fake projective planes a fake projective plane is a complex surface different from but has the same betti numbers as the complex projective plane it is a complex hyperbolic space form and has the smallest euler poincare characteristic among smooth surfaces of general type the first example was constructed by mumford later on two more examples were found by ishida and kato a fourth possible one was recently proposed by keum in a recent joint work with gopal prasad we showed that there are seventeen non empty classes of fake projective planes and there can at most be four more specific classes higher dimensional analogues and examples were also obtained the main purpose of the talk is to explain the joint work with prasad and other related results such as arithmeticity of the lattices involved obtained earlier by klingler and yeung independently

the toledo invariant on smooth varieties of general type we propose a definition of the toledo invariant for representations of fundamental groups of smooth varieties of general type into semisimple lie groups of hermitian type this definition allows to generalize the results known in the classical case of representations of complex hyperbolic lattices to this new setting assuming that the rank of the target lie group is not greater than two we prove that the toledo invariant satisfies a milnor wood type inequality and we characterize the corresponding maximal representations

the synthesis of universal feedback pursuit strategies in differential games we show how any generalized supersolution of the hamilton jacobi equation can be used to construct a feedback pursuit strategy which guarantees to any given tolerance a capture time not exceeding the solution s value if the supersolution is the value function then a near optimal pursuit strategy is obtained in this way an important feature of the construction is its universal nature i e the fact that the feedback law is uniformly effective on compact sets of initial conditions this implies in particular that the feedback construction is one that exploits nonoptimal behavior on the part of the evader

on the finite time stabilization of uncertain nonlinear systems with relative degree three this technical note discusses a finite time stabilization problem in a nonlinear uncertain environment the main result of the present note is presented making reference to a triple integrator affected by bounded uncertainties and subject to ldquohardrdquo measurement nonlinearities such that the sign of the state variables is the only reliable information available for feedback we propose a discontinuous control scheme guaranteeing the practical finite time annihilation of the three state variables constructive proof and computer simulations as well as guidelines for practical implementations are provided throughout this technical note

asymptotic controllability implies feedback stabilization it is shown that every asymptotically controllable system can be globally stabilized by means of some discontinuous feedback law the stabilizing strategy is based on pointwise optimization of a smoothed version of a control lyapunov function iteratively sending trajectories into smaller and smaller neighborhoods of a desired equilibrium a major technical problem and one of the contributions of the present paper concerns the precise meaning of solution when using a discontinuous controller

on the shimura variety having mumford s fake projective plane as a connected component we give an explicit construction of a unitary shimura surface that has mumford s fake projective plane as one of its connected components moreover as a byproduct of the construction we show that mumford s fake projective place has a model defined over the 7th cyclotomic field

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

quotients of products of curves new surfaces with pg 0 and their fundamental groups we construct many new surfaces of general type with q pg 0 whose canonical model is the quotient of the product of two curves by the action of a finite group g constructing in this way many new interesting fundamental groups which distinguish connected components of the moduli space of surfaces of general type we indeed classify all such surfaces whose canonical model is singular the smooth case was classified in an earlier work as an important tool we prove a structure theorem giving a precise description of the fundamental group of quotients of products of curves by the action of a finite group g

a novel higher order sliding mode control scheme a higher order sliding mode control algorithm is proposed for a class of uncertain multi input multi output nonlinear systems this problem can be viewed as the flnite time stabilization of a higher order input output dynamic system with bounded uncertainties the developed control scheme is based on geometric homogeneity and sliding mode control the proposed procedure provides explicit conditions on the controller parameters and guarantees robustness against uncertainties an illustrative example of a hovercraft vessel control demonstrates the advantages of the strategy

feedback stabilization and lyapunov functions given a locally defined nondifferentiable but lipschitz lyapunov function we employ it in order to construct a discontinuous feedback law which stabilizes the underlying system to any given tolerance a converse result shows that suitable lyapunov functions of this type exist under mild assumptions we also establish that the feedback in question possesses a robustness property relative to measurement error despite the fact that it may not be continuous

sliding order and sliding accuracy in sliding mode control the synthesis of a control algorithm that stirs a nonlinear system to a given manifold and keeps it within this constraint is considered usually what is called sliding mode is employed in such synthesis this sliding mode is characterized in practice by a high frequency switching of the control it turns out that the deviation of the system from its prescribed constraints sliding accuracy is proportional to the switching time delay a new class of sliding modes and algorithms is presented and the concept of sliding mode order is introduced these algorithms feature a bounded control continuously depending on time with discontinuities only in the control derivative it is also shown that the sliding accuracy is proportional to the square of the switching time delay

on the projective normality of double coverings over a rational surface abstract we study the projective normality of a minimal surface x which is a ramified double covering over a rational surface s with dim k s 1 in particular horikawa surfaces the minimal surfaces of general type with k x 2 2 p g x 4 are of this type up to resolution of singularities let be the covering map from x to s we show that the z 2 invariant adjoint divisors k x r a are normally generated where the integer r 3 and a is an ample divisor on s

mean value inequalities in hilbert space we establish a new mean value theorem applicable to lower semicontinuous functions on hilbert space a novel feature of the result is its multidirectionality it compares the value of a function at a point to its values on a set we then discuss some refinements and consequences of the theorem including applications to calculus flow invariance and generalized solutions to partial differential equations rtsumt on etablit un nouveau theoreme de la valeur moyenne qui s applique aux fonctions semicontinues inferieurement sur un espace de hilbert on deduit plusieurs consequences du resultat portant par exemple sur les fonctions monotones et sur les solutions generalisees des equations aux derivees partielles

a sharp lower bound for the canonical volume of 3 folds of general type let v be a smooth projective 3 fold of general type denote by k 3 a rational number the self intersection of the canonical sheaf of any minimal model of v one defines k 3 as the canonical volume of v assume p g v 2 we show that k 3 ge frac 1 3 which is a sharp lower bound then we classify those v with small volume we also give some new examples with p g 2 which have maximal canonical stability index finally we give an application to 4 folds of general type

on finite group actions on reductive groups and buildings let h be a connected reductive group over a non archimedean local field k and let f autk h be a finite group of order not divisible by p the residual characteristic of k let g h f be the identity component of the subgroup of h consisting of points fixed by f the main theorem of this paper asserts that the bruhat tits building b g of g can be identified with the set of f fixed points of b h several special cases of this theorem have been known previously when e k is a finite totally ramified galois extension h rese k g and f gal e k the condition p f is simply that e k is tamely ramified in this case our main theorem is a well known but unpublished theorem of g rousseau recently one of us g p found a simple proof of this theorem p when g is a classical group realized in the standard way as the identity component of the group of fixed points of an involution of a general linear group h bruhat and tits bt4 have given a description of b g as a subset of b h in particular it follows from their description that b g b h when p 2 recently this fact has been rediscovered by j kim and a moy km and independently a simple proof has been given by gy let h be the split form of spin 8 then one can choose an outer automorphism aut h of order 3 such that g h is of type g2 from a joint work gy by w t gan and one of us j y it is known that b g b h is true for arbitrary p

on the adjunction mapping of very ample vector bundles of corank one let e be a very ample vector bundle of rank n 1 over a smooth complex projective variety x of dimension n 3 the structure of x e being known when k x det e 0 we investigate the structure of the adjunction mapping when 0 k x det e n

homogeneity approach to high order sliding mode design it is shown that a general uncertain single input single output regulation problem is solvable only by means of discontinuous control laws giving rise to the so called high order sliding modes the homogeneity properties of the corresponding controllers yield a number of practically important features in particular finite time convergence is proved and asymptotic accuracy is calculated in a very general way in the presence of input noises discrete measurements and switching delays a robust homogeneous differentiator is included in the control structure thus yielding robust output feedback controllers with finite time convergence it is demonstrated that homogeneity features significantly simplify the design and investigation of a new family of high order sliding mode controllers finally simulation results are presented

arithmetic fake projective spaces and arithmetic fake grassmannians in a recent paper we have classified fake projective planes natural higher dimensional generalization of these surfaces are arithmetic fake bf p n 1 c and arithmetic fake bf gr m n in this paper we show that arithmetic fake bf p n 1 c can exist only if n 3 5 and an arithmetic fake bf gr m n can exist with n3 odd only if n 5 here we construct four distinct arithmetic fake bf p 4 c and four distinct fake arithmetic bf gr 2 5 furthermore we use certain results and computations of py to exhibit five irreducible arithmetic fake bf p 2 c times bf p 2 c all these are connected smooth complex projective shimura varieties

derived categories of keum s fake projective planes abstract we conjecture that derived categories of coherent sheaves on fake projective n spaces have a semi orthogonal decomposition into a collection of n 1 exceptional objects and a category with vanishing hochschild homology we prove this for fake projective planes with non abelian automorphism group such as keum s surface then by passing to equivariant categories we construct new examples of phantom categories with both hochschild homology and grothendieck group vanishing

universal single input single output siso sliding mode controllers with finite time convergence an universal controller is constructed formulated in input output terms only which causes the output of any uncertain smooth single input single output siso minimum phase dynamic system with known relative degree to vanish in finite time this allows exact tracking of arbitrary real time smooth signals only one parameter is to be adjusted since the approach is based on higher order finite time convergence sliding modes the control can be made arbitrarily smooth providing for the arbitrarily high tracking accuracy order with respect to the sampling step

arithmetic structure of cmsz fake projective planes abstract we show that the fake projective planes that are constructed from dyadic discrete subgroups discovered by cartwright mantero steger and zappa are realized as connected components of certain unitary shimura surfaces as a corollary we show that these fake projective planes have models defined over the number field q 3 5

examples of mori dream surfaces of general type with pg 0 abstract in this paper we study effective nef and semiample cones of minimal surfaces of general type with p g 0 we provide examples of minimal surfaces of general type with p g 0 and 2 k 2 9 which are mori dream spaces on these examples we also give explicit description of their effective cones with all negative curves we also present non minimal surfaces of general type with p g 0 that are not mori dream surfaces

symmetric differentials kahler groups and ball quotients while margulis superrigidity theorem completely describes the finite dimensional linear representations of lattices of higher rank simple real lie groups almost nothing is known concerning the representation theory of complex hyperbolic lattices the main result of this paper theorem 1 3 is a strong rigidity theorem for a certain class of cocompact arithmetic complex hyperbolic lattices it relies on the following two ingredients theorem 1 6 showing that the representations of the topological fundamental group of a compact kahler manifold x are controlled by the global symmetric differentials on x an arithmetic vanishing theorem for global symmetric differentials on certain compact ball quotients using automorphic forms in particular deep results of clozel on base change theorem 1 11

sur la rigidite de certains groupes fondamentaux l arithmeticite des reseaux hyperboliques complexes et les faux plans projectifs the motivation of this work comes from the study of lattices in real simple lie groups the famous margulis s superrigidity theorem claims that finite dimensional reductive representations of any lattice of a real simple lie group of real rank 2 are superrigid as a corollary such a lattice is arithmetic these results extend to the real rank one case for lattices in sp n 1 and f4 20 by the work of corlette and gromov schoen on the other hand mostow and deligne mostow exhibited arithmetic lattices with non superrigid representations as well as non arithmetic lattices in the unitary group pu 2 1 a natural question is then to find simple sufficient conditions for superrigidity or arithmeticity of lattices in pu 2 1 rogawski conjectured the following let be a torsion free cocompact lattice in pu 2 1 such that the hyperbolic quotient m b2 verifies the cohomogical conditions b1 m 0 and h1 1 m h2 m then is arithmetic in this paper we consider a smooth complex projective surface m verifying the above cohomological assumptions and study zariski dense representations of the fundamental group 1 m in a simple k group h of k rank 2 where k denotes a local field our main result states that there are strong restrictions on such representations especially when k is non archimedean theorem 5 we then consider some cocompact lattices in pu 2 1 of special geometric interest recall that a fake p2 is a smooth complex surface distinct from p2 having the same betti numbers as p2 fake p2 exist by a result of mumford and are complex hyperbolic quotients h2 by yau s proof of the calabi conjecture they obviously verify the hypotheses of rogawski s conjecture in this case we prove that every zariski dense representation of in pgl 3 is superrigid in the sense of margulis theorem 3 as a corollary every fake p2 is an arithmetic quotient of the ball b2

fake quadrics from irreducible lattices acting on the product of upper half planes in the present article we provide examples of fake quadrics that is minimal complex surfaces of general type with the same numerical invariants as the smooth quadric in mathbb p 3 which are quotients of the bidisc by an irreducible lattice of automorphisms moreover we list classes of arithmetic lattices over a real quadratic number field which define a fake quadric and give general results towards a classification of all such fake quadrics

finite time stability and robust control synthesis of uncertain switched systems stability analysis is developed for uncertain nonlinear switched systems while being asymptotically stable and homogeneous of degree q 0 these systems are shown to approach the equilibrium point in finite time restricted to second order systems this feature is additionally demonstrated to persist regardless of inhomogeneous perturbations based on this fundamental property switched control algorithms are then developed to globally stabilize uncertain minimum phase systems of uniform m vector relative degree 2 2 t the controllers constructed do not rely on the generation of sliding motions while providing robustness features similar to those possessed by their sliding mode counterparts the proposed synthesis procedure is illustrated via application to a friction servo motor

differential games of generalized pursuit and evasion differential games of generalized pursuit and evasion are studied by comparing them with differential games of fixed duration for which a theory already has been established it is shown that if the isaacs condition holds and the data satisfy reasonable hypotheses then the games have values which are continuous functions of the initial time and state if the data satisfy appropriate lipschitz conditions then the value is lipschitz continuous and satisfies the isaacs equation at all points of differentiability of the value

new non arithmetic complex hyperbolic lattices we produce a family of new non arithmetic lattices in mathrm pu 2 1 all previously known examples were commensurable with lattices constructed by picard mostow and deligne mostow and fell into nine commensurability classes our groups produce five new distinct commensurability classes most of the techniques are completely general and provide efficient geometric and computational tools for constructing fundamental domains for discrete groups acting on the complex hyperbolic plane

pursuit evasion problems and viscosity solutions of isaacs equations the dirichlet problem for first order hamilton jacobi equations arising in differential games of pursuit and evasion is studied local and global sub and superoptimality principles are stated for respectively viscosity sub and supersolutions these results are applied to obtain a general existence theorem and to prove the existence of the value of the game the main application concerns the problem of stability terminability of a dynamical system with two competitive controls and the opposite one of evadability from a general closed set the approach used in this paper allows lyapunov functions satisfying the usual condition in the weak sense of viscosity solutions

explicit equations of a fake projective plane fake projective planes are smooth complex surfaces of general type with betti numbers equal to those of the usual projective plane they come in complex conjugate pairs and have been classified as quotients of the 2 dimensional ball by explicitly written arithmetic subgroups in the following we find equations of a projective model of a conjugate pair of fake projective planes by studying the geometry of the quotient of such surface by an order 7 automorphism