algebraic or geometric interest it is easy to prove that such enhancements exist, it is usually more difficult to understand if they are also "unique". Actually there are two different natural notions of uniqueness for enhancements, called, respectively, uniqueness and strong uniqueness. I will report on some new results, obtained in collaboration with Amnon Neeman and Paolo Stellari, related both to uniqueness and to strong uniqueness. They include a generalization of a classic theorem by Rickard about derived equivalences for rings.
• Francesco Cattafi: PB-groupoids vs VB-groupoids
  
  It is well known that the collection of linear frames of a smooth n-manifold M defines a principal GL(n,R)-bundle over M (called the frame bundle); more generally, this construction makes sense for any vector bundle over M. Conversely, any principal bundle together with a representation induces an associated vector bundle; these processes establish therefore a correspondence between vector bundles on one side, and principal bundles with representations on the other side.
  In differential geometry there are several natural instances where diagrams of Lie groupoids and vector bundles, together with suitable compatibilities, appear. They are known as vector bundle groupoids (VB-groupoids) and their theory has been fairly developed in the past decades, with applications e.g. to representations up to homotopy and deformation theory. On the other hand, little is known about the principal bundle counterpart of these objects.
  In this talk, I will recall all the notions mentioned above, and then introduce a special class of frames of VB-groupoids which interact nicely with the groupoid structure. I will then use them to associate to any given VB-groupoid a diagram of Lie groupoids and principal bundles, together with the action of a (strict) Lie 2-groupoid GL(l,k); this will lead to the general notion of a principal bundle groupoid (PB-groupoid). Moreover, I will sketch how to generalise the standard correspondence between vector bundles and principal bundles to a correspondence between VB-groupoids and PB-groupoids. I will conclude discussing a few examples and future applications.
  This is joint work with Alfonso Garmendia.
• Rosa Marchesini: TBA
  
  coming soon
• Francesca Pratali: Higher structures on homology groups
  
  We dualise the classical fact that an operad with multiplication leads to cohomology groups which form a Gerstenhaber algebra to the context of cooperads: as a result, a cooperad with comultiplication induces a homology theory that is endowed with the structure of a Gerstenhaber coalgebra, that is, it comes with a (graded cocommutative) coproduct which is compatible with a cobracket in a dual Leibniz sense. As an application, one obtains Gerstenhaber coalgebra structures on Tor groups over bialgebras or Hopf algebras, as well as on Hochschild homology for Frobenius algebras. This is joint work with Niels Kowalzig.
• Stefano Ronchi: Higher cotangent groupoids
  
  The cotangent bundle of a manifold is a central object in mathematics and physics. There is a similar construction for Lie groupoids. In this talk I will explain how to extend this to Lie n-groupoids and give some fundamental applications. This is joint work with Miquel Cueca and Chenchang Zhu.
• Jonas Schnitzer: Quantization of momentum maps via g-adapted formalities
  
  If a Lie group acts on a Poisson manifold by Hamiltonian symmetries there is a well-understood way to get rid of unnecessary degrees of freedom and pass to a Poisson manifold of a lower dimension. This procedure is known as Poisson-Hamiltonian reduction. There is a similar construction for invariant star products admitting a quantum momentum map, which leads to a deformation quantization of the Poisson-Hamiltonian reduction of the classical limit.
  The existence of quantum momentum maps is only known in very few cases, like linear Poisson structures and symplectic manifolds. The aim of this talk is to fill this gap and show that there is a universal way to find quantized momentum maps using so-called adapted formality morphisms which exist, if one considers nice enough Lie group actions. This is a work in progress with Chiara Esposito, Ryszard Nest and Boris Tsygan.
• Alfonso Tortorella: A rigidity result for coisotropic submanifolds in contact geometry
  
  In this talk, based on joint work with Stephane Geudens, we study coisotropic deformations of a compact regular coisotropic submanifold C in a contact manifold (M,H). Our main result states that C is rigid among nearby coisotropic submanifolds whose characteristic foliation is diffeomorphic to that of C. When combined with a classical rigidity result for foliations, this yields conditions under which C is rigid among all nearby coisotropic submanifolds.
• Ping Xu: TBA
  
  coming soon