Department of Mathematics

Abstract: We study the lattice of submonoids of the uniform block permutation monoid containing the symmetric group (which is its group of units). We prove that this lattice is distributive under union and intersection by relating the submonoids containing the symmetric group to downsets in a new partial order on integer partitions. Furthermore, we show that the sizes of the $$\mathcal{J}$$ J -classes of the uniform block permutation monoid are sums of squares of dimensions of irreducible modules of the monoid algebra.

Abstract: We introduce the immersion poset $$(\mathcal{P}(n),{⩽}_I)$$ ( P ( n ) , ⩽ I ) on partitions, defined by $$\lambda {⩽}_I\mu$$ λ ⩽ I μ if and only if $$s_{\mu }(x_1,\dots ,x_N)-s_{\lambda }(x_1,\dots ,x_N)$$ s μ ( x 1 , … , x N ) - s λ ( x 1 , … , x N ) is monomial-positive. Relations in the immersion poset determine when irreducible polynomial representations of $$G{L}_N(\mathbb{C})$$ G L N ( C ) form an immersion pair, as defined by Prasad and Raghunathan [7]. We develop injections $$\mathsf{\text{SSYT}}(\lambda ,u)\hookrightarrow \mathsf{\text{SSYT}}(\mu ,u)$$ SSYT ( λ , ν ) ↪ SSYT ( μ , ν ) on semistandard Young tableaux given constraints on the shape of $$\lambda$$ λ , and present results on immersion relations among hook and two column partitions. The standard immersion poset $$(\mathcal{P}(n),{⩽}_{std})$$ ( P ( n ) , ⩽ std ) is a refinement of the immersion poset, defined by $$\lambda {⩽}_{std}\mu$$ λ ⩽ std μ if and only if $$\lambda {⩽}_D\mu$$ λ ⩽ D μ in dominance order and $$f^{\lambda }⩽f^{\mu }$$ f λ ⩽ f μ , where $$f^u$$ f ν is the number of standard Young tableaux of shape $$u$$ ν . We classify maximal elements of certain shapes in the standard immersion poset using the hook length formula. Finally, we prove Schur-positivity of power sum symmetric functions on conjectured lower intervals in the immersion poset, addressing questions posed by Sundaram [12].

Abstract: We study classes of domains in with sufficiently flat boundaries that admit a decomposition or covering of bounded overlap by Lipschitz graph domains with controlled total surface area. This study is motivated by the following result proved by Peter Jones as a piece of his proof of the Analyst's Traveling Salesman Theorem in the complex plane: Any simply connected domain with finite boundary length can be decomposed into Lipschitz graph domains with total boundary length at most for some independent of . In this paper, we prove an analogous Lipschitz decomposition result in higher dimensions for domains with Reifenberg flat boundaries satisfying a uniform beta‐squared sum bound. We use similar techniques to show that domains with general Reifenberg flat or uniformly rectifiable boundaries admit similar Lipschitz decompositions while allowing the constituent domains to have bounded overlaps rather than be disjoint.

We study the structure of the set of all possible affine hyperplane sections of a convex polytope. We present two different cell decompositions of this set, induced by hyperplane arrangements. Using our decomposition, we bound the number of possible combinatorial types of sections and craft algorithms that compute optimal sections of the polytope according to various combinatorial and metric criteria, including sections that maximize the number of k-dimensional faces, maximize the volume, and maximize the integral of a polynomial. Our optimization algorithms run in polynomial time in fixed dimension, but the same problems show computational complexity hardness otherwise. Our tools can be extended to intersection with halfspaces and projections onto hyperplanes. Finally, we present several experiments illustrating our theorems and algorithms on famous polytopes.

R-loops are a class of non-canonical nucleic acid structures that typically form during transcription when the nascent RNA hybridizes the DNA template strand, leaving the non-template DNA strand unpaired. These structures are abundant in nature and play important physiological and pathological roles. Recent research shows that DNA sequence and topology affect R-loops, yet it remains unclear how these and other factors contribute to R-loop formation. In this work, we investigate the link between nascent RNA folding and the formation of R-loops. We introduce tree-polynomials, a new class of representations of RNA secondary structures. A tree-polynomial representation consists of a rooted tree associated with an RNA secondary structure together with a polynomial that is uniquely identified with the rooted tree. Tree-polynomials enable accurate, interpretable and efficient data analysis of RNA secondary structures without pseudoknots. We develop a computational pipeline for investigating and predicting R-loop formation from a genomic sequence. The pipeline obtains nascent RNA secondary structures from a co-transcriptional RNA folding software, and computes the tree-polynomial representations of the structures. By applying this pipeline to plasmid sequences that contain R-loop forming genes, we establish a strong correlation between the coefficient sums of tree-polynomials and the experimental probability of R-loop formation. Such strong correlation indicates that the pipeline can be used for accurate R-loop prediction. Furthermore, the interpretability of tree-polynomials allows us to characterize the features of RNA secondary structure associated with R-loop formation. In particular, we identify that branches with short stems separated by bulges and interior loops are associated with R-loops.

We prove nonemptyness of domains of proper discontinuity of Anosov groups of affine Lorentzian transformations of Rn.

Our previous multiscale graph basis dictionaries/graph signal transforms—Generalized Haar-Walsh Transform (GHWT); Hierarchical Graph Laplacian Eigen Transform (HGLET); Natural Graph Wavelet Packets (NGWPs); and their relatives—were developed for analyzing data recorded on vertices of a given graph. In this article, we propose their generalization for analyzing data recorded on edges, faces (i.e., triangles), or more generally κ -dimensional simplices of a simplicial complex (e.g., a triangle mesh of a manifold). The key idea is to use the Hodge Laplacians and their variants for hierarchical partitioning of a set of κ -dimensional simplices in a given simplicial complex, and then build localized basis functions on these partitioned subsets. We demonstrate their usefulness for data representation on both illustrative synthetic examples and real-world simplicial complexes generated from a co-authorship/citation dataset and an ocean current/flow dataset.

Many microorganisms propel themselves through complex media by deforming their flagella. The beat is thought to emerge from interactions between forces of the surrounding fluid, the passive elastic response from deformations of the flagellum and active forces from internal molecular motors. The beat varies in response to changes in the fluid rheology, including elasticity, but there are limited data on how systematic changes in elasticity alter the beat. This work analyses a related problem with fixed-strength driving force: the emergence of beating of an elastic planar filament driven by a follower force at the tip of a viscoelastic fluid. This analysis examines how the onset of oscillations depends on the strength of the force and viscoelastic parameters. Compared to a Newtonian fluid, it takes more force to induce the instability in viscoelastic fluids, and the frequency of the oscillation is higher. The linear analysis predicts that the frequency increases with the fluid relaxation time. Using numerical simulations, the model predictions are compared with experimental data on frequency changes in the bi-flagellated alga Chlamydomonas reinhardtii. The model shows the same trends in response to changes in both fluid viscosity and Deborah number and thus provides a possible mechanistic understanding of the experimental observations.