Communications in Contemporary Mathematics

Let $V$ be a finite-dimensional representation of the group ${SL}_2$ of $2\times 2$ matrices with complex coefficients and determinant one. Let $R=ℂ{[V]}^{{SL}_2}$ be the algebra of ${SL}_2$-invariant polynomials on $V$. We present a calculation of the Hilbert series ${\mathrm{\text{Hilb}}}_R(t)={\sum }_{n\ge 0}dim(R_n)t^n$ as well as formulas for the first four coefficients of the Laurent expansion of ${\mathrm{\text{Hilb}}}_R(t)$ at $t=1$.

We present some exact controllability results for parabolic equations in the context of hierarchic control using Stackelberg–Nash strategies. We analyze two cases: in the first one, the main control (the leader) acts in the interior of the domain and the secondary controls (the followers) act on small parts of the boundary; in the second one, we consider a leader acting on the boundary while the followers are of the distributed kind. In both cases, for each leader, an associated Nash equilibrium pair is found; then, we obtain a leader that leads the system exactly to a prescribed (but arbitrary) trajectory. We consider linear and semilinear problems.

We study a coupled system of a complex Ginzburg–Landau equation with a quasilinear conservation law

In this paper, we establish fine properties of functions of bounded higher variation in the framework of fractional Sobolev spaces. In particular, inspired by the recent work of Brezis–Nguyen on the distributional Jacobian, we extend the definition of functions of bounded higher variation, which defined by Jerrard–Soner in $W^{1,N-1}\cap L^{\infty }(\mathrm{\Omega },{ℝ}^N)$, to the fractional Sobolev space $W^{1-\frac{1}{N},N}(\mathrm{\Omega },{ℝ}^N)$, and apply Cartesian currents theory to establishing general versions of coarea formula, chain rule and decomposition property.

Motivated by the goal of detecting minimal surfaces in hyperbolic manifolds, we study geometric flows in complete hyperbolic $3$-manifolds. In general, the flows might develop singularities at some finite time. In this paper, we investigate the mean curvature flow in a class of complete hyperbolic $3$-manifolds (Fuchsian manifolds) which are warped products of a closed surface of genus at least two and $ℝ$. We show that for a large class of closed initial surfaces, which are graphs over the totally geodesic surface $\mathrm{\Sigma }$, the mean curvature flow exists for all time and converges to $\mathrm{\Sigma }$. This is among the first examples of converging mean curvature flows starting from closed hypersurfaces in Riemannian manifolds. We also provide calculations for the general warped product setting which will be useful for further works.

Let ${\xi }_0,{\xi }_1,\dots$ be i.i.d. random variables with zero mean and unit variance. Consider a random Taylor series of the form $f(z)={\sum }_{k=0}^{\infty }{\xi }_k{c}_k{z}^k,$ where $c_0,c_1,\dots$ is a real sequence such that $c_n^2$ is regularly varying with index $\gamma -1$, where $\gamma >0$. We prove that

In analogy with the Gopakumar–Vafa (GV) conjecture on Calabi–Yau (CY) 3-folds, Klemm and Pandharipande defined GV type invariants on Calabi–Yau 4-folds using Gromov–Witten theory and conjectured their integrality. In a joint work with Maulik and Toda, the author conjectured their genus zero invariants are $\mathrm{\text{DT˙4}}$ invariants of one-dimensional stable sheaves. In this paper, we study this conjecture on the total space of canonical bundle of a Fano 3-fold $Y$, which reduces to a relation between twisted GW and $\mathrm{\text{DT˙3}}$ invariants on $Y$. Examples are computed for both compact and non-compact Fano 3-folds to support our conjecture.

In this paper, we describe domain walls appearing in a thin, nematic liquid crystal sample subject to an external field with intensity close to the Fréedericksz transition threshold. Using the gradient theory of the phase transition adapted to this situation, we show that depending on the parameters of the system, domain walls occur in the bistable region or at the border between the bistable and the monostable region.

We study the $\mathrm{\Gamma }$-convergence of a family of non-local, non-convex functionals in $L^p(I)$ for $p\ge 1$, where $I$ is an open interval. We show that the limit is a multiple of the $W^{1,p}(I)$ semi-norm to the power $p$ when $p>1$ (respectively, the $BV(I)$ semi-norm when $p=1$). In dimension one, this extends earlier results which required a monotonicity condition.

We prove that the integral Hodge conjecture holds for 1-cycles on irreducible holomorphic symplectic varieties of $K{3}^{[n]}$-type and of generalized Kummer type. As an application, we give a new proof of the integral Hodge conjecture for cubic fourfolds.

In this paper, we give a detailed proof of a result due to Torsten Ekedahl, describing complex tori admitting a rigid group action and showing explicitly their projectivity and their structure in terms of CM-fields. In the appendix, joint with Claudon, we show, using a method of Green-Voisin, that all group actions on complex tori deform to projective ones.

Hecke–Kiselman monoids ${\mathrm{\text{HK}}}_{\mathrm{\Theta }}$ and their algebras $K[{\mathrm{\text{HK}}}_{\mathrm{\Theta }}]$, over a field $K$, associated to finite oriented graphs $\mathrm{\Theta }$ are studied. In the case $\mathrm{\Theta }$ is a cycle of length $n\ge 3$, a hierarchy of certain unexpected structures of matrix type is discovered within the monoid $C_n={\mathrm{\text{HK}}}_{\mathrm{\Theta }}$ and this hierarchy is used to describe the structure and the properties of the algebra $K[C_n]$. In particular, it is shown that $K[C_n]$ is a right and left Noetherian algebra, while it has been known that it is a PI-algebra of Gelfand–Kirillov dimension one. This is used to characterize all Noetherian algebras $K[{\mathrm{\text{HK}}}_{\mathrm{\Theta }}]$ in terms of the graphs $\mathrm{\Theta }$. The strategy of our approach is based on the crucial role played by submonoids of the form $C_n$ in combinatorics and structure of arbitrary Hecke–Kiselman monoids ${\mathrm{\text{HK}}}_{\mathrm{\Theta }}$.