Events

Title: Operahedron Lattices

 Abstract: Two classical lattices are the Tamari lattice on bracketings of a word and the weak order on permutations. The Hasse diagram of each of these lattices is the oriented 1-skeleton of a polytope, theassociahedron and the permutohedron respectively. We examine a poset on bracketings of rooted trees whose Hasse diagram is the oriented 1-skeleton of a polytope called th operahedron. We show this poset is a lattice which answers question of Laplante-Anfossi. These lattices provide an extremelynatural generalization of both the Tamari lattice and the weak order.

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

Title:Benign Optimization Landscape of Formulations for Time-of-Arrival-Based Source Localization Problem

Abstract: : In this talk, we will address the maximum-likelihood (ML) formulation and a least-squares (LS) formulation of the time-of-arrival (TOA)-based source localization problem. Although both formulations are generally non-convex, we will show that they both possess benign optimization landscape. First, we consider the ML formulation of the TOA-based source localization problem. Under standard assumptions on the TOA measurement model, we will show a bound on the distance between an optimal solution and the true target position and establish the local strong convexity of the ML function at its global minima. Second, we consider the LS formulation of the TOA-based source localization problem. We will show that the LS formulation is globally strongly convex under certain condition on the geometric configuration of the anchors and the source and on the measurement noise. We will then derive a characterization of the critical points of the LS formulation, which leads to a bound on the maximum number of critical points under a very mild assumption on the measurement noise and a sufficient condition for the critical points of the LS formulation to be isolated. The said characterization also leads to an algorithm that can find a global optimum of the LS formulation by searching through all critical points. Lastly, we will discuss some possible future directions.

Title: Covers of the complete graph, equiangular lines, and the absolute bound

Abstract: We discuss equiangular lines and covers of the complete graph. The relationship between these objects dates to at least 1992, when Godsil and Hensel showed that any distance-regular antipodal cover of the complete graph (DRACKN) produces an ensemble of equi-isoclinic subspaces. In the case of a regular abelian DRACKN, this produces equiangular lines.
In the first part of the talk, we combine Godsil and Hensel's theorem with a 2017 observation of Waldron to explain why (with a single exception) there DO NOT exist regular abelian DRACKNs that would create d^2 equiangular lines in d-dimensional complex space, to achieve Gerzon's "absolute bound". This rules out a family of otherwise feasible DRACKN parameters that were enunciated in a 2016 paper of Coutinho, Godsil, Shirazi, and Zhan.
In the second part of the talk, we introduce "roux", a slight generalization of regular abelian DRACKNs. Roux are covers of the complete graph that produce equiangular lines. They come up naturally in the classification of doubly transitive lines, all of which arise from roux. Keeping hope alive for the present, we enunciate an infinite family of feasible roux parameters that would produce equiangular lines achieving Gerzon's absolute bound.
Based on joint work with Dustin Mixon.

Title: Sequential Contracts on Matroids

Abstract: First, I will talk about the well-known pandora’s box problem, and then I will introduce the generalization of pandora’s box to matroids. We call this problem “sequential contracts on matroids” and we will discuss some recent results about this problem. In particular, we will look at complexity results of the problem. This is joint work with Kanstantsin Pashkovich and Jacob Skitsko for my URA project in Spring 2024.

Title: Positivity of P-Recursive Sequences Satisfying Linear Recurrences

 Abstract: Whether it is decidable to determine when sequences satisfying linear recurrences with constant coefficients have all positive terms is a notorious problem in enumerative combinatorics that has essentially been open for around 90 years. Nevertheless, a "meta-principle" states that all such sequences arising from combinatorial counting problems belong to a special class where positivity (and more general asymptotic

behaviour) is decidable. Here we discuss new software for determining positivity for sequences satisfying linear recurrences with *polynomial* coefficients. Originally motivated by a novel approach to proving genus one solution uniqueness for the Canham model for biomembrane shapes, our algorithm combines rigorous numeric analytic continuation of functions satisfying linear ODEs with singularity analysis techniques from analytic combinatorics. The main talk will be presented using a live Sage Jupyter notebook, and audience members who have access to Sage with a recent version of the ore_algebra package installed (available at

https://github.com/mkauers/ore_algebra) will be able to follow along and play with the package during the talk.

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

Title:A bound on the quantum value of all compiled nonlocal games

Abstract: Nonlocal games provide valuable insights into quantum entanglement and even enable a classical verifier to confirm and control the behavior of entangled quantum provers. However, an issue with this approach has always been the necessity of two non-communicating quantum provers. To address this issue, a group of researchers recently introduced a "compilation procedure" that reduces the need for multiple provers and enforces non-communication through cryptographic methods. In this talk, we will show that even in this single prover "compiled setting," the prover remains fundamentally constrained. Specifically, we show that any polynomial-time quantum prover cannot win the "compiled game" with a higher probability than any quantum commuting provers could win the original nonlocal game. Our result is derived through a novel combination of techniques from cryptography and operator algebras and allows us to recover several important self-testing results in the "compiled setting".