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Let G be the simple, simply connected algebraic group SL_3 defined over an algebraically closed field K of characteristic p>0. In this paper, we find H^2(G,V) for any irreducible G-module V. When p>7 we also find H^2(G(q),V) for any…

We show that the cohomology of the Regge complex in three dimensions is isomorphic to $\mathcal{H}^{{\scriptscriptstyle \bullet}}_{dR}(\Omega)\otimes\mathcal{RM}$, the infinitesimal-rigid-body-motion-valued de~Rham cohomology. Based on an…

We discuss some applications of the Morse-Novikov theory to some problems in modern physics, where appears a non-exact closed 1-form $\omega$ (a multi-valued functional). We focus mainly our attention to the cohomology of the de Rham…

We compute the Hochschild cohomology ring of the algebras $A= k\langle X, Y\rangle/ (X^a, XY-qYX, Y^a)$ over a field $k$ where $a\geq 2$ and where $q\in k$ is a primitive $a$-th root of unity. We find the the dimension of $\mathrm{HH}^n(A)$…

For $X$ a smooth projective variety, the quantum cohomology ring $QH^*(X)$ is a deformation of the usual cohomology ring $H^*(X)$, where the product structure is modified to incorporate quantum corrections. These correction terms are…

The Hamiltonian approach to isomonodromic deformation systems for generic rational covariant derivative operators on the Riemann sphere, having any matrix dimension $r$ and any number of isolated singularities of arbitrary Poincar\'e rank,…

A noncommutative algebra of the complex $q$-twistors and their differentials is considered on the basis of the quantum $GL_q (4)\times SL_q (2)$ group. Real and pseudoreal $q$-twistors are discussed too. We consider the quantum-group…

Building on the now established presentation of the standard Podles sphere as an example of a noncommutative complex structure, we investigate how its classical Kahler geometry behaves under $q$-deformation. Discussed are noncommutative…

In recent work with Bhatt and Morrow, we defined a new integral p-adic cohomology theory interpolating between etale and de Rham cohomology. An unexpected feature of this cohomology is that in coordinates, it can be computed by a…

We compute the moduli of endomorphisms of the de Rham and crystalline cohomology functors, viewed as a cohomology theory on smooth schemes over truncated Witt vectors. As applications of our result, we deduce Drinfeld's refinement of the…

The study of differential forms that are closed but not exact reveals important information about the global topology of a manifold, encoded in the de Rham cohomology groups $H^k(M)$, named after Georges de Rham (1903-1990). This expository…

Let $G$ be the simple algebraic group $\mathrm{SL}_2$ defined over an algebraically closed field $k$ of characteristic $p > 0$. Using results of A. Parker, we develop a method which gives, for any $q \in \mathbb{N}$, a closed form…

Given an $N$-dimensional compact manifold $M$ and a field $\bk$, F. Cohen and L. Taylor have constructed a spectral sequence, $\cE(M,n,\bk)$, converging to the cohomology of the space of ordered configurations of $n$ points in $M$. The…

Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the space of ordered pairs of distinct points…

SL_q(2) at odd roots of unity q^l =1 is studied as a quantum cover of the complex rotation group SO(3,C), in terms of the associated Hopf algebras of (quantum) polynomial functions. We work out the irreducible corepresentations, the…

q-Deformed harmonic oscillator algebra for real and root of unity values of the deformation parameter is discussed by using an extension of the number concept proposed by Gauss, namely the Q-numbers. A study of the reducibility of the Fock…

Owing to the presence of exceptional points (EPs), non-Hermitian (NH) systems can display intriguing topological phenomena without Hermitian analogs. However, experimental characterizations of exceptional topological invariants have been…

For any compact almost complex manifold $(M,J)$, the last two authors defined two subgroups $H_J^+(M)$, $H_J^-(M)$ of the degree 2 real de Rham cohomology group $H^2(M, \mathbb{R})$ in arXiv:0708.2520. These are the sets of cohomology…

The $C^*$-algebra of continuous functions on the quantum quaternion sphere $H_q^{2n}$ can be identified with the quotient algebra $C(SP_q(2n)/SP_q(2n-2))$. In commutative case i.e. for $q=1$, the topological space $SP(2n)/SP(2n-2)$ is…

A noncommutative *-algebra that generalizes the canonical commutation relations and that is covariant under the quantum groups SOq(3) or SOq(1,3) is introduced. The generating elements of this algebra are hermitean and can be identified…