We develop a theory of Thom series for contact function singularities. Quadratic stabilization σ_q of a contact singularity η∈ J^k(n,1) gives contact singularities σ_q^iη∈ J^k(n+i,1). We study the stable Thom polynomials (i.e. the Thom polynomial in quotient variables) of σ_q^iη as i increases. To this end, we define the Thom series of a contact function singularity η, which for any given i is a linear combination of Schur polynomials s_λ. For large enough i the value of the Thom series at 1-(n+i) is the stable Thom polynomial of σ_q^iη and for small i it determines its coefficients outside the kernel of a specialization map. We prove that the Thom series has two main properties: 1) as i increases, the partitions λ follow a simple stabilization pattern, and there is a finite set of λ which generates the support of the entire Thom series via this stabilization; 2) the coefficients of the s_λ are polynomials in i with explicit degree bounds. We describe the precise relationship between unstable and stable Thom polynomials of contact function singularities and Legendre Thom polynomials and we carry out computations of all three, as follows. We compute the complete family of stable Thom polynomials for function singularities with γ≤ 6. We compute unstable and Legendre Thom polynomials for several families of binary and ternary singularities. We define a class of multi-binary singularities, and compute their Thom polynomials. We discuss second order Thom-Boardman classes, and indicate difficulties that arise beyond function singularities. The results of the paper will be used in a companion paper, where Thom polynomials will be applied to problems in enumerative geometry.

We study the symmetric polynomial ∏_α∈ A_n,d(1+α_1 x_1+⋯+α_n x_n) where A_n,d:={α∈ℤ_≥ 0^n:|α|=d}, which is the total Chern class of Sym^d(ℂ^n), viewed as a torus representation whose Chern roots are the weights α_1 x_1+⋯+α_n x_n for α∈ A_n,d. Its homogeneous degree-k part c_k(n,d) is the k-th Chern class of Sym^d(ℂ^n). These Chern classes, together with their coefficients in various symmetric function bases, play a central role in enumerative geometry. Despite their simple definition, general closed formulas for their coefficients are subtle, and many structural properties of these classes have remained poorly understood. In this paper we prove several conjectures concerning their structure, establish explicit formulas, and study log-concavity properties for both the Chern classes and their K-theoretic analogue. In rank two, passing to the Schur basis and expanding the Schur coefficients in the binomial basis of d, we uncover a new binomial log-concavity phenomenon and prove refined positivity results. The paper demonstrates a novel methodology: we combine several AI systems with human mathematical insight in a coordinated workflow, deploying each tool according to its strengths in experimental discovery, conjecture formation, symbolic proof construction, and verification. To our knowledge, this is one of the first detailed case studies of orchestrating multiple AI tools to make substantial progress on a coherent mathematical research project.

For a singularity type eta, let the eta-avoiding number of an n-dimensional manifold M be the lowest k for which there is a map M -> Rn+k without i type singular points. For instance, the case of eta = Sigma(1) is the case of immersions, which has been extensively studied in the case of real projective spaces. In this paper, we study the i-avoiding number for other singularity types. Our results come in two levels: first we give an abstract reasoning that a nonzero cohomology class is supported on the singularity locus eta(f), proving that eta(f) cannot be empty. Second, we interpret this obstruction as a nonzero invariant of the singularity locus eta(f) for generic f. The main technique that we employ is Sullivan's Stiefel-Whitney classes, which are mod 2, real analogues of the Chern-Schwartz-MacPherson classes. We introduce the Segre-Stiefel-Whitney classes of a singularity s(eta)(sw) whose lowest degree term is the mod 2 Thom polynomial of eta. Using these techniques, we compute some universal formulae for the Euler characteristic of a singularity locus.

We introduce a geometric method to study additive combinatorial problems. Using equivariant cohomology we reprove the Dias da Silva-Hamidoune theorem. We improve a result of Sun on the linear extension of the Erdős-Heilbronn conjecture. We generalize a theorem of G. Kós (the Grashopper problem) which in some sense is a simultaneous generalization of the Erdős-Heilbronn conjecture. We also prove a signed version of the Erdős-Heilbronn conjecture and the Grashopper problem. Most identities used are based on calculating the projective degree of an algebraic variety in two different ways.

We develop the theory of halving spaces to obtain lower bounds in real enumerative geometry. Halving spaces are topological spaces with an action of a Lie group $\Gamma$ with additional cohomological properties. For $\Gamma=\mathbb{Z}_2$ we recover the conjugation spaces of Hausmann, Holm and Puppe. For $\Gamma=\mathrm{U}(1)$ we obtain the circle spaces. We show that real even and quaternionic partial flag manifolds are circle spaces leading to non-trivial lower bounds for even real and quaternionic Schubert problems. To prove that a given space is a halving space, we generalize results of Borel and Haefliger on the cohomology classes of real subvarieties and their complexifications. The novelty is that we are able to obtain results in rational cohomology instead of modulo 2. The equivariant extension of the theory of circle spaces leads to generalizations of the results of Borel and Haefliger on Thom polynomials.

We study a K-theoretic characteristic class of singular varieties, namely the equivariant motivic Chern class. We prove that the motivic Chern class is characterized by an axiom system inspired by that of "K-theoretic stable envelopes," recently defined by Okounkov and studied in relation with quantum group actions on the K-theory algebra of moduli spaces. We also give explicit formulas for the equivariant motivic Chern classes of Schubert cells and matrix Schubert cells. Lastly, we calculate the equivariant motivic Chern class of the orbits of the A2 quiver representation, which yields formulas for the motivic Chern classes of determinantal varieties and more general degeneracy loci.

We study motivic Chern classes of cones. First we show examples of projective cones of smooth curves such that their various $K$-classes (sheaf theoretic, push-forward and motivic) are all different. Then we show connections between the torus equivariant motivic Chern class of a projective variety and of its affine cone, generalizing results on projective Thom polynomials.

Consider a complex algebraic group $G$ acting on a smooth variety $M$ with finitely many orbits, and let $\Omega$ be an orbit. The following three invariants of $\Omega\subset M$ can be characterized axiomatically: (1) the equivariant fundamental class $[\overline{\Omega}, M]\in H^*_G(M)$, (2) the equivariant Chern-Schwartz-MacPherson class $c(\Omega, M)\in H^*_G(M)$, and (3) the equivariant motivic Chern class $mC(\Omega, M) \in K_G(M)[y]$. The axioms for Chern-Schwartz-MacPherson and motivic Chern classes are motivated by the axioms for cohomological and K-theoretic stable envelopes of Okounkov and his coauthors. For $M$ a flag variety and $\Omega$ a Schubert cell---an orbit of the Borel group acting---this implies that CSM and MC classes coincide with the weight functions studied by Rimanyi-Tarasov-Varchenko. In this paper we review the general theory and illustrate it with examples.

. Thorn polynomials compute the cohomology classes of degen eracy loci. In this paper we use a simple example to review the core ideas in different-mostly recently found-methods of comput ing Thorn polynomials. Our goal is to show the underlying topol ogy I geometry I algebra without involving combinatorics.

We consider a smooth algebraic variety with an action of a linear algebraic group acting with finitely many orbits. We study equivariant characteristic classes of the orbits, namely the equivariant motivic Chern classes, in the K-theory of the ambient space. We prove that the motivic Chern class satisfies the axiom system inspired by that of “K-theoretic stable envelopes”, recently defined by Okounkov and studied in relation with quantum group actions on the Ktheory algebra of moduli spaces. We also give explicit formulas for the equivariant motivic Chern classes of Schubert cells and matrix Schubert cells. Lastly, we calculate the equivariant motivic Chern class of the orbits of the A2 quiver representation, which yields formulas for the motivic Chern classes of determinantal varieties and more general degeneracy loci.

We study some global aspects of differential complex 2and 3-forms on complex manifolds. We compute the cohomology classes represented by the sets of points on a manifold where such a form degenerates in various senses, together with other similar cohomological obstructions. Based on these results and a formula for projective representations, we calculate the degree of the projectivization of certain orbits of the representation ΛC.

The Chern-Schwartz-MacPherson class (CSM) and the Segre-Schwartz-MacPherson class (SSM) are deformations of the fundamental class of an algebraic variety. They encode finer enumerative invariants of the variety than its fundamental class. In this paper we offer three contributions to the theory of equivariant CSM/SSM classes. First, we prove an interpolation characterization for CSM classes of certain representations. This method---inspired by recent works of Maulik-Okounkov and Gorbounov-Rimanyi-Tarasov-Varchenko---does not require a resolution of singularities and often produces explicit (not sieve) formulas for CSM classes. Second, using the interpolation characterization we prove explicit formulas---including residue generating sequences---for the CSM and SSM classes of matrix Schubert varieties. Third, we suggest that a stable version of the SSM class of matrix Schubert varieties will serve as the building block of equivariant SSM theory, similarly to how the Schur functions are the building blocks of fundamental class theory. We illustrate these phenomena, and related stability and (2-step) positivity properties for some relevant representations.

We study a 2-parameter family of enumerative problems over the reals. Over the complex field, these problems can be solved by Schubert calculus. In the real case the number of solutions can be different on the distinct connected components of the configuration space, resulting in a solution function. The cohomology calculation in the real case only gives the signed sum of the solutions, therefore in general it only gives a lower bound on the range of the solution function. We calculate the solution function for the 2-parameter family and we show that in the even cases the solution function is constant modulo 4. We show how to determine the sign of a solution and describe the connected components of the configuration space. We translate the problem to the language of quivers and also give a geometric interpretation of the sign. Finally, we discuss what aspects might be considered when solving other real enumerative problems.

Two parameter families of plane conics are called nets of conics. There is a natural group action on the vector space of nets of conics, namely the product of the group reparametrizing the underlying plane, and the group reparametrizing the parameter space of the family. We calculate equivariant fundamental classes of orbit closures. Based on this calculation we develop the invariant theory of nets of conics. As an application we determine Thom polynomials of contact singularities of type (3,3). We also show how enumerative problems, in particular the intersection multiplicities of the determinant map from nets of conics to plane cubics, can be solved studying equivariant classes of orbit closures.

Consider an integer associated with every subset of the set of columns of an $n\times k$ matrix. The collection of those matrices for which the rank of a union of columns is the predescribed integer for every subset, will be denoted by $X_C$. We study the equivariant cohomology class represented by the Zariski closure $Y_C$ of this set. We show that the coefficients of this class are solutions to problems in enumerative geometry, which are natural generalization of the linear Gromov-Witten invariants of projective spaces. We also show how to calculate these classes and present their basic properties.

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.

Two parameter families of plane conics are called nets of conics. There is a natural group action on the vector space of nets of conics, namely the product of the group reparametrizing the underlying plane, and the group reparametrizing the parameter space of the family. We calculate equivariant fundamental classes of orbit closures. Based on this calculation we develop the invariant theory of nets of conics. As an application we determine Thom polynomials of contact singularities of type (3,3). We also show how enumerative problems---in particular the intersection multiplicities of the determinant map from nets of conics to plane cubics---can be solved studying equivariant classes of orbit closures.

R. Rimányi defined the incidence class of two singularities X and Y as [X]|_Y, the restriction of the Thom polynomial of X to Y. He conjectured that (under mild conditions) the incidence is not zero if and only if Y is in the closure of X. Generalizing this notion we define the incidence class of two orbits X and Y of a representation. We give a sufficient condition (positivity) for Y to have the property that the incidence class [X]|_Y is not zero if and only if Y is in the closure of X for any other orbit X. We show that for many interesting cases, e.g. the quiver representations of Dynkin type positivity holds for all orbits. In other words in these cases the incidence classes completely determine the hierarchy of the orbits. We also study the case of singularities where positivity doesn't hold for all orbits.