Abstract:
We study cubical sets without degeneracies, which we call square sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a square set C has an infinite family of associated square sets J^i(C), i=1,2,..., which we call James complexes. There are mock bundle projections p_i:|J^i(C)|-->|C| (which we call James bundles) defining classes in unstable cohomotopy which generalise the classical James--Hopf invariants of Omega(S^2). The algebra of these classes mimics the algebra of the cohomotopy of Omega(S^2) and the reduction to cohomology defines a sequence of natural characteristic classes for a square set. An associated map to BO leads to a generalised cohomology theory with geometric interpretation similar to that for Mahowald orientation [M Mahowald, Ring Spectra which are Thom complexes, Duke Math. J. 46 (1979) 549--559] and [B Sanderson, The geometry of Mahowald orientations, SLN 763 (1978) 152--174].

Abstract:
We describe the positivity of Thom polynomials of singularities of maps, Lagrangian Thom polynomials and Legendrian Thom polynomials. We show that these positivities come from Schubert calculus.

Abstract:
The popular show Supernatural, in particular the character of Dean Winchester, provides an interesting examination of freedom of choice. In fact Supernatural proves itself amenable to an existentialist reading of law, in particular the existentialism of Jean Paul Sartre. After a brief introduction to the show, the elements of Sartre’s existentialism I will be developing in this paper include freedom, choice and authenticity. These elements combine to demonstrate the existentialist law favoured by Dean, whereby Dean’s scepticism of God allows for an authenticity that furthers his own autonomy. The patterning trope of two brothers is essential to Dean developing his own law and morality separate from the divine one that is imposed on him throughout the show, with natural law showcasing that there is more than one kind of existentialist law to choose from.

Abstract:
We analyze the functorial and multiplicative properties of the Thom spectrum functor in the setting of symmetric spectra, and we establish the relevant homotopy invariance.

Abstract:
Thom polynomials of singularities express the cohomology classes dual to singularity submanifolds. A stabilization property of Thom polynomials is known classically, namely that trivial unfolding does not change the Thom polynomial. In this paper we show that this is a special case of a product rule. The product rule enables us to calculate the Thom polynomials of singularities if we know the Thom polynomial of the product singularity. As a special case of the product rule we define a formal power series (Thom series, Ts_Q) associated with a commutative, complex, finite dimensional local algebra Q, such that the Thom polynomial of {\em every} singularity with local algebra Q can be recovered from Ts_Q.

Abstract:
In this paper we identify conditions under which the cohomology $H^*(\Omega M\xi;\k)$ for the loop space $\Omega M\xi$ of the Thom space $M\xi$ of a spherical fibration $\xi\downarrow B$ can be a polynomial ring. We use the Eilenberg-Moore spectral sequence which has a particularly simple form when the Euler class $e(\xi)\in H^n(B;\k)$ vanishes, or equivalently when an orientation class for the Thom space has trivial square. As a consequence of our homological calculations we are able to show that the suspension spectrum $\Sigma^\infty\Omega M\xi$ has a local splitting replacing the James splitting of $\Sigma\Omega M\xi$ when $M\xi$ is a suspension.

Abstract:
Thom polynomials measure how global topology forces singularities. The power of Thom polynomials predestine them to be a useful tool not only in differential topology, but also in algebraic geometry (enumerative geometry, moduli spaces) and algebraic combinatorics. The main obstacle of their widespread application is that only a few, sporadic Thom polynomials have been known explicitly. In this paper we develop a general method for calculating Thom polynomials of contact singularities. Along the way, relations with the equivariant geometry of (punctual, local) Hilbert schemes, and with iterated residue identities are revealed.

Abstract:
We study Legendrian singularities arising in complex contact geometry. We define a one-parameter family of bases in the ring of Legendrian characteristic classes such that any Legendrian Thom polynomial has nonnegative coefficients when expanded in these bases. The method uses a suitable Lagrange Grassmann bundle on the product of projective spaces. This is an extension of a nonnegativity result for Lagrangian Thom polynomials obtained by the authors previously. For a fixed pecialization, other specializations of the parameter lead to upper bounds for the coefficients of the given basis. One gets also upper bounds of the coefficients from the positivity of classical Thom polynomials (for mappings), obtained previously by the last two authors.

Abstract:
We prove a formula for Thom polynomials of Morin (or A_d) singularities in any codimension. We use a combination of the test-curve method of Porteous, and the localization methods in equivariant cohomology. Our formulas are independent of the codimension, and they are computationally efficient for d less than 7.