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##
Seminari periodici

DIPARTIMENTO DI MATEMATICA

# Logic, Categories, and Applications Seminar

**Organizzato da:**Martino Lupini

Lunedì

11 novembre

Giuseppe Rosolini

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, logica, teoria delle categorie

ore
14:00

presso Seminario II

The notion of ultracategory was introduced by Michael Makkai in a paper in APAL in 1990 for the characterisation of categories of models of pretoposes, an ample extension to (intuitionistic) first order theories of
Stone duality for Boolean algebras, providing a kind of Stone duality for first order theories -- aka conceptual completeness. Recently, Jacob Lurie refined that notion in unpublished notes producing another approach to the duality for pretoposes -- the two notions of ultracategory appear to be different, though no separating example has been produced yet.
In the talk, we shall give intuitions about Makkai's and Lurie's notions, providing examples and applications. Then we shall introduce an algebraic notion of structured category which subsumes the two kinds of ultracategories mentioned above -- technically, the "ultracompletion" 2-functor on the 2-category of small categories, and extend it to a pseudomonad. Next we show how it relates to the two existing notions.
This is joint work with Richard Garner.

Venerdì

15 novembre

Matthew Di Meglio

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, interdisciplinare, logica, teoria delle categorie

ore
14:00

presso Seminario II

The notion of abelian category is an elegant distillation of the fundamental properties of the category of abelian groups, comprising a few simple axioms about products and kernels. While the categories of real and complex Hilbert spaces and bounded linear maps are not abelian, they satisfy almost all of the abelian category axioms. Heunen and Kornell’s recent characterisation (https://doi.org/10.1073/pnas.2117024119) of these categories of Hilbert spaces is reminiscent of the Freyd–Mitchell embedding theorem, which says that every abelian category has a full, faithful and exact embedding into the category of modules over a ring. The axioms are similar, but incorporate the extra structure of a dagger—an identity-on-objects involutive contravariant endofunctor—which encodes adjoints of bounded linear maps. By keeping only the axioms that directly parallel the ones for abelian categories, we arrive at a nice class of dagger categories, which I call rational dagger categories, that enjoy many of the same properties as the categories of Hilbert spaces mentioned above. The name alludes to their unique enrichment in the category of rational vector spaces.
In this talk, I will give a gentle introduction to rational dagger categories, highlighting the parallels with abelian categories. I will not assume prior familiarity with dagger categories, instead introducing the relevant concepts as needed. This talk is based on a recent preprint (https://arxiv.org/abs/2312.02883).

Lunedì

18 novembre

Anna De Mase

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, interdisciplinare, logica

ore
14:00

presso Seminario II

Explicit constructions of models of the theory of a valued field are useful tools for understanding its model theory. Since Kaplansky’s work, it has been a topic of interest to characterize value fields in terms of fields of power series. In particular, Kaplansky proved that, under certain assumptions, an equicharacteristic valued field is isomorphic to a Hahn field. In this talk, we show that in the mixed characteristic case, assuming the Continuum Hypothesis, we can provide a characterization, in terms of power series, of pseudo-complete finitely ramified valued fields with a fixed residue field k and valued in a Z-group G, using a Hahn-like construction with coefficients in a finite extension of the Cohen field C(k) of k. In this construction, the elements of the field are “twisted” power series, i.e. powers series whose product is defined by having an extra factor, given by the cross-section and a 2-cocycle determined via the value group. This generalizes a result by Ax and Kochen, who characterize pseudo-complete valued fields elementarily equivalent to the field of p-adic numbers Q_p. If time permits, we will see some consequences of this characterization regarding the problem of lifting automorphisms of the residue field and the value group to automorphisms of the valued field in the mixed characteristic case.

Venerdì

22 novembre

Mario Fuentes

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, logica

ore
14:00

presso Seminario II

The rational homotopy type of simply connected spaces is fully captured by its Quillen model, a differential graded Lie algebra constructed from the space. Conversely, any positively graded differential Lie algebra can be "realized" as a topological space, with rational homotopical and homological invariants preserved by these two functors.
However, these constructions are inherently limited to connected and simply connected spaces. To remove these constraints, we must move to the category of complete Lie algebras. Within this category, there exists a cosimplicial object that gives rise to a pair of adjoint functors between the categories of complete Lie algebras and topological spaces.
In this talk, we will explore the
construction of this pair of functors and some important properties. Concretely, we will show that composing both of them results in the Bousfield-Kan $\mathbb{Q}$-completion. Additionally, we will discuss how this framework can be extended to curved Lie algebras, leading to a unpointed theory.

Lunedì

02 dicembre

Luca Marchiori

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, logica

ore
14:00

presso Seminario I

Enriching traditional algebraic structures such as abelian groups with a topology runs into some issues from the Homological Algebra point of view, as the categories thus obtained are not, in general, classical abelian categories, but instead only satisfy the weaker notion of being quasi-abelian. The additional homological work required to deal with topological abelian groups has traditionally been carried out for Locally Compact Groups by Moskowitz with extensive use of the Pontryagin duality. However, this work has shown that Locally Compact Groups do not satisfy some useful homological properties. In this talk I will introduce the category of pro-Lie Polish groups, a recently proposed generalization of Polish Locally Compact Groups, and discuss why they are better suited as an object of study of Homological Algebra.

Venerdì

13 dicembre

Noa Lavi

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

ore
14:00

presso Seminario I

A classical tool in the study of real closed fields are the fields K((G)) of generalized power series (i.e., formal sums with well-ordered support) with coefficients in a field K of characteristic 0 and exponents in an ordered abelian group G. We generalize previous results about irreducible elements and unique factorization in the subring K((G≤0)).

Lunedì

16 dicembre

Francesco Paolo Gallinaro

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, logica

ore
14:00

presso Seminario I

A valued difference field is a field equipped with a valuation and an automorphism which preserves the valuation ring setwise. In this talk I will discuss various results on these objects, concerning the existence of a section of the valuation compatible with the automorphism and of extensions of the valuation to difference fields extension, the solvability of amalgamation problems of valued difference fields, and the classification of the theory of valued difference fields in the sense of positive model theory. This is joint work with Jan Dobrowolski and Rosario Mennuni.

Mercoledì

18 dicembre

Linus Richter

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, interdisciplinare, logica, teoria delle categorie

ore
14:00

presso - Aula Da Stabilire -

Lunedì

20 gennaio

Lorenzo Luperi Baglini

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di logica

ore
14:00

presso Seminario II

We introduce the concept of Ramsey pairs, and show how they can be prove several infinitary results in combinatorics.

Martedì

25 febbraio

Claudio Agostini

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, logica

ore
14:00

presso Seminario II

Venerdì

28 marzo

Alessio Savini

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, logica

ore
14:00

presso Seminario II

Given a measured groupoid G, together with Filippo Sarti, we defined a cohomology theory which generalizes the measurable bounded cohomology of a locally compact group. In the particular case of a groupoid associated to a measure preserving action, our cohomology boils down to the usual bounded cohomology of the group with twisted coefficients. We will discuss the possible applications of this result to orbit equivalence.

## Seminari passati

2024

04 novembre

Sebastian Eterovich

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, logica, teoria delle categorie

The j-function is a central player in the study of elliptic curves. It satisfies many interesting algebraic properties, but there many things still unknown. In this talk we will discuss the problem of trying to solve polynomial equations that involve the j-function and its first two derivatives, and we will discuss some important cases that can be solved. This is joint work with Vahagn Aslanyan and Vincenzo Mantova.

2024

29 ottobre

Nicola Carissimi

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, logica, teoria delle categorie

Two main generalizations of categories are bicategories and enriched categories. The first one allows morphisms one level up, the other one allows morphisms to be much more general objects rather than just sets. This talk will try to explain what happens if we do the two at the same time. In particular, we will explore the main available results and tools with which enriched bicategories can be tamed. Among the results we have strictification theorems, for what concerns the tools, most notably, the extremely powerful language of string diagrams. Time permitting, we will see the combination of the two in action.

2024

27 maggio

Luca Motto Ros

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario interdisciplinare

We show that the natural operation of connected sum for graphs can be used to prove at once most of the universality results from the literature concerning graph homomorphism. In doing so, we significantly improve many existing theorems and solve some natural open problems. Despite its simplicity, our technique unexpectedly leads to applications in quite diverse areas of mathematics, such as category theory, combinatorics, classical descriptive set theory, generalized descriptive set theory, model theory, and theoretical computer science. (Joint work with S. Scamperti)

2024

18 aprile

Riccardo Camerlo

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario interdisciplinare

The Wadge preorder is a tool to compare the complexity of subsets of topological spaces: if $A,B$ are subsets of the topological spaces $X,Y$, respectively, $A$ \emph{Wadge reduces} to $B$ if there exists a continuous function $f:A\to B$ such that $A=f^{-1}(B)$.
While most of the earlier work on the Wadge preorder concerned zero-dimensional Polish spaces, recent investigations have involved more general kinds of spaces. This talk surveys some of the results and presents a few open problems and perspectives in the field.

2024

23 febbraio

Mauro Di Nasso

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, logica

In recent years there has been a growing interest in Ramsey theory and related
problems in combinatorics of numbers. Historically, the earliest results in this field
are Schur's Theorem ("In every finite coloring of the naturals there exists a monochromatic triple a, b, a+b")
and van der Waerden's Theorem ("In every finite coloring of the naturals there exist monochromatic
arithmetic progressions of arbitrary length").
A peculiar aspect of this area of research is the wide variety of methods used:
in addition to the tools of elementary combinatorics, also methods of discrete Fourier analysis,
ergodic theory, and ultrafilter space algebra have been successfully applied.
Recently, a further line of research has been undertaken, in which combinatorial properties
of sets of integers are studied by methods of nonstandard analysis.
In this seminar I will discuss these methods and present some examples of their applications.

2024

22 febbraio

Nicholas Meadows

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, interdisciplinare, logica, teoria delle categorie

Monads and algebraic theories are two categorical approach to universal algebra. In his book Higher Algebra, Jacob Lurie established a relatively comprehensive theory of monads on infinity categories. However, his approach can be difficult in practice to use due to its highly technical nature.
In this talk, we will describe a version of generalized algebraic theories in the $\infty$-categorical setting, and show that it is recovers Lurie's theory for nice monads. As an application, we will prove several structural results about monads in the $\infty$-categorical setting. We will also use our result to describe the algebraic theories of E_1, E_2, and E_\infty algebras.

2024

16 febbraio

Hilbert geometries have been introduced as a generalization of hyperbolic geometry, and provide a family of metric spaces where the Euclidean straight lines are geodesics. A Hilbert geometry is said to be divisible if it admits a group of isometries that acts cocompactly on the space. The aim of this talk is to introduce the class of divisible Hilbert geometries and to look at a characterization of hyperbolicity in this class.

2024

07 febbraio

Aristotelis Panagiotopoulos

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, analisi matematica, logica, sistemi dinamici

A Polish group is TSI if it admits a two-side invariant metric. It is CLI if it admits complete and left-invariant metric. The class of CLI groups contains every TSI group but there are many CLI groups that fail to be TSI. In this talk we will introduce the class of α-balanced Polish groups where α ranges over all countable ordinals. We will show that these classes completely stratify the space between TSI and CLI. We will also introduce "generic α-unbalancedness": a turbulence-like obstruction to classification by actions of α-balanced Polish groups. Finally, for each α we will provide an action of an α-balanced Polish group whose orbit equivalence relation is not classifiable by actions of any β-balanced Polish group with β<α. This is joint work with Shaun Allison.

2024

26 gennaio

Antongiulio Fornasiero

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, interdisciplinare, logica

Let d be a finite tuple of commuting derivations on a field K.
A classical result allows us to associate a numerical polynomial to d (the Kolchin polynomial), measuring the "growth rate" of d.
We show that we can abstract from the setting of fields with derivations, and consider instead a matroid with a tuple d of commuting (quasi)-endomorphisms.
In this setting too there exists a polynomial measuring the growth rate of d.
Joint work with E. Kaplan

2024

19 gennaio

Elena Bogliolo

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, interdisciplinare, logica, teoria delle categorie

Bounded cohomology of groups is a variant of group cohomology that, given a group and a Banach coefficient module over such group, gives graded semi-normed vector spaces.
A major role in the theory of bounded cohomology is played by amenable groups and amenable actions as they provide vanishing conditions for bounded cohomology. The goal of this talk is to introduce bounded cohomology of groups and look into its realtion with amenability.

2024

08 gennaio

Ivan Di Liberti

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, interdisciplinare, logica, teoria delle categorie

2023

15 dicembre

Nicola Carissimi

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, interdisciplinare, logica, teoria delle categorie

In this talk we will give a brief introduction to the theory of bicategories, in order to present a bicategorical gadget, called Mackey 2-functor, axiomatizing an ubiquitous phenomena in finite equivariant mathematics, especially in representation theory of finite groups and equivariant topology. The key point of this theory so far is the existence of a universal bicategory (the Mackey 2-motives) encoding the properties of such 2-functors, and allowing furthers constructions.

2023

11 dicembre

Matteo Casarosa

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di analisi matematica, interdisciplinare, logica

In this talk, after reviewing the main concepts related to forcing and giving some examples of frequently used forcing notions (i.e. partially ordered sets deployed for this technique), we discuss some "concrete" mathematical statements that can be shown to be undecidable. In particular, we will show that the existence of a Suslin Line is independent of ZFC.

2023

11 dicembre

Filippo Calderoni

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, interdisciplinare, logica

In this talk we shall discuss condensed points in the Polish space of left-orderings of a fixed left-orderable groups. We will describe new techniques to show that the conjugacy relation on the space of left-orderings is not smooth. We discuss how these methods apply to a large class of left-orderable groups, and they shed light on spaces of left-orderings with low Borel complexity. This is joint work with Adam Clay.

2023

05 dicembre

Matteo Casarosa

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di analisi matematica, interdisciplinare, logica

In this talk, we introduce some set-theoretic tools to prove consistency results. More precisely, the presentation will cover Goedel's Constructible Universe as well as Cohen's method of forcing. No previous knowledge on this subject will be assumed.

2023

13 novembre

Daniele Mundici

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, analisi matematica, logica, teoria delle categorie

2023

09 novembre

Ilaria Castellano

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, interdisciplinare, logica

With the solution of Hilbert’s fifth problem, our understanding of connected locally compact groups has significantly increased. Therefore, the contemporary structure problem on locally compact groups concerns the class of totally disconnected locally compact (= t.d.l.c.) groups. The investigation of the class of t.d.l.c. groups can be made more
manageable by dividing the infinity of objects under investigation into classes of types with “similar structure”. To this end we introduce the rational discrete cohomology for t.d.l.c. groups and discuss some of the invariants that it produces. For example, the rational discrete cohomological dimension, the number of ends, finiteness properties FP_n and F_n, and the Euler-Poincaré characteristic.

2023

30 ottobre

Riccardo Treglia

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario interdisciplinare

Moving from the abstract definition of monads, we introduce a version of the call-by-value computational λ-calculus based on Wadler’s variant. We call the calculus computational core and study its reduction, prove it confluent, and study its operational properties on two crucial properties: returning a value and having a normal form. The cornerstone of our analysis is factorization results.
In the second part, we study a Curry-style type assignment system for the computational core. We introduce an intersection type system inspired by Barendregt, Coppo, and Dezani system for ordinary untyped λ-calculus, establishing type invariance under conversion. Finally, we introduce a notion of convergence, which is precisely related to reduction, and characterizes convergent terms via their types.
For greater accessibility, the presentation will begin with a brief introduction to lambda calculus, monads, and intersection types.

2023

15 settembre

Joost Hooyman

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, interdisciplinare, teoria delle categorie

A well-known shortcoming of the category of smooth manifolds is its lack of arbitrary pullbacks. A pullback of manifolds, and in particular an intersection of submanifolds, exists only along maps which are transversal. This problem can be overcome by passing to the larger category of derived smooth manifolds. The construction of this category combines ideas from algebraic geometry, homotopy theory and of course differential topology.
We can describe this construction in several steps. Firstly, we consider the relation between manifolds and schemes. Here, we employ the so-called C^\infty-rings, which are algebraic objects encoding the structure of the collection of smooth functions on R^n beyond that of an R-algebra. By the general philosophy of algebraic geometry, their duals give rise to geometric objects, called C^\infty-schemes. These geometric objects are primarily studied as models for synthetic differential geometry.
Secondly, we introduce homotopy theory into the picture. This step adapts the ideas of derived algebraic geometry to the setting of C^\infty-schemes. Our approach replaces the algebraic objects involved by their simplicial counterparts. In this context, the main objective is to develop a homotopy theory of presheaves which allows us to work with sheaf axioms 'up to homotopy'.
Succinctly, a derived smooth manifold can be described as a homotopical C^\infty scheme of finite type. In my talk, I will highlight some steps of the rather intricate construction described above. Hopefully, this will give the audience a perspective from which to think further about these exciting interactions between algebraic geometry, homotopy theory and differential topology.

2023

14 settembre

Joost Hooyman

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Seminario di algebra e geometria, interdisciplinare, teoria delle categorie