There is a general notion of the magnitude of an enriched category, defined subject to hypotheses. In topological and geometric contexts, magnitude is already known to be closely related to classical invariants such as Euler characteristic and dimension. Here we establish its significance in an algebraic context. Specifically, in the representation theory of an associative algebra A, a central ...

We compare Green’s function g on an infinite volume, hyperbolic Riemann surface X with an analogous discrete function gdisc on a graphical caricature Γ of X. The main result, modulo technical hypotheses, is that g and gdisc differ by at most an additive constant C which depends only on the Euler characteristic of X. In particular, the estimate of g by gdisc remains uniform as the geometry (i.e....

We discuss an algorithm computing the push-forward to projective space of several classes associated to a (possibly singular, reducible, nonreduced) projective scheme. For example, the algorithm yields the topological Euler characteristic of the support of a projective scheme S, given the homogeneous ideal of S. The algorithm has been implemented in Macaulay2.

It is well known that a three dimensional (closed, connected and compact) manifold is obtained by identifying boundary faces from a stellar ball a ⋆ S. The study of S/ ∼, two dimensional stellar sphere S with 2-simplexes identified in pairs leads us to the following conclusion: either a three dimensional manifold is homeomorphic to a sphere or to a stellar ball a⋆S with its boundary 2-simplexes...

1. Review of Differential Geometry 2 2. Definition of an Elliptic Operator 5 3. Properties of Elliptic Operators 7 4. Example of an Elliptic Operator 9 5. Example: The Euler Characteristic 12 6. Example: The Signature Invariant 14 7. A Theorem of Atiyah, Frank and Mayer 18 8. Clifford Algebras 20 9. A Diversion: Constructing Vector Fields on Spheres using Clifford Algebras 23 10. Topological In...

We define motivic Milnor fibre of cyclic L∞-algebras of dimension three using the method of Denef and Loeser of motivic integration. It is proved by Nicaise and Sebag that the topological Euler characteristic of the motivic Milnor fiber is equal to the Euler characteristic of the étale cohomology of the analytic Milnor fiber. We prove that the value of Behrend function on the germ moduli space ...

If a real value invariant of closed combinatorial manifolds admits a local formula that depends only on the f -vector of the link of each vertex, then the invariant must be proportional to the Euler characteristic.

We will show that any complex minimal surface of general type with c1 = 2χ− 1 having non-trivial 2-torsion divisors, where c 2 1 and χ are the first Chern number and the Euler characteristic of the structure sheaf respectively, has the Euler characteristic χ not exceeding 4. We will also give a complete description for the surfaces of the case χ = 4.

For a partition λ = {λ1 ≥ λ2 ≥ λ3 ≥ 0} of non-negative integers, we calculate the Euler characteristic of the local system Vλ on the moduli space of genus 3 hyperelliptic curves using a suitable stratification. For some λ of low degree, we make a guess for the motivic Euler characteristic of Vλ using counting curves over finite fields.

We have initiated the study of topology of the space of coverings on grid domains. The space has the following constraint: while all the covering agents can move freely (we allow overlapping) on the domain, their union must cover the whole domain. A minimal number N of the covering agents is required for a successful covering of the domain. In this paper, we demonstrate beautiful topological st...