Abstract:
Consider the Cohomological Hall Algebra as defined by Kontsevich and Soibelman, associated with a Dynkin quiver. We reinterpret the geometry behind the multiplication map in the COHA, and give an iterated residue formula for it. We show natural subalgebras whose product is the whole COHA (except in the $E_8$ case). The dimension count version of this statement is an identity for quantum dilogarithm series, first proved by Reineke. We also show that natural structure constants of the COHA are universal polynomials representing degeneracy loci, a.k.a. quiver polynomials.

Abstract:
Maps between manifolds $M^m\to N^{m+\ell}$ ($\ell>0$) have multiple points, and more generally, multisingularities. The closure of the set of points where the map has a particular multisingularity is called the multisingularity locus. There are universal relations among the cohomology classes represented by multisingularity loci, and the characteristic classes of the manifolds. These relations include the celebrated Thom polynomials of monosingularities. For multisingularities, however, only the form of these relations is clear in general (due to Kazarian), the concrete polynomials occurring in the relations are much less known. In the present paper we prove the first general such relation outside the region of Morin-maps: the general quadruple point formula. We apply this formula in enumerative geometry by computing the number of 4-secant linear spaces to smooth projective varieties. Some other multisingularity formulas are also studied, namely 5, 6, 7 tuple point formulas, and one corresponding to $\Sigma^2\Sigma^0$ multisingularities.

Abstract:
We consider the rational dynamical quantum group $E_y(gl_2)$ and introduce an $E_y(gl_2)$-module structure on $\oplus_{k=0}^n H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$, where $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$ is the equivariant cohomology algebra $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))$ of the cotangent bundle of the Grassmannian $\Gr(k,n)$ with coefficients extended by a suitable ring of rational functions in an additional variable $\lambda$. We consider the dynamical Gelfand-Zetlin algebra which is a commutative algebra of difference operators in $\lambda$. We show that the action of the Gelfand-Zetlin algebra on $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$ is the natural action of the algebra $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))\otimes \C[\delta^{\pm1}]$ on $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$, where $\delta : \zeta(\lambda)\to\zeta(\lambda+y)$ is the shift operator. The $E_y(gl_2)$-module structure on $\oplus_{k=0}^n H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$ is introduced with the help of dynamical stable envelope maps which are dynamical analogs of the stable envelope maps introduced by Maulik and Okounkov. The dynamical stable envelope maps are defined in terms of the rational dynamical weight functions introduced in [FTV] to construct q-hypergeometric solutions of rational qKZB equations. The cohomology classes in $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$ induced by the weight functions are dynamical variants of Chern-Schwartz-MacPherson classes of Schubert cells.

Abstract:
Consider the natural torus action on a partial flag manifold $Fl$. Let $\Omega_I\subset Fl$ be an open Schubert variety, and let $c^{sm}(\Omega_I)\in H_T^*(Fl)$ be its torus equivariant Chern-Schwartz-MacPherson class. We show a set of interpolation properties that uniquely determine $c^{sm}(\Omega_I)$, as well as a formula, of `localization type', for $c^{sm}(\Omega_I)$. In fact, we proved similar results for a class $\kappa_I\in H_T^*(Fl)$ --- in the context of quantum group actions on the equivariant cohomology groups of partial flag varieties. In this note we show that $c^{SM}(\Omega_I)=\kappa_I$.

Abstract:
Grothendieck polynomials are important objects in the study of the K-theory of flag varieties. Their many remarkable properties have been studied in the context of algebraic geometry and tableaux combinatorics. We explore a new tool, similar to generating sequences, which we call the iterated residue technique. We prove multiplication and comultiplication formulas for Grothendieck polynomials in terms iterated residues and other results useful for computations. Moreover, we show the effectiveness of the method with new proofs that the Pieri rule for stable Grothendieck polynomials and their Schur expansions exhibit alternating signs.

Abstract:
Using equivariant localization formulas we give a formula for conformal blocks at level one on the sphere as suitable polynomials. Using this presentation we give a generating set in the space of conformal blocks at any level if the marked points on the sphere are generic.

Abstract:
In this paper we propose a systematic study of Thom polynomials for group actions defined by M. Kazarian. On one hand we show that Thom polynomials are first obstructions for the existence of a section and are connected to several problems of topology, global geometry and enumerative algebraic geometry. On the other hand we describe a way to calculate Thom polynomials: the method of restriction equations. It turned out that though the idea is quite simple the method is very powerful. We reproduced and improved earlier result in several directions (singularities, Schubert calculus, quivers). However a proper introduction to the basic theorems was missing. In this paper we try to pay this debt as well as we present the connections with obstruction theory and equivariant cohomology. We give some new results and outline possible generalizations and problems.

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:
We prove a positive combinatorial formula for the equivariant class of an orbit closure in the space of representations of an arbitrary quiver of type $A$. Our formula expresses this class as a sum of products of Schubert polynomials indexed by a generalization of the minimal lace diagrams of Knutson, Miller, and Shimozono. The proof is based on the interpolation method of Feh\'er and Rim\'anyi. We also conjecture a more general formula for the equivariant Grothendieck class of an orbit closure.

Abstract:
We study the canonical U(n-)-valued differential form, whose projections to different Kac-Moody algebras are key ingredients of the hypergeometric integral solutions of KZ-type differential equations and Bethe ansatz constructions. We explicitly determine the coefficients of the projections in the simple Lie albegras A_r, B_r, C_r, D_r in a conviniently chosen Poincare-Birkhoff-Witt basis. As a byproduct we obtain results on the combinatorics of rational functions, namely non-trivial identities are proved between certain rational functions with partial symmetries.