Future students

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: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:Two-coloured lines in finite geometry

Abstract: Given a colouring of the points of a projective plane, when is it true that every line contains at most two colours? I will discuss recent generalizations of classical results in this area, and a surprising link with a famous question in graph theory.

Title: : Precise Eigenvalue Location for Random Regular Graphs

Abstract:The spectral theory of regular graphs has broad applications in theoretical computer science, statistical physics, and other areas of mathematics. Graphs with optimally large spectral gap are known as Ramanujan graphs. Previous constructions of Ramanujan graphs are based on number theory and have specific constraints on the degree and number of vertices. In this talk, we show that, in fact, most regular graphs are Ramanujan; specifically, a randomly selected regular graph has a probability of 69% of being Ramanujan. We establish this through a rigorous analysis of the Green’s function of the adjacency operator, focusing on its behavior under random edge switches.

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: Combinatorial Contract Theory

AbstractFirst, I will introduce the contract theory setting. Afterwards, we'll survey some approximation and hardness of approximation results in the land of combinatorial contract theory. We'll talk briefly about the flavour of techniques used in such results. This should exemplify a few typical approaches taken in this setting. With any luck, we'll all find the level of detail pleasant. The main results will be from Multi-Agent Contracts (STOC 23) and On The (In)approximability of Combinatorial Contracts (ITCS 24). We may take a dip into results from Multi-Agent Combinatorial Contracts (SODA 25) or other papers, time permitting.

Title:What is New in Join-Aggregate Query Processing?

Abstract: Join-aggregate queries defined over commutative semirings subsume a wide variety of common algorithmic problems, such as graph pattern matching, graph colorability, matrix multiplication, and constraint satisfaction problems. Developing efficient algorithms for computing join-aggregate queries in the conventional RAM model has been a holy grail in database theory. One of the most celebrated results in this area is the Yannakakis algorithm dating back to 1981. Despite its prominence as a textbook solution, no improvements in its complexity have been made over the past 40 years. In this talk, I will present the first algorithm that improves upon Yannakakis for computing acyclic join-aggregate queries. Moreover, this algorithm is proven to be output-optimal among all combinatorial algorithms. One application is an output-optimal algorithm for chain matrix multiplication over sparse matrices. Beyond combinatorial algorithms, I will also show how fast matrix multiplication can further speed up the processing of conjunctive queries, a critical subclass of join-aggregate queries. Finally, I will highlight a few interesting open problems in this area.

Title: Spherical friezes

 Abstract: A fundamental problem in spherical distance geometry aims to recover an $n$-tuple of points on a 2-sphere in $\mathbb{R}^3$, viewed up to oriented isometry, from $O(n)$ input measurements. This talk will discuss an algebraic solution to this problem using only the four arithmetic operations. We will show how a new type of frieze pattern can be employed to arrange the measurement data. These friezes exhibit glide symmetry and a version of the Laurent phenomenon.

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

Title: Structure of tournaments with a forbidden subtournament

Abstract: For a tournament $S$, a tournament is $S$-free if it has no subtournament isomorphic to $S$. Until now, there have been only a small number of tournaments $S$ such that the complete structure of $S$-free tournaments is known. 

Let $\triangle(1, 2, 2)$ be a tournament obtained from the cyclic triangle by substituting two-vertex tournaments for two of its vertices. In this talk, we present a structure theorem for $\triangle(1, 2, 2)$-free tournaments, which was previously unknown. As an application, we provide tight bounds for the chromatic number as well as the size of the largest transitive subtournament for such tournaments.

This talk is based on joint work with Taite LaGrange, Mathieu Rundström, Arpan Sadhukhan, and Sophie Spirkl.

Title:Accuracy Aware Minimally Invasive Data Exploration For Decision Support

Abstract: Decision-support (DS) applications, crucial for timely and informed decision-making, often analyze sensitive data, raising significant privacy concerns. While privacy-preserving randomized mechanisms can mitigate these concerns, they introduce the risk of both false positives and false negatives. Critically, in DS applications, the number of false negatives often needs to be strictly controlled. Existing privacy-preserving techniques like differential privacy, even when adapted, struggle to meet this requirement without substantial privacy leakage, particularly when data distributions are skewed. This talk introduces a novel approach to minimally invasive data exploration for decision support. Our method minimizes privacy loss while guaranteeing a bound on false negatives by dynamically adapting privacy levels based on the underlying data distribution. We further extend this approach to handle complex DS queries, which may involve multiple conditions on diverse aggregate statistics combined through logical disjunction and conjunction. Specifically, we define complex DS queries and their associated accuracy requirements, and present algorithms that strategically allocate a privacy budget to minimize overall privacy loss while satisfying the bounded accuracy guarantee.