ALGEBRAIC MODELS, DUALITY, AND RESONANCE. Alex Suciu. Topology Seminar. MIT March 5, Northeastern University

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1 ALGEBRAIC MODELS, DUALITY, AND RESONANCE Alex Suciu Northeastern University Topology Seminar MIT March 5, 2018 ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 1 / 24

2 DUALITY PROPERTIES POINCARÉ DUALITY ALGEBRAS POINCARÉ DUALITY ALGEBRAS Let A be a graded, graded-commutative algebra over a field k. A = À iě0 Ai, where A i are k-vector spaces. : A i b A j Ñ A i+j. ab = ( 1) ij ba for all a P A i, b P B j. We will assume that A is connected (A 0 = k 1), and locally finite (all the Betti numbers b i (A) := dim k A i are finite). A is a Poincaré duality k-algebra of dimension n if there is a k-linear map ε : A n Ñ k (called an orientation) such that all the bilinear forms A i b k A n i Ñ k, a b b ÞÑ ε(ab) are non-singular. Consequently, b i (A) = b n i (A), and A i = 0 for i ą n. ε is an isomorphism. The maps PD: A i Ñ (A n i ), PD(a)(b) = ε(ab) are isomorphisms. Each a P A i has a Poincaré dual, a _ P A n i, such that ε(aa _ ) = 1. The orientation class is defined as ω A = 1 _, so that ε(ω A ) = 1. ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 2 / 24

3 DUALITY PROPERTIES POINCARÉ DUALITY ALGEBRAS THE ASSOCIATED ALTERNATING FORM Associated to a k-pd n algebra there is an alternating n-form, µ A : Źn A 1 Ñ k, µ A (a 1 ^ ^ a n ) = ε(a 1 a n ). Assume now that n = 3, and set r = b 1 (A). Fix a basis te 1,..., e r u for A 1, and let te _ 1,..., e_ r u be the dual basis for A 2. The multiplication in A, then, is given on basis elements by rÿ e i e j = µ ijk e k _, e ie j _ = δ ij ω, k=1 where µ ijk = µ(e i ^ e j ^ e k ). Alternatively, let A i = (A i ), and let e i P A 1 be the (Kronecker) dual of e i. We may then view µ dually as a trivector, µ = ÿ µ ijk e i ^ e j ^ e k P Ź3 A 1, which encodes the algebra structure of A. ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 3 / 24

4 DUALITY PROPERTIES POINCARÉ DUALITY ALGEBRAS POINCARÉ DUALITY IN ORIENTABLE MANIFOLDS If M is a compact, connected, orientable, n-dimensional manifold, then the cohomology ring A = H. (M, k) is a PD n algebra over k. Sullivan (1975): for every finite-dimensional Q-vector space V and every alternating 3-form µ P Ź3 V, there is a closed 3-manifold M with H 1 (M, Q) = V and cup-product form µ M = µ. Such a 3-manifold can be constructed via Borromean surgery." If M bounds an oriented 4-manifold W such that the cup-product pairing on H 2 (W, M) is non-degenerate (e.g., if M is the link of an isolated surface singularity), then µ M = 0. ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 4 / 24

5 DUALITY PROPERTIES DUALITY SPACES DUALITY SPACES A more general notion of duality is due to Bieri and Eckmann (1978). Let X be a connected, finite-type CW-complex, and set π = π 1 (X, x 0 ). X is a duality space of dimension n if H i (X, Zπ) = 0 for i n and H n (X, Zπ) 0 and torsion-free. Let D = H n (X, Zπ) be the dualizing Zπ-module. Given any Zπ-module A, we have H i (X, A) H n i (X, D b A). If D = Z, with trivial Zπ-action, then X is a Poincaré duality space. If X = K (π, 1) is a duality space, then π is a duality group. ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 5 / 24

6 DUALITY PROPERTIES ABELIAN DUALITY SPACES ABELIAN DUALITY SPACES We introduce in [Denham S. Yuzvinsky 2016/17] an analogous notion, by replacing π π ab. X is an abelian duality space of dimension n if H i (X, Zπ ab ) = 0 for i n and H n (X, Zπ ab ) 0 and torsion-free. Let B = H n (X, Zπ ab ) be the dualizing Zπ ab -module. Given any Zπ ab -module A, we have H i (X, A) H n i (X, B b A). The two notions of duality are independent: EXAMPLE Surface groups of genus at least 2 are not abelian duality groups, though they are (Poincaré) duality groups. Let π = Z 2 G, where G = xx 1,..., x 4 x 2 1 x 2x 1 x 1 2,..., x 2 4 x 1x 4 x 1 1 y is Higman s acyclic group. Then π is an abelian duality group (of dimension 2), but not a duality group. ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 6 / 24

7 DUALITY PROPERTIES ARRANGEMENTS OF SMOOTH HYPERSURFACES THEOREM (DSY) Let X be an abelian duality space of dimension n. Then: b 1 (X ) ě n 1. b i (X ) 0, for 0 ď i ď n and b i (X ) = 0 for i ą n. ( 1) n χ(x ) ě 0. THEOREM (DENHAM S. 2017) Let U be a connected, smooth, complex quasi-projective variety of dimension n. Suppose U has a smooth compactification Y for which 1 Components of Y zu form an arrangement of hypersurfaces A; 2 For each submanifold X in the intersection poset L(A), the complement of the restriction of A to X is a Stein manifold. Then U is both a duality space and an abelian duality space of dimension n. ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 7 / 24

8 DUALITY PROPERTIES ARRANGEMENTS OF SMOOTH HYPERSURFACES LINEAR, ELLIPTIC, AND TORIC ARRANGEMENTS THEOREM (DS17) Suppose that A is one of the following: 1 An affine-linear arrangement in C n, or a hyperplane arrangement in CP n ; 2 A non-empty elliptic arrangement in E n ; 3 A toric arrangement in (C ) n. Then the complement M(A) is both a duality space and an abelian duality space of dimension n r, n + r, and n, respectively, where r is the corank of the arrangement. This theorem extends several previous results: 1 Davis, Januszkiewicz, Leary, and Okun (2011); 2 Levin and Varchenko (2012); 3 Davis and Settepanella (2013), Esterov and Takeuchi (2014). ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 8 / 24

9 ALGEBRAIC MODELS AND RESONANCE VARIETIES COMMUTATIVE DIFFERENTIAL GRADED ALGEBRAS COMMUTATIVE DIFFERENTIAL GRADED ALGEBRAS Let A = (A, d) be a commutative, differential graded algebra over a field k of characteristic 0. That is: A = À iě0 Ai, where A i are k-vector spaces. The multiplication : A i b A j Ñ A i+j is graded-commutative, i.e., ab = ( 1) a b ba for all homogeneous a and b. The differential d: A i Ñ A i+1 satisfies the graded Leibnitz rule, i.e., d(ab) = d(a)b + ( 1) a a d(b). A CDGA A is of finite-type (or q-finite) if it is connected (i.e., A 0 = k 1) and dim A i ă 8 for all i ď q. H (A) inherits an algebra structure from A. A cdga morphism ϕ : A Ñ B is both an algebra map and a cochain map. Hence, it induces a morphism ϕ : H (A) Ñ H (B). ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 9 / 24

10 ALGEBRAIC MODELS AND RESONANCE VARIETIES COMMUTATIVE DIFFERENTIAL GRADED ALGEBRAS A map ϕ : A Ñ B is a quasi-isomorphism if ϕ is an isomorphism. Likewise, ϕ is a q-quasi-isomorphism (for some q ě 1) if ϕ is an isomorphism in degrees ď q and is injective in degree q + 1. Two cdgas, A and B, are (q-)equivalent (» q ) if there is a zig-zag of (q-)quasi-isomorphisms connecting A to B. A cdga A is formal (or just q-formal) if it is (q-)equivalent to (H (A), d = 0). A CDGA is q-minimal if it is of the form ( Ź V, d), where the differential structure is the inductive limit of a sequence of Hirsch extensions of increasing degrees, and V i = 0 for i ą q. Every CDGA A with H 0 (A) = k admits a q-minimal model, M q (A) (i.e., a q-equivalence M q (A) Ñ A with M q (A) = ( Ź V, d) a q-minimal cdga), unique up to iso. ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 10 / 24

11 ALGEBRAIC MODELS AND RESONANCE VARIETIES ALGEBRAIC MODELS FOR SPACES ALGEBRAIC MODELS FOR SPACES Given any (path-connected) space X, there is an associated Sullivan Q-cdga, A PL (X ), such that H (A PL (X )) = H (X, Q). An algebraic (q-)model (over k) for X is a k-cgda (A, d) which is (q-) equivalent to A PL (X ) b Q k. If M is a smooth manifold, then Ω dr (M) is a model for M (over R). Examples of spaces having finite-type models include: Formal spaces (such as compact Kähler manifolds, hyperplane arrangement complements, toric spaces, etc). Smooth quasi-projective varieties, compact solvmanifolds, Sasakian manifolds, etc. ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 11 / 24

12 ALGEBRAIC MODELS AND RESONANCE VARIETIES RESONANCE VARIETIES OF A CDGA RESONANCE VARIETIES OF A CDGA Let A = (A, d) be a connected, finite-type CDGA over k = C. For each a P Z 1 (A) H 1 (A), we have a cochain complex, (A, δ a ) : A 0 δ 0 a A 1 δ 1 a A 2 δ 2 a, with differentials δ i a(u) = a u + d u, for all u P A i. The resonance varieties of A are the affine varieties R i s(a) = ta P H 1 (A) dim H i (A, δ a ) ě su. If A is a CGA (that is, d = 0), the resonance varieties R i s(a) are homogeneous subvarieties of A 1. If X is a connected, finite-type CW-complex, we set R i s(x ) := R i s(h (X, C)). ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 12 / 24

13 ALGEBRAIC MODELS AND RESONANCE VARIETIES RESONANCE VARIETIES OF A CDGA Fix a k-basis te 1,..., e r u for A 1, and let tx 1,..., x r u be the dual basis for A 1 = (A 1 ). Identify Sym(A 1 ) with S = k[x 1,..., x r ], the coordinate ring of the affine space A 1. Define a cochain complex of free S-modules, L(A) := (A b S, δ), where A i b S δi A i+1 b S δi+1 A i+2 b S, δ i (u b f ) = ř n j=1 e ju b f x j + d u b f. The specialization of (A b S, δ) at a P A 1 coincides with (A, δ a ). Hence, R i s(a) is the zero-set of the ideal generated by all minors of size b i s + 1 of the block-matrix δ i+1 δ i. In particular, R 1 s(a) = V (I r s (δ 1 )), the zero-set of the ideal of codimension s minors of δ 1. ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 13 / 24

14 ALGEBRAIC MODELS AND RESONANCE VARIETIES RESONANCE VARIETIES OF A CDGA RESONANCE VARIETIES OF PD-ALGEBRAS Let A be a PD n algebra. For all 0 ď i ď n and all a P A 1, the square (A n i ) (δn i 1 a ) PD A i δ i a commutes up to a sign of ( 1) i. Consequently, Hence, for all i and s, In particular, R n 1 (A) = t0u. (A n i 1 ) PD A i+1 ( H i (A, δ a )) H n i (A, δ a ). R i s(a) = R n i s (A). ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 14 / 24

15 ALGEBRAIC MODELS AND RESONANCE VARIETIES 3-DIMENSIONAL POINCARÉ DUALITY ALGEBRAS 3-DIMENSIONAL POINCARÉ DUALITY ALGEBRAS Let A be a PD 3 -algebra with b 1 (A) = r ą 0. Then R 3 1 (A) = R0 1 (A) = t0u. R 2 s(a) = R 1 s(a) for 1 ď s ď r. R i s(a) = H, otherwise. Write R s (A) = R 1 s(a). Then R 2k (A) = R 2k+1 (A) if r is even. R 2k 1 (A) = R 2k (A) if r is odd. If µ A has rank r ě 3, then R r 2 (A) = R r 1 (A) = R r (A) = t0u. If r ě 4, then dim R 1 (A) ě null(µ A ) ě 2. Ź Here, the rank of a form µ : 3 V Ñ k is the minimum dimension of a linear subspace W Ă V such that µ factors through Ź3 W. The nullity of µ is the maximum dimension of a subspace U Ă V such that µ(a ^ b ^ c) = 0 for all a, b P U and c P V. ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 15 / 24

16 ALGEBRAIC MODELS AND RESONANCE VARIETIES 3-DIMENSIONAL POINCARÉ DUALITY ALGEBRAS If r is even, then R 1 (A) = R 0 (A) = A 1. If r is odd ą 1, then R 1 (A) A 1 if and only if µ A is generic," that is, there is a c P A 1 such that the 2-form γ c P Ź2 A 1 given by γ c (a ^ b) = µ A (a ^ b ^ c) has rank 2g, i.e., γ g c 0 in Ź 2g A 1. In that case, R 1 (A) is the hypersurface Pf(µ A ) = 0, where pf(δ 1 (i; i)) = ( 1) i+1 x i Pf(µ A ). EXAMPLE Let M = S 1 ˆ Σ g, where g ě 2. Then µ M = ř g i=1 a ib i c is generic, and Pf(µ M ) = x g 1 2g+1. Hence, R 1 = = R 2g 2 = tx 2g+1 = 0u and R 2g 1 = R 2g = R 2g+1 = t0u. ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 16 / 24

17 ALGEBRAIC MODELS AND RESONANCE VARIETIES 3-DIMENSIONAL POINCARÉ DUALITY ALGEBRAS RESONANCE VARIETIES OF 3-FORMS OF LOW RANK n µ R n µ R 1 = R 2 R tx 5 = 0u 0 n µ R 1 R 2 = R 3 R C 6 tx 1 = x 2 = x 3 = 0u Y tx 4 = x 5 = x 6 = 0u C 6 tx 3 = x 5 = x 6 = 0u 0 n µ R 1 = R 2 R 3 = R 4 R tx 7 = 0u tx 7 = 0u tx 7 = 0u tx 4 = x 5 = x 6 = x 7 = 0u tx 1 = 0u Y tx 4 = 0u tx 1 = x 2 = x 3 = x 4 = 0u Y tx 1 = x 4 = x 5 = x 6 = 0u tx 1 x 4 + x 2 x 5 = 0u tx 1 = x 2 = x 4 = x 5 = x7 2 x 3x 6 = 0u tx 1 x 4 + x 2 x 5 + x 3 x 6 = x7 2u 0 0 n µ R 1 R 2 = R 3 R 4 = R C 8 tx 7 = 0u tx 3 =x 5 =x 7 =x 8 =0uYtx 1 =x 3 =x 4 =x 5 =x 7 =0u C 8 tx 5 = x 7 = 0u tx 3 = x 4 = x 5 = x 7 = x 1 x 8 + x6 2 = 0u C 8 tx 1 = x 5 = 0u Y tx 3 = x 4 = 0u tx 1 = x 3 = x 4 = x 5 = x 2 x 6 + x 7 x 8 = 0u C 8 tx 1 = x 5 = 0u Y tx 3 = x 4 = x 5 = 0u tx 1 = x 2 = x 3 = x 4 = x 5 = x 7 = 0u C 8 tx 3 = x 5 = x 1 x 4 x7 2 = 0u tx 1 = x 2 = x 3 = x 4 = x 5 = x 6 = x 7 = 0u C 8 tx 1 = x 4 = x 7 = 0u Y tx 8 = 0u tx 1 =x 4 =x 7 =x 8 =0uYtx 2 =x 3 =x 5 =x 6 =x 8 =0u C 8 L 1 Y L 2 Y L 3 L 1 Y L C 8 C 1 Y C 2 L 1 Y L C 8 L 1 Y L 2 Y L 3 Y L 4 L 1 1 Y L1 2 Y L C 8 C 1 Y C 2 Y C 3 L 1 Y L 2 Y L C 8 C 1 Y C 2 L C 8 tf 1 = = f 20 = 0u C 8 tg 1 = = g 20 = 0u 0 ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 17 / 24

18 ALGEBRAIC MODELS AND RESONANCE VARIETIES PROPAGATION OF RESONANCE PROPAGATION OF RESONANCE We say that the resonance varieties of a graded algebra A = À n i=0 Ai propagate if R 1 1 (A) Ď Ď Rn 1 (A). (Eisenbud Popescu Yuzvinsky 2003) If X is the complement of a hyperplane arrangement, then its resonance varieties propagate. THEOREM (DSY 2016/17) Suppose the k-dual of A has a linear free resolution over E = Ź A 1. Then the resonance varieties of A propagate. Let X be a formal, abelian duality space. Then the resonance varieties of X propagate. Let M be a closed, orientable 3-manifold. If b 1 (M) is even and non-zero, then the resonance varieties of M do not propagate. ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 18 / 24

19 CHARACTERISTIC VARIETIES CHARACTERISTIC VARIETIES CHARACTERISTIC VARIETIES Let X be a connected, finite-type CW-complex. Then π = π 1 (X, x 0 ) is a finitely presented group, with π ab H 1 (X, Z). The ring R = C[π ab ] is the coordinate ring of the character group, Char(X ) = Hom(π, C ) (C ) r ˆ Tors(π ab ), where r = b 1 (X ). The characteristic varieties of X are the homology jump loci V i s(x ) = tρ P Char(X ) dim H i (X, C ρ ) ě su. These varieties are homotopy-type invariants of X, with V 1 s (X ) depending only on π = π 1 (X ). Set V 1 (π) := V 1 1 (K (π, 1)); then V 1(π) = V 1 (π/π 2 ). ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 19 / 24

20 CHARACTERISTIC VARIETIES ABELIAN DUALITY AND PROPAGATION OF CHARACTERISTIC VARIETIES EXAMPLE Let f P Z[t 1 1,..., t 1 n ] be a Laurent polynomial, f (1) = 0. There is then a finitely presented group π with π ab = Z n such that V 1 (π) = V(f ). THEOREM (DSY) Let X be an abelian duality space of dimension n. If ρ : π 1 (X ) Ñ C satisfies H i (X, C ρ ) 0, then H j (X, C ρ ) 0, for all i ď j ď n. COROLLARY Let X be an abelian duality space of dimension n. Then The characteristic varieties propagate, i.e., V1 1(X ) Ď Ď V 1 n (X ). ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 20 / 24

21 CHARACTERISTIC VARIETIES INFINITESIMAL FINITENESS OBSTRUCTIONS INFINITESIMAL FINITENESS OBSTRUCTIONS QUESTION Let X be a connected CW-complex with finite q-skeleton. Does X admit a q-finite q-model A? THEOREM If X is as above, then, for all i ď q and all s: (Dimca Papadima 2014) V i s(x ) (1) R i s(a) (0). In particular, if X is q-formal, then V i s(x ) (1) R i s(x ) (0). (Macinic, Papadima, Popescu, S. 2017) TC 0 (R i s(a)) Ď R i s(x ). (Budur Wang 2017) All the irreducible components of V i (X ) passing through the origin of H 1 (X, C ) are algebraic subtori. EXAMPLE Let π be a finitely presented group with π ab = Z n and V 1 (π) = tt P (C ) n ř n i=1 t i = nu. Then π admits no 1-finite 1-model. ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 21 / 24

22 CHARACTERISTIC VARIETIES INFINITESIMAL FINITENESS OBSTRUCTIONS THEOREM (PAPADIMA S. 2017) Suppose X is (q + 1) finite, or X admits a q-finite q-model. Then b i (M q (X )) ă 8, for all i ď q + 1. COROLLARY Let π be a finitely generated group. Assume that either π is finitely presented, or π has a 1-finite 1-model. Then b 2 (M 1 (π)) ă 8. EXAMPLE Consider the free metabelian group π = F n / F 2 n with n ě 2. We have V 1 (π) = V 1 (F n ) = (C ) n, and so π passes the Budur Wang test. But b 2 (M 1 (π)) = 8, and so π admits no 1-finite 1-model (and is not finitely presented). ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 22 / 24

23 CHARACTERISTIC VARIETIES A TANGENT CONE THEOREM FOR 3-MANIFOLDS A TANGENT CONE THEOREM FOR 3-MANIFOLDS THEOREM Let M be a closed, orientable, 3-dimensional manifold. Suppose b 1 (M) is odd and µ M is generic. Then TC 1 (V 1 1 (M)) = R1 1 (M). If b 1 (M) is even, the conclusion may or may not hold: Let M = S 1 ˆ S 2 #S 1 ˆ S 2 ; then V 1 1 (M) = Char(M) = (C ) 2, and so TC 1 (V 1 1 (M)) = R1 1 (M) = C2. Let M be the Heisenberg nilmanifold; then TC 1 (V1 1 (M)) = t0u, whereas R 1 1 (M) = C2. Let M be a closed, orientable 3-manifold with b 1 = 7 and µ = e 1 e 3 e 5 + e 1 e 4 e 7 + e 2 e 5 e 7 + e 3 e 6 e 7 + e 4 e 5 e 6. Then µ is generic and Pf(µ) = (x x 2 7 )2. Hence, R 1 1 (M) = tx x 2 7 = 0u splits as a union of two hyperplanes over C, but not over Q. ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 23 / 24

24 REFERENCES REFERENCES G. Denham, A.I. Suciu, and S. Yuzvinsky, Combinatorial covers and vanishing of cohomology, Selecta Math. 22 (2016), no. 2, G. Denham, A.I. Suciu, and S. Yuzvinsky, Abelian duality and propagation of resonance, Selecta Math. 23 (2017), no. 4, G. Denham, A.I. Suciu, Local systems on arrangements of smooth, complex algebraic hypersurfaces, arxiv: S. Papadima, A.I. Suciu, Infinitesimal finiteness obstructions, arxiv: A.I. Suciu, Cohomology jump loci of 3-manifolds, preprint. A.I. Suciu, Poincaré duality and resonance varieties, preprint. ALEX SUCIU (NORTHEASTERN) MODELS, DUALITY, AND RESONANCE MIT TOPOLOGY SEMINAR 24 / 24