Recent ucdavismath items

Crystal for stable Grothendieck polynomials

Wed, 12 Aug 2026 00:00:00 +0000

We introduce a type A crystal structure on decreasing factorizations on 321-avoiding elements in the 0-Hecke monoid which we call *-crystal. This crystal is a K-theoretic generalization of the crystal on decreasing factorizations in the symmetric group of the first and last author. We prove that under the residue map the *-crystal intertwines with the crystal on set-valued tableaux recently introduced by Monical, Pechenik and Scrimshaw. We also define a new insertion from decreasing factorization to pairs of semistandard Young tableaux and prove several properties, such as its relation to the Hecke insertion and the uncrowding algorithm. The new insertion also intertwines with the crystal operators.

Richard Stanley through a crystal lens and from a random angle

Wed, 12 Aug 2026 00:00:00 +0000

Richard Stanley through a crystal lens and from a random angle

Markov chains for promotion operators

Wed, 12 Aug 2026 00:00:00 +0000

We consider generalizations of Schützenberger’s promotion operator on the set ℒ of linear extensions of a finite poset. This gives rise to a strongly connected graph on ℒ. In earlier work (Ayyer et al., J. Algebraic Combinatorics 39(4), 853–881 (2014)), we studied promotion-based Markov chains on these linear extensions which generalizes results on the Tsetlin library. We used the theory of ℛ-trivial monoids in an essential way to obtain explicitly the eigenvalues of the transition matrix in general when the poset is a rooted forest. We first survey these results and then present explicit bounds on the mixing time and conjecture eigenvalue formulas for more general posets. We also present a generalization of promotion to arbitrary subsets of the symmetric group.

Quantum Lakshmibai-Seshadri paths and root operators

Wed, 12 Aug 2026 00:00:00 +0000

We give an explicit description of the image of a quantum LS path, regarded as a rational path, under the action of root operators, and show that the set of quantum LS paths is stable under the action of the root operators. As a by-product, we obtain a new proof of the fact that a projected level-zero LS path is just a quantum LS path.

Eight quick tips for data-model integration in ecology

Thu, 30 Jul 2026 00:00:00 +0000

Eight quick tips for data-model integration in ecology

Quantification of flagellar gait changes with combined shape mode analysis and swimming simulations

Wed, 29 Jul 2026 00:00:00 +0000

Many different microswimmers propel themselves using flagella that beat periodically. The shape of the flagellar beat and swimming speed have been observed to change with fluid rheology. We quantify changes in the flagellar waveforms of Chlamydomonas reinhardtii in response to changes in fluid viscosity using (i) shape mode analysis and (ii) a full swimmer simulation to analyse how shape changes affect the swimming speed and to explore the dimensionality of the shape space. By decomposing the gait into the time-independent mean shape and the time-varying stroke, we find that the flagellar mean shape substantially changes in response to viscosity, while the changes in the time-varying stroke are more subtle. Using the swimmer simulation, we quantify how the swimming speed is affected by the dimensionality of the flagellar shape reconstruction, and we show that the observed change in swimming speed with viscosity is explained by the variations in mean flagellar shape and...

Optimizing metachronal paddling with reinforcement learning at low Reynolds number

Wed, 29 Jul 2026 00:00:00 +0000

Metachronal paddling is a swimming strategy in which an organism oscillates sets of adjacent limbs with a constant phase lag, propagating a metachronal wave through its limbs and propelling it forward. This limb coordination strategy is utilized by swimmers across a wide range of Reynolds numbers, which suggests that this metachronal rhythm was selected for its optimality of swimming performance. In this study, we apply reinforcement learning to a swimmer at zero Reynolds number and investigate whether the learning algorithm selects this metachronal rhythm, or if other coordination patterns emerge. We design the swimmer agent with an elongated body and pairs of straight, inflexible paddles placed along the body for various fixed paddle spacings. Based on paddle spacing, the swimmer agent learns qualitatively different coordination patterns. At tight spacings, a back-to-front metachronal wave-like stroke emerges which resembles the commonly observed biological rhythm, but at wide...

Pumping and steady streaming driven by two-frequency oscillations of a cylinder

Wed, 29 Jul 2026 00:00:00 +0000

The classical problem of steady streaming induced by an oscillating object has been studied extensively, but prior work has focused almost exclusively on single-frequency oscillations, which result in symmetric, quadrupole-like flows. Here we demonstrate that dual-frequency oscillations induce asymmetric steady streaming with a non-zero net flux in a direction determined by the polarity of the oscillation-the oscillator serves as a pump. We use numerical simulations and asymptotic analysis at small amplitude to examine two-dimensional steady streaming around a cylinder, first focusing on frequency ratio 2. The computational experiments show asymmetrical streaming and pumping, i.e. net flux downstream. It is well known from asymptotic analysis that steady streaming is second order in amplitude, and we show that pumping occurs at third order. We then extend the analysis to general frequency ratios, where we give necessary conditions for pumping, and predict the order in amplitude...

Flag Gromov-Witten invariants via crystals

Wed, 15 Jul 2026 00:00:00 +0000

We apply ideas from crystal theory to affine Schubert calculus and flag Gromov-Witten invariants. By defining operators on certain decompositions of elements in the type-A affine Weyl group, we produce a crystal reflecting the internal structure of Specht modules associated to permutation diagrams. We show how this crystal framework can be applied to study the product of a Schur function with a k-Schur function. Consequently, we prove that a subclass of 3-point Gromov-Witten invariants of complete flag varieties for Cⁿ enumerate the highest weight elements under these operators.

Affine charge and the k-bounded Pieri rule

Wed, 15 Jul 2026 00:00:00 +0000

We provide a new description of the Pieri rule of the homology of the affine Grassmannian and an affine analogue of the charge statistics in terms of bounded partitions. This makes it possible to extend the formulation of the Kostka–Foulkes polynomials in terms of solvable lattice models by Nakayashiki and Yamada to the affine setting.

The mystery of plethysm coefficients

Wed, 15 Jul 2026 00:00:00 +0000

Composing two representations of the general linear groups gives rise to Littlewood’s (outer) plethysm. On the level of characters, this poses the question of finding the Schur expansion of the plethysm of two Schur functions. A combinatorial interpretation for the Schur expansion coefficients of the plethysm of two Schur functions is, in general, still an open problem. We identify a proof technique of combinatorial representation theory, which we call the “ -perp trick”, and point out several examples in the literature where this idea is used. We use the -perp trick to give algorithms for computing monomial and Schur expansions of symmetric functions. In several special cases, these algorithms are more efficient than those currently implemented in SageMath .

Combinatorial Characterization of Queer Supercrystals (Survey)

Wed, 15 Jul 2026 00:00:00 +0000

We provide a characterization of the crystal bases for the quantum queer superalgebra recently introduced by Grantcharov et al. This characterization is a combination of local queer axioms generalizing Stembridge’s local axioms for crystal bases for simply-laced root systems, which were recently introduced by Assaf and Oguz, with further axioms and a new graph G characterizing the relations of the type A components of the queer crystal. We provide a counterexample to Assaf’s and Oguz’ conjecture that the local queer axioms uniquely characterize the queer supercrystal. We obtain a combinatorial description of the graph G on the type A components by providing explicit combinatorial rules for the odd queer operators on certain highest weight elements.

Dimension of Splines on Graphs in the Case of Degree Two and Smoothness One in Two Variables

Wed, 15 Jul 2026 00:00:00 +0000

Dimension of Splines on Graphs in the Case of Degree Two and Smoothness One in Two Variables

A minimaj-preserving crystal structure on ordered multiset partitions

Wed, 15 Jul 2026 00:00:00 +0000

We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected components of the crystal. In particular, this yields another proof of the Schur positivity of the graded Frobenius series of the generalization Rn,k due to Haglund, Rhoades and Shimozono of the coinvariant algebra Rn. The crystal structure also yields a bijective proof of the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions.

Queer supercrystals in SageMath

Wed, 15 Jul 2026 00:00:00 +0000

We describe the implementation of queer supercrystals. Our code is documented with test suites and has been integrated into SageMath. Through computer explorations using this implementation, we provide a counterexample to Assaf's and Oguz' conjecture that their local queer axioms uniquely characterize the queer supercrystal.

Structural classification of locally stationary time series based on second-order characteristics

Fri, 10 Jul 2026 00:00:00 +0000

Abstract Time series classification is crucial for numerous scientific and engineering applications. In this article, we present a numerically efficient, practically competitive, and theoretically rigorous classification method for distinguishing between two classes of locally stationary time series based on their time-domain, second-order characteristics. Our approach builds on the autoregressive approximation for locally stationary time series, imposes no requirement on the training sample size, and is shown to achieve zero misclassification error rate asymptotically when the underlying time series differ only mildly in their second-order characteristics. The new method is demonstrated to outperform a variety of state-of-the-art solutions, including wavelet-based, tree-based, convolution-based methods, as well as modern deep learning methods, through intensive numerical simulations and a real electroencephalography data analysis for epilepsy classification.

Multiscale Hodge scattering networks for data analysis

Thu, 26 Mar 2026 00:00:00 +0000

We propose new scattering networks for signals measured on simplicial complexes, which we call Multiscale Hodge Scattering Networks (MHSNs). Our construction builds on multiscale basis dictionaries on simplicial complexes—namely, the κ-GHWT and κ-HGLET—which we recently developed for simplices of dimension κ ∈ N in a given simplicial complex by generalizing the node-based Generalized Haar–Walsh Transform (GHWT) and Hierarchical Graph Laplacian Eigen Transform (HGLET). Both the κ-GHWT and the κ-HGLET form redundant sets (i.e., dictionaries) of multiscale basis vectors and the corresponding expansion coefficients of a given signal. Our MHSNs adopt a layered structure analogous to a convolutional neural network (CNN), cascading the moments of the modulus of the dictionary coefficients. The resulting features are invariant to reordering of the simplices (i.e., node permutation of the underlying graphs). Importantly, the use of multiscale basis dictionaries in our MHSNs admits a natural...

Explainable and Class-Revealing Signal Feature Extraction via Scattering Transform and Constrained Zeroth-Order Optimization

Thu, 26 Mar 2026 00:00:00 +0000

We propose a new method to extract discriminant and explainable features from a particular machine learning model, i.e., a combination of the scattering transform and the multiclass logistic regression. Although this model is well-known for its ability to learn various signal classes with high classification rate, it remains elusive to understand why it can generate such successful classification, mainly due to the nonlinearity of the scattering transform. In order to uncover the meaning of the scattering transform coefficients selected by the multiclass logistic regression (with the Lasso penalty), we adopt zeroth-order optimization algorithms to search an input pattern that maximizes the class probability of a class of interest given the learned model. In order to do so, it turns out that imposing sparsity and smoothness of input patterns is important. We demonstrate the effectiveness of our proposed method using a couple of synthetic time-series classification problems.

Explainable and Class-Revealing Signal Feature Extraction via Scattering Transform and Constrained Zeroth-Order Optimization

Thu, 26 Mar 2026 00:00:00 +0000

Plabic tangles and cluster promotion maps

Wed, 11 Mar 2026 00:00:00 +0000

Inspired by the BCFW recurrence for tilings of the amplituhedron, we introduce the general framework of plabic tangles that utilizes plabic graphs to define rational maps between products of Grassmannians called promotions. The central conjecture of the paper is that promotion maps are quasi-cluster homomorphisms, which we prove for several classes of promotions. In order to define promotion maps, we utilize m-vector-relation configurations (m-VRCs) on plabic graphs. We relate m-VRCs to the degree (a.k.a ‘intersection number’) of the amplituhedron map on positroid varieties and characterize all plabic trees with intersection number one and their VRCs. Finally, we show that promotion maps admit an operad structure and, supported by the class of 4-mass box promotions, we point at new positivity properties for non-rational maps beyond cluster algebras. Promotion maps have important connections to the geometry and cluster structure of the amplituhedron and singularities of scattering...

Hook-Valued Tableau Uncrowding and Tableau Switching

Wed, 11 Mar 2026 00:00:00 +0000

Refined canonical stable Grothendieck polynomials were introduced by Hwang et al. There exist two combinatorial models for these polynomials: one using hook-valued tableaux and the other using pairs of a semistandard Young tableau and (what we call) an exquisite tableau. An uncrowding algorithm on hook-valued tableaux was introduced by Pan et al. In this paper, we discover a novel connection between the two models via the uncrowding and Goulden and Greene's jeude taquin algorithms, using a classical result of Benkart, Sottile, and Stroomer on tableau switching. This connection reveals a symmetry of the uncrowding algorithm defined on hook-valued tableaux. As a corollary, we obtain another combinatorial model for the refined canonical stable Grothendieckpolynomials in terms of biflagged tableaux, which naturally appear in the characterization of theimage of the uncrowding map.

Braid variety cluster structures, II: general type

Wed, 11 Mar 2026 00:00:00 +0000

We show that braid varieties for any complex simple algebraic group G$$G$$ are cluster varieties. This includes open Richardson varieties inside the flag variety G/B$$G/B$$.

The symplectic isotopy problem for rational cuspidal curves

Wed, 25 Feb 2026 00:00:00 +0000

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology.

Rational cuspidal curves and symplectic fillings

Wed, 25 Feb 2026 00:00:00 +0000

Rational cuspidal curves and symplectic fillings

Equivariant Perturbation in Gomory and Johnson’s Infinite Group Problem. VII. Inverse Semigroup Theory, Closures, Decomposition of Perturbations

Sat, 14 Feb 2026 00:00:00 +0000

In this self-contained paper, we present a theory of the piecewise linear minimal valid functions for the 1-row Gomory–Johnson infinite group problem. The non-extreme minimal valid functions are those that admit effective perturbations. We give a precise description of the space of these perturbations as a direct sum of certain finite- and infinite-dimensional subspaces. The infinite-dimensional subspaces have partial symmetries; to describe them, we develop a theory of inverse semigroups of partial bijections, interacting with the functional equations satisfied by the perturbations. Our paper provides the foundation for grid-free algorithms for the Gomory–Johnson model, in particular for testing extremality of piecewise linear functions whose breakpoints are rational numbers with huge denominators.

Dual-feasible functions for integer programming and combinatorial optimization: Algorithms, characterizations, and approximations

Sat, 14 Feb 2026 00:00:00 +0000

Within the framework of the superadditive duality theory of integer programming, we study two types of dual-feasible functions of a single real variable (Alves et al., 2016). We introduce software that automates testing piecewise linear functions for maximality and extremality, enabling a computer-based search. We build a connection to cut-generating functions in the Gomory–Johnson and related models, complete the characterization of maximal functions, and prove analogues of the Gomory–Johnson 2-slope theorem and the Basu–Hildebrand–Molinaro approximation theorem.

Integer points in arbitrary convex cones: the case of the PSD and SOC cones

Wed, 11 Feb 2026 00:00:00 +0000

We investigate the semigroup of integer points inside a convex cone. We extend classical results in integer linear programming to integer conic programming. We show that the semigroup associated with nonpolyhedral cones can sometimes have a notion of finite generating set with the help of a group action. We show this is true for the cone of positive semidefinite matrices (PSD) and the second-order cone (SOC). Both cones have a finite generating set of integer points, similar in spirit to Hilbert bases, under the action of a finitely generated group. We also extend notions of total dual integrality, Gomory-Chvátal closure, and Carathéodory rank to integer points in arbitrary cones.

On contact 3‐manifolds that admit a nonfree toric action

Wed, 28 Jan 2026 00:00:00 +0000

Abstract We classify all contact structures on 3‐manifolds that admit a nonfree toric action, up to contactomorphism, and present them through explicit topological descriptions. Our classification is based on Lerman's classification of toric contact 3‐manifolds up to equivariant contactomorphism [Lerman, J. Symplectic Geom. 1 (2003), 785–828]. We also prove that every contact 3‐manifold with a nonfree toric action arises as the concave boundary of a toric linear plumbing over spheres inspired by Marinković et al. As a corollary of both results, we classify which contact 3‐manifolds arise as the concave boundary of a linear plumbing of spheres.

An extended Vinogradov’s mean value theorem

Sun, 7 Dec 2025 00:00:00 +0000

In this paper, we provide novel mean value estimates for exponential sums related to the extended main conjecture of Vinogradov’s mean value theorem, by developing the Hardy-Littlewood circle method together with a refined shifting variables argument.

Let be a natural number and . Define the exponential sum For , consider mean values of the exponential sums where we wrote . By making use of the aforementioned tools, we obtain the sharp upper bound for , for and . Furthermore, for , we obtain analogous results depending on a small cap decoupling inequality for the moment curves in .

The contact cut graph and a Weinstein L$\mathcal {L}$‐invariant

Mon, 1 Dec 2025 00:00:00 +0000

Abstract We define and study the contact cut graph which is an analogue of Hatcher and Thurston's cut graph for contact geometry, inspired by contact Heegaard splittings (Giroux, Proceedings of the International Congress of Mathematicians , Beijing, 2002; Torisu, Internat. Math. Res. Notices (2000), 441–454). We show how oriented paths in the contact cut graph correspond to Lefschetz fibrations and multisection with divides diagrams. We also give a correspondence for achiral Lefschetz fibrations. We use these correspondences to define a new invariant of Weinstein domains, the Weinstein ‐invariant , that is a symplectic analogue of the Kirby–Thompson's ‐invariant of smooth 4‐manifolds. We discuss the relation of Lefschetz stabilization with the Weinstein ‐invariant. We present topological and geometric constraints of Weinstein domains with . We also give two families of examples of multisections with divides that have arbitrarily large ‐invariant.

Morse actions of discrete groups on symmetric spaces: local-to-global principle

Wed, 24 Sep 2025 00:00:00 +0000

Our main result is a local-to-global principle for Morse quasigeodesics, maps and actions. As an application of our techniques we show algorithmic recognizability of Morse actions and construct Morse “Schottky subgroups” of higher-rank semisimple Lie groups via arguments not based on Tits pingpong. Our argument is purely geometric and proceeds by constructing equivariant Morse quasiisometric embeddings of trees into higher-rank symmetric spaces.

The Combinatorial Theory Flip

Wed, 27 Aug 2025 00:00:00 +0000

The Combinatorial Theory Flip

Interview with Anne Schilling

Wed, 27 Aug 2025 00:00:00 +0000

Interview with Anne Schilling

Complexity of Finite Semigroups: History and Decidability

Wed, 13 Aug 2025 00:00:00 +0000

Margolis, Rhodes and Schilling recently submitted two papers that proved that the complexity of a finite semigroup is computable. The purpose of this paper is to survey the basic results of Krohn–Rhodes complexity of finite semigroups and to outline the proof of its computability.

Weinstein Handlebodies for Complements of Smoothed Toric Divisors

Thu, 17 Jul 2025 00:00:00 +0000

We study the interactions between toric manifolds and Weinstein handlebodies. We define a partially-centeredness condition on a Delzant polytope, which we prove ensures that the complement of a corresponding partial smoothing of the total toric divisor supports an explicit Weinstein structure. Many examples which fail this condition also fail to have Weinstein (or even exact) complement to the partially smoothed divisor. We investigate the combinatorial possibilities of Delzant polytopes that realize such Weinstein domain complements. We also develop an algorithm to construct a Weinstein handlebody diagram in Gompf standard form for the complement of such a partially smoothed total toric divisor. The algorithm we develop more generally outputs a Weinstein handlebody diagram for any Weinstein 4-manifold constructed by attaching 2-handles to the disk cotangent bundle of any surface , where the 2-handles are attached along the co-oriented conormal lifts of curves on ...

Optimization tools for computing colorings of [ 1 , … , n ] with few monochromatic solutions on 3-variable linear equations

Sun, 25 May 2025 00:00:00 +0000

A famous result in arithmetic Ramsey theory says that for many linear homogeneous equations E there is a threshold value Rk(E) (the Rado number of E) such that for any k-coloring of the integers in the interval [1,n], with n≥Rk(E), there exists at least one monochromatic solution. But one can further ask, how many monochromatic solutions is the minimum possible in terms of n? Several authors have estimated this function before, here we offer new tools from integer and semidefinite optimization that help find either optimal or near optimal 2-colorings minimizing the number of monochromatic solutions of several families of 3-variable non-regular homogeneous linear equations. In the last part of the paper we further extend to three and more colors for the Schur equation, improving earlier work.

Lipschitz decompositions of domains with bilaterally flat boundaries

Mon, 28 Apr 2025 00:00:00 +0000

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.

The geometry of colors in van Gogh’s Sunflowers

Thu, 10 Apr 2025 00:00:00 +0000

“Paintings fade like flowers”: van Gogh’s prediction on the impact of age on paintings came true for most of his paintings. We have studied the consequences of this aging on the Sunflowers in a vase with a yellow background series, namely its original, F454, currently in London, and two replicates, F457, in Tokyo, and F458, in Amsterdam, which van Gogh painted using the original as a model. The background and flower renditions in those paintings have faded and turned brown, making them less vibrant that van Gogh had most likely intended. We have attempted to restore van Gogh’s intent using a computational approach based on data science. After identifications of regions of interest (ROI) within the three paintings F454, F457, and F458 that capture the flowers, stems of the flowers, and background, respectively, we studied the geometry of the color space (in RGB representation) occupied by those ROIs. By comparing those color spaces with those occupied by similar ROIs in photographs...

Continued fractions and lines across the Stern–Brocot diagram

Wed, 9 Apr 2025 00:00:00 +0000

Continued fractions and lines across the Stern–Brocot diagram

Farey recursive functions

Wed, 9 Apr 2025 00:00:00 +0000

Farey recursive functions

The lattice of submonoids of the uniform block permutations containing the symmetric group

Wed, 26 Mar 2025 00:00:00 +0000

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 J$$\mathscr {J}$$-classes of the uniform block permutation monoid are sums of squares of dimensions of irreducible modules of the monoid algebra.

The immersion poset on partitions

Wed, 12 Mar 2025 00:00:00 +0000

Abstract: We introduce the immersion poset $$({\mathcal {P}}(n), \leqslant _I)$$ ( P ( n ) , ⩽ I ) on partitions, defined by $$\lambda \leqslant _I \mu $$ λ ⩽ I μ if and only if $$s_\mu (x_1, \ldots , x_N) - s_\lambda (x_1, \ldots , x_N)$$ ...

Corrigendum: A Demazure crystal construction for Schubert polynomials

Wed, 26 Feb 2025 00:00:00 +0000

We give a corrigendum to our paper [4] entitled “A Demazure crystal construction for Schubert polynomials”.

The strong Haken theorem via sphere complexes

Wed, 29 Jan 2025 00:00:00 +0000

The strong Haken theorem via sphere complexes

Tree polynomials identify a link between co-transcriptional R-loops and nascent RNA folding

Mon, 20 Jan 2025 00:00:00 +0000

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...

THE BEST WAYS TO SLICE A POLYTOPE

Fri, 17 Jan 2025 00:00:00 +0000

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.

Weighted Ehrhart theory: Extending Stanley's nonnegativity theorem

Fri, 17 Jan 2025 00:00:00 +0000

We generalize R. P. Stanley's celebrated theorem that the h⁎-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to hold for weighted Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope. We also show nonnegativity of the h⁎-polynomial as a real-valued function for a larger family of weights. We explore the case when the weight function is the square of a single (arbitrary) linear form. We show stronger results for two-dimensional convex lattice polygons and give concrete examples showing tightness of the hypotheses. As an application, we construct a counterexample to a conjecture by Berg, Jochemko, and Silverstein on Ehrhart tensor polynomials.

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

Fri, 17 Jan 2025 00:00:00 +0000

We discuss five fundamental results of discrete mathematics: the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carathéodory, Helly, and Tverberg from combinatorial geometry. We explore their connections and emphasize their broad impact in application areas such as data science, game theory, graph theory, mathematical optimization, computational geometry, etc.

Klein–Maskit combination theorem for Anosov subgroups: Amalgams

Wed, 20 Nov 2024 00:00:00 +0000

Abstract: The classical Klein–Maskit combination theorems provide sufficient conditions to construct new Kleinian groups using old ones. There are two distinct but closely related combination theorems: the first deals with amalgamated free products, whereas the second deals with HNN extensions. This article gives analogs of both combination theorems for Anosov subgroups.

From Quasi-Symmetric to Schur Expansions with Applications to Symmetric Chain Decompositions and Plethysm

Wed, 20 Nov 2024 00:00:00 +0000

It is an important problem in algebraic combinatorics to deduce the Schur function expansion of a symmetric function whose expansion in terms of the fundamental quasisymmetric function is known. For example, formulas are known for the fundamental expansion of a Macdonald symmetric function and for the plethysm of two Schur functions, while the Schur expansions of these expressions are still elusive. Based on work of Egge, Loehr and Warrington, Garsia and Remmel provided a method to obtain the Schur expansion from the fundamental expansion by replacing each quasisymmetric function by a Schur function (not necessarily indexed by a partition) and using straightening rules to obtain the Schur expansion. Here we provide a new method that only involves the coefficients of the quasisymmetric functions indexed by partitions and the quasi-Kostka matrix. As an application, we identify the lexicographically largest term in the Schur expansion of the plethysm of two Schur functions. We provide...

Fair Data Representation for Machine Learning at the Pareto Frontier.

Wed, 13 Nov 2024 00:00:00 +0000

As machine learning powered decision-making becomes increasingly important in our daily lives, it is imperative to strive for fairness in the underlying data processing. We propose a pre-processing algorithm for fair data representation via which -objective supervised learning results in estimations of the Pareto frontier between prediction error and statistical disparity. Particularly, the present work applies the optimal affine transport to approach the post-processing Wasserstein barycenter characterization of the optimal fair -objective supervised learning via a pre-processing data deformation. Furthermore, we show that the Wasserstein geodesics from learning outcome marginals to their barycenter characterizes the Pareto frontier between -loss and total Wasserstein distance among the marginals. Numerical simulations underscore the advantages: (1) the pre-processing step is compositive with arbitrary conditional expectation estimation supervised learning methods and unseen...

Complete characterizations of hyperbolic Coxeter groups with Sierpiński curve boundary and with Menger curve boundary

Wed, 23 Oct 2024 00:00:00 +0000

Complete characterizations of hyperbolic Coxeter groups with Sierpiński curve boundary and with Menger curve boundary

A multiscale predictive digital twin for neurocardiac modulation

Mon, 16 Sep 2024 00:00:00 +0000

Cardiac function is tightly regulated by the autonomic nervous system (ANS). Activation of the sympathetic nervous system increases cardiac output by increasing heart rate and stroke volume, while parasympathetic nerve stimulation instantly slows heart rate. Importantly, imbalance in autonomic control of the heart has been implicated in the development of arrhythmias and heart failure. Understanding of the mechanisms and effects of autonomic stimulation is a major challenge because synapses in different regions of the heart result in multiple changes to heart function. For example, nerve synapses on the sinoatrial node (SAN) impact pacemaking, while synapses on contractile cells alter contraction and arrhythmia vulnerability. Here, we present a multiscale neurocardiac modelling and simulator tool that predicts the effect of efferent stimulation of the sympathetic and parasympathetic branches of the ANS on the cardiac SAN and ventricular myocardium. The model includes a layered...

Domains of discontinuity of Lorentzian affine group actions

Wed, 14 Aug 2024 00:00:00 +0000

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

Klein–Maskit combination theorem for Anosov subgroups: free products

Wed, 14 Aug 2024 00:00:00 +0000

We prove a generalization of the classical Klein–Maskit combination theorem, in the free product case, in the setting of Anosov subgroups. Namely, if Γ A and Γ B are Anosov subgroups of a semisimple Lie group G of noncompact type, then under suitable topological assumptions, the group generated by Γ A and Γ B in G is again Anosov, and is naturally isomorphic to the free product Γ A∗ Γ B . Such a generalization was conjectured in Dey et al. (Math Z 293(1–2):551–578, 2019).

Affine Springer Fibers and Generalized Haiman Ideals (with an Appendix by Eugene Gorsky and Joshua P. Turner)

Thu, 25 Jul 2024 00:00:00 +0000

Abstract: We compute the Borel–Moore homology of unramified affine Springer fibers for $\textrm{Gr}_n$ under the assumption that they are equivariantly formal and relate them to certain ideals discussed by Haiman. For $n=3$, we give an explicit description of these ideals, compute their Hilbert series, generators, and relations, and compare them to generalized $(q,t)$-Catalan numbers. We also compare the homology to the Khovanov–Rozansky homology of the associated link, and prove a version of a conjecture of Oblomkov, Rasmussen, and Shende in this case.

Effect of fluid elasticity on the emergence of oscillations in an active elastic filament

Tue, 25 Jun 2024 00:00:00 +0000

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...

Reproducibility of 3D chromatin configuration reconstructions

Mon, 17 Jun 2024 00:00:00 +0000

It is widely recognized that the three-dimensional (3D) architecture of eukaryotic chromatin plays an important role in processes such as gene regulation and cancer-driving gene fusions. Observing or inferring this 3D structure at even modest resolutions had been problematic, since genomes are highly condensed and traditional assays are coarse. However, recently devised high-throughput molecular techniques have changed this situation. Notably, the development of a suite of chromatin conformation capture (CCC) assays has enabled elicitation of contacts-spatially close chromosomal loci-which have provided insights into chromatin architecture. Most analysis of CCC data has focused on the contact level, with less effort directed toward obtaining 3D reconstructions and evaluating the accuracy and reproducibility thereof. While questions of accuracy must be addressed experimentally, questions of reproducibility can be addressed statistically-the purpose of this paper. We use a constrained...

The ENIGMA Consortium: large-scale collaborative analyses of neuroimaging and genetic data

Sun, 16 Jun 2024 00:00:00 +0000

The Enhancing NeuroImaging Genetics through Meta-Analysis (ENIGMA) Consortium is a collaborative network of researchers working together on a range of large-scale studies that integrate data from 70 institutions worldwide. Organized into Working Groups that tackle questions in neuroscience, genetics, and medicine, ENIGMA studies have analyzed neuroimaging data from over 12,826 subjects. In addition, data from 12,171 individuals were provided by the CHARGE consortium for replication of findings, in a total of 24,997 subjects. By meta-analyzing results from many sites, ENIGMA has detected factors that affect the brain that no individual site could detect on its own, and that require larger numbers of subjects than any individual neuroimaging study has currently collected. ENIGMA’s first project was a genome-wide association study identifying common variants in the genome associated with hippocampal volume or intracranial volume. Continuing work is exploring genetic associations...

Private measures, random walks, and synthetic data

Wed, 12 Jun 2024 00:00:00 +0000

Differential privacy is a mathematical concept that provides an information-theoretic security guarantee. While differential privacy has emerged as a de facto standard for guaranteeing privacy in data sharing, the known mechanisms to achieve it come with some serious limitations. Utility guarantees are usually provided only for a fixed, a priori specified set of queries. Moreover, there are no utility guarantees for more complex—but very common—machine learning tasks such as clustering or classification. In this paper we overcome some of these limitations. Working with metric privacy, a powerful generalization of differential privacy, we develop a polynomial-time algorithm that creates a private measure from a data set. This private measure allows us to efficiently construct private synthetic data that are accurate for a wide range of statistical analysis tools. Moreover, we prove an asymptotically sharp min-max result for private measures and synthetic data in general compact...

Early career Latinas in STEM: Challenges and solutions

Thu, 9 May 2024 00:00:00 +0000

Mexican, Puerto Rican, and Central American Ancestry (MPRCA) individuals represent 82% of US Latinos. An intergenerational group of MPRCA women and allies met to discuss persistent underrepresentation of MPRCA women in STEM, identifying multi-level challenges and solutions. Implementation of these solutions is important and will benefit MPRCA women and the entire academic community.

Multiscale transforms for signals on simplicial complexes

Wed, 8 May 2024 00:00:00 +0000

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$$\kappa $$-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$$\kappa $$-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.

Computational complexity and 3–manifolds and zombies

Wed, 24 Apr 2024 00:00:00 +0000

Computational complexity and 3–manifolds and zombies

The Cartan–Hadamard conjecture and the Little Prince

Wed, 24 Apr 2024 00:00:00 +0000

The Cartan–Hadamard conjecture and the Little Prince

Identifying lens spaces in polynomial time

Wed, 24 Apr 2024 00:00:00 +0000

Identifying lens spaces in polynomial time

Coloring invariants of knots and links are often intractable

Wed, 24 Apr 2024 00:00:00 +0000

Coloring invariants of knots and links are often intractable

CANONICAL REPRESENTATIVES FOR DIVISOR CLASSES ON TROPICAL CURVES AND THE MATRIX–TREE THEOREM

Wed, 24 Apr 2024 00:00:00 +0000

Abstract: Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\Gamma $ be a compact tropical curve (or metric graph) of genus $g$. Using the theory of tropical theta functions, Mikhalkin and Zharkov proved that there is a canonical effective representative (called a break divisor) for each linear equivalence class of divisors of degree $g$ on $\Gamma $. We present a new combinatorial proof of the fact that there is a unique break divisor in each equivalence class, establishing in the process an ‘integral’ version of this result which is of independent interest. As an application, we provide a ‘geometric proof’ of (a dual version of) Kirchhoff’s celebrated matrix–tree theorem. Indeed, we show that each weighted graph model $G$ for $\Gamma $ gives rise to a canonical polyhedral decomposition of the $g$-dimensional...

Modeling time-varying phytoplankton subsidy reveals at-risk species in a Chilean intertidal ecosystem

Mon, 1 Apr 2024 00:00:00 +0000

The allometric trophic network (ATN) framework for modeling population dynamics has provided numerous insights into ecosystem functioning in recent years. Herein we extend ATN modeling of the intertidal ecosystem off central Chile to include empirical data on pelagic chlorophyll-a concentration. This intertidal community requires subsidy of primary productivity to support its rich ecosystem. Previous work models this subsidy using a constant rate of phytoplankton input to the system. However, data shows pelagic subsidies exhibit highly variable, pulse-like behavior. The primary contribution of our work is incorporating this variable input into ATN modeling to simulate how this ecosystem may respond to pulses of pelagic phytoplankton. Our model results show that: (1) closely related sea snails respond differently to phytoplankton variability, which is explained by the underlying network structure of the food web; (2) increasing the rate of pelagic-intertidal mixing increases fluctuations...

Markov Bases: A 25 Year Update