Combinatorics of the double-dimer model

We prove that the partition function for tripartite double-dimer configurations of a planar bipartite graph satisfies a recurrence related to the Desnanot-Jacobi identity from linear algebra. A similar identity for the dimer partition function was established nearly 20 years ago by Kuo and has applications to random tiling theory and the theory of cluster algebras. This work was motivated in part by the potential for applications in these areas. Additionally, we discuss an application to Donaldson-Thomas and Pandharipande-Thomas theory which will be the subject of a forthcoming paper. The proof of our recurrence requires generalizing work of Kenyon and Wilson; specifically, lifting their assumption that the nodes of the graph are black and odd or white and even.Comment: 60 pages, 16 figure

Double-dimer condensation and the PT-DT correspondence

We resolve an open conjecture from algebraic geometry, which states that two generating functions for plane partition-like objects (the "box-counting" formulae for the Calabi-Yau topological vertices in Donaldson-Thomas theory and Pandharipande-Thomas theory) are equal up to a factor of MacMahon's generating function for plane partitions. The main tools in our proof are a Desnanot-Jacobi-type condensation identity, and a novel application of the tripartite double-dimer model of Kenyon-Wilson.Comment: 91 pages, 15 figures. This is the full version of the FPSAC extended abstract arXiv:2012.0848

Using Higher-Order Moments to Assess the Quality of GAN-generated Image Features

The rapid advancement of Generative Adversarial Networks (GANs) necessitates the need to robustly evaluate these models. Among the established evaluation criteria, the Fr\'{e}chet Inception Distance (FID) has been widely adopted due to its conceptual simplicity, fast computation time, and strong correlation with human perception. However, FID has inherent limitations, mainly stemming from its assumption that feature embeddings follow a Gaussian distribution, and therefore can be defined by their first two moments. As this does not hold in practice, in this paper we explore the importance of third-moments in image feature data and use this information to define a new measure, which we call the Skew Inception Distance (SID). We prove that SID is a pseudometric on probability distributions, show how it extends FID, and present a practical method for its computation. Our numerical experiments support that SID either tracks with FID or, in some cases, aligns more closely with human perception when evaluating image features of ImageNet data

Stepping out of Flatland: Discovering Behavior Patterns as Topological Structures in Cyber Hypergraphs

Data breaches and ransomware attacks occur so often that they have become part of our daily news cycle. This is due to a myriad of factors, including the increasing number of internet-of-things devices, shift to remote work during the pandemic, and advancement in adversarial techniques, which all contribute to the increase in both the complexity of data captured and the challenge of protecting our networks. At the same time, cyber research has made strides, leveraging advances in machine learning and natural language processing to focus on identifying sophisticated attacks that are known to evade conventional measures. While successful, the shortcomings of these methods, particularly the lack of interpretability, are inherent and difficult to overcome. Consequently, there is an ever-increasing need to develop new tools for analyzing cyber data to enable more effective attack detection. In this paper, we present a novel framework based in the theory of hypergraphs and topology to understand data from cyber networks through topological signatures, which are both flexible and can be traced back to the log data. While our approach's mathematical grounding requires some technical development, this pays off in interpretability, which we will demonstrate with concrete examples in a large-scale cyber network dataset. These examples are an introduction to the broader possibilities that lie ahead; our goal is to demonstrate the value of applying methods from the burgeoning fields of hypernetwork science and applied topology to understand relationships among behaviors in cyber data.Comment: 18 pages, 11 figures. This paper is written for a general audienc

Therapeutic targeting of cathepsin C::from pathophysiology to treatment

Cathepsin C (CatC) is a highly conserved tetrameric lysosomal cysteine dipeptidyl aminopeptidase. The best characterized physiological function of CatC is the activation of pro-inflammatory granule-associated serine proteases. These proteases are synthesized as inactive zymogens containing an N-terminal pro-dipeptide, which maintains the zymogen in its inactive conformation and prevents premature activation, which is potentially toxic to the cell. The activation of serine protease zymogens occurs through cleavage of the N-terminal dipeptide by CatC during cell maturation in the bone marrow. In vivo data suggest that pharmacological inhibition of pro-inflammatory serine proteases would suppress or attenuate deleterious effects of inflammatory/auto-immune disorders mediated by these proteases. The pathological deficiency in CatC is associated with Papillon-Lefèvre syndrome. The patients however do not present marked immunodeficiency despite the absence of active serine proteases in immune defense cells. Hence, the transitory pharmacological blockade of CatC activity in the precursor cells of the bone marrow may represent an attractive therapeutic strategy to regulate activity of serine proteases in inflammatory and immunologic conditions. A variety of CatC inhibitors have been developed both by pharmaceutical companies and academic investigators, some of which are currently being employed and evaluated in preclinical/clinical trials

Combinatorics of the Double-Dimer Model

We prove that the partition function for tripartite double-dimer configurations of a planar bipartite graph satisfies a recurrence related to the Desnanot-Jacobi identity from linear algebra. A similar identity for the dimer partition function was established nearly 20 years ago by Kuo. This work was motivated in part by the potential for applications, including a problem in Donaldson-Thomas and Pandharipande-Thomas theory, which we will discuss. The proof of our recurrence requires generalizing work of Kenyon and Wilson; specifically, lifting their assumption that the nodes of the graph be black and odd or white and even

Lattice walks confined to an octant in dimension 3: (non-)rationality of the second critical exponent

23 pages, 5 figuresIn the field of enumeration of walks in cones, it is known how to compute asymptotically the number of excursions (finite paths in the cone with fixed length, starting and ending points, using jumps from a given step set). As it turns out, the associated critical exponent is related to the eigenvalues of a certain Dirichlet problem on a spherical domain. An important underlying question is to decide whether this asymptotic exponent is a (non-)rational number, as this has important consequences on the algebraic nature of the associated generating function. In this paper, we ask whether such an excursion sequence might admit an asymptotic expansion with a first rational exponent and a second non-rational exponent. While the current state of the art does not give any access to such many-term expansions, we look at the associated continuous problem, involving Brownian motion in cones. Our main result is to prove that in dimension three, there exists a cone such that the heat kernel (the continuous analogue of the excursion sequence) has the desired rational/non-rational asymptotic property. Our techniques come from spectral theory and perturbation theory. More specifically, our main tool is a new Hadamard formula, which has an independent interest and allows us to compute the derivative of eigenvalues of spherical triangles along infinitesimal variations of the angles

The Combinatorial PT-DT Correspondence

We resolve an open conjecture from algebraic geometry, which states that two generating functions for plane partition-like objects (the "box-counting" formulae for the Calabi-Yau topological vertices in Donaldson-Thomas theory and Pandharipande-Thomas theory) are equal up to a factor of MacMahon's generating function for plane partitions. The main tools in our proof are a Desnanot-Jacobi-type condensation identity, and a novel application of the tripartite double-dimer model of Kenyon-Wilson

Letters From Camp

What if there was a love so great that it could keep two people together through the worst times imaginable? The year is 1944. The place is Eastern Europe. The skies are black with ash, and the ground is stained with blood and tears. A husband and wife are separated by the Nazi Regime and sent off to the Concentration Camps, likely to never see each other again. Yet in the face of unimaginable trauma, they refuse to let go of each other. Andor Vermes risked everything to keep a diary in the Concentration Camps, not to write an accurate account of what he was experiencing, but to write to his love, Magdolna, in the hopes that someday they will be reunited. These letters are real. The events they describe are gruesome, horrifying and heart wrenching, but they cannot be denied. This was Europe in 1944 for the Jewish people. This was Andor and Magdolna’s third year of marriage. This was life as they knew it. What would you say to the one you love if it might be the last letter you ever get to write?https://spiral.lynn.edu/facbooks/1039/thumbnail.jp