Towards a Combinatorial Proof Theory

International audienceThe main part of a classical combinatorial proof is a skew fi-bration, which precisely captures the behavior of weakening and contraction. Relaxing the presence of these two rules leads to certain substruc-tural logics and substructural proof theory. In this paper we investigate what happens if we replace the skew fibration by other kinds of graph homomorphism. This leads us to new logics and proof systems that we call combinatorial

Lecture hall graphs and the Askey scheme

We establish, for every family of orthogonal polynomials in the $q$-Askey scheme and the Askey scheme, a combinatorial model for mixed moments and coefficients in terms of paths on the lecture hall graph. This generalizes the previous results of Corteel and Kim for the little $q$-Jacobi polynomials. We build these combinatorial models by bootstrapping, beginning with polynomials at the bottom and working towards Askey-Wilson polynomials which sit at the top of the $q$-Askey scheme. As an application of the theory, we provide the first combinatorial proof of the symmetries in the parameters of the Askey-Wilson polynomials.Comment: 43 pages, 23 figure

On the Polytopal Generalization of Sperner’s Lemma

We introduce and prove Sperner’s lemma, the well known combinatorial analogue of the Brouwer fixed point theorem, and then attempt to gain a better understanding of the polytopal generalization of Sperner’s lemma conjectured in Atanassov (1996) and proven in De Loera et al. (2002). After explaining the polytopal generalization and providing examples, we present a new, simpler proof of a slightly weaker result that helps us better understand the result and why it is correct. Some ideas for how to generalize this proof to the complete result are discussed. In the last two chapters we provide a brief introduction to the basics of matroid theory before generalizing a matroid generalization of Sperner’s lemma proven in Lovász (1980) to polytopes. At the end we present some partial progress towards proving the polytopal generalization of Sperner’s lemma using this matroid generalization

The Borsuk-Ulam-property, Tucker-property and constructive proofs in combinatorics

This article is concerned with a general scheme on how to obtain constructive proofs for combinatorial theorems that have topological proofs so far. To this end the combinatorial concept of Tucker-property of a finite group $G$ is introduced and its relation to the topological Borsuk-Ulam-property is discussed. Applications of the Tucker-property in combinatorics are demonstrated.Comment: 12 pages, 0 figure

Towards a Theory of Logarithmic GLSM Moduli Spaces

In this article, we establish foundations for a logarithmic compactification of general GLSM moduli spaces via the theory of stable log maps. We then illustrate our method via the key example of Witten's $r$-spin class. In the subsequent articles, we will push the technique to the general situation. One novelty of our theory is that such a compactification admits two virtual cycles, a usual virtual cycle and a "reduced virtual cycle". A key result of this article is that the reduced virtual cycle in the $r$-spin case equals to the r-spin virtual cycle as defined using cosection localization by Chang--Li--Li. The reduced virtual cycle has the advantage of being $\mathbb{C}^*$-equivariant for a non-trivial $\mathbb{C}^*$-action. The localization formula has a variety of applications such as computing higher genus Gromov--Witten invariants of quintic threefolds and the class of the locus of holomorphic differentials

The Jacobian Conjecture as a Problem of Perturbative Quantum Field Theory

The Jacobian conjecture is an old unsolved problem in mathematics, which has been unsuccessfully attacked from many different angles. We add here another point of view pertaining to the so called formal inverse approach, that of perturbative quantum field theory.Comment: 22 pages, 13 diagram