Let ρ be a representation of a knot group (or more generally, the fundamental group of a tangle complement) into SL2(C) expressed in terms of the Wirtinger generators of a diagram D. This diagram also determines an ideal triangulation of the complement called the octahedral decomposition. ρ induces… read more about this publication »
Ramsey’s theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. In 1962, Erdős conjectured that the random 2-edge-coloring minimizes the number of monochromatic copies of Kk, and the conjecture was extended by Burr… read more about this publication »
We consider equations of the type: (Formula presented) , for general linear operators R in any spatial dimension. We prove that such equations almost always exhibit finite-time singularities for smooth and localised solutions. Singularities can even form in settings where solutions dissipate an… read more about this publication »
We study the mixing time of a recently proposed efficiently implementable Lindbladian designed to prepare the Gibbs states in the setting of weakly interacting fermionic systems. We show that at any temperature, the Lindbladian spectral gap for even parity observables is lower bounded by a constant… read more about this publication »
Chemotactic singularity formation in the context of the Patlak-Keller-Segel equation is an extensively studied phenomenon. In recent years, it has been shown that the presence of fluid advection can arrest the singularity formation given that the fluid flow possesses mixing or diffusion enhancing… read more about this publication »
In this paper, we investigate the instability of growing tumors by employing both analytical and numerical techniques to validate previous results and extend the analytical findings presented in a prior study by Feng et al. (Z Angew Math Phys 74:107, 2023). Building upon the insights derived from… read more about this publication »
The accurate quantification of symmetry is a key goal in biological inquiries because symmetry can affect biological performance and can reveal insights into development and evolutionary history. Recently, we proposed a versatile measure of symmetry, transformation information (TI), which provides… read more about this publication »
Many fundamental problems in extremal combinatorics are equivalent to proving certain polynomial inequalities in graph homomorphism densities. In 2011, a breakthrough result by Hatami and Norine showed that it is undecidable to verify polynomial inequalities in graph homomorphism densities.… read more about this publication »
Hamiltonian simulation becomes more challenging as the underlying unitary becomes more oscillatory. In such cases, an algorithm with commutator scaling and a weak dependence, such as logarithmic, on the derivatives of the Hamiltonian is desired. We introduce a new time-dependent Hamiltonian… read more about this publication »
We obtain strong upper bounds for the Betti numbers of compact complex-hyperbolic manifolds. We use the unitary holonomy to improve the results given by the most direct application of the techniques of [DS22]. We also provide effective upper bounds for Betti numbers of compact quaternionic- and… read more about this publication »
We derive a PDE that models the behavior of a boundary layer solution to the incompressible porous media (IPM) equation posed on the 2D periodic half-plane. This 1D IPM model is a transport equation with a non-local velocity similar to the well-known Córdoba-Córdoba-Fontelos (CCF) equation. We… read more about this publication »
We consider the advection-diffusion equation on T2 with a Lipschitz and time-periodic velocity field that alternates between two piecewise linear shear flows. We prove enhanced dissipation on the timescale |log υ|, where υ is the diffusivity parameter. This is the optimal decay rate as υ → 0 for… read more about this publication »
We provide a classification of curvature homogeneous hypersurfaces in space forms by classifying the ones in and . In higher dimensions, besides the isoparametric and the constant curvature ones, there is a single one in . Besides the obvious examples, we show that there exists an isolated… read more about this publication »
MotivationFeature selection is a critical task in machine learning and statistics. However, existing feature selection methods either (i) rely on parametric methods such as linear or generalized linear models, (ii) lack theoretical false discovery control, or (iii) identify few true positives.… read more about this publication »
Studies on quantum algorithms for ground-state energy estimation often assume perfect ground-state preparation; however, in reality the initial state will have imperfect overlap with the true ground state. Here, we address that problem in two ways: by faster preparation of matrix-product-state (MPS… read more about this publication »
We consider the patch problem for the (Formula presented.) -(surface quasi-geostrophic) SQG system with the values (Formula presented.) and (Formula presented.) being the 2D Euler and the SQG equations respectively. It is well-known that the Euler patches are globally wellposed in non-endpoint… read more about this publication »
We prove a uniqueness result for asymptotically conical (AC) gradient shrinking solitons for the Laplacian flow of closed G2-structures: If two gradient shrinking solitons to Laplacian flow are asymptotic to the same closed G2-cone, then their G2-structures are equivalent, and in particular, the… read more about this publication »
We prove a Poisson summation formula for the zero locus of a quadratic form in an even number of variables with no assumption on the support of the functions involved. The key novelty in the formula is that all “boundary terms” are given either by constants or sums over smaller quadrics related to… read more about this publication »
Understanding the mixing of open quantum systems is a fundamental problem in physics and quantum information science. Existing approaches for estimating the mixing time often rely on the spectral gap estimation of the Lindbladian generator, which can be challenging to obtain in practice. We propose… read more about this publication »
A canonical minimal free resolution of an arbitrary co-artinian lattice ideal over the polynomial ring is constructed over any field whose characteristic is 0 or any but finitely many positive primes. The differential has a closed-form combinatorial description as a sum over lattice paths in Zn of… read more about this publication »
The symmetric spaces that appear as moduli spaces in string theory and supergravity can be decomposed with explicit metrics using parabolic subgroups. The resulting isometry between the original moduli space and this decomposition can be used to find parametrizations of the moduli. One application… read more about this publication »
We study the relation between the coregularity, the index of log Calabi-Yau pairs and the complements of Fano varieties. We show that the index of a log Calabi-Yau pair of coregularity is at most, where is the Weil index of. This extends a recent result due to Filipazzi, Mauri and Moraga. We prove… read more about this publication »
The Homomorphism Preservation Theorem (HPT) of classical model theory states that a first-order sentence is preserved under homomorphisms if, and only if, it is equivalent to an existential-positive sentence. This theorem remains valid when restricted to finite structures, as demonstrated by the… read more about this publication »
We consider steady states of the two-dimensional incompressible Euler equations on T2 and construct smooth and singular steady states around a particular singular steady state. More precisely, we construct families of smooth and singular steady solutions that converge to the Bahouri–Chemin patch. read more about this publication »