Mathematical modeling and quantitative study of biological motility (in particular, of motility at microscopic scales) is producing new biophysical insight and is offering opportunities for new discoveries at the level of both fundamental science and technology. These range from the explanation of how complex behavior at the level of a single organism emerges from body architecture, to the understanding of collective phenomena in groups of organisms and tissues, and of how these forms of swarm intelligence can be controlled and harnessed in engineering applications, to the elucidation of processes of fundamental biological relevance at the cellular and sub-cellular level. In this paper, some of the most exciting new developments in the fields of locomotion of unicellular organisms, of soft adhesive locomotion across scales, of the study of pore translocation properties of knotted DNA, of the development of synthetic active solid sheets, of the mechanics of the unjamming transition in dense cell collectives, of the mechanics of cell sheet folding in volvocalean algae, and of the self-propulsion of topological defects in active matter are discussed. For each of these topics, we provide a brief state of the art, an example of recent achievements, and some directions for future research.

10aactive matter10aadhesive locomotion10acell motility10acell sheet folding10aknotted DNA10atopological defects10aunicellular swimmers10aunjamming transition1 aAgostinelli, Daniele1 aCerbino, Roberto1 aDel Alamo, Juan, C1 aDeSimone, Antonio1 aHöhn, Stephanie1 aMicheletti, Cristian1 aNoselli, Giovanni1 aSharon, Eran1 aYeomans, Julia uhttp://dx.doi.org/10.3934/mine.202001100806nas a2200181 4500008004100000022001400041245009400055210006900149260000800218300001400226490000700240520023200247100002900479700002900508700002300537700001800560856004600578 2018 eng d a1424-066100aLp-Boundedness of Wave Operators for the Three-Dimensional Multi-Centre Point Interaction0 aLpBoundedness of Wave Operators for the ThreeDimensional MultiCe cJan a283–3220 v193 aWe prove that, for arbitrary centres and strengths, the wave operators for three-dimensional Schrödinger operators with multi-centre local point interactions are bounded in Lp(R3)for 1<p<3 and unbounded otherwise.

1 aDell'Antonio, Gianfausto1 aMichelangeli, Alessandro1 aScandone, Raffaele1 aYajima, Kenji uhttps://doi.org/10.1007/s00023-017-0628-400506nas a2200145 4500008004100000022001400041245011400055210006900169300001200238490000800250100001900258700002000277700001300297856005000310 2016 eng d a0167-278900aCorrelation functions of the KdV hierarchy and applications to intersection numbers over $\overline\CalM_g,n$0 aCorrelation functions of the KdV hierarchy and applications to i a30–570 v3271 aBertola, Marco1 aDubrovin, Boris1 aYang, Di uhttp://dx.doi.org/10.1016/j.physd.2016.04.00800397nas a2200121 4500008004100000245004500041210004500086490000900131100001900140700002000159700001300179856008300192 2016 eng d00aSimple Lie Algebras and Topological ODEs0 aSimple Lie Algebras and Topological ODEs0 v20161 aBertola, Marco1 aDubrovin, Boris1 aYang, Di uhttps://www.math.sissa.it/publication/simple-lie-algebras-and-topological-odes00596nas a2200181 4500008004100000245003700041210003000078520010700108100001700215700001800232700001700250700001700267700002400284700002000308700001700328700001800345856005100363 2015 en d00aThe deal.II Library, Version 8.20 adealII Library Version 823 aThis paper provides an overview of the new features of the finite element library deal.II version 8.21 aBangerth, W.1 aHeister, Timo1 aHeltai, Luca1 aKanschat, G.1 aKronbichler, Martin1 aMaier, Matthias1 aTurcksin, B.1 aYoung, T., D. uhttp://urania.sissa.it/xmlui/handle/1963/3446400453nas a2200133 4500008004100000022001400041245008800055210006900143300001500212490000700227100001900234700001300253856005300266 2015 eng d a1751-811300aThe partition function of the extended $r$-reduced Kadomtsev-Petviashvili hierarchy0 apartition function of the extended rreduced KadomtsevPetviashvil a195205, 200 v481 aBertola, Marco1 aYang, Di uhttp://dx.doi.org/10.1088/1751-8113/48/19/19520501332nas a2200121 4500008004100000245014300041210006900184260002100253520085000274100002301124700001201147856005101159 2014 en d00aGlobal Structure of Admissible BV Solutions to Piecewise Genuinely Nonlinear, Strictly Hyperbolic Conservation Laws in One Space Dimension0 aGlobal Structure of Admissible BV Solutions to Piecewise Genuine bTaylor & Francis3 aThe paper describes the qualitative structure of an admissible BV solution to a strictly hyperbolic system of conservation laws whose characteristic families are piecewise genuinely nonlinear. More precisely, we prove that there are a countable set of points Θ and a countable family of Lipschitz curves T{script} such that outside T{script} ∪ Θ the solution is continuous, and for all points in T{script}{set minus}Θ the solution has left and right limit. This extends the corresponding structural result in [7] for genuinely nonlinear systems. An application of this result is the stability of the wave structure of solution w.r.t. -convergence. The proof is based on the introduction of subdiscontinuities of a shock, whose behavior is qualitatively analogous to the discontinuities of the solution to genuinely nonlinear systems.

1 aBianchini, Stefano1 aYu, Lei uhttp://urania.sissa.it/xmlui/handle/1963/3469400484nas a2200133 4500008004100000245009400041210006900135260001900204300001200223490000800235100002300243700001200266856007200278 2014 en d00aStructure of entropy solutions to general scalar conservation laws in one space dimension0 aStructure of entropy solutions to general scalar conservation la bSISSAc08/2015 a356-3860 v4281 aBianchini, Stefano1 aYu, Lei uhttps://www.sciencedirect.com/science/article/pii/S0022247X1500221800604nas a2200193 4500008004100000245003700041210003000078260001000108520010800118100001700226700001800243700001700261700001700278700002400295700002000319700001700339700001800356856003600374 2013 en d00aThe deal.II Library, Version 8.10 adealII Library Version 81 bSISSA3 aThis paper provides an overview of the new features of the finite element library deal.II version 8.0.1 aBangerth, W.1 aHeister, Timo1 aHeltai, Luca1 aKanschat, G.1 aKronbichler, Martin1 aMaier, Matthias1 aTurcksin, B.1 aYoung, T., D. uhttp://hdl.handle.net/1963/723601436nas a2200145 4500008004100000245008000041210006900121260001000190520096800200100002001168700002401188700002401212700001801236856003601254 2013 en d00aN=2 gauge theories on toric singularities, blow-up formulae and W-algebrae0 aN2 gauge theories on toric singularities blowup formulae and Wal bSISSA3 aWe compute the Nekrasov partition function of gauge theories on the\r\n(resolved) toric singularities C^2/\\Gamma in terms of blow-up formulae. We\r\ndiscuss the expansion of the partition function in the \\epsilon_1,\\epsilon_2\r\n\\to 0 limit along with its modular properties and how to derive them from the\r\nM-theory perspective. On the two-dimensional conformal field theory side, our\r\nresults can be interpreted in terms of representations of the direct sum of\r\nHeisenberg plus W_N-algebrae with suitable central charges, which can be\r\ncomputed from the fan of the resolved toric variety.We provide a check of this\r\ncorrespondence by computing the central charge of the two-dimensional theory\r\nfrom the anomaly polynomial of M5-brane theory. Upon using the AGT\r\ncorrespondence our results provide a candidate for the conformal blocks and\r\nthree-point functions of a class of the two-dimensional CFTs which includes\r\nparafermionic theories.1 aBonelli, Giulio1 aMaruyoshi, Kazunobu1 aTanzini, Alessandro1 aYagi, Futoshi uhttp://hdl.handle.net/1963/657700932nas a2200109 4500008004100000245011500041210006900156260001000225520044500235100001200680856013000692 2013 en d00aThe structure and regularity of admissible BV solutions to hyperbolic conservation laws in one space dimension0 astructure and regularity of admissible BV solutions to hyperboli bSISSA3 aThis thesis is devoted to the study of the qualitative properties of admissible BV solutions to the strictly hyperbolic conservation laws in one space dimension by using wave-front tracking approximation. This thesis consists of two parts: • SBV-like regularity of vanishing viscosity BV solutions to strict hyperbolic systems of conservation laws. • Global structure of admissible BV solutions to strict hyperbolic conservation laws.1 aYu, Lei uhttps://www.math.sissa.it/publication/structure-and-regularity-admissible-bv-solutions-hyperbolic-conservation-laws-one-space01152nas a2200145 4500008004100000245009700041210006900138260001000207520067600217100002200893700001500915700002000930700002000950856003600970 2012 en d00aA Codazzi-like equation and the singular set for C1 smooth surfaces in the Heisenberg group.0 aCodazzilike equation and the singular set for C1 smooth surfaces bSISSA3 aIn this paper, we study the structure of the singular set for a C 1 smooth surface in the 3-dimensional Heisenberg group ℍ 1. We discover a Codazzi-like equation for the p-area element along the characteristic curves on the surface. Information obtained from this ordinary differential equation helps us to analyze the local configuration of the singular set and the characteristic curves. In particular, we can estimate the size and obtain the regularity of the singular set. We understand the global structure of the singular set through a Hopf-type index theorem. We also justify the Codazzi-like equation by proving a fundamental theorem for local surfaces in ℍ 11 aMalchiodi, Andrea1 aYang, Paul1 aCheng, Jih-Hsin1 aHwang, JennFang uhttp://hdl.handle.net/1963/655600509nas a2200121 4500008004100000245009900041210006900140300001400209490000700223100002300230700001200253856012200265 2012 eng d00aSBV-like regularity for general hyperbolic systems of conservation laws in one space dimension0 aSBVlike regularity for general hyperbolic systems of conservatio a439–4720 v441 aBianchini, Stefano1 aYu, Lei uhttps://www.math.sissa.it/publication/sbv-regularity-general-hyperbolic-systems-conservation-laws-one-space-dimension00772nas a2200157 4500008004300000245008600043210007000129260003400199520023700233653003600470100001900506700002200525700001600547700001500563856003600578 2011 en_Ud 00aAxial symmetry of some steady state solutions to nonlinear Schrödinger equations0 aAxial symmetry of some steady state solutions to nonlinear Schrö bAmerican Mathematical Society3 aIn this note, we show the axial symmetry of steady state solutions of nonlinear Schrodinger equations when the exponent of the nonlinearity is between the critical Sobolev exponent of n dimensional space and n - 1 dimensional space.10aNonlinear Schrödinger equation1 aGui, Changfeng1 aMalchiodi, Andrea1 aXu, Haoyuan1 aYang, Paul uhttp://hdl.handle.net/1963/410001229nas a2200169 4500008004100000245006500041210006200106260001000168520073800178100001600916700003100932700001500963700001200978700001400990700001901004856003601023 2011 en d00aD-branes, surface operators, and ADHM quiver representations0 aDbranes surface operators and ADHM quiver representations bSISSA3 aA supersymmetric quantum mechanical model is constructed for BPS states bound to surface operators in five dimensional SU(r) gauge theories using D-brane engineering. This model represents the effective action of a certain D2-brane configuration, and is naturally obtained by dimensional reduction of a quiver $(0,2)$ gauged linear sigma model. In a special stability chamber, the resulting moduli space of quiver representations is shown to be smooth and isomorphic to a moduli space of framed quotients on the projective plane. A precise conjecture relating a K-theoretic partition function of this moduli space to refined open string invariants of toric lagrangian branes is formulated for conifold and local P^1 x P^1 geometries.1 aBruzzo, Ugo1 aDiaconescu, Duiliu-Emanuel1 aYardim, M.1 aPan, G.1 aZhang, Yi1 aWu-yen, Chuang uhttp://hdl.handle.net/1963/413301134nas a2200145 4500008004100000245006700041210006700108260001000175520068000185100002000865700002400885700002400909700001900933856003600952 2011 en d00aGeneralized matrix models and AGT correspondence at all genera0 aGeneralized matrix models and AGT correspondence at all genera bSISSA3 aWe study generalized matrix models corresponding to n-point Virasoro\r\nconformal blocks on Riemann surfaces with arbitrary genus g. Upon AGT\r\ncorrespondence, these describe four dimensional N=2 SU(2)^{n+3g-3} gauge\r\ntheories with generalized quiver diagrams. We obtain the generalized matrix\r\nmodels from the perturbative evaluation of the Liouville correlation functions\r\nand verify the consistency of the description with respect to degenerations of\r\nthe Riemann surface. Moreover, we derive the Seiberg-Witten curve for the N=2\r\ngauge theory as the spectral curve of the generalized matrix model, thus\r\nproviding a check of AGT correspondence at all genera.1 aBonelli, Giulio1 aMaruyoshi, Kazunobu1 aTanzini, Alessandro1 aYagib, Futoshi uhttp://hdl.handle.net/1963/656800701nas a2200121 4500008004300000245009900043210006900142520027900211100002000490700001500510700001800525856003600543 2008 en_Ud 00aFrobenius Manifolds and Central Invariants for the Drinfeld - Sokolov Bihamiltonian Structures0 aFrobenius Manifolds and Central Invariants for the Drinfeld Soko3 aThe Drinfeld - Sokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine Lie algebra. We compute the complete set of invariants of the related bihamiltonian structures with respect to the group of Miura type transformations.1 aDubrovin, Boris1 aSi-Qi, Liu1 aYoujin, Zhang uhttp://hdl.handle.net/1963/252301700nas a2200121 4500008004300000245004200043210004200085520136200127100002101489700001601510700001601526856003601542 2007 en_Ud 00aAsymptotic variational wave equations0 aAsymptotic variational wave equations3 aWe investigate the equation $(u_t + (f(u))_x)_x = f\\\'\\\'(u) (u_x)^2/2$ where $f(u)$ is a given smooth function. Typically $f(u)= u^2/2$ or $u^3/3$. This equation models unidirectional and weakly nonlinear waves for the variational wave equation $u_{tt} - c(u) (c(u)u_x)_x =0$ which models some liquid crystals with a natural sinusoidal $c$. The equation itself is also the Euler-Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.\\nWe establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function $f$ has Lipschitz continuous second-order derivative. In the case where $f$ is convex, the Cauchy problem is well-posed also within the class of dissipative solutions. However, when $f$ is not convex, we show that the dissipative solutions do not depend continuously on the initial data.1 aBressan, Alberto1 aPing, Zhang1 aYuxi, Zheng uhttp://hdl.handle.net/1963/218200638nas a2200109 4500008004300000245006800043210006800111520027600179100002100455700001600476856003600492 2006 en_Ud 00aConservative Solutions to a Nonlinear Variational Wave Equation0 aConservative Solutions to a Nonlinear Variational Wave Equation3 aWe establish the existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation $u_{tt} - c(u)(c(u)u_x)_x=0$, for initial data of finite energy. Here $c(\\\\cdot)$ is any smooth function with uniformly positive bounded values.1 aBressan, Alberto1 aYuxi, Zheng uhttp://hdl.handle.net/1963/218400687nas a2200121 4500008004300000245006200043210005900105520031100164100002000475700001800495700001600513856003600529 2006 en_Ud 00aExtended affine Weyl groups and Frobenius manifolds -- II0 aExtended affine Weyl groups and Frobenius manifolds II3 aFor the root system of type $B_l$ and $C_l$, we generalize the result of \\\\cite{DZ1998} by showing the existence of a Frobenius manifold structure on the orbit space of the extended affine Weyl group that corresponds to any vertex of the Dynkin diagram instead of a particular choice of \\\\cite{DZ1998}.1 aDubrovin, Boris1 aYoujin, Zhang1 aDafeng, Zuo uhttp://hdl.handle.net/1963/178701265nas a2200121 4500008004300000245012600043210006900169520081600238100002001054700001501074700001801089856003601107 2006 en_Ud 00aOn Hamiltonian perturbations of hyperbolic systems of conservation laws I: quasitriviality of bihamiltonian perturbations0 aHamiltonian perturbations of hyperbolic systems of conservation 3 aWe study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEs. Under certain genericity assumptions it is proved that any bihamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives. The main tools is in constructing of the so-called quasi-Miura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bihamiltonian structure of hydrodynamic type (the quasitriviality theorem). We also describe, following \\\\cite{LZ1}, the invariants of such bihamiltonian structures with respect to the group of Miura-type transformations depending polynomially on the derivatives.1 aDubrovin, Boris1 aSi-Qi, Liu1 aYoujin, Zhang uhttp://hdl.handle.net/1963/253501933nas a2200145 4500008004100000245004900041210004900090260002900139520150600168100002001674700002001694700002201714700001501736856003601751 2005 en d00aMinimal surfaces in pseudohermitian geometry0 aMinimal surfaces in pseudohermitian geometry bScuola Normale Superiore3 aWe consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate some {\em extension} theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and to solve a Bernstein-type problem (for graphs over the $xy$-plane) in the Heisenberg group $H_1$. In $H_{1}$, identified with the Euclidean space $R^{3}$, the p-minimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set in two dimensions, and generalize to higher dimensions without any size control condition. We also show that there are no closed, connected, $C^{2}$ smoothly immersed constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard $S^{3}.$ This fact continues to hold when $S^{3}$ is replaced by a general spherical pseudohermitian 3-manifold.1 aCheng, Jih-Hsin1 aHwang, JennFang1 aMalchiodi, Andrea1 aYang, Paul uhttp://hdl.handle.net/1963/457901240nas a2200121 4500008004300000245006600043210005900109260001000168520086800178100002101046700001501067856003601082 2004 en_Ud 00aOn the convergence rate of vanishing viscosity approximations0 aconvergence rate of vanishing viscosity approximations bWiley3 aGiven a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound $\\\\big\\\\|u(t,\\\\cdot)-u^\\\\ve(t,\\\\cdot)\\\\big\\\\|_{\\\\L^1}= \\\\O(1)(1+t)\\\\cdot \\\\sqrt\\\\ve|\\\\ln\\\\ve|$ on the distance between an exact BV solution $u$ and a viscous approximation $u^\\\\ve$, letting the viscosity coefficient $\\\\ve\\\\to 0$. In the proof, starting from $u$ we construct an approximation of the viscous solution $u^\\\\ve$ by taking a mollification $u*\\\\phi_{\\\\strut \\\\sqrt\\\\ve}$ and inserting viscous shock profiles at the locations of finitely many large shocks, for each fixed $\\\\ve$. Error estimates are then obtained by introducing new Lyapunov functionals which control shock interactions, interactions between waves of different families and by using sharp decay estimates for positive nonlinear waves.1 aBressan, Alberto1 aYang, Tong uhttp://hdl.handle.net/1963/291500305nas a2200109 4500008004300000245003200043210002800075100001800103700002000121700001800141856003600159 2004 en_Ud 00aThe Extended Toda Hierarchy0 aExtended Toda Hierarchy1 aCarlet, Guido1 aDubrovin, Boris1 aYoujin, Zhang uhttp://hdl.handle.net/1963/254200741nas a2200121 4500008004300000245005600043210005400099260000900153520038500162100002100547700001500568856003600583 2004 en_Ud 00aA sharp decay estimate for positive nonlinear waves0 asharp decay estimate for positive nonlinear waves bSIAM3 aWe consider a strictly hyperbolic, genuinely nonlinear system of conservation laws in one space dimension. A sharp decay estimate is proved for the positive waves in an entropy weak solution. The result is stated in terms of a partial ordering among positive measures, using symmetric rearrangements and a comparison with a solution of Burgers\\\' equation with impulsive sources.1 aBressan, Alberto1 aYang, Tong uhttp://hdl.handle.net/1963/291601068nas a2200109 4500008004300000245005500043210005500098520073100153100002000884700001800904856003600922 2004 en_Ud 00aVirasoro Symmetries of the Extended Toda Hierarchy0 aVirasoro Symmetries of the Extended Toda Hierarchy3 aWe prove that the extended Toda hierarchy of \\\\cite{CDZ} admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators $L_m$, $m\\\\geq -1$ of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau function of a generic solution to the extended Toda hierarchy is annihilated by a combination of the Virasoro operators and the flows of the hierarchy. As an application we show that the validity of the Virasoro constraints for the $CP^1$ Gromov-Witten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended Toda hierarchy.1 aDubrovin, Boris1 aYoujin, Zhang uhttp://hdl.handle.net/1963/254400433nas a2200121 4500008004100000245008600041210006900127260001800196100002400214700001500238700002200253856003600275 2002 en d00aOn the Yamabe problem and the scalar curvature problems under boundary conditions0 aYamabe problem and the scalar curvature problems under boundary bSISSA Library1 aAmbrosetti, Antonio1 aYanYan, Li1 aMalchiodi, Andrea uhttp://hdl.handle.net/1963/151000420nas a2200121 4500008004100000245007300041210006900114260001800183100002400201700001500225700002200240856003600262 2000 en d00aA note on the scalar curvature problem in the presence of symmetries0 anote on the scalar curvature problem in the presence of symmetri bSISSA Library1 aAmbrosetti, Antonio1 aYanYan, Li1 aMalchiodi, Andrea uhttp://hdl.handle.net/1963/136500372nas a2200121 4500008004100000245004700041210004700088260001800135100002400153700001500177700002200192856003600214 2000 en d00aScalar curvature under boundary conditions0 aScalar curvature under boundary conditions bSISSA Library1 aAmbrosetti, Antonio1 aYanYan, Li1 aMalchiodi, Andrea uhttp://hdl.handle.net/1963/150600734nas a2200121 4500008004300000245004900043210004900092260001300141520038400154100002000538700001800558856003600576 1999 en_Ud 00aFrobenius manifolds and Virasoro constraints0 aFrobenius manifolds and Virasoro constraints bSpringer3 aFor an arbitrary Frobenius manifold a system of Virasoro constraints is constructed. In the semisimple case these constraints are proved to hold true in the genus one approximation. Particularly, the genus $\\\\leq 1$ Virasoro conjecture of T.Eguchi, K.Hori, M.Jinzenji, and C.-S.Xiong and of S.Katz is proved for smooth projective varieties having semisimple quantum cohomology.1 aDubrovin, Boris1 aYoujin, Zhang uhttp://hdl.handle.net/1963/288301453nas a2200133 4500008004300000245005600043210005500099260001300154520106200167100002101229700001801250700001501268856003601283 1999 en_Ud 00aL-1 stability estimates for n x n conservation laws0 aL1 stability estimates for n x n conservation laws bSpringer3 aLet $u_t+f(u)_x=0$ be a strictly hyperbolic $n\\\\times n$ system of conservation laws, each characteristic field being linearly degenerate or genuinely nonlinear. In this paper we explicitly define a functional $\\\\Phi=\\\\Phi(u,v)$, equivalent to the $L^1$ distance, which is `almost decreasing\\\', i.e., $\\\\Phi(u(t),v(t))-\\\\Phi(u(s),v(s))\\\\leq\\\\break O (\\\\epsilon)·(t-s)$ for all $t>s\\\\geq 0$, for every pair of $\\\\epsilon$-approximate solutions $u,v$ with small total variation, generated by a wave-front-tracking algorithm. The small parameter $\\\\epsilon$ here controls the errors in the wave speeds, the maximum size of rarefaction fronts and the total strength of all non-physical waves in $u$ and in $v$. From the above estimate, it follows that front-tracking approximations converge to a unique limit solution, depending Lipschitz continuously on the initial data, in the $L^1$ norm. This provides a new proof of the existence of the standard Riemann semigroup generated by an $n\\\\times n$ system of conservation laws.\\\'\\\'1 aBressan, Alberto1 aLiu, Tai-Ping1 aYang, Tong uhttp://hdl.handle.net/1963/337300862nas a2200121 4500008004300000245008700043210006900130260001300199520045400212100002000666700001800686856003600704 1998 en_Ud 00aBihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation0 aBihamiltonian Hierarchies in 2D Topological Field Theory At OneL bSpringer3 aWe compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical W-algebra; we compute the central charge of this algebra. We also express the generating function of elliptic Gromov - Witten invariants via tau-function of the isomonodromy deformation problem arising in the theory of WDVV equations of associativity.1 aDubrovin, Boris1 aYoujin, Zhang uhttp://hdl.handle.net/1963/3696