Abstract: Singular cohomology of a space carries a natural CDGA structure induced by cup product at the level of cochains. While cup product is a well-defined operation on singular cochains, commutativity only holds after passing to cohomology, due to the presence of non-trivial Steenrod operations. This fact will be the motivating example for introducing E_∞-algebras and to explain how the E_∞-structure on cochains encodes this phenomenon in a precise way. Since the theory of E_∞-algebras needs the notion of operad, general recollections on the basics of operad theory will be provided.

This talk is part of the activity of the MIUR Excellence Department Projects MatMod@TOV.

Abstract: The Reeb graph R(f) of a C¹-function f from M to the real numbers with isolated critical points is a quotient object by the identification of connected components of function levels which has a natural structure of graph. The quotient map p from M to R(f) induces a homomorphism p_* from the fundamental group of M to the fundamental group of R(f) which is equal to F_r , the free group of r generators. This leads to the natural question whether every epimorphism from a finitely presented group G to F_r can be represented as the Reeb epimorphism p_* for a suitable Reeb (or even Morse) function f. We present a positive answer to this question. This is done by use of a construction of correspondence between epimorphisms from the fundamental group of M to F_r and systems of r framed non-separating hypersurfaces in M, which induces a bijection onto their framed cobordism classes. As applications we provide new purely geometrical-topological proofs of some algebraic facts.

This talk is part of the activity of the MIUR Excellence Department Projects MatMod@TOV.

Abstract: The goal of this talk is to algebraically model the Sr-equivariant homotopy type of the configuration space of r labeled and distinct points in d-dimensional Euclidean space. We will present and compare two models: the Barratt-Eccles simplicial set and the multisimplicial set of 'surjections'. Moreover, we will introduce multisimplicial sets and discuss their connection to more well-known simplicial sets. Multisimplicial sets can model homotopy types using fewer cells, making them a highly useful tool. Following this, we will explore in detail how to recognize configuration spaces in the aforementioned models by playing with a graph poset. An explicit relationship between the models will also be presented. This is joint work with Anibal M. Medina-Mardones and Paolo Salvatore.

This talk is part of the activity of the MIUR Excellence Department Projects MatMod@TOV.

Abstract: Persistent homology, a popular method in Topological Data Analysis, encodes geometric information of data into algebraic objects called persistence modules. Invoking a decomposition theorem, these algebraic objects are usually represented as multisets of points in the plane, called persistence diagrams, which can be fruitfully used in data analysis in combination with statistical or machine learning methods. Wasserstein distances between persistence diagrams are a common way to compare the outputs of the persistent homology pipeline. In this talk, I will explain how a notion of p-norm for persistence modules leads to an algebraic version of Wasserstein distances which fit into a general framework for producing distances between persistence modules. I will then present stable invariants of persistence modules which depend on Wasserstein distances and can be computed efficiently. The use of these invariants in a supervised learning context will be illustrated with some examples.

This talk is part of the activity of the MIUR Excellence Department Projects MatMod@TOV.

Abstract: The induced character formula in classical representation theory can be used, among other things, to describe the dimension of coinvariants of a representation in terms of its character. In this talk, I will explain how this formula is related to the multiplicativity of Euler characteristics in algebraic topology, and, in a more homotopy-coherent context, to the composability of so-called Becker-Gottlieb transfers, which are "wrong-way maps" in singular (co)homology; by describing a general formula to compute "dimensions of homotopy colimits". If time permits, I will discuss the most general case in which composability of Becker-Gottlieb transfers is now known. This is based on joint works with Carmeli-Cnossen-Yanovski, Klein-Malkiewich (and the last part with Volpe-Wolf).

This talk is part of the activity of the MIUR Excellence Department Projects MatMod@TOV.

Abstract: Operads are algebraic structures capturing multi-ary operations. The little disks operads, encoding operations on iterated loop spaces, are fundamental examples. Voronov introduced Swiss-Cheese operads, which generalize little disks to relative loop spaces of pairs of spaces. While the little disks operads are formal (their cohomology determines their rational homotopy type), the standard Swiss-Cheese operads are not. We will discuss why higher-codimensional Swiss-Cheese operads are formal, and why Voronov's original version is not. This non-formality result, stemming from joint work with R. V. Vieira, leads to questions about truncations and Massey products.

This talk is part of the activity of the MIUR Excellence Department Projects MatMod@TOV.

Abstract: For a framed n-manifold M one can produce an explicit pairing between M and its one point point compactification M⁺, taking values in Sⁿ, which on homology induces the Poincaré duality pairing. We show that this can be lifted to the level of operads to produce a stable equivalence between E_n, the little n-disks operad, and a shift of its Koszul dual. This gives a proof of the same celebrated result of Ching--Salvatore, but manages to avoid using technical results in geometry, homotopy theory, and even basic analysis that appear in their proof.

This talk is part of the activity of the MIUR Excellence Department Projects MatMod@TOV.

Abstract: Bounded cohomology of groups is a variant of ordinary group cohomology introduced by Johnson in the 70s in the context of Banach algebras and then intensively studied by Gromov in his seminal paper "Volume and bounded cohomology" in relation to geometry and topology of manifolds. Since the 80s bounded cohomology has then grown up as an independent and active research field. On the other hand, it is notoriously hard to compute bounded cohomology. For this reason it is natural to first investigate groups with trivial bounded cohomology groups. During this talk we survey recent advances around "acyclicity" in bounded cohomology and we will introduce a new algebraic criterion for the vanishing of bounded cohomology. This is part of a joint work with Caterina Campagnolo, Francesco Fournier-Facio and Yash Lodha.

This talk is part of the activity of the MIUR Excellence Department Projects MatMod@TOV.

Abstract: Knots form an infinite and complex data set, with topological invariants that are often intertwined in ways not yet fully understood. Many foundational challenges in knot theory and low-dimensional topology can be recast as problems in reinforcement learning and generative machine learning. A key decision in approaching knot theory through an ML lens is determining how to represent knots in a machine-readable format, which can be thought of as selecting a suitable prior distribution over the space of all knots. In this talk, I will explore the challenges of representing knots for ML applications and showcase recent examples where machine learning has been successfully applied to problems in low-dimensional topology.

This talk is part of the activity of the MIUR Excellence Department Projects MatMod@TOV.

Abstract: TBA

This talk is part of the activity of the MIUR Excellence Department Projects MatMod@TOV.

Abstract: Beilinson-Drinfeld's Grassmannian on an algebraic curve is an important object in Representation Theory and in the Geometric Langlands Program. I will describe some analogs of this construction when the curve is replaced by a surface, together with related preliminary results. This is partly a joint work with Benjamin Hennion (Orsay) and Valerio Melani (Florence) , and partly a joint work in progress with Andrea Maffei (Pisa) and Valerio Melani (Florence).

This talk is part of the activity of the MIUR Excellence Department Projects MatMod@TOV.

Abstract: For many known families of algebro-geometric objets indexed by natural numbers (the symmetric groups, the braid groups,...) a phenomenon known as homological stability happens : their homology H_d(X_n) becomes stationary when n goes to ∞. Sometimes, these objects are equipped with actions of the symmetric group Σ_n on X_n : in this context, the right notion of stability is that of representation stability introduced by Church and Farb in 2010. One main example is the family of ordered configuration spaces of manifolds, whose cohomology is known to be representation stable with explicit ranges. I will review in this talk the main ideas of representation stability and explain how we can take a homotopy-theoretic point of view on representation stability to generalize such stability theorems.

This talk is part of the activity of the MIUR Excellence Department Projects MatMod@TOV.

Abstract: Characterizing intrinsic and extrinsic transitions in cortical activity can provide an understanding of cognition, e.g., how the ebbs and flows of cognition are anchored in the transitions of neural activity. Further, such anchoring could facilitate better models for psychiatric disorders and provide novel avenues for cognitive enhancement. This talk explores how noninvasive neuroimaging, despite its inherent limitations, can be leveraged to anchor cognitive performance and psychiatric nosology into rich spatiotemporal dynamics. We propose using tools from Topological Data Analysis (TDA), especially Mapper, to tackle the inherent noise in noninvasive neuroimaging devices and for capturing and characterizing brain dynamics in healthy and patient populations.

This talk is part of the activity of the MIUR Excellence Department Projects MatMod@TOV.