Events

Title:Strongly connected orientations and integer lattices

Abstract: Let D = (V, A) be a digraph whose underlying graph is 2-edge-connected, and let P be the polytope whose vertices are the incidence vectors of arc sets whose reversal makes D strongly connected. We study the lattice theoretic properties of the integer points contained in a proper 'slanted' face F of P. We prove under a mild necessary condition that the 0,1 points in F contain an integral basis B, i.e., B is linearly independent, and every integral vector in the linear of span of F is an integral linear combination of B. This result is surprising as the integer points in F do not necessarily form a Hilbert basis.

Our result has consequences for head-disjoint strong orientations in hypergraphs, and also to a famous conjecture by Woodall that the minimum size of a dicut of D, say k, is equal to the maximum number of disjoint dijoins. We prove a relaxation of this conjecture, by finding for any odd prime number p, a p-adic packing of dijoins of value k and of support size at most 2|A|. We also prove that the all-ones vector belongs to the lattice generated by the 0,1 points in F, where F is the face of P satisfying x(C) = 1 for every minimum dicut C.

This is based on joint work with Gerard Cornuejols, Siyue Liu, and Olha Silina.

Title:Arrangements and Likelihood

Abstract: In this talk, we establish connections between hypersurface arrangements and likelihood geometry. The central object is the likelihood correspondence which captures the dependence between data and critical points of the likelihood function of a statistical model parametrized by the polynomials defining the arrangement. In particle physics, this same object is known as the scattering correspondence. The connection to hypersurface arrangements leads to a new description of the prime ideal of the likelihood correspondence, which is often computationally advantageous. This description is based on the Rees algebra of the likelihood module of the arrangement, a module closely related to the module of logarithmic derivations. We present results for generic and graphic arrangements.

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

Title:Faster linear algebra using treewidth

Abstract:

: We look at the complexity of solving sparse linear systems as a function of the treewidth of the instance. That is, the sparse matrix is associated with a sparse graph, and solutions can be found faster when that graph has low treewidth. We give a parameterized algorithm in system size and treewidth achieving the conjectured optimal performance.

This is joint work with Daniel Grier.

Title: Learning from Coarse Samples

Abstract:Coarsening occurs when the exact value of a sample is not observed.
Instead, only a subset of the sample space containing the exact value is known.
Coarse data naturally arises in diverse fields, including Economics, Engineering, Medical and Biological Sciences, and all areas of the Physical Sciences.
One of the simplest forms of coarsening is rounding, where data values are mapped to the nearest point on a specified lattice.

We survey applications of coarse learning to regression with self-selection bias, regression with second-price auction data, and present details of an SGD-based algorithm for coarse Gaussian mean estimation.

Based on joint work with Alkis Kalavasis and Anay Mehrotra (https://arxiv.org/abs/2504.07133)

Title:Source characterization of the hypegraphic posets

Abstract: For a hypergraph $\mathbb{H}$ on $[n]$, the hypergraphic poset $P_\mathbb{H}$ is the transitive closure of the oriented $1$-skeleton of the hypergraphic polytope $\Delta_\mathbb{H}$, which is the Minkowski sum of the standard simplices $\Delta_H$ for each hyperedge $H \in \mathbb{H}$. In 2019, C. Benedetti, N. Bergeron, and J. Machacek established a remarkable correspondence between the transitive closure of the oriented $1$-skeleton of $\Delta_\mathbb{H}$ and the flip graph on acyclic orientations of $\mathbb{H}$. Viewing an orientation of $\mathbb{H}$ as a map $A$ from $\mathbb{H}$ to $[n]$, we define the sources of the acyclic orientations as the values $A(H)$ for each hyperedge $H \in \mathbb{H}$. In a recent paper, N. Bergeron and V.

Pilaud provided a characterization of $P_\mathbb{H}$ based on the sources of acyclic orientations for interval hypergraphs. Specifically, two distinct acyclic orientations $A$ and $B$ of $\mathbb{H}$ are comparable in $P_\mathbb{H}$ if and only if their sources satisfy $A(H) \leq B(H)$ for all hyperedges $H\in \HH$. The goal of this work is to extend this source characterization of $P_\mathbb{H}$ to arbitrary hypergraphs on $[n]$.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:30pm,

Title:Constraining moduli space cohomology by counting graphs

Abstract: In 1992, Kontsevich defined complexes spanned by graphs. These 
complexes are increasingly prominent in algebraic topology, geometric 
group theory and mathematical physics. For instance, a 2021 theorem by 
Chan-Galatius and Payne implies that the top-weight cohomology of the 
moduli space of curves of genus g is equal to the homology of a specific 
graph complex. I will present a new theorem on the asymptotic growth 
rate of the Euler characteristic of this graph complex and explain its 
implication on the cohomology of the moduli space of curves. The proof 
involves solving a specific graph counting problem.

Title: An O(log log n)-approximate budget-feasible mechanism for subadditive valuations

Abstract:In the setting of budget feasible mechanism design, a buyer wants to purchase items from a set of agents. Each agent holds one item, and incurs a cost of c_i upon supplying the item to the buyer. The buyer wants to maximize the value of the set of items that are bought from the sellers. The buyer has a budget B on the total payments made to the sellers. The cost c_i is private information that the buyer doesn't have access to. The goal is to design a mechanism that is truthful, in the sense that the sellers do not have incentive to deviate from reporting their true costs, and budget feasible, in the sense that the total payments made to the sellers is within some budget B, and that outputs a set whose value is a good approximation to the algorithmic optimum, OPT = max{v(S) : c(S)<=B}.

In this talk, I will present our recent work that obtains an O(log log n)-approximate budget-feasible mechanism when the valuation function is subadditive. This is joint work with Kanstantsin Pashkovich and Chaitanya Swamy.

Title:Splitting algorithms for monotone inclusions with minimal dimension

Abstract: Many situations in convex optimization can be modeled as the problem of finding a zero of a monotone operator, which can be regarded as a generalization of the gradient of a differentiable convex function. In order to numerically address this monotone inclusion problem, it is vital to be able to exploit the inherent structure of the operator defining it. The algorithms in the family of the splitting methods achieve this by iteratively solving simpler subtasks that are defined by separately using some parts of the original problem.

In the first part of this talk, we will introduce some of the most relevant monotone inclusion problems and present their applications to optimization. Subsequently, we will draw our attention to a common anomaly that has persisted in the design of methods in this family: the dimension of the underlying space —which we denote as lifting— of the algorithms abnormally increases as the problem size grows. This has direct implications on the computational performance of the method as a result of the increase of memory requirements. In this framework, we characterize the minimal lifting that can be obtained by splitting algorithms adept at solving certain general monotone inclusions. Moreover, we present splitting methods matching these lifting bounds, and thus having minimal lifting.

Title:A Toric Analogue for Greene's Rational Function of a Poset

Abstract: Given a finite poset, Greene introduced a rational function obtained by summing certain rational functions over the linear extensions of the poset. This function has interesting interpretations, and for certain families of posets, it simplifies surprisingly. In particular, Greene evaluated this rational function for strongly planar posets in his work on the Murnaghan–Nakayama formula. Develin, Macauley, and Reiner introduced toric posets, which combinatorially are equivalence classes of posets (or rather acyclic quivers) under the operation of flipping maximum elements into minimum elements and vice versa. In this work, we introduce a toric analogue of Greene's rational function for toric posets, and study its properties. In addition, we use toric posets to show that the Kleiss–Kuijf relations, which appear in scattering amplitudes, are equivalent to a specific instance of Greene's evaluation of his rational function for strongly planar posets. Also in this work, we give an algorithm for finding the set of toric total extensions of a toric poset.

Title:Shifted Parking function

Abstract:Stanley recently introduced the shifted parking function symmetric function as a shifted analogue of the parking function symmetric function and posed the question of what the corresponding combinatorial objects are. This talk will answer that question and explain how the answer connects to projective representations of the symmetric group. Based on joint work with Zach Hamaker.