TAG Seminar

Motivated by applications to the theory of error-correcting codes, we give an algorithmic method for computing a generating set for the ideal generated by $\beta$-graded polynomials vanishing on a subset of a simplicial complete toric variety $X$ over a finite field $\mathbb{F}_q$, parameterized by rational functions, where $\beta$ is a $d\times r$ matrix whose columns generate a subsemigroup $\mathbb{N}\beta$ of $\mathbb{N}^d$. We also give a method for computing the vanishing ideal of the set of $\mathbb{F}_q$-rational points of $X$. We talk about some of its algebraic invariants related to basic parameters of the corresponding evaluation code. When $\beta=[w_1 \cdots w_r]$ is a row matrix corresponding to a numerical semigroup $\mathbb{N}\beta=\langle w_1,\dots,w_r \rangle$, $X$ is a weighted projective space and generators of its vanishing ideal is related to the generators of the defining (toric) ideals of some numerical semigroup rings corresponding to semigroups generated by subsets of $\{w_1,\dots,w_r\}$.

Note: This talk will be on Zoom. Please contact the organizers for the link.

Although logic and set theory are commonly associated with discussions about the foundations of mathematics, another no less important rôle that they fulfill is as a framework for rigurously studying the combinatorics of infinite structures. In this talk, I will illustrate the flavour of this kind of mathematics by exhibiting some examples, primarily stemming from the branch of combinatorics known as Ramsey theory, where one can appreciate the subtle variations in behaviour of the objects of study as one analyzes them from different logical viewpoints (countable objects, uncountable objects, what happens without the Axiom of Choice, etc.).

Note: These talks will be on Zoom. Please contact the organizers for the link.

Fundamental notions on rings

Introduction to coding theory over finite rings

Symmetric, frobenius, and quasi-frobenius

Classification of finite rings

Sept.

6th

A \emph{digital image} is a finite subset of the integer lattice in $\mathbb{R}^n$ together with an adjacency relation.  For instance, a 2-dimensional digital image is an abstraction of an actual digital image consisting of pixels.  Then \emph{digital topology} refers to the many and ongoing efforts to use notions from (algebraic) topology to  study and analyze digital images.

As part of a broad program to bring notions of homotopy theory into the setting of digital topology, we have defined a fundamental group for an arbitrary digital image.  Our construction of this digital fundamental group is intrinsic, in that it is defined in an essentially combinatorial way directly in terms of the digital image; it also differs in key aspects from previously defined versions of the fundamental group in digital topology.

We show that our fundamental group of a digital image is isomorphic to the edge group of the clique complex of the digital image considered as a graph. The clique complex is a simplicial complex.  As such, its edge group is isomorphic to the ordinary (topological) fundamental group of its geometric realization.  These identifications mean that many familiar facts about the ordinary topological fundamental group may be translated into their counterparts for our digital fundamental group.   We conclude that the digital fundamental group of any digital circle is $\Z$. We deduce the Seifert-Van Kampen Theorem in the setting of digital topology.  We show that the (digital) fundamental group of every 2-dimensional digital image is a free group.   These results have the potential to be useful in topology-related image processing operations such as thinning and feature extraction. 

Model categories were created by Daniel Quillen to provide a general environment in which to do homotopy theory.  The factorizations asked for by the axioms (for example, factoring a morphism into a weak cofibration followed by a fibration) are nowadays assumed to be functorial, or natural.  Emily Riehl showed that, if the fibration half of the factorization had a certain algebraic structure (and the cofibrations a coalgebraic structure), then the solutions to the lifting problems demanded by the model structure can also be made functorial.  Furthermore, she showed that any cofibrantly generated model category (such as the category of chain algebras) has such an algebraic system.

We provide an explicit construction of such an algebraic system for chain algebras, using relatively classical techniques.  

This is joint work with P.-E. Parent (U. Ottawa) and K. Hess (EPFL).

March

29

Kaplansky's

Conjecture