controllability for semilinear functional differential equations with unbounded delays in this paper the approximate controllability of the semilinear functional differential equations with unbounded delays is first investigated then the regularity of solutions of the given system is addressed finally a simple example to which our main result can be applied is given

the geometry of the moduli space of curves of genus 23 the problem of describing the birational geometry of the moduli space mg of complex curves of genus g has a long history severi already knew in 1915 thatmg is unirational for g 10 cf sev see also ac1 for a modern proof in the same paper severi conjectured thatmg is unirational for all genera g then for a long period this problem seemed intractable mumford writes in mu p 51 whether more mg s g 11 are unirational or not is a very interesting problem but one which looks very hard too especially if g is quite large the breakthrough came in the eighties when eisenbud harris and mumford proved that mg is of general type as soon as g 24 and that the kodaira dimension of m23 is 1 see hm eh3 we note that mg is rational for g 6 see dol for problems concerning the rationality of various moduli spaces severi s proof of the unirationality of mg for small g was based on representing a general curve of genus g as a plane curve of degree d with nodes this is possible when d 2g 3 2 when the number of nodes is small i e d 1 d 2 6 the dominant map from the variety of plane curves of degree d and genus g to mg yields a rational parametrization of the moduli space the two conditions involving d and can be satisfied only when g 10 so severi s argument cannot be extended for other genera however using much more subtle ideas chang ran and sernesi proved the unirationality of mg for g 11 12 13 see cr1 se1 while for g 15 16 they proved that the kodaira dimension is see cr2 4 the remaining cases g 14 and 17 g 23 are still quite mysterious harris and morrison conjectured in hmo that mg is uniruled precisely when g 23 all these facts indicate that m23 is a very interesting transition case our main result is the following

tour of bordered floer theory heegaard floer theory is a kind of topological quantum field theory tqft assigning graded groups to closed connected oriented 3 manifolds and group homomorphisms to smooth oriented four dimensional cobordisms bordered heegaard floer homology is an extension of heegaard floer homology to 3 manifolds with boundary with extended tqft type gluing properties in this survey we explain the formal structure and construction of bordered floer homology and sketch how it can be used to compute some aspects of heegaard floer theory

brill noether loci and the gonality stratification of m g for an irreducible smooth projective complex curve c of genus g the gonality defined as gon c min d z 1 there exists a gd on c is perhaps the second most natural invariant it gives an indication of how far c is from being rational in a way different from what the genus does for g 3 we consider the stratification of the moduli space mg of smooth curves of genus g given by gonality mg 2 m 1 g 3 m 1 g k mg

on slicing invariants of knots the slicing number of a knot u s k is the minimum number of crossing changes required to convert k to a slice knot this invariant is bounded above by the unknotting number and below by the slice genus g s k we show that for many knots previous bounds on the unknotting number obtained by ozsvath and szabo and by the author in fact give bounds on the slicing number livingston defined another invariant u s k which takes into account signs of crossings changed to get a slice knot and which is bounded above by the slicing number and below by the slice genus we exhibit an infinite family of knots k n with slice genus n and livingston invariant greater than n our bounds are based on restrictions using donaldson s diagonalisation theorem or heegaard floer homology on the intersection forms of four manifolds bounded by the double branched cover of a knot

rational homology spheres and the four ball genus of knots using the heegaard floer homology of ozsvath and szabo we investigate obstructions to a rational homology sphere bounding a four manifold with a definite intersection pairing as an application we obtain new lower bounds for the four ball genus of montesinos links

on the existence of certain families of curves introduction 25 w 1 generalities on families of curves in ip 28 w the map 0 d 33 w 3 the number of moduli of a family 40 w 4 plane curves existence of families with the expected number of moduli 43 w 5 smoothing certain reducible curves in ip r r 3 47 w 6 curves in ip r 3 existence of families with the expected number of moduli 52 w curves in ip3 a more general existence theorem 55

l2 regularity for parabolic partial integrodifferential equations with delay in the highest order derivatives abstract a partial integrodifferential equation is studied in which the derivatives of highest order also contain a discrete and a distributed delay by means of abstract regularity results global existence and uniqueness of a strict solution are obtained moreover a characterization of the infinitesimal generator of the solution operator is given

the gonality of general smooth curves with a prescribed plane nodal model let c be a smooth curve of genus g it is well known that c possesses a linear system g l if q l g 2n g 2 0 moreover if c is general and q g 2 k l and 6 kd k l 2 3 theorem 2 1 in loc cit and for each d this bound on 6 is sharp examples 4 1 and 4 2 in loc cit in this paper we consider the question for a general choice of f since the space parametrizing integral plane nodal curves with prescribed d and 6 is irreducible famous conjecture of severi proved by harris 9 the meaning of general becomes represented by a point of a particular nonempty zariski open subset of that parameter space note that qn x 3 g 0 if and only if 6 d 2 7d 18 2

linear systems on k3 sections on etudie la presence d un g d r special sur une section d hyperplan donnee c d une surface k 3 et les consequences sur l existence d un tel g d r sur toute courbe non singuliere c e c

controllability of semilinear control systems dominated by the linear part while various equivalent conditions for controllability have been obtained in the case of linear control systems controllability problems of semilinear control systems usually require some complicated and limited assumptions in this paper we show the approximate controllability of an abstract semilinear control system under the assumption which has a simple form and can be easily checked in many examples

absolutely graded floer homologies and intersection forms for four manifolds with boundary abstract in ozsvath and szabo holomorphic triangles and invariants for smooth four manifolds math sg 0110169 2001 we introduced absolute gradings on the three manifold invariants developed in ozsvath and szabo holomorphic disks and topological invariants for closed three manifolds math sg 0101206 ann of math 2001 to appear coupled with the surgery long exact sequences we obtain a number of three and four dimensional applications of this absolute grading including strengthenings of the complexity bounds derived in ozsvath and szabo holomorphic disks and three manifold invariants properties and applications math sg 0105202 ann of math 2001 to appear restrictions on knots whose surgeries give rise to lens spaces and calculations of hf for a variety of three manifolds moreover we show how the structure of hf constrains the exoticness of definite intersection forms for smooth four manifolds which bound a given three manifold in addition to these new applications the techniques also provide alternate proofs of donaldson s diagonalizability theorem and the thom conjecture for cp 2

donaldson s theorem heegaard floer homology and knots with unknotting number one abstract we establish an obstruction to unknotting an alternating knot by a single crossing change the obstruction is lattice theoretic in nature and combines donaldson s diagonalization theorem with an obstruction developed by ozsvath and szabo using heegaard floer homology as an application we enumerate the alternating 3 braid knots with unknotting number one and show that each has an unknotting crossing in its standard alternating diagram

alternating knots with unknotting number one abstract we show that an alternating knot with unknotting number one has an unknotting crossing in any alternating diagram we also prove that an alternating knot has unknotting number one if and only if its branched double cover arises as half integer surgery on a knot in s 3 thus establishing a converse to the montesinos trick along the way we reprove a characterisation of almost alternating diagrams of the unknot originally due to tsukamoto these results are established using the obstruction to unknotting number one developed by greene

approximate controllability for a class of semilinear abstract equations in this paper a class of control systems governed by the semilinear abstract equation dot y ay f y bv is considered a sufficient condition for the approximate controllability is obtained which improves j henry s results on a nonlinear parabolic control systems with some more serious restrictions it is suitable not only to the infinite dimensional case but also to the finite dimensional case two examples are given to explain the applications of the theory

approximate controllability of semilinear functional equations in hilbert spaces in this paper approximate and complete controllability for semilinear functional differential systems is studied in hilbert spaces sufficient conditions are established for each of these types of controllability the results address the limitation that linear systems in infinite dimensional spaces with compact semigroup cannot be completely controllable the conditions are obtained by using the schauder fixed point theorem when the semigroup is compact and the banach fixed point theorem when the semigroup is not compact

unknotting information from heegaard floer homology we use heegaard floer homology to obtain bounds on unknotting numbers this is a generalisation of ozsvath and szabo s obstruction to unknotting number one we determine the unknotting numbers of 910 913 935 938 1053 10101 and 10120 this completes the table of unknotting numbers for prime knots with crossing number nine or less our obstruction uses a refined version of montesinos theorem which gives a dehn surgery description of the branched double cover of a knot