Topological Methods in Combinatorics
... the elements of A have a common neighbor. NW = NW (G) is defined analogously. Lemma 2.7 The two neighborhood complexes of a bipartite graph are homotopy equivalent. Proof. For every simplex S ∈ NU , let f (S) ⊆ W denote the set common neighbors of nodes in S. Clearly f (S) ∈ NW , so f : NU → NW . We ...
... the elements of A have a common neighbor. NW = NW (G) is defined analogously. Lemma 2.7 The two neighborhood complexes of a bipartite graph are homotopy equivalent. Proof. For every simplex S ∈ NU , let f (S) ⊆ W denote the set common neighbors of nodes in S. Clearly f (S) ∈ NW , so f : NU → NW . We ...
4. Morphisms
... Exercise 4.13. Let X ⊂ A2 be the zero locus of a single polynomial ∑i+ j≤d ai, j x1i x2j of degree at most d. Show that: (a) Any line in A2 (i.e. any zero locus of a single polynomial of degree 1) not contained in X intersects X in at most d points. (b) Any affine conic (as in Exercise 4.12 over a f ...
... Exercise 4.13. Let X ⊂ A2 be the zero locus of a single polynomial ∑i+ j≤d ai, j x1i x2j of degree at most d. Show that: (a) Any line in A2 (i.e. any zero locus of a single polynomial of degree 1) not contained in X intersects X in at most d points. (b) Any affine conic (as in Exercise 4.12 over a f ...
Abstracts of Papers
... X/T (R). A topology τ on X is quasi-spectral when it satisfies all of Hochster’s spectral conditions except for requiring T0 . We show that (X, R) is spectral if and only if (X/T (R), ≤R ) is a spectral ordered space. (There is an order compatible quasi-spectral topology on X.) We extend this result ...
... X/T (R). A topology τ on X is quasi-spectral when it satisfies all of Hochster’s spectral conditions except for requiring T0 . We show that (X, R) is spectral if and only if (X/T (R), ≤R ) is a spectral ordered space. (There is an order compatible quasi-spectral topology on X.) We extend this result ...
Chapter 2 (as PDF)
... but: ρ is a linear map and [x, y]ρ = [xρ, yρ] = (xρ) · (yρ) − (yρ) · (xρ) for all x, y ∈ L. Two representations ρ : L → Lie(End(V )) and ρ 0 : L → Lie(End(V 0 )) of degree n are called equivalent, if there is an invertible linear map T : V → V 0 such that (xρ) · T = T · (xρ 0 ) for all x ∈ L (the do ...
... but: ρ is a linear map and [x, y]ρ = [xρ, yρ] = (xρ) · (yρ) − (yρ) · (xρ) for all x, y ∈ L. Two representations ρ : L → Lie(End(V )) and ρ 0 : L → Lie(End(V 0 )) of degree n are called equivalent, if there is an invertible linear map T : V → V 0 such that (xρ) · T = T · (xρ 0 ) for all x ∈ L (the do ...
Thèse de doctorat - IMJ-PRG
... Further, the second chapter contains an appendix by Sergey Neshveyev who showed that our arithmetic model is essentially unique. The last chapter is concerned with functoriality properties of BC-systems. More precisely, in the context of Endomotives, we will construct an algebraic refinement of a fu ...
... Further, the second chapter contains an appendix by Sergey Neshveyev who showed that our arithmetic model is essentially unique. The last chapter is concerned with functoriality properties of BC-systems. More precisely, in the context of Endomotives, we will construct an algebraic refinement of a fu ...
LECTURE NOTES OF INTRODUCTION TO LIE GROUPS
... Q: What we are interested in is Lie groups, then why do we study topological groups, even though some of them are not Lie groups? Example : natural example of topological group. Let X be a topological space1. Let Hemeo(X) denote the group of all homeomorphisms of X. Claim: Hemeo(X) with the topology ...
... Q: What we are interested in is Lie groups, then why do we study topological groups, even though some of them are not Lie groups? Example : natural example of topological group. Let X be a topological space1. Let Hemeo(X) denote the group of all homeomorphisms of X. Claim: Hemeo(X) with the topology ...
Frobenius algebras and 2D topological quantum field theories (short
... in which one can do calculations and gain experience before embarking on the quest for the full-fledged theory. Roughly, the closed manifolds represent space, while the cobordisms represent space-time. The associated vector spaces are then the state spaces, and an operator associated to a space-tim ...
... in which one can do calculations and gain experience before embarking on the quest for the full-fledged theory. Roughly, the closed manifolds represent space, while the cobordisms represent space-time. The associated vector spaces are then the state spaces, and an operator associated to a space-tim ...
Lecture 4 Super Lie groups
... is a super diffeomorphism of W ′ × H onto the open sub-supermanifold of G with reduced manifold the open subset |W ′ ||H| of |G|. Proof. The map γ in question is the informal description of the map µ◦(iW ′ × iH ) where iM refers to the canonical inclusion M ֒→ G of a sub-supermanifold of G into G, a ...
... is a super diffeomorphism of W ′ × H onto the open sub-supermanifold of G with reduced manifold the open subset |W ′ ||H| of |G|. Proof. The map γ in question is the informal description of the map µ◦(iW ′ × iH ) where iM refers to the canonical inclusion M ֒→ G of a sub-supermanifold of G into G, a ...
Homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. The development of homological algebra was closely intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of chain complexes, which can be studied both through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences.From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes.