REFINMENTS OF HOMOLOGY. Homological spectral package.
|
|
- Beatrice Hunter
- 5 years ago
- Views:
Transcription
1 REFINMENTS OF HOMOLOGY (homological spectral package). Dan Burghelea Department of Mathematics Ohio State University, Columbus, OH
2 In analogy with linear algebra over C we propose REFINEMENTS of the simplest topological invariants HOMOLOGY of X NOVIKOV HOMOLOY / L 2 HOMOLOGY of (X, ξ H 1 (X, Z)), associated to a continuous REAL or ANGLE VALUED MAP This work is : implicit in joint work with S.Haller (Vienna) influenced by joint work with Tamal Dey (OSU- Columbus) and Du Dong (Shanghai)
3 HOMOLOGY for a fixed field Homology : Space X : (1.) H r (X), Novikov homology : Pair (X; ξ H 1 (X; Z)) : β r (X) = dim H r (X) (2.) H N r (X, ξ), β N r (X, ξ) = rank H N r (X, ξ) (X, ξ) X X infinite cyclic cover associated to ξ. X compact ANR H r ( X) a f.g. κ[t 1, t] module X compact ANR TH r ( X) a κ[t 1, t] module which is a f.d. κ vector space. X compact ANR H N (X; ξ) := H r ( X)/TH r ( X) a f.g. free κ[t 1, t] module
4 L 2 homology If κ = C by "completion" C[t 1, t] L (S 1 ) a von Neumann algebra. Hr N (X : ξ) H L 2 r ( X) finite type L (S 1 ) Hilbert module and β N (X; ξ) = dim vn (H L 2 r ( X)).
5 New invariants - configurations Configurations of points in Y with multiplicities in N Conf n (Y ) := {δ : Y N 0 (supp δ) < } = Y n /Σ n Conf n (Y ) := {δ : Y N 0 y δ(y) = n} = Y n /Σ n Configurations of points in Y indexing subspaces of V Conf vect (Y ) := {ˆδ : Y Vect (supp ˆδ) < } Conf V (Y ) := {ˆδ : Y S(V ) y ˆδ(y) = V } Y = C Conf n (Y ) = C n and identifies with degree n monic polynomials. Y = C \ 0 Conf n (Y ) = C n 1 (C \ 0) and identifies with degree n monic polynomials with nonzero free coefficient.
6 SPECTRAL PACKAGE in LINEAR ALGEBRA SYSTHEM (V, T : V V ) { V f.d. complex vector space. T : V V linear map = SPECTRAL PACKAGE dim V = n z 1, z 2,, z k 1, z k n 1, n 2,, n k 1, n k V 1, V 2,, V k 1, V k N C; eigenvalues N; multiplicities V ; generalized eigenspaces PROPERTIES : dim V = n i, dim V i = n i, V = V i
7 geometrization δ T : { z1, z 2,, z k 1, z k n 1, n 2,, n k 1, n k finite configurations of points with multiplicities s.t. δ T (z i ) = dim V i δ T P T (z) = (z z 1 ) n 1(z z 2 ) n 2 (z z k ) n k the characteristic polynomial, n = dim V. ˆδ T : { z1, z 2,, z k 1, z k V 1, V 2,, V k 1, V k finite configurations of points indexinig disjoint subspaces of V s.t. i ˆδ T (z i ) = V δ T Conf n (C), ˆδ T Conf V (C)
8 basic properties 1 (Stability) L(V, V ) T δ T (z) = P T (z) C n is continuous 2 (Duality) δ T = δ T 3 For an open and dense set of T L(V, V ), δ T (z) = 0 or 1 4 (Computability) P T (z) can be calculated with arbitrary accuracy. One regards δ T P T (z) as a refinement of dim V, One regards ˆδ T as an implementation of the refinement δ T.
9 TOPOLOGY ; HOMOLOGICAL SPECTRAL PACKAGE (real valued map) SYSTHEM (X, f : X R) X a compact ANR, f continuous, κ a field, H r (X) := H r (X; κ), r N 0. HOMOLOGICAL SPECTRAL PACKAGE dim H r (X) = β r (X) ; Betti number z 1, z 2,, z k 1, z k C; barcodes n 1, n 2,, n k 1, n k N; multiplicities V 1, V 2,, V k 1, V k ; quotients of subspaces of V. β r = i n i, dim V i = n i, V V i
10 specifications V i = L i /L i are quotients of subspaces of H r (X), L i L i H r (X), essentially disjoint If H r (X) has an inner product then V i is canonically realizable as a subspace of H r (X) with V i V j, i j.
11 geometrization δ f r : { z1, z 2,, z k 1, z k n 1, n 2,, n k 1, n k finite configurations of points with multiplicities s.t. δr(z f i ) = β r (X) δ f r P f r (z) = (z z 1 ) n 1(z z 2 ) n 2 (z z k ) n k the homological characteristic polynomial. ˆδ f r : { z1, z 2,, z k 1, z k V 1, V 2,, V k 1, V k δ f r Conf βr (X)(C), ˆδ f r Conf Hr (X)(C) finite configurations of points indexing subspaces of H r (X) s.t. ˆδ r f (z i ) = H r (X)
12
13 TOPOLOGY ; HOMOLOGICAL SPECTRAL PACKAGE (angle valued map) SYSTHEM (X, f : X S 1 ) (X, f : X S 1 ) ξ f H 1 (X; Z) X a compact ANR, f continuous, κ a field, Hr N (X; ξ f ) := H r ( X)/TH r ( X) r N 0. f : X R denotes the infinite cyclic cover of f
14 HOMOLOGICAL SPECTRAL PACKAGE rank H N r (X; ξ f ) = βr N (X; ξ f ) ; Novikov Betti number z 1, z 2,, z k 1, z k C \ 0; exponentiated barcodes n 1, n 2,, n k 1, n k N; multiplicities V 1, V 2,, V k 1, V k ; free κ[t 1, t] modules. V i = L i /L i quotients of free κ[t 1, t] submodules of H N r (X; ξ f ), L i L i H r (X), essentially disjoint β N r = i n i, rank V i = n i, H N r (X; ξ f ) V i
15 If κ = C and a C[t 1, t] inner product by completion Hr N (X; ξ) is replaced by the Hilbert module H L 2 r ( X) and the configuration of free C[t 1, t] by configurations of mutually orthogonal closed Hilbert L (S 1 ) submodules.
16 geometrization δ f r : { finite configurations of z1, z 2,, z k 1, z k points with multiplicities in n 1, n 2,, n k 1, n k C \ 0 s.t. δr(z f i ) = βr N (X; ξ) δ f r P f r (z) = (z z 1 ) n 1(z z 2 ) n 2 (z z k ) n k the homological characteristic polynomial of degree β N r (X; ξ). ˆδ f r : { z1, z 2,, z k 1, z k V 1, V 2,, V k 1, V k finite configurations of points indexing submodules of δr f Conf β N r (X;ξ f ) (C \ 0), ˆδ r f Conf H N r (X;ξ f )(C \ 0) H N r (X; ξ)s.t. ˆδ f r (z i ) = H N r (X; ξ)
17
18 Definitions of δ f r and ˆδ f r For f : X R a proper map a, b R, κ a field Denote: X a := f 1 ((, a]) sub-level X b := f 1 ([b, )) over-level Define: I a (r) := img(h r (X a ) H r (X)) I b (r) := img(h r (X b ) H r (X)) F r (a, b) := I a (r) I b (r)
19 Observe a a, b b imply F r (a, b ) F r (a, b). Prove F r (a, b) finite dimensional. Define F r (a, b, ɛ) := F r (a, b)/f r (a ɛ, b) + F r (a, b + ɛ) for ɛ > 0 Observe that ɛ > ɛ induces a surjective map F r (a, b; ɛ ) F r (a, b; ɛ ). Definition ˆd r f (a, b) := lim F r (a, b, ɛ) ɛ 0 d f r (a, b) := dim ˆd f r (a, b)
20 The case of f : X R, X compact. Define δ f r (z) = d f r (a, b), z = a + ib ˆδ f r (z) = ˆd f r (a, b), z = a + ib.
21 The case of f : X S 1, X compact. Consider f : X R an infinite cyclic cover of f. Observe that 1 t : ˆd f r (a, b) ˆd f r (a + 2π, b + 2π) is an isomorphism and 2 dr f (a, b) = d f r (a + 2π, b + 2π) Define δ f r (z) = d f r (a, b), z = e (b a)+ia ˆδ f r (z) = k ˆd f r (a + 2πk, b + 2πk), z = e (b a)+ia.
22 Main results The values t R is CRITICAL if the homology of the levels of f changes at t. Cr(f ) = the set of critical values of f. Theorem 1. suppδ f r β r (X), z suppδ f r δf r (z) = β f r 2. For an open sense set of maps f, δ f r (z) = 0 or δ f r (z) = 1.
23 Proposition 1 If z = a + ib suppδr f then both a, b Cr(f ). 2 z = (a + ib) suppδr f above or on diagonal implies [a.b] is a closed r bar code. 3 z = (a + ib) suppδr f below diagonal implies (b, a) is an open (r 1) bar code.
24 Theorem The assignment C(X, R) f δ f r C βr is continuous. Theorem If M n is a closed κ orientable topological n dimensional manifold then δ f r (z) = δ f n r ( iz) The same remains true for the configuration ˆδ f r
25 Suppose f is an angle valued map Theorem 1. suppδr f βr N (X), z suppδr f δf r (z) = βr N (X; ξ) 2. For an open sense set of maps f, δr f (z) = 0 or δr f (z) = 1. Proposition 1 If z = e ia+(b a) suppδr f then both a, b Cr( f ) = {t e it Cr(f ). 2 z outside or on the unit circle implies [a, b] is a closed r bar code. 3 z inside the unit circle implies (b, a) is an open (r 1) bar code.
26 Theorem The assignment C(X, S 1 ) f δ f r C βn r 1 C \ 0 is continuous Theorem If M n is a closed κ orientable topological n dimensional manifold then δ f r (z) = δ f n r ( iz) The same remains true for the configuration ˆδ f r
27 The proofs are done in two steps. Step 1: Establish the results for X a finite simplicial complex and f a simplicial map Step 2. Use results about the topology of Hilbert cube manifolds and show that the statements are true for f iff are true for f Q = f p X, p X : X Q X, Q the Hilbert cube I.
28 Step 1 Introduce the concepts: 1 tame map, 2 homological critical value and for a tame map, 3 the number ɛ(f ) = inf c c c, c different homological critical values Verify that for f : X R proper tame map 1 For f proper tame map d f r (a, b) 0 a, b critical values, 2 For a box B = (a, b] [c, d) and f proper tame map F f r (B) = (a b ) B suppδ ˆd f r (a, b ) Let f a tame map, X compact and ɛ < ɛ(f ). Then g : X R proper map with g f < ɛ the support of δ g is in an 2ɛ neighborhood of the support of δ f and of equal cardinality as suppδ f.
29 Applications In view of effective computability of the configuration δ f r the result can be used to : Applications in topology: Calculation of Betti numbers, Novikov Betti numbers Refinements of Morse inequalities. Applications in data analysis: Homological recognition of shapes which can be manifolds. Homological differentiations of shapes. Applications in geometric analysis: Refinement of Hodge de Rham theorem on compact Riemannian maniflods, Canonical base in the space of Harmonic forms Applications in dynamics: for dynamics of flows which admit an action
30 References 1.D. Burghelea and T. K. Dey, Persistence for circle valued maps. Discrete and Computational Geometry, Vol , pp Dan Burghelea, Stefan Haller, Topology of angle valued maps, bar codes and Jordan blocks arxiv: Dan Burghelea, A refinement of Betti numbers in the presence of a continuous function ( I ), arxiv: Dan Burghelea, A refinement of Betti numbers in the presence of a continuous function ( II ), to be posted soon
31 Persistent homology PH r (X; f ) (Persistent homology) are If f real valued κ vector spaces If f angle valued κ[t 1, t] free modules / L (S 1 ) Hilbert modules. Refinments δ P,f r and ˆδ P,f r (Configurations) For f real valued configurations on C \ For f angle valued configurations on C \ {0 S 1 } Based on Fredholm cross ratio of α : A B, β : B C, γ : C D Fredholm maps.
32 Monodromy of f : X S 1 Isomorphism classes of pairs (T r, V r ) V r = TH r ( X), T r = t ) Invariants JORDAN CELLS J r (f ) := {(λ r i, nr i ), i = 1 k r λ r i κ, n r i N} Based on Linear relations.
A (computer friendly) alternative to Morse Novikov theory for real and angle valued maps
A (computer friendly) alternative to Morse Novikov theory for real and angle valued maps Dan Burghelea Department of mathematics Ohio State University, Columbus, OH based on joint work with Stefan Haller
More informationBarcodes for closed one form; Alternative to Novikov theory
Barcodes for closed one form; Alternative to Novikov theory Dan Burghelea 1 Department of mathematics Ohio State University, Columbus, OH Tel Aviv April 2018 Classical MN-theory Input: Output M n smooth
More informationGraphs Representations and the Topology of Real and Angle valued maps (an alternative approach to Morse-Novikiov theory)
Graphs Representations and the Topology of Real and Angle valued maps (an alternative approach to Morse-Novikiov theory) Dan Burghelea Department of mathematics Ohio State University, Columbus, OH Tianjin,
More informationA COMPUTATIONAL ALTERNATIVE TO MORSE-NOVIKOV THEORY
A COMPUTATIONAL ALTERNATIVE TO MORSE-NOVIKOV THEORY Dan Burghelea Department of Mathematics Ohio State University, Columbus, OH Shanghai, July 2017 summary A brief summary of what AMN-theory does The AMN
More informationarxiv: v1 [math.at] 4 Apr 2019
Comparison between Parametrized homology via zigzag persistence and Refined homology in the presence of a real-valued continuous function arxiv:1904.02626v1 [math.at] 4 Apr 2019 Dan Burghelea Abstract
More informationWitten Deformation and analytic continuation
Witten Deformation and analytic continuation Dan Burghelea Department of Mathematics Ohio State University, Columbus, OH Bonn, June 27, 2013 Based on joint work with STEFAN HALLER Departments of Mathematics
More informationTOEPLITZ OPERATORS. Toeplitz studied infinite matrices with NW-SE diagonals constant. f e C :
TOEPLITZ OPERATORS EFTON PARK 1. Introduction to Toeplitz Operators Otto Toeplitz lived from 1881-1940 in Goettingen, and it was pretty rough there, so he eventually went to Palestine and eventually contracted
More informationContents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2
Contents Preface for the Instructor xi Preface for the Student xv Acknowledgments xvii 1 Vector Spaces 1 1.A R n and C n 2 Complex Numbers 2 Lists 5 F n 6 Digression on Fields 10 Exercises 1.A 11 1.B Definition
More informationHandlebody Decomposition of a Manifold
Handlebody Decomposition of a Manifold Mahuya Datta Statistics and Mathematics Unit Indian Statistical Institute, Kolkata mahuya@isical.ac.in January 12, 2012 contents Introduction What is a handlebody
More informationMORSE THEORY DAN BURGHELEA. Department of Mathematics The Ohio State University
MORSE THEORY DAN BURGHELEA Department of Mathematics The Ohio State University 1 Morse Theory begins with the modest task of understanding the local maxima, minima and sadle points of a smooth function
More informationDYNAMICS, Topology and Spectral Geometry
DYNAMICS, Topology and Spectral Geometry Dan Burghelea Department of Mathematics Ohio State University, Columbuus, OH Iasi, June 21 Prior work on the topic: S. Smale, D.Fried, S.P.Novikov, A.V. Pajitnov,
More informationCOUNTEREXAMPLES TO THE COARSE BAUM-CONNES CONJECTURE. Nigel Higson. Unpublished Note, 1999
COUNTEREXAMPLES TO THE COARSE BAUM-CONNES CONJECTURE Nigel Higson Unpublished Note, 1999 1. Introduction Let X be a discrete, bounded geometry metric space. 1 Associated to X is a C -algebra C (X) which
More informationTopological Persistence for Circle Valued Maps
Topological Persistence for Circle Valued Maps Dan Burghelea Tamal K. Dey Abstract We study circle valued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection
More informationAn overview of D-modules: holonomic D-modules, b-functions, and V -filtrations
An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations Mircea Mustaţă University of Michigan Mainz July 9, 2018 Mircea Mustaţă () An overview of D-modules Mainz July 9, 2018 1 The
More informationQUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday 10 February 2004 (Day 1)
Tuesday 10 February 2004 (Day 1) 1a. Prove the following theorem of Banach and Saks: Theorem. Given in L 2 a sequence {f n } which weakly converges to 0, we can select a subsequence {f nk } such that the
More informationMATRIX LIE GROUPS AND LIE GROUPS
MATRIX LIE GROUPS AND LIE GROUPS Steven Sy December 7, 2005 I MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in, and for some, then either
More informationLecture 5 - Hausdorff and Gromov-Hausdorff Distance
Lecture 5 - Hausdorff and Gromov-Hausdorff Distance August 1, 2011 1 Definition and Basic Properties Given a metric space X, the set of closed sets of X supports a metric, the Hausdorff metric. If A is
More informationThe semantics of non-commutative geometry and quantum mechanics.
The semantics of non-commutative geometry and quantum mechanics. B. Zilber University of Oxford July 30, 2014 1 Dualities in logic and geometry 2 The Weyl-Heisenberg algebra 3 Calculations Tarskian duality
More informationQuasi Riemann surfaces II. Questions, comments, speculations
Quasi Riemann surfaces II. Questions, comments, speculations Daniel Friedan New High Energy Theory Center, Rutgers University and Natural Science Institute, The University of Iceland dfriedan@gmail.com
More informationElliptic Regularity. Throughout we assume all vector bundles are smooth bundles with metrics over a Riemannian manifold X n.
Elliptic Regularity Throughout we assume all vector bundles are smooth bundles with metrics over a Riemannian manifold X n. 1 Review of Hodge Theory In this note I outline the proof of the following Fundamental
More informationMatthew L. Wright. St. Olaf College. June 15, 2017
Matthew L. Wright St. Olaf College June 15, 217 Persistent homology detects topological features of data. For this data, the persistence barcode reveals one significant hole in the point cloud. Problem:
More informationN-WEAKLY SUPERCYCLIC MATRICES
N-WEAKLY SUPERCYCLIC MATRICES NATHAN S. FELDMAN Abstract. We define an operator to n-weakly hypercyclic if it has an orbit that has a dense projection onto every n-dimensional subspace. Similarly, an operator
More informationTopology of Nonarchimedean Analytic Spaces
Topology of Nonarchimedean Analytic Spaces AMS Current Events Bulletin Sam Payne January 11, 2013 Complex algebraic geometry Let X C n be an algebraic set, the common solutions of a system of polynomial
More informationThe Gram matrix in inner product modules over C -algebras
The Gram matrix in inner product modules over C -algebras Ljiljana Arambašić (joint work with D. Bakić and M.S. Moslehian) Department of Mathematics University of Zagreb Applied Linear Algebra May 24 28,
More informationLECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES
LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 1. Vector Bundles In general, smooth manifolds are very non-linear. However, there exist many smooth manifolds which admit very nice partial linear structures.
More informationDefinition (T -invariant subspace) Example. Example
Eigenvalues, Eigenvectors, Similarity, and Diagonalization We now turn our attention to linear transformations of the form T : V V. To better understand the effect of T on the vector space V, we begin
More informationAnalysis Preliminary Exam Workshop: Hilbert Spaces
Analysis Preliminary Exam Workshop: Hilbert Spaces 1. Hilbert spaces A Hilbert space H is a complete real or complex inner product space. Consider complex Hilbert spaces for definiteness. If (, ) : H H
More informationLECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori
LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.
More informationUnbounded KK-theory and KK-bordisms
Unbounded KK-theory and KK-bordisms joint work with Robin Deeley and Bram Mesland University of Gothenburg 161026 Warzaw 1 Introduction 2 3 4 Introduction In NCG, a noncommutative manifold is a spectral
More informationNOTES ON THE ARTICLE: A CURRENT APPROACH TO MORSE AND NOVIKOV THEORIES
Rendiconti di Matematica, Serie VII Volume 36, Roma (2015), 89 94 NOTES ON THE ARTICLE: A CURRENT APPROACH TO MORSE AND NOVIKOV THEORIES REESE HARVEY BLAINE LAWSON The following article, A current approach
More informationCOMPACT OPERATORS. 1. Definitions
COMPACT OPERATORS. Definitions S:defi An operator M : X Y, X, Y Banach, is compact if M(B X (0, )) is relatively compact, i.e. it has compact closure. We denote { E:kk (.) K(X, Y ) = M L(X, Y ), M compact
More informationCompact operators on Banach spaces
Compact operators on Banach spaces Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto November 12, 2017 1 Introduction In this note I prove several things about compact
More informationPreliminaries on von Neumann algebras and operator spaces. Magdalena Musat University of Copenhagen. Copenhagen, January 25, 2010
Preliminaries on von Neumann algebras and operator spaces Magdalena Musat University of Copenhagen Copenhagen, January 25, 2010 1 Von Neumann algebras were introduced by John von Neumann in 1929-1930 as
More informationExtensions of Stanley-Reisner theory: Cell complexes and be
: Cell complexes and beyond February 1, 2012 Polyhedral cell complexes Γ a bounded polyhedral complex (or more generally: a finite regular cell complex with the intersection property). Polyhedral cell
More informationCOMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY
COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY BRIAN OSSERMAN Classical algebraic geometers studied algebraic varieties over the complex numbers. In this setting, they didn t have to worry about the Zariski
More informationSPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS
SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS TSOGTGEREL GANTUMUR Abstract. After establishing discrete spectra for a large class of elliptic operators, we present some fundamental spectral properties
More informationDEFINABLE OPERATORS ON HILBERT SPACES
DEFINABLE OPERATORS ON HILBERT SPACES ISAAC GOLDBRING Abstract. Let H be an infinite-dimensional (real or complex) Hilbert space, viewed as a metric structure in its natural signature. We characterize
More informationIr O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )
Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O
More informationLinear Algebra problems
Linear Algebra problems 1. Show that the set F = ({1, 0}, +,.) is a field where + and. are defined as 1+1=0, 0+0=0, 0+1=1+0=1, 0.0=0.1=1.0=0, 1.1=1.. Let X be a non-empty set and F be any field. Let X
More informationFunctional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...
Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................
More informationA new proof of Gromov s theorem on groups of polynomial growth
A new proof of Gromov s theorem on groups of polynomial growth Bruce Kleiner Courant Institute NYU Groups as geometric objects Let G a group with a finite generating set S G. Assume that S is symmetric:
More information7.3 Singular Homology Groups
184 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups 7.3.1 Cycles, Boundaries and Homology Groups We can define the singular p-chains with coefficients in a field K. Furthermore, we can define the
More informationRIEMANN S INEQUALITY AND RIEMANN-ROCH
RIEMANN S INEQUALITY AND RIEMANN-ROCH DONU ARAPURA Fix a compact connected Riemann surface X of genus g. Riemann s inequality gives a sufficient condition to construct meromorphic functions with prescribed
More informationBifurcation and symmetry breaking in geometric variational problems
in geometric variational problems Joint work with M. Koiso and B. Palmer Departamento de Matemática Instituto de Matemática e Estatística Universidade de São Paulo EIMAN I ENCONTRO INTERNACIONAL DE MATEMÁTICA
More informationTranscendental L 2 -Betti numbers Atiyah s question
Transcendental L 2 -Betti numbers Atiyah s question Thomas Schick Göttingen OA Chennai 2010 Thomas Schick (Göttingen) Transcendental L 2 -Betti numbers Atiyah s question OA Chennai 2010 1 / 24 Analytic
More informationHonors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35
Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime
More informationALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP. Contents 1. Introduction 1
ALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP HONG GYUN KIM Abstract. I studied the construction of an algebraically trivial, but topologically non-trivial map by Hopf map p : S 3 S 2 and a
More informationJune 2014 Written Certification Exam. Algebra
June 2014 Written Certification Exam Algebra 1. Let R be a commutative ring. An R-module P is projective if for all R-module homomorphisms v : M N and f : P N with v surjective, there exists an R-module
More informationhere, this space is in fact infinite-dimensional, so t σ ess. Exercise Let T B(H) be a self-adjoint operator on an infinitedimensional
15. Perturbations by compact operators In this chapter, we study the stability (or lack thereof) of various spectral properties under small perturbations. Here s the type of situation we have in mind:
More informationMath 25a Practice Final #1 Solutions
Math 25a Practice Final #1 Solutions Problem 1. Suppose U and W are subspaces of V such that V = U W. Suppose also that u 1,..., u m is a basis of U and w 1,..., w n is a basis of W. Prove that is a basis
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationMATH 241B FUNCTIONAL ANALYSIS - NOTES SPECTRAL THEOREM
MATH 241B FUNCTIONAL ANALYSIS - NOTES SPECTRAL THEOREM We present the material in a slightly different order than it is usually done (such as e.g. in the course book). Here we prefer to start out with
More informationFunctional Analysis HW #1
Functional Analysis HW #1 Sangchul Lee October 9, 2015 1 Solutions Solution of #1.1. Suppose that X
More informationLecture 21: The decomposition theorem into generalized eigenspaces; multiplicity of eigenvalues and upper-triangular matrices (1)
Lecture 21: The decomposition theorem into generalized eigenspaces; multiplicity of eigenvalues and upper-triangular matrices (1) Travis Schedler Tue, Nov 29, 2011 (version: Tue, Nov 29, 1:00 PM) Goals
More informationHodge theory for combinatorial geometries
Hodge theory for combinatorial geometries June Huh with Karim Adiprasito and Eric Katz June Huh 1 / 48 Three fundamental ideas: June Huh 2 / 48 Three fundamental ideas: The idea of Bernd Sturmfels that
More informationCompact symetric bilinear forms
Compact symetric bilinear forms Mihai Mathematics Department UC Santa Barbara IWOTA 2006 IWOTA 2006 Compact forms [1] joint work with: J. Danciger (Stanford) S. Garcia (Pomona
More informationThe geometrical semantics of algebraic quantum mechanics
The geometrical semantics of algebraic quantum mechanics B. Zilber University of Oxford November 29, 2016 B.Zilber, The semantics of the canonical commutation relations arxiv.org/abs/1604.07745 The geometrical
More informationPoisson configuration spaces, von Neumann algebras, and harmonic forms
J. of Nonlinear Math. Phys. Volume 11, Supplement (2004), 179 184 Bialowieza XXI, XXII Poisson configuration spaces, von Neumann algebras, and harmonic forms Alexei DALETSKII School of Computing and Technology
More information5 Compact linear operators
5 Compact linear operators One of the most important results of Linear Algebra is that for every selfadjoint linear map A on a finite-dimensional space, there exists a basis consisting of eigenvectors.
More informationCHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS. Contents
CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS Contents 1. The Lefschetz hyperplane theorem 1 2. The Hodge decomposition 4 3. Hodge numbers in smooth families 6 4. Birationally
More informationTHE HODGE DECOMPOSITION
THE HODGE DECOMPOSITION KELLER VANDEBOGERT 1. The Musical Isomorphisms and induced metrics Given a smooth Riemannian manifold (X, g), T X will denote the tangent bundle; T X the cotangent bundle. The additional
More informationSpectral Measures, the Spectral Theorem, and Ergodic Theory
Spectral Measures, the Spectral Theorem, and Ergodic Theory Sam Ziegler The spectral theorem for unitary operators The presentation given here largely follows [4]. will refer to the unit circle throughout.
More informationDifferential Topology Final Exam With Solutions
Differential Topology Final Exam With Solutions Instructor: W. D. Gillam Date: Friday, May 20, 2016, 13:00 (1) Let X be a subset of R n, Y a subset of R m. Give the definitions of... (a) smooth function
More informationGorenstein rings through face rings of manifolds.
Gorenstein rings through face rings of manifolds. Isabella Novik Department of Mathematics, Box 354350 University of Washington, Seattle, WA 98195-4350, USA, novik@math.washington.edu Ed Swartz Department
More informationGRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.
GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. Linear Algebra Standard matrix manipulation to compute the kernel, intersection of subspaces, column spaces,
More informationNotes by Maksim Maydanskiy.
SPECTRAL FLOW IN MORSE THEORY. 1 Introduction Notes by Maksim Maydanskiy. Spectral flow is a general formula or computing the Fredholm index of an operator d ds +A(s) : L1,2 (R, H) L 2 (R, H) for a family
More informationWITTEN HELLFER SJÖSTRAND THEORY. 1.DeRham Hodge Theorem. 2.WHS-Theorem. 3.Mathematics behind WHS-Theorem. 4.WHS-Theorem in the presence of.
WITTEN HELLFER SJÖSTRAND THEORY 1.DeRham Hodge Theorem 2.WHS-Theorem 3.Mathematics behind WHS-Theorem 4.WHS-Theorem in the presence of symmetry 1 2 Notations M closed smooth manifold, (V, V ) euclidean
More informationPseudogroups of foliations Algebraic invariants of solenoids The discriminant group Stable actions Wild actions Future work and references
Wild solenoids Olga Lukina University of Illinois at Chicago Joint work with Steven Hurder March 25, 2017 1 / 25 Cantor laminations Let M be a compact connected metrizable topological space with a foliation
More informationBulletin of the Iranian Mathematical Society
ISSN: 1017-060X (Print) ISSN: 1735-8515 (Online) Special Issue of the Bulletin of the Iranian Mathematical Society in Honor of Professor Heydar Radjavi s 80th Birthday Vol 41 (2015), No 7, pp 155 173 Title:
More informationAlgebra Exam Syllabus
Algebra Exam Syllabus The Algebra comprehensive exam covers four broad areas of algebra: (1) Groups; (2) Rings; (3) Modules; and (4) Linear Algebra. These topics are all covered in the first semester graduate
More informationMath 209B Homework 2
Math 29B Homework 2 Edward Burkard Note: All vector spaces are over the field F = R or C 4.6. Two Compactness Theorems. 4. Point Set Topology Exercise 6 The product of countably many sequentally compact
More informationBetti numbers of abelian covers
Betti numbers of abelian covers Alex Suciu Northeastern University Geometry and Topology Seminar University of Wisconsin May 6, 2011 Alex Suciu (Northeastern University) Betti numbers of abelian covers
More informationMorse-Forman-Conley theory for combinatorial multivector fields
1 Discrete, Computational and lgebraic Topology Copenhagen, 12th November 2014 Morse-Forman-Conley theory for combinatorial multivector fields (research in progress) P Q L M N O G H I J K C D E F Marian
More informationLECTURE VI: SELF-ADJOINT AND UNITARY OPERATORS MAT FALL 2006 PRINCETON UNIVERSITY
LECTURE VI: SELF-ADJOINT AND UNITARY OPERATORS MAT 204 - FALL 2006 PRINCETON UNIVERSITY ALFONSO SORRENTINO 1 Adjoint of a linear operator Note: In these notes, V will denote a n-dimensional euclidean vector
More informationWEIGHTED SHIFTS OF FINITE MULTIPLICITY. Dan Sievewright, Ph.D. Western Michigan University, 2013
WEIGHTED SHIFTS OF FINITE MULTIPLICITY Dan Sievewright, Ph.D. Western Michigan University, 2013 We will discuss the structure of weighted shift operators on a separable, infinite dimensional, complex Hilbert
More informationLinear Algebra. Min Yan
Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................
More informationLinear Algebra: Matrix Eigenvalue Problems
CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given
More informationShort note on compact operators - Monday 24 th March, Sylvester Eriksson-Bique
Short note on compact operators - Monday 24 th March, 2014 Sylvester Eriksson-Bique 1 Introduction In this note I will give a short outline about the structure theory of compact operators. I restrict attention
More informationProblem Set 6: Solutions Math 201A: Fall a n x n,
Problem Set 6: Solutions Math 201A: Fall 2016 Problem 1. Is (x n ) n=0 a Schauder basis of C([0, 1])? No. If f(x) = a n x n, n=0 where the series converges uniformly on [0, 1], then f has a power series
More informationNONCOMMUTATIVE LOCALIZATION IN ALGEBRA AND TOPOLOGY Andrew Ranicki (Edinburgh) aar. Heidelberg, 17th December, 2008
1 NONCOMMUTATIVE LOCALIZATION IN ALGEBRA AND TOPOLOGY Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/ aar Heidelberg, 17th December, 2008 Noncommutative localization Localizations of noncommutative
More informationarxiv: v1 [math.ag] 13 Mar 2019
THE CONSTRUCTION PROBLEM FOR HODGE NUMBERS MODULO AN INTEGER MATTHIAS PAULSEN AND STEFAN SCHREIEDER arxiv:1903.05430v1 [math.ag] 13 Mar 2019 Abstract. For any integer m 2 and any dimension n 1, we show
More informationEXPLICIT l 1 -EFFICIENT CYCLES AND AMENABLE NORMAL SUBGROUPS
EXPLICIT l 1 -EFFICIENT CYCLES AND AMENABLE NORMAL SUBGROUPS CLARA LÖH Abstract. By Gromov s mapping theorem for bounded cohomology, the projection of a group to the quotient by an amenable normal subgroup
More informationLinear Algebra and Dirac Notation, Pt. 2
Linear Algebra and Dirac Notation, Pt. 2 PHYS 500 - Southern Illinois University February 1, 2017 PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 2 February 1, 2017 1 / 14
More informationFUNCTIONAL ANALYSIS-NORMED SPACE
MAT641- MSC Mathematics, MNIT Jaipur FUNCTIONAL ANALYSIS-NORMED SPACE DR. RITU AGARWAL MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY JAIPUR 1. Normed space Norm generalizes the concept of length in an arbitrary
More informationMATH 8253 ALGEBRAIC GEOMETRY WEEK 12
MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f
More informationLecture 7: Positive Semidefinite Matrices
Lecture 7: Positive Semidefinite Matrices Rajat Mittal IIT Kanpur The main aim of this lecture note is to prepare your background for semidefinite programming. We have already seen some linear algebra.
More informationInfinite-Dimensional Triangularization
Infinite-Dimensional Triangularization Zachary Mesyan March 11, 2018 Abstract The goal of this paper is to generalize the theory of triangularizing matrices to linear transformations of an arbitrary vector
More informationLECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE
LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE JOHANNES EBERT 1.1. October 11th. 1. Recapitulation from differential topology Definition 1.1. Let M m, N n, be two smooth manifolds
More informationCohomology jump loci of quasi-projective varieties
Cohomology jump loci of quasi-projective varieties Botong Wang joint work with Nero Budur University of Notre Dame June 27 2013 Motivation What topological spaces are homeomorphic (or homotopy equivalent)
More informationQUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday September 21, 2004 (Day 1)
QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday September 21, 2004 (Day 1) Each of the six questions is worth 10 points. 1) Let H be a (real or complex) Hilbert space. We say
More informationMATHEMATICS. Course Syllabus. Section A: Linear Algebra. Subject Code: MA. Course Structure. Ordinary Differential Equations
MATHEMATICS Subject Code: MA Course Structure Sections/Units Section A Section B Section C Linear Algebra Complex Analysis Real Analysis Topics Section D Section E Section F Section G Section H Section
More informationGEOMETRIC QUANTIZATION
GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical
More informationLECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS
LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS WEIMIN CHEN, UMASS, SPRING 07 1. Basic elements of J-holomorphic curve theory Let (M, ω) be a symplectic manifold of dimension 2n, and let J J (M, ω) be
More informationCourse Description - Master in of Mathematics Comprehensive exam& Thesis Tracks
Course Description - Master in of Mathematics Comprehensive exam& Thesis Tracks 1309701 Theory of ordinary differential equations Review of ODEs, existence and uniqueness of solutions for ODEs, existence
More informationAlgebraic Varieties. Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra
Algebraic Varieties Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra Algebraic varieties represent solutions of a system of polynomial
More informationTHIRD SEMESTER M. Sc. DEGREE (MATHEMATICS) EXAMINATION (CUSS PG 2010) MODEL QUESTION PAPER MT3C11: COMPLEX ANALYSIS
THIRD SEMESTER M. Sc. DEGREE (MATHEMATICS) EXAMINATION (CUSS PG 2010) MODEL QUESTION PAPER MT3C11: COMPLEX ANALYSIS TIME:3 HOURS Maximum weightage:36 PART A (Short Answer Type Question 1-14) Answer All
More informationOn the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem
On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem Bertram Kostant, MIT Conference on Representations of Reductive Groups Salt Lake City, Utah July 10, 2013
More informationSPECTRAL THEORY EVAN JENKINS
SPECTRAL THEORY EVAN JENKINS Abstract. These are notes from two lectures given in MATH 27200, Basic Functional Analysis, at the University of Chicago in March 2010. The proof of the spectral theorem for
More informationChapter 7. Canonical Forms. 7.1 Eigenvalues and Eigenvectors
Chapter 7 Canonical Forms 7.1 Eigenvalues and Eigenvectors Definition 7.1.1. Let V be a vector space over the field F and let T be a linear operator on V. An eigenvalue of T is a scalar λ F such that there
More informationarxiv: v2 [math.na] 8 Sep 2015
OMPLEXES OF DISRETE DISTRIBUTIONAL DIFFERENTIAL FORMS AND THEIR HOMOLOGY THEORY MARTIN WERNER LIHT arxiv:1410.6354v2 [math.na] 8 Sep 2015 Abstract. omplexes of discrete distributional differential forms
More information