Seminar

Title: Statistical Fluctuations in the Causal Set-Continuum Correspondence

 Abstract: Causal set theory is an approach to quantum gravity that proposes that spacetime is fundamentally discrete and the causal relations among the discrete elements play a prominent role in the physics. Progress has been made in recognizing and understanding how some continuumlike features can emerge from causal sets at macroscopic scales, i.e., when the number of elements is large. An important result in this context is that a causal set is well approximated by a continuum spacetime if there is a number-volume correspondence between the causal set and spacetime.

This occurs when the number of elements within an arbitrary spacetime region is proportional to its volume. Such a correspondence is known to be best achieved when the number of causal set elements is randomly distributed according to the Poisson distribution. I will discuss the Poisson distribution and the statistical fluctuations it induces in the causal set-continuum correspondence, highlighting why it is important and interesting. I will also discuss new tools and techniques that facilitate such analyses.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,

Title: : Algebraic bounds for sum-rank-metric codes

Abstract:The sum-rank metric is a generalization of the well-known Hamming and rank metrics. In this talk we introduce two new bounds on the maximal cardinality of the sum-rank-metric code with a given minimum distance. One of the bounds exploits a connection between such a code and a (d-1)-independent set in a graph defined for the sum-rank-metric space. We then use the eigenvalues of the graph to deduce the bound. The second bound is derived from the Delsarte's LP method, which has been previously obtained for Hamming metric, rank metric, Lee metric, and others, but the sum-rank-metric case remained open. To derive the new LP bound we propose a way to construct an association scheme for the sum-rank metric, since the approach used in the Hamming and the rank-metric cases fails due to the associated graph not being distance-regular in general. Based on computational experiments on relatively small instances, we observe that the obtained bounds often outperform the bounds previously known for sum-rank-metric codes.

This talk is based on a joint work with A. Abiad, A.L. Gavrilyuk, A. Ravagnani, and I. Ponomarenko.

Title: A New Complexity Analysis of Primal-Dual Interior-Point Methods with Applications to Hyperbolic Cone Programming

Abstract: Primal-dual interior-point methods stand at the forefront of convex optimization, distinguished by their theoretical efficiency and practical robustness. After presenting the fundamental 
structure of these algorithms, I will introduce a complexity measure for a large family of primal-dual algorithms. A consequence of the new analyses of this complexity measure is a substantial improvement of the iteration 
complexity bounds for these primal-dual algorithms on hyperbolic cone programming problems.

This talk is based on joint work with Joachim Dahl and Lieven Vandenberghe.

Title: Entropy and the growth rate of universal covering trees

Abstract:This work studies the relation between two graph parameters, rho and Lambda. For an undirected graph G,  rho(G) is the growth rate of its universal covering tree, while Lambda(G) is a weighted geometric average of the vertex degree minus one, corresponding to the rate of entropy growth for the non-backtracking random walk (NBRW). It is well known that rho >= Lambda for all graphs, and that graphs with rho = Lambda exhibit some special properties. In this work we derive an easy to check, necessary and sufficient condition for the equality to hold. Furthermore, we show that the variance of the number of random bits used by a length l NBRW is O(1) if rho = Lambda and Omega(l) if rho > Lambda. As a consequence we exhibit infinitely many non-trivial examples of graphs with rho = Lambda.

Joint work with Idan Eisner, Tel-Hai College, Israel.

Title:Evolution of random graph orders and their dimensions

Abstract: A poset is a set X equipped with a partial order. In this talk I will briefly review the literature of different models on random orders. Then we discuss a particular model called the random graph order. The random graph order is classical model to generate a random causal set, which was introduced in physics to model and analyse the space-time universe. 

We will focus on an open problem proposed by Erdos, Kierstad and Trotter on the evolution of the dimensions of random graph orders. This problem has been studied by Albert and Frieze, and by Erdos, Kierstad and Trotter around 1990. Better bounds on the dimensions were obtained by Bollobas and Brightwell in 1997, for “non-sparse” random graph orders. We study the last piece of the puzzle, in the bipartite case, by investigating “a transition phase” that was predicted to occur in the sparse regime by Bollobas and Brightwel, and we prove a negative result to their prediction. We expect that a similar phenomenon would occur in the nonbipartite case.

This talk is based on a collaborated work with Arnav Kumar. 

Title: Combinatorial interpretation of the coefficients of the causal set theory d'Alembertian

 Abstract: Causal set theory is an approach to quantum gravity where the  underlying spacetime is a locally finite poset.  This opens up many interesting combinatorial questions on posets that are either useful to the physics or that are asked by the physics but wouldn't necessarily be asked from a purer perspective.  This talk is about one of the latter questions. Glaser gave a formula for the causal set theory analogue of the d'Alembertian in general dimension (growing out of previous work of Sorkin, Benincasa and Dowker, and Dowker and Glaser).  The formula

contains integer coefficients.  Who can resist trying to find something that they count -- not me! -- so I will tell you about such a something.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,

Title: Equivariant quasisymmetry

 Abstract: We introduce equivariant quasisymmetry, a version of quasisymmetry for polynomials in two sets of variables. Using this definition we define double fundamental polynomials and double forest polynomials, a quasisymmetric generalizations of the theory of double Schur and double Schubert polynomials, where the subset of noncrossing permutations play the role of $S_n$.This combinatorics is governed by the quasisymmetric flag variety, a new geometric construction which plays the role for equivariant quasisymmetry what the usual flag variety plays in the classical story.

In this talk, I will focus on the combinatorial aspect first, and with the remaining time discuss the geometrical implications.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,

Title: New perspectives on quasisymmetry via divided differences, flag varieties, etc.

 Abstract: In this talk we will introduce a new combinatorial approach to understanding quasisymmetric polynomials via a quasisymmetric analogue of "divided differences". The resulting combinatorial theory is paired with a geometric theory that closely follows the classical development of Schubert calculus and Schubert varieties. Next week Nantel Bergeron will continue with the torus-equivariant story.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,

Title: Well-quasi-ordering and connectivity

Abstract: Connectivity systems provide an abstract framework for studying various connectivity notions on graphs, matroids, and other combinatorial structures. In joint work with Rutger Campbell, we prove a novel well-quasi-ordering result on certain connectivity systems which gives new excluded minor results in matroid theory and gives new proofs for several results in graph theory.

Title: Cayley incidence graphs

Abstract: Evra, Feigon, Maurischat, and Parzanchevksi defined Cayley incidence graphs (which they refer to as Cayley bigraphs), a class of biregular graphs used to construct bipartite expander graphs. These graphs are defined using a group G and a set of cells satisfying certain properties. Alternatively, they can be described as the incidence graphs of uniform and regular linear hypergraphs with a group G acting regularly on the vertices. In this talk, I will explore some basic properties of Cayley incidence graphs, as well as their connections to other combinatorial objects such as designs, coset geometries, and difference sets.

This talk is based on joint work with Arnbjörg Soffía Árnadóttir, Alexey Gordeev, Sabrina Lato, and Tovohery Randrianarisoa.