When auditing a redistricting plan, a persuasive method is to compare the plan with an ensemble of neutrally drawn redistricting plans. Ensembles are generated via algorithms that sample distributions on balanced graph partitions. To audit the partisan difference between the ensemble and a given… read more about this publication »
We study the period map of configurations of n points on the projective line constructed via a cyclic cover branching along these points. By considering the decomposition of its Hodge structure into eigenspaces, we establish the codimension of the locus where the eigenperiod map is still pure.… read more about this publication »
Numerical semigroups with multiplicity m are parametrized by integer points in a polyhedral cone Cm, according to Kunz. For the toric ideal of any such semigroup, the main result here constructs a free resolution whose overall structure is identical for all semigroups parametrized by the relative… read more about this publication »
We show that if a composite θ-curve has (proper rational) unknotting number one, then it is the order 2 sum of a (proper rational) unknotting number one knot and a trivial θ-curve. We also prove similar results for 2-strand tangles and knotoids read more about this publication »
When studying out-of-equilibrium systems, one often excites the dynamics in some degrees of freedom while removing the excitation in others through damping. In order for the system to converge to a statistical steady state, the dynamics must transfer the energy from the excited modes to the… read more about this publication »
We consider nonnegative solutions of the quasilinear heat equation in one dimension. Our solutions may vanish and may be unbounded. The equation is then degenerate, and weak solutions are generally nonunique. We introduce a notion of strong solution that ensures uniqueness. For suitable initial… read more about this publication »
We show that elliptic Calabi–Yau threefolds form a bounded family. We also show that the same result holds for minimal terminal threefolds of Kodaira dimension 2, upon fixing the rate of growth of pluricanonical forms and the degree of a multisection of the Iitaka fibration. Both of these… read more about this publication »
In this note we describe basic geometric properties of p-harmonic forms and p-coclosed forms and use them to reprove vanishing theorems of Pansu and new injectivity theorems for the Lp-cohomology of simply connected, pinched negatively curved manifolds. We also provide a partial resolution of a… read more about this publication »
We prove that a geometrically integral smooth projective 3-fold (Formula presented.) with nef anti-canonical class and negative Kodaira dimension over a finite field (Formula presented.) of characteristic (Formula presented.) and cardinality (Formula presented.) has a rational point. Additionally,… read more about this publication »
We study the index of log Calabi–Yau pairs (X, B) of coregularity 0. We show that 2λ(KX + B) ∼ 0, where λ is the Weil index of (X, B). This is in contrast to the case of klt Calabi–Yau varieties, where the index can grow doubly exponentially with the dimension. Our sharp bound on the index extends… read more about this publication »
For the one dimensional Burgers equation with a random and periodic forcing, it is well-known that there exists a family of invariant measures, each corresponding to a different average velocity. In this paper, we consider the coupled invariant measures and study how they change as the velocity… read more about this publication »
Linear response theory is a fundamental framework studying the macroscopic response of a physical system to an external perturbation. This paper focuses on the rigorous mathematical justification of linear response theory for Langevin dynamics. We give some equivalent characterizations for… read more about this publication »
We construct a divergence-free velocity field u:[0,T]×T2→R2 satisfying (Formula presented.) such that the corresponding drift-diffusion equation exhibits anomalous dissipation for all smooth initial data. We also show that, given any α0<1, the flow can be modified such that it is uniformly… read more about this publication »
We show that, under mild assumptions, the spectrum of a sum of independent random matrices is close to that of the Gaussian random matrix whose entries have the same mean and covariance. This nonasymptotic universality principle yields sharp matrix concentration inequalities for general sums of… read more about this publication »
Bi-stochastic normalization provides an alternative normalization of graph Laplacians in graph-based data analysis and can be computed efficiently by Sinkhorn-Knopp (SK) iterations. This paper proves the convergence of bi-stochastically normalized graph Laplacian to manifold (weighted-)Laplacian… read more about this publication »
We develop a protocol for learning a class of interacting bosonic Hamiltonians from dynamics with Heisenberg-limited scaling. For Hamiltonians with an underlying bounded-degree graph structure, we can learn all parameters with root mean square error ϵ using O(1/ϵ) total evolution time, which is… read more about this publication »
We present a form of stratified MCMC algorithm built with non-reversible stochastic dynamics in mind. It can also be viewed as a generalization of the exact milestoning method or form of NEUS. We prove the convergence of the method under certain assumptions, with expressions for the convergence… read more about this publication »
We study the perfect conductivity problem with closely spaced perfect conductors embedded in a homogeneous matrix where the current-electric field relation is the power law J=σ|E|p-2E. The gradient of solutions may be arbitrarily large as ε, the distance between inclusions, approaches to 0. To… read more about this publication »
The diffusion maps embedding of data lying on a manifold has shown success in tasks such as dimensionality reduction, clustering, and data visualization. In this work, we consider embedding data sets that were sampled from a manifold which is closed under the action of a continuous matrix group. An… read more about this publication »
A new mathematical model of melatonin synthesis in pineal cells is created and connected to a slightly modified previously created model of the circadian clock in the suprachiasmatic nucleus (SCN). The SCN influences the production of melatonin by upregulating two key enzymes in the pineal. The… read more about this publication »
In this paper, we consider topological featurizations of data defined over simplicial complexes, like images and labeled graphs, obtained by convolving this data with various filters before computing persistence. Viewing a convolution filter as a local motif, the persistence diagram of the… read more about this publication »
We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version of the Hardy-Littlewood circle method over number fields. read more about this publication »