Logic, Categories, and Applications Seminar

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.