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# LIN ALG & DIFF EQNS MATH 054

Kavon Feest

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Date Created: 10/22/15
RowReduction Boot Camp Constantin Teleman Math 541 Fall 09 1 Notation and refreshers Unless otherwise speci ed A is an m x n real matrix and vectors x are column vectors Row vectors are written as xT where x is the corresponding column vector The nullspace of A NullA C R is the space of solutions of the homogeneous system Ax 0 the column space ColA C R is the span of the columns There are two more subspaces associated to A this time consisting of row uectors the row space RowA C R the span of the rows in A and the left nullspace LNullA C R that we ll meet below A quick and dirty de nition is as the nullspace of the ipped or transposed matrix AT this is the matrix with switched indexing ATM A74 Caution These four subspaces are distinct in general with no obvious relation among them We will learn some subtle relations later The reduced row echelon form of A is the matrix rrefA produced from elementary row oper ations with the properties that o All zero rows are at the bottom 0 Every row which is not all zero starts with a 1 called the pivot or leading 1 0 Every pivot is strictly to the right of all pivots in the rows above it and o All entries above and below a pivot are zero The columns containing pivots are called piuot columns the others are the free columns The nullspace of A agrees with that of rrefA and can be parametrized as follows the free variables can be chosen freely and each equation in the reduced row system can then be used to solve for the corresponding pivot variable A similar story applies to the inhomogeneous system Ax b it has the same general solution as the reduced system rrefAx b where rrefAlb is the reduced form of the augmented matrix Alb The general solution can be parametrized by the same procedure We will learn a strong uniqueness property of rrefA it is completely determined by nullspace of A Similarly the reduced form of the augmented matrix Alb is determined uniquely from the affine space of solutions of Ax b A collection of vectors V1 V is linearly independent if the only expression of 0 as a linear combination of the Vl39 is the one with 0 weights that is kzv1kvo k1k2mk0 A collection w1 ws spans the linear subspace L C R if every wi lies in L and every vector in L can be expressed as a linear combination of the w A linearly independent ordered collection of vectors which spans L is called a basis Each vector in L can be uniquely expressed as a linear combination of the basis elements that is the weights are uniquely determined The main example is the standard basis 31 en of R the unit vectors on the coordinate axes 2 Meaning of the four subspaces 0 We ve discussed the nullspace of A the set of solutions of the homogeneous system c The column space is the space of vectors b for which the system AX b is solvable o A vector CT 01 02 on lies in the row space RowA if and only if the equation Clm1onn0 21 holds identically for all X E NullA Seeing this in one direction is easy a vector CT in the row space must be a linear combina tion of the rows of A therefore the equation 21 is a consequence of the equations in the homogeneous system and must hold for any solution To see the other direction start with an equation 21 and subtract suitable multiples of the equations in the row reduced system so as to cancel the coef cients of the pivot variables The equation we now get involves only the free variables But the free variables can be chosen freely so this cannot hold identically on NullA unless it is the trivial equation 0 0 So the original equation can only hold if it can be converted to 0 0 by subtracting rows of rrefA that is if CT 6 RowA A vector dT 11 d2 dm lies in the left nullspace if and only if the equation 11131 dmbm 0 2392 holds for all vectors b E ColA That is LNullA is the space of homogeneous linear equations which hold on all vectors b for which the system AX b is solvable As above one can show a converse every common solution b of all equations in the left nullspace does in fact belong to the column space So a basis of LNullA allows us to check solvability by testing the linear equations in the basis on b without performing a row reduction Of course nding a basis in the rst place does require a row reduction see below The row space and left nullspace are naturally spaces of homogeneous linear equations and they are best regarded as row vectors 3 Bases for the four subspaces 31 Column space A basis for ColA is given by the pivot columns of A They form a basis because any b E ColA leads to a solvable system AX b and then to one unique solution with zero values for the free variables but this gives a unique expression for b as a combination of the pivot columns Row space A basis for the row space of A is given by the nonzero rows in rrefA Linear independence applies because the pivots are in distinct position so in any linear combi nation each weight can be read off in the corresponding pivot entry These vectors span the row space because every row of A can be recovered as a linear combination of rows in rrefA this just requires tracing back through the row reduction algorithm Nullspace NullA has a basis of fundamental homogeneous solutions of Ax 0 Each fundamental solution vector f is associated to one free variable it is obtained by setting that free variable to 1 all other free variables to O and then solving for the pivot variables These vectors are independent in any linear combination the weight of f will be displayed in the ith entry of the sum They also span the nullspace because we can linearly combine them to get any values for the free variables More precisely every solution to Ax 0 has the form x E zifi summing only over free variables x Comment In working out examples you will notice that except for the obligatory zeroes and ones the fundamental solution vectors contains exactly the sign changed entries of rrefA but in somewhat different positions There is a mechanical procedure for writing down the fundamental solution from rrefA I do not recommend that you commit that to memory the important point is that the procedure is reversible and you can write down rrefA from the fundamental solutions 34 Left nullspace Let rrefAlM be the reduced echelon form of the matrix AlIm with an m x in identity block appended Then for each zero row in rrefA the corresponding row of M gives a basis vector for the left nullspace This magical method will be clearer when we discuss the LU factorization However try to do the forward pass of reduction for some small say 3 x 2 example You will discover that the entries appearing in M record precisely the multipliers which you use in the reduction process A zero row in rrefA stems from a vanishing linear combination of rows of A and the associated row of M records the weights of this combination This gives an equation which holds on all columns of A 35 Schubert basis The bases that we got for the row space and left nullspace have the special virtue that when writing the basis vectors as rows the resulting matrix is in reduced row echelon form Let us call such a basis of R a Schubert basis The fundamental basis of NullA also has a virtue namely written as rows the vectors produce a matrix in reduced lower echelon form Every row ends with a trailing 1 followed by a string of zeroes entries below and above trailing 1 s vanish and trailing 1 s move down and to the right Write some examples One can of course convert a reduced lower echelon matrix into the familiar upper echelon form by reversing the order of the rows and of the columns 36 Theorem Every linear subspace L C R has a Schubert basis and this basis is unique Same statement holds for a reverse Schubert basis This gives a new meaning to row reduction as is an algorithm for producing the Schubert basis of the row space RowA We also conclude that the basis fundamental solutions is unique Proof Here is how you construct a Schubert basis Note you will produce the basis elements in reverse order Choose a vector f1 6 L with the longest string of leading zeroes and scale it so that the rst non zero entry is 1 the pivot This fl is unique otherwise the difference between two of them would have a longer string of leading zeroes lf f1 does not span L we move on among the non zero vectors in L whose entries in the pivot position of f1 vanish choose one with the longest string of leading 0 s and scale it so that its leading entry is 1 This new vector f2 is unique for the same reason which applied to f1 We can continue with this method at each stage we take care to consider vectors which have zeroes in the pivot positions As the pivots move forward at each step the process terminates after at most n steps with a basis of L A reverse Schubert basis can be produced similarly with trailing zeroes and 1 s Caution The Schubert basis while unique depends critically on the ordering of the coordinates A reordering will produce a completely different basis of L 4 Checking equality and inclusion relations for subspaces 41 Equality How can we check whether two subspaces L and L of R are equal There are several methods depending how L and L have been described o If L L are given by homogeneous equations that is as solution sets of systems Ax 0 and A x 0 then L L if and only if rrefA rrefA this is because the fundamental solutions reverse Schubert bases and rrefs determine each other If L and L are spaces of row vectors and are described by two spanning collections you can View them as row spaces of two matrices B B Equality is equivalent to rrefB rrefB by uniqueness of the Schubert basis If L and L are spaces of column vectors and are described by two spanning collections this realizes them them as columns spaces of two matrices C C In that case L L if and only if the left nullspaces of C and Cquot agree which happens if and only if their Schubert bases constructed in 34 agree You can choose between the row column pictures which is best depends on which row reduction is most useful according to what else you need to nd out about L L or the respective matrices o If L is given by equations and L by a spanning set its easy to check one inclusion by verifying that the equations hold on the spanning vectors If you happen to know that the dimensions agree then you are done but if not more work is needed For example you can convert to an equation description of L by nding the left nullspace basis for the matrix B whose columns span L or you can compute the fundamental solution basis for L and compare to the reverse Schubert basis for L which you can compute by a row reduction 42 Inclusions The Schubert basis does not help for checking a containment relation such as L D L even when the latter holds the Schubert basis for L will usually not contain that for L Instead you can use several methods depending how L L have been described o If L L are given by homogeneous equations you can just check whether each of the funda mental solutions of L veri es the equations de ning L o If L L are the column spaces of two matrices C C then L D L if and only if the matrix O C has no pivots in the C column Explain this We already know the case when Cquot consists of a single column o If L is the column space of C and L the nullspace of A you must determine either the left nullspace of C or the fundamental basis of L and then apply the appropriate method above Homework problem For each of the following two matrices nd bases of the four subspaces List all containment or equalities that hold between all pairs of subspaces of the same kind Topological Field Theories in 2 dimensions Constantin Teleman UC Berkeley Amsterdam 14 July 2008 Constantin Teleman Topological Field Theories The notion of a Topological Field Theory TFT was formalised by Atiyah and Witten N 1990 and modelled on Graeme Segal39s notion of 2 dimesional Conformal Field Theory Constantin Teleman 2D Topological Field Theories The notion of a Topological Field Theory TFT was formalised by Atiyah and Witten N 1990 and modelled on Graeme Segal39s notion of 2 dimesional Conformal Field Theory This was to provide a framework for the new topological invariants of the 198039s 4D Donaldson theory 3D Chern Simons theory Constantin Teleman 2D Topological Field Theories The notion of a Topological Field Theory TFT was formalised by Atiyah and Witten N 1990 and modelled on Graeme Segal39s notion of 2 dimesional Conformal Field Theory This was to provide a framework for the new topological invariants of the 198039s 4D Donaldson theory 3D Chern Simons theory The distinguishing feature of the new invariants is multiplicativity under unions rather than the additivity common to algebraic topology eg characteristic numbers Additivity comes from the Mayer Vietoris sequence Constantin Teleman 2D Topological Field Theories The notion of a Topological Field Theory TFT was formalised by Atiyah and Witten N 1990 and modelled on Graeme Segal39s notion of 2 dimesional Conformal Field Theory This was to provide a framework for the new topological invariants of the 198039s 4D Donaldson theory 3D Chern Simons theory The distinguishing feature of the new invariants is multiplicativity under unions rather than the additivity common to algebraic topology eg characteristic numbers Additivity comes from the Mayer Vietoris sequence Quantum field theory explains this behaviour heuristically the invariants of a manifold X are integrals not over X but over a space of fields on X maps to another fixed space This space of fields is multiplicative in pieces of X Constantin Teleman 2D Topological Field Theories An n dimensional topological field theory is a strong symmetric monoidal functor from the category of n dimensional oriented bordisms to that of complex vector spaces The monoidal structures are disjoint union and tensor product respectively Constantin Teleman Topological Field Theories An n dimensional topological field theory is a strong symmetric monoidal functor from the category of n dimensional oriented bordisms to that of complex vector spaces The monoidal structures are disjoint union and tensor product respectively This means that to each closed oriented n 71 manifold we assign a vector space to disjoint unions we assign tensor products to a bordism we assign linear maps between the boundary spaces and the gluing of bordisms corresponds to the composition of linear maps Constantin Teleman Topological Field Theories An n dimensional topological field theory is a strong symmetric monoidal functor from the category of n dimensional oriented bordisms to that of complex vector spaces The monoidal structures are disjoint union and tensor product respectively This means that to each closed oriented n 71 manifold we assign a vector space to disjoint unions we assign tensor products to a bordism we assign linear maps between the boundary spaces and the gluing of bordisms corresponds to the composition of linear maps PICTURE GOES HERE SOME DAY Constantin Teleman Topological Field Theories An n dimensional topological field theory is a strong symmetric monoidal functor from the category of n dimensional oriented bordisms to that of complex vector spaces The monoidal structures are disjoint union and tensor product respectively This means that to each closed oriented n 71 manifold we assign a vector space to disjoint unions we assign tensor products to a bordism we assign linear maps between the boundary spaces and the gluing of bordisms corresponds to the composition of linear maps PICTURE GOES HERE SOME DAY There are variations of this definitions in the case of surfaces n 2 they are substantial Cohomological Field Theories Open Closed Theories Constantin Teleman Topological Field Theories Two dimensions The classification of compact connected oriented topological surfaces has long been known The only invariants are the number of components and the Euler characteristic TFT39s in dimension 2 were initially studied as a toy model not as a source of invariants Their structure was understood early on Constantin Teleman Topological Field Theories Two dimensions The classification of compact connected oriented topological surfaces has long been known The only invariants are the number of components and the Euler characteristic TFT39s in dimension 2 were initially studied as a toy model not as a source of invariants Their structure was understood early on Theorem folklore A 2 dimensional oriented TFT over C is equivalent to the datum of a commutative Frobenius algebra A over C Constantin Teleman Topological Field Theories Two dimensions The classification of compact connected oriented topological surfaces has long been known The only invariants are the number of components and the Euler characteristic TFT39s in dimension 2 were initially studied as a toy model not as a source of invariants Their structure was understood early on Theorem folklore A 2 dimensional oriented TFT over C is equivalent to the datum of a commutative Frobenius algebra A over C Recall that an associative algebra A is Frobenius if it comes equipped with a trace 9 A a C for which a7 b l gt 0a b gives a perfect symmetric paring In particular dimA lt oo Constantin Teleman Topological Field Theories Two dimensions The classification of compact connected oriented topological surfaces has long been known The only invariants are the number of components and the Euler characteristic TFT39s in dimension 2 were initially studied as a toy model not as a source of invariants Their structure was understood early on Theorem folklore A 2 dimensional oriented TFT over C is equivalent to the datum of a commutative Frobenius algebra A over C Recall that an associative algebra A is Frobenius if it comes equipped with a trace 9 A a C for which a7 b l gt 0a b gives a perfect symmetric paring In particular dimA lt 00 Yet divertingly enough it is in 2D that the notion of TFT and its variations has seen the most powerful applications Constantin Teleman Topological Field Theories mov Witten theory My application is to Gromov Witten theory which generalises a classical and delicate question in enumerative geometry counting algebraic curves in a projective manifold with prescribed degree and intersection conditions Constantin Teleman Topological Field Theories mov Witten theory My application is to Gromov Witten theory which generalises a classical and delicate question in enumerative geometry counting algebraic curves in a projective manifold with prescribed degree and intersection conditions For example there is a unique linear map P1 a l sending 01700 to three general position linear subspaces of total dimension n 71 Constantin Teleman Topological Field Theories Gromov Witten theory My application is to Gromov Witten theory which generalises a classical and delicate question in enumerative geometry counting algebraic curves in a projective manifold with prescribed degree and intersection conditions For example there is a unique linear map P1 a l sending 0100 to three general position linear subspaces of total dimension n 71 GW theory encodes this by deforming the cohomology algebra W09quot CMWM into the quantum cohomology algebra parametrised by q E C QH 19 CMW 1 7 qgt Constantin Teleman 2D Topological Field Theories Gromov Witten theory My application is to Gromov Witten theory which generalises a classical and delicate question in enumerative geometry counting algebraic curves in a projective manifold with prescribed degree and intersection conditions For example there is a unique linear map P1 a l sending 0100 to three general position linear subspaces of total dimension n 71 GW theory encodes this by deforming the cohomology algebra W09quot CMWM into the quantum cohomology algebra parametrised by q E C QWW CMW 1 7 qgt The coefficient 1 of q is the uniqueness and its exponent 1 is the degree of a straight line q E C exp H2 Constantin Teleman 2D Topological Field Theories Frobenius structure on quantum cohomology The quantum cohomology of a projective manifold X like the ordinary cohomology is a Frobenius algebra trace integration Thus we get a family of 2D TFT39s parametrised by H2XC Constantin Teleman 2D Topological Field Theories Frobenius structure on quantum cohomology The quantum cohomology of a projective manifold X like the ordinary cohomology is a Frobenius algebra trace integration Thus we get a family of 2D TFT39s parametrised by H2XC There is a general method to extend the space of parameters to the rest of H5VX C The properties of the resulting structure were abstracted into the notion of a Frobenius manifold Dubrovin Givental Manin This includes the grading broken by quantum multiplication it is restored by grading the parameter space using the Euler vector field one grades H2 using 6100 and the rest of cohomology by the normalised degree deg 2 71 Constantin Teleman 2D Topological Field Theories Frobenius structure on quantum cohomology The quantum cohomology of a projective manifold X like the ordinary cohomology is a Frobenius algebra trace integration Thus we get a family of 2D TFT39s parametrised by H2XC There is a general method to extend the space of parameters to the rest of H5VX C The properties of the resulting structure were abstracted into the notion of a Frobenius manifold Dubrovin Givental Manin This includes the grading broken by quantum multiplication it is restored by grading the parameter space using the Euler vector field one grades H2 using 6100 and the rest of cohomology by the normalised degree deg 2 71 The Frobenius manifold contain almost all answers to enumerative questions about rational curves in X The geometric explanation lies in a theorem that genus zero cohomological Field theories in two dimensions are equivalent to germs of Frobenius manifolds Constantin Teleman 2D Topological Field Theories Givental s Reconstruction Conjecture Gromov Witten theory extends quantum cohomology to curves of any genus A fundamental result Ruan Tian Li McDufF Salomon ensures that GW invariants are goverened by the structure of a all genus Cohomological Field theory l39ll focus on one important consequence of this structure Constantin Teleman Topological Field Theories Givental s Reconstruction Conjecture Gromov Witten theory extends quantum cohomology to curves of any genus A fundamental result Ruan Tian Li McDufF Salomon ensures that GW invariants are goverened by the structure of a all genus Cohomological Field theory l39ll focus on one important consequence of this structure Conjecture Givental 1999 For a compact symplectic manifold X Whose quantum cohomology ring is generically semi simple all Gromov Witten invariants are determined from genus zero information Constantin Teleman Topological Field Theories Givental s Reconstruction Conjecture Gromov Witten theory extends quantum cohomology to curves of any genus A fundamental result Ruan Tian Li McDufF Salomon ensures that GW invariants are goverened by the structure of a all genus Cohomological Field theory l39ll focus on one important consequence of this structure Conjecture Givental 1999 For a compact symplectic manifold X Whose quantum cohomology ring is generically semi simple all Gromov Witten invariants are determined from genus zero information Remark gt Loosely speaking counting rational curves determines the answer to enumerative questions for curves of all genera Constantin Teleman Topological Field Theories Givental s Reconstruction Conjecture Gromov Witten theory extends quantum cohomology to curves of any genus A fundamental result Ruan Tian Li McDuff Salomon ensures that GW invariants are goverened by the structure of a all genus Cohomological Field theory l39ll focus on one important consequence of this structure Conjecture Givental 1999 For a compact symplectic manifold X Whose quantum cohomology ring is generically semi simple all Gromov Witten invariants are determined from genus zero information Remark gt Loosely speaking counting rational curves determines the answer to enumerative questions for curves of all genera gt Givental gave a formula for the generating function of GW invariants in terms of quantised quadratic Hamiltonians Constantin Teleman Topological Field Theories Classification of semi simple theories Theorem Givental39s conjecture holds More precisely the GW descendent invariants are determined by a recursive relation from the quantum multiplication at a single but generic value of the parameter Constantin Teleman Topological Field Theories Classification of semi simple theories Theorem Givental39s conjecture holds More precisely the GW descendent invariants are determined by a recursive relation from the quantum multiplication at a single but generic value of the parameter This theorem follows from a structural classification Theorem A CthT based on a semi simple Frobenius algebra A is determined by a power series Rz E EndAz R Id mod 2 subject to Givental39s symplectic constraint RzR7z E Id Constantin Teleman 2D Topological Field Theories Classification of semi simple theories Theorem Givental39s conjecture holds More precisely the GW descendent invariants are determined by a recursive relation from the quantum multiplication at a single but generic value of the parameter This theorem follows from a structural classification Theorem A CthT based on a semi simple Frobenius algebra A is determined by a power series Rz E EndAz R Id mod 2 subject to Givental39s symplectic constraint RzR7z E Id Remark gt Givental describes R from the Frobenius manifold But one sufficiently generic quantum multiplication suffices Constantin Teleman 2D Topological Field Theories Classification of semi simple theories Theorem Givental39s conjecture holds More precisely the GW descendent invariants are determined by a recursive relation from the quantum multiplication at a single but generic value of the parameter This theorem follows from a structural classification Theorem A CthT based on a semi simple Frobenius algebra A is determined by a power series Rz E EndAz R Id mod 2 subject to Givental39s symplectic constraint RzR7z E Id Remark gt Givental describes R from the Frobenius manifold But one sufficiently generic quantum multiplication suffices gt The essential input in the classification theorem is the Mumford conjecture Madsen Weiss 2002 Constantin Teleman 2D Topological Field Theories When does the theorem apply Semi simplicity of quantum cohomology is a very strong condition Constantin Teleman Topological Field Theories When does the theorem apply Semi simplicity of quantum cohomology is a very strong condition gt Projective spaces Grassmannians and many toric Fanos work Constantin Teleman Topological Field Theories When does the theorem apply Semi simplicity of quantum cohomology is a very strong condition gt Projective spaces Grassmannians and many toric Fanos work gt Toric manifolds always have semi simple deformations to their torus equivariant cohomology This was used by Givental in computing their GW theory verifying his conjecture there Constantin Teleman Topological Field Theories When does the theorem apply Semi simplicity of quantum cohomology is a very strong condition gt Projective spaces Grassmannians and many toric Fanos work gt Toric manifolds always have semi simple deformations to their torus equivariant cohomology This was used by Givental in computing their GW theory verifying his conjecture there gt Semi simplicity is preserved by blowing up points Bayer 2004 in particular there exist non Fano examples Constantin Teleman Topological Field Theories When does the theorem apply Semi simplicity of quantum cohomology is a very strong condition gt Projective spaces Grassmannians and many toric Fanos work gt Toric manifolds always have semi simple deformations to their torus equivariant cohomology This was used by Givental in computing their GW theory verifying his conjecture there V Semi simplicity is preserved by blowing up points Bayer 2004 in particular there exist non Fano examples gt 36 of the 59 families of 3D Fanos with no odd cohomology have been checked Ancona Maggesi 2002 Ciolli 2004 Constantin Teleman 2D Topological Field Theories When does the theorem apply Semi simplicity of quantum cohomology is a very strong condition gt Projective spaces Grassmannians and many toric Fanos work gt Toric manifolds always have semi simple deformations to their torus equivariant cohomology This was used by Givental in computing their GW theory verifying his conjecture there V Semi simplicity is preserved by blowing up points Bayer 2004 in particular there exist non Fano examples gt 36 of the 59 families of 3D Fanos with no odd cohomology have been checked Ancona Maggesi 2002 Ciolli 2004 gt On the negative side if the even part of quantum cohomology is semi simple then the manifold has even cohomology only and in the algebraic case p7 p cohomology only Bayer and Manin 2004 Manin Hertling T 2008 This contradicts claims in the literature about complete intersection Fanos Constantin Teleman 2D Topological Field Theories Dubrovin s conjecture A remarkable conjecture has gathered experimental support Recall that an ordered collection of objects in a triangulated C linear category is exceptional if Extigii C in degree 0 while if j gt i Extkc jc 07Vk The collection is complete if it generates the triangulated category Constantin Teleman 2D Topological Field Theories Dubrovin s conjecture A remarkable conjecture has gathered experimental support Recall that an ordered collection of objects in a triangulated C linear category is exceptional if Extigii C in degree 0 while if j gt i Extkc jc 07Vk The collection is complete if it generates the triangulated category Conjecture Dubrovin A projective manifold has semi simple quantum cohomology iff its derived category of coherent sheaves contains a complete exceptional collection Constantin Teleman 2D Topological Field Theories Dubrovin s conjecture A remarkable conjecture has gathered experimental support Recall that an ordered collection of objects in a triangulated C linear category is exceptional if E3tc 7 C in degree 0 while if j gt i Extkc jc 07Vk The collection is complete if it generates the triangulated category Conjecture Dubrovin A projective manifold has semi simple quantum cohomology iff its derived category of coherent sheaves contains a complete exceptional collection Remark gt Ciolli 2004 checks this for 36 families of 3D Fanos Constantin Teleman 2D Topological Field Theories Dubrovin s conjecture A remarkable conjecture has gathered experimental support Recall that an ordered collection of objects in a triangulated C linear category is exceptional if E3tc 7 C in degree 0 while if j gt i Extkc jc 07Vk The collection is complete if it generates the triangulated category Conjecture Dubrovin A projective manifold has semi simple quantum cohomology iff its derived category of coherent sheaves contains a complete exceptional collection Remark gt Ciolli 2004 checks this for 36 families of 3D Fanos gt Dubrovin also relates the Ext Euler characteristics to quantum cohomology data This would be a consequence of some formulations of Mirror symmetry Constantin Teleman 2D Topological Field Theories Related results of K ntsevich Constantin Teleman Topological Field Theories Related results of K ntsevich In the mid 199039s Kontsevich initiated a programme Homological Mirror Symmetry which among others should give a far reaching adaptation of Givental39s reconstruction conjecture This preceded Givental39s cited work but only converged with it later For a recent update see Katzarkov Kontsevich Pantev 2008 Constantin Teleman Topological Field Theories Related results of Kontsevich In the mid 199039s Kontsevich initiated a programme Homological Mirror Symmetry which among others should give a far reaching adaptation of Givental39s reconstruction conjecture This preceded Givental39s cited work but only converged with it later For a recent update see Katzarkov Kontsevich Pantev 2008 A key step is to replace the notion of cohomological field theory with that of chain level open closed field theory See also Costello for an implementation of these ideas Constantin Teleman 2D Topological Field Theories Related results of Kontsevich In the mid 199039s Kontsevich initiated a programme Homological Mirror Symmetry which among others should give a far reaching adaptation of Givental39s reconstruction conjecture This preceded Givental39s cited work but only converged with it later For a recent update see Katzarkov Kontsevich Pantev 2008 A key step is to replace the notion of cohomological field theory with that of chain level open closed field theory See also Costello for an implementation of these ideas This is required by the fact that cohomological field theories seem unclassifiable with our limited understanding of Deligne Mumford spaces The semi simple classification was a pleasant surprise Constantin Teleman 2D Topological Field Theories Related results of Kontsevich In the mid 199039s Kontsevich initiated a programme Homological Mirror Symmetry which among others should give a far reaching adaptation of Givental39s reconstruction conjecture This preceded Givental39s cited work but only converged with it later For a recent update see Katzarkov Kontsevich Pantev 2008 A key step is to replace the notion of cohomological field theory with that of chain level open closed field theory See also Costello for an implementation of these ideas This is required by the fact that cohomological field theories seem unclassifiable with our limited understanding of Deligne Mumford spaces The semi simple classification was a pleasant surprise However applying this programme to Gromov Witten theory requires the construction a good Fukaya category for a symplectic manifold This is not yet within reach Constantin Teleman 2D Topological Field Theories mological Field Theory CthT Constantin Teleman Topological Field Theories Cohomological Field Theory CthT gt Defined by Kontsevich and Manin for application to GW theory but related notions Segal39s Topological Conformal Field theory had a parallel development Constantin Teleman 2D Topological Field Theories Cohomological Field Theory CthT gt Defined by Kontsevich and Manin for application to GW theory but related notions Segal39s Topological Conformal Field theory had a parallel development gt Is a version of TFT for families ofsurfaces taking values in the cohomology of the parameter space instead of numbers Constantin Teleman 2D Topological Field Theories Cohomological Field Theory CthT gt Defined by Kontsevich and Manin for application to GW theory but related notions Segal39s Topological Conformal Field theory had a parallel development gt Is a version of TFT for families ofsurfaces taking values in the cohomology of the parameter space instead of numbers gt A family of closed surfaces over B gives a class in HBC Constantin Teleman 2D Topological Field Theories Cohomological Field Theory CthT gt Defined by Kontsevich and Manin for application to GW theory but related notions Segal39s Topological Conformal Field theory had a parallel development gt Is a version of TFT for families ofsurfaces taking values in the cohomology of the parameter space instead of numbers gt A family of closed surfaces over B gives a class in HBC gt A family of surfaces with m input and n output boundaries gives a class in HBHomA m A Constantin Teleman 2D Topological Field Theories Cohomological Field Theory CthT gt Defined by Kontsevich and Manin for application to GW theory but related notions Segal39s Topological Conformal Field theory had a parallel development gt Is a version of TFT for families ofsurfaces taking values in the cohomology of the parameter space instead of numbers gt A family of closed surfaces over B gives a class in HBC gt A family of surfaces with m input and n output boundaries gives a class in HBHomA m A gt Gluing composition applies in families Constantin Teleman 2D Topological Field Theories Cohomological Field Theory CthT gt Defined by Kontsevich and Manin for application to GW theory but related notions Segal39s Topological Conformal Field theory had a parallel development gt Is a version of TFT for families ofsurfaces taking values in the cohomology of the parameter space instead of numbers gt A family of closed surfaces over B gives a class in HBC gt A family of surfaces with m input and n output boundaries gives a class in HBHomA m A gt Gluing composition applies in families gt Nodal degenerations Lefschetz fibrations are allowed Constantin Teleman 2D Topological Field Theories Cohomological Field Theory CthT gt Defined by Kontsevich and Manin for application to GW theory but related notions Segal39s Topological Conformal Field theory had a parallel development gt Is a version of TFT for families ofsurfaces taking values in the cohomology of the parameter space instead of numbers V A family of closed surfaces over B gives a class in HBC V A family of surfaces with m input and n output boundaries gives a class in HBHomA m A Gluing composition applies in families V V Nodal degenerations Lefschetz fibrations are allowed All this is functorial in the base B V Constantin Teleman 2D Topological Field Theories Cohomological Field Theory CthT Defined by Kontsevich and Manin for application to GW theory but related notions Segal39s Topological Conformal Field theory had a parallel development V V Is a version of TFT for families ofsurfaces taking values in the cohomology of the parameter space instead of numbers V A family of closed surfaces over B gives a class in HBC V A family of surfaces with m input and n output boundaries gives a class in HBHomA m A Gluing composition applies in families Nodal degenerations Lefschetz fibrations are allowed All this is functorial in the base B VVVV t suffices to specify the classes for the universal Lefschetz fibrations over the Deligne Mumford spaces of stable curves Constantin Teleman 2D Topological Field Theories Cohomological Field Theory CthT gt Defined by Kontsevich and Manin for application to GW theory but related notions Segal39s Topological Conformal Field theory had a parallel development gt Is a version of TFT for families ofsurfaces taking values in the cohomology of the parameter space instead of numbers V A family of closed surfaces over B gives a class in HBC V A family of surfaces with m input and n output boundaries gives a class in HBHomA m A Gluing composition applies in families Nodal degenerations Lefschetz fibrations are allowed All this is functorial in the base B VVVV t suffices to specify the classes for the universal Lefschetz fibrations over the Deligne Mumford spaces of stable curves V Have skipped some details stability flat identity Constantin Teleman 2D Topological Field Theories Details I What makes the Classification work Constantin Telelnan Topological Field Theories Details I What makes the Classification work The Euler class of a Frobenius algebra A is the product of the co product of 1 1gt gt A X A gt gt A Pictorially this is represented by a torus with one outgoing boundary Constantin Teleman 2D Topological Field Theories Details I What makes the Classification work The Euler class of a Frobenius algebra A is the product of the co product of 1 1gt gt A X A gt gt A Pictorially this is represented by a torus with one outgoing boundary For the cohomology ring of a manifold this is the usual Euler class However the quantum Euler class can be inveritble this happens ifF the quantum multiplication is semi simple Constantin Teleman 2D Topological Field Theories Details I What makes the classification work The Euler class of a Frobenius algebra A is the product of the co product of 1 1gt gt A X A gt gt A Pictorially this is represented by a torus with one outgoing boundary For the cohomology ring of a manifold this is the usual Euler class However the quantum Euler class can be inveritble this happens ifF the quantum multiplication is semi simple Hence in the semi simple case one can increase the genus of surfaces without loss of information in the CthT Constantin Teleman Topological Field Theories Details I What makes the classification work The Euler class of a Frobenius algebra A is the product of the co product of 1 ll gt A X A l gt A Pictorially this is represented by a torus with one outgoing boundary For the cohomology ring of a manifold this is the usual Euler class However the quantum Euler class can be inveritble this happens iff the quantum multiplication is semi simple Hence in the semi simple case one can increase the genus of surfaces without loss of information in the CthT The Mumford conjecture Madsen Weiss describes the complex cohomology of the open moduli space M of smooth curves in the g gt oo limit as a free C algebra in the tautological classes mi12 From here we can classify the Mg part of semi simple CthT39s Constantin Teleman 2D Topological Field Theories Details I What makes the classification work The Euler class of a Frobenius algebra A is the product of the co product of 1 ll gt A X A l gt A Pictorially this is represented by a torus with one outgoing boundary For the cohomology ring of a manifold this is the usual Euler class However the quantum Euler class can be inveritble this happens iff the quantum multiplication is semi simple Hence in the semi simple case one can increase the genus of surfaces without loss of information in the CthT The Mumford conjecture Madsen Weiss describes the complex cohomology of the open moduli space M of smooth curves in the g gt oo limit as a free C algebra in the tautological classes mi12 From here we can classify the Mg part of semi simple CthT39s Finally in large g the boundary divisors of V have Euler classes which are not zero divisors This controls the problem of extending cohomology classes to the boundary Constantin Teleman 2D Topological Field Theories Details ll Moral Meaning of Rz The Frobenius algebra A is associated to the circle in a 2D field theory Heuristically it should be viewed as the cohomology of a space Y with circle action In all known applications A is the Hochschild cohomology of a category so its chain level model has an algebraic circle action Constantin Teleman Topological Field Theories Details ll Moral Meaning of Rz The Frobenius algebra A is associated to the circle in a 2D field theory Heuristically it should be viewed as the cohomology of a space Y with circle action In all known applications A is the Hochschild cohomology of a category so its chain level model has an algebraic circle action The series R should give a splitting of Sl equivariant cohomology 210 HWY Clzll where z is the generator of H1point Constantin Teleman 2D Topological Field Theories Details ll Moral Meaning of Rz The Frobenius algebra A is associated to the circle in a 2D field theory Heuristically it should be viewed as the cohomology of a space Y with circle action In all known applications A is the Hochschild cohomology of a category so its chain level model has an algebraic circle action The series R should give a splitting of Sl equivariant cohomology 310 g HWY Clzllv where z is the generator of H1point The existence of such a splitting is necessary for the extension of a cohomological field theory from the open moduli space Mg over its Deligne Mumford boundary Fairly easy The core of the classification is that a choice of splitting determines this extension Constantin Teleman 2D Topological Field Theories Details lll Construction of the theory Constantin Teleman Topological Field Theories Details lll Construction of the theory A cohomological field theory based on A assigns to each allowed pair mg a class in WW A These classes make A into an algebra over the modular operad WW C Restriction to boundary divisors is subject to gluing rules Constantin Teleman 2D Topological Field Theories Details lll Construction of the theory A cohomological field theory based on A assigns to each allowed pair mg a class in WW A These classes make A into an algebra over the modular operad WW C Restriction to boundary divisors is subject to gluing rules Theories can be constructed using the Morita Mumford Miller tautological classes Start with exp ajnj 6 WW A co multiplied out to A The aj E A are determined from R Twist each output by R1J with the w class at the respective marked point Finally add recursively for all boundary strata terms of the form Id 7 R Rw l 11 contracted with the classes already constructed on the boundary stratum and pushed forward by the Thom class Constantin Teleman 2D Topological Field Theories Details lll Construction of the theory A cohomological field theory based on A assigns to each allowed pair mg a class in WW A These classes make A into an algebra over the modular operad WW C Restriction to boundary divisors is subject to gluing rules Theories can be constructed using the Morita Mumford Miller tautological classes Start with exp ajnj 6 WW A co multiplied out to A The aj E A are determined from R Twist each output by R1J with the w class at the respective marked point Finally add recursively for all boundary strata terms of the form Id 7 R Rw l 11 contracted with the classes already constructed on the boundary stratum and pushed forward by the Thom class This construction can be captured by a certain action of matrices Rz on the cohomology of Deligne Mumford spaces Constantin Teleman 2D Topological Field Theories Open Questions gt Degeneration Semi simple theories come in families with non semi simple degenerations classical cohomology for GW theory the Jacobian ring for the Landau Ginzburg B model potential with an isolated critical point The Givental data for semi simple theories degenerates at such a classical point Nonetheless some theories are continuous Constantin Teleman Topological Field Theories Open Questions gt Degeneration Semi simple theories come in families with non semi simple degenerations classical cohomology for GW theory the Jacobian ring for the Landau Ginzburg B model potential with an isolated critical point The Givental data for semi simple theories degenerates at such a classical point Nonetheless some theories are continuous Problem Understand this phenomenon Constantin Teleman Topological Field Theories Open Questions gt Degeneration Semi simple theories come in families with non semi simple degenerations classical cohomology for GW theory the Jacobian ring for the Landau Ginzburg B model potential with an isolated critical point The Givental data for semi simple theories degenerates at such a classical point Nonetheless some theories are continuous Problem Understand this phenomenon gt Formalty GW theory can be defined at chain level A cohomological classification leaves open the possibility of higher operations Massey products Constantin Teleman Topological Field Theories Open Questions gt Degeneration Semi simple theories come in families with non semi simple degenerations classical cohomology for GW theory the Jacobian ring for the Landau Ginzburg B model potential with an isolated critical point The Givental data for semi simple theories degenerates at such a classical point Nonetheless some theories are continuous Problem Understand this phenomenon gt Formalty GW theory can be defined at chain level A cohomological classification leaves open the possibility of higher operations Massey products Question Can this happen in semi simple field theories Constantin Teleman Topological Field Theories Open Questions gt Degeneration Semi simple theories come in families with non semi simple degenerations classical cohomology for GW theory the Jacobian ring for the Landau Ginzburg B model potential with an isolated critical point The Givental data for semi simple theories degenerates at such a classical point Nonetheless some theories are continuous Problem Understand this phenomenon V Formalty GW theory can be defined at chain level A cohomological classification leaves open the possibility of higher operations Massey products Question Can this happen in semi simple field theories The expected answer is no because of formality of DeligneMumford spaces Constantin Teleman Topological Field Theories Open Questions gt Twisted Frobenius manifolds Gromov Witten K theory and the twisted Gromov Witten invariants Coates Givental as well as other example do not fit the standard definition of Frobenius manifolds Variations of the notion have been studied by Dubrovin and Manin Constantin Teleman Topological Field Theories Open Questions gt Twisted Frobenius manifolds Gromov Witten K theory and the twisted Gromov Witten invariants Coates Givental as well as other example do not fit the standard definition of Frobenius manifolds Variations of the notion have been studied by Dubrovin and Manin Problem Describe Givental39s higher genus reconstruction in this more general setting Constantin Teleman Topological Field Theories Open Questions gt Twisted Frobenius manifolds Gromov Witten K theory and the twisted Gromov Witten invariants Coates Givental as well as other example do not fit the standard definition of Frobenius manifolds Variations of the notion have been studied by Dubrovin and Manin Problem Describe Givental39s higher genus reconstruction in this more general setting gt Thank you for your patience Constantin Teleman Topological Field Theories

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