Related papers: Discrete Quantum Field Theories and the Intersecti…
We consider the Boltzmann equation with the soft potential and angular cutoff. Inspired by the methods from dispersive PDEs, we establish its sharp local well-posedness and ill-posedness in $H^{s}$ Sobolev space. We find the…
We consider a simple and natural coboundary operator, on the Lie algebra valued differential forms on a manifold, which in the abelian case reduces to usual exterior derivative of such forms. Using the corresponding de Rham cohomology Lie…
We give an alternative proof of the mod $p$ vanishing theorem by F.Fang of Seiberg-Witten invariants under a cyclic group action of prime order, and generalize it to the case when $b_1>0$. Although we also use the finite dimensional…
We use the Dold-Whitney theorem classifying $SO(3)$-bundles over a 4-complex to give a mod 4 obstruction to a 2-component link of trivial linking number being slice. It turns out that this coincides with the reduction of the Sato-Levine…
The purpose of this talk is to present an (apparently) new way to look at the intersection complex of a singular variety over a finite field, or, more generally, at the intermediate extension functor on pure perverse sheaves, and an…
We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincar\'e duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain proofs of Dowling and Wilson's Top-Heavy…
A nonlinear Poisson--Boltzmann equation with transmission boundary conditions at the interface between two materials is investigated. The model describes the electrostatic potential generated by a vector of ion concentrations in a periodic…
We introduce a new version of non-linear electrodynamics which is produced by a spontaneous symmetry breaking of Lorentz invariance induced by the non-zero expectation value of the electromagnetic field strength. The symmetry breaking…
We prove that mod-$p$ congruences between polynomials in $\mathbb{Z}_p[X]$ are equivalent to deeper $p$-power congruences between power-sum functions of their roots. This result generalizes to torsion-free $\mathbb{Z}_{(p)}$-algebras modulo…
We present explicit formulas for the intersection pairing in the intersection cohomology of the moduli space $M_0(r)$ of rank-$r$, degree-$0$ semistable bundles on a Riemann surface. The key idea is to realize this intersection cohomology…
We devise a lattice Boltzmann method (LBM) for a matrix-valued quantum Boltzmann equation, with the classical Maxwell distribution replaced by Fermi-Dirac functions. To accommodate the spin density matrix, the distribution functions become…
Starting from the defining transformations of complex matrices for the $SO(4,R)$ group, we construct the fundamental representation and the tensor and spinor representations of the group $SO(4,R)$. Given the commutation relations for the…
All indecomposable finite-dimensional representations of the homogeneous Galilei group which when restricted to the rotation subgroup are decomposed to spin 0, 1/2 and 1 representations are constructed and classified. These representations…
A gauge theory with an underlying SU_q(2) quantum group symmetry is introduced, and its properties examined. With suitable assumptions, this model is found to have many similarities with the usual SU(2)\times U(1) Standard Model,…
The Dicke model, which describes the dipolar coupling between N two-level atoms and a quantized electromagnetic field, seemingly violates gauge invariance in the presence of ultrastrong light-matter coupling, a regime that is now…
We build free, bigraded bidifferential algebra models for the forms on a complex manifold, with respect to a strong notion of quasi-isomorphism and compatible with the conjugation symmetry. This answers a question of Sullivan. The resulting…
We consider the problem of uniform interpolation of functions with values in a complex inner product space of finite dimension. This problem can be casted within a modified weighted pluripotential theoretic framework. Indeed, in the…
We give simple explicit formulas for deformation quantization of Poisson-Lie groups and of similar Poisson manifolds which can be represented as moduli spaces of flat connections on surfaces. The star products depend on a choice of…
A coupling-constant definition is given based on the compositeness property of some particle states with respect to the elementary states of other particles. It is applied in the context of the vector-spin-1/2-particle interaction vertices…
A new model for calculating the structure of bound states of interacting particles is considered. The model takes into account the noncommutativity of the space and impulse operators plus the correlation equations for the indeterminacy of…