I will discuss a classic result of Tutte from 1963 that shows an elegant connection between graph theory, physics, geometry, and linear algebra. Tutte addressed the problem of how to draw a planar graph (a collection of vertices and edges that can be drawn without crossing edges in the plane) so that every edge is a straight line and the resulting faces are all convex polygons.

With the advantage of hindsight, I will also show how this result has influenced many modern ideas in graph theory and graph drawing. Despite covering a lot of ground, the talk should be widely accessible.

Abstract: By diagrammatically reformulating the notion of a regular conditional probability (RCP), we are able to abstract the idea to a variety of categories. However, the interpretation of the RCP may be different from our usual probabilistic intuition and typically depends on the specific category. We will specialize this abstraction of a RCP to the category of finite-dimensional $$C^*$$-algebras and completely positive maps, a setting for non-commutative spaces and quantum information theory. Our main result is a necessary and sufficient condition for the existence and uniqueness of non-commutative RCPs. As a corollary, we reproduce a familiar theorem in the commutative (classical) setting. This talk should be accessible to a wide (mathematically-minded) audience. This is joint work with Benjamin Russo at SUNY Farmingdale.

Join us for a talk by Adrian Brasoveanu (UC Santa Cruz) in the Logic Colloquium!


"Computational Cognitive Modeling for Syntax and Semantics"

Adrian Brasoveanu (joint work with Jakub Dotlacil)

Abstract: I introduce a typical experimental task in psycholinguistics -- self-paced reading -- and show how to build end-to-end simulations of a human participant in such an experiment; end-to-end means that we model visual and motor processes together with specifically linguistic processes (syntactic and semantic parsing) in a complete model of the experimental task. The model embeds theoretical hypotheses about linguistic representations and parsing processes in an independently motivated cognitive architecture (ACT-R). In turn, the resulting cognitive models can be embedded in Bayesian models to fit them to experimental data, estimate their parameters and perform quantitative model comparison for qualitative theories.


All welcome!

https://logic.uconn.edu/

Abstract: A quasiregular (qr) mapping folds one space over, with limited distortion of topology and of relative distances. The theory of qr mappings in Euclidean spaces is rich with examples and connections complex analysis, dynamical systems, and PDEs. In the early 2000s, Heinonen and Rickman initiated a broader study of qr mappings in more general metric spaces, such as the Heisenberg group. In this talk, I will give an introduction to the field, and discuss some recent results on building and forbidding qr mappings between sub-Riemannian manifolds.

TBA

Each year, the UConn Logic Group bestows the title of Scholar of Consequence to an eminent logician whose work shaped the field.

The 2019 Scholar of Consequence is Sara Negri (Helsinki, Finland).

Professor Negri will give the following public the Annual Logic Lecture:

A three-fold method for non-classical logics

Abstract:

There are three established ways of analysing the validity of a logical argument: axiomatic, semantic, and inferential; they correspond to what could be called the normative, the descriptive, and the deductive face of a logical system and are tightly related to each other by fundamental meta-theorems.

For non-classical logics, each of the three aspects has drawbacks when considered in isolation. The situation changes, however, drastically when the three aspects are developed in parallel. Through the labelled formalism, by now is a well-developed methodology, models can be considered as purely mathematical objects with no ontological assumptions upon them: the semantics is turned into an essential component in the syntax of sequent calculi with invertible rules. Such calculi not only provide a tool for the automatisation of reasoning, but can also be used to establish very general properties of logical systems.

All welcome!

https://logic.uconn.edu/annual-logic-lecture/

Join is for the Logic Colloquium given by the 2019 Scholar of Consequence!

Sara Negri (Helsinki, Finland)

"Unveiling the constructive core of classical theories: A contribution to 90 years of Glivenko’s theorem"

Abstract:

Glivenko’s well known result of 1929 established that a negated propositional formula provable in classical logic is even provable intuitionistically. Similar later transfers from classical to intuitionistic provability therefore fall under the nomenclature of Glivenko-style results: these are results about classes of formulas for which classical provability yields intuitionistic provability. The interest in isolating such classes lies in the fact that it may be easier to prove theorems by the use of classical rather than intuitionistic logic. Further, since a proof in intuitionistic logic can be associated to a lambda term and thus obtain a computational meaning, such results have more recently been gathered together under the conceptual umbrella “computational content of classical theories.” They also belong to a more general shift of perspective in foundations: rather than developing constructive mathematics separately, as in Brouwer’s program, one studies which parts of classical mathematics can be directly translated into constructive terms.

We shall survey how Glivenko-style results can be easily obtained by the choice of suitable sequent calculi for classical and intuitionistic logic, by the conversion of axioms into inference rules, and by the procedure of geometrization of first order logic.

All welcome!

https://logic.uconn.edu/calendar/

This is the soft introduction to the very recent paper (https://arxiv.org/abs/1904.08568) which is joint work with a prof. F. Baudoin. From Hurwitz's theorem, all normed division algebra over real numbers up to isomorphism is the algebras of real numbers, complex numbers C, quaternions H, and octonions O. One way to understand those division algebras is by investigating so-called the Hopf fibration. We will try to see which geometric structures and probability tools play a role to understand each division algebras. This talk would be accessible to a general math audience.

Abstract: We study nonlocal operators that generate anisotropic jump processes, such as a jump process that behaves like a stable process in each direction but with a different index of stability. Its generator is the sum of one-dimensional fractional Laplace operators with different orders of differentiability. We study such operators in the general framework of bounded measurable coefficients.
The objective of this talk is to provide regularity results for weak solutions to corresponding integro-differential equations.
Joint work with Moritz Kassmann.

Abstract: TBA