Department of Mathematics

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/

All are invited to celebrate our undergraduate and graduate students! Ceremony will be followed by a talk from Dr. Steven Miller (Williams College), entitled "The German Tank Problem: Math/Stats At War!"

Abstract: During World War II the German army used tanks to devastating advantage. The Allies needed accurate estimates of their tank production and deployment. They used two approaches to find these values: spies, and statistics. In this talk we describe the statistical approach and its generalization. Assuming the tanks are labeled consecutively starting at 1, if we observe $k$ serial numbers from an unknown number $N$ of tanks, with the maximum observed value $m$, what is the best estimate for $N$? This is now known as the German Tank Problem, and is a terrific example of the applicability of mathematics and statistics in the real world. We quickly review some needed combinatorial identities (which is why we are able to obtain clean, closed form expressions), give the proof for the standard problem, discuss the generalization, and show how if we were unable to do the algebra we could guess the formula by an application of linear regression, thus highlighting its power and applicability. Most of the talk only uses basic algebra and elementary knowledge of WWII.

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.