We study the maximum set coverage problem in the massively parallel model. In this setting, m sets that are subsets of a universe of n elements are distributed among m machines.  In each round, these machines can communicate with each other, subject to the memory constraint that no machine may use more than  \Tilde{O}(n) memory. The objective is to find the  sets whose coverage is maximized. We consider the regime where k = \Omega(m), m = O(n), and each machine has  memory. 

We study the densest subgraph problem and its variant through the lens of learning-augmented algorithms. More specifically, we show that given a partial solution-i.e., one produced by a machine learning classifier that capture at least (1-\epsilon)-fraction of nodes in the optimal subgraph-it is possible to design an extremely simple linear-time algorithm that achieves a provable (1-\epsilon)-approximation. Our approach also naturally extends to the directed densest subgraph problem and several NP-hard variants.

In this work, we provide a complete classification of small roots and low elements for a large class of rank 3 hyperbolic Coxeter groups. Our approach relies on the projective geometric representation and the normalized isotropic cone associated with these groups to characterize these elements systematically.

Have you ever woken up in the middle of the night and wondered to what extent the geometry of the Kunz fan informs us about Betti numbers and free resolutions of toric ideals associated to numerical semigroups living up there? We propose an answer to this question for a class of numerical semigroups and are currently conducting a case study on how free resolutions for hyperplane sections behave across faces of a chamber coming from the Kunz fan of a special family of staircase nilsemigroups.