The two Rogers-Ramanujan $q$-series \[ \sum_{n=0}^{\infty}\frac{q^{n(n+\sigma)}}{(1-q)\cdots (1-q^n)}, \] where $\sigma=0,1$, play many roles in mathematics and physics. By the Rogers-Ramanujan identities, they are essentially modular functions. Their quotient, the Rogers-Ramanujan continued fraction, has the special property that its singular values are algebraic integral units. We find a framework which extends the Rogers-Ramanujan identities to doubly-infinite families of $q$-series identities. If $a\in\{1,2\}$ and $m,n\geq 1$, then we have \[ \sum_{\substack{\lambda \lambda_1\leq m}} q^{a|\lambda|} P_{2\lambda}(1,q,q^2,\dots;q^n) =\textrm{"infinite product modular function"}, \] where the $P_{\lambda}(x_1,x_2,\dots;q)$ are Hall-Littlewood polynomials. These $q$-series are specialized characters of affine Kac--Moody algebras. Generalizing the Rogers-Ramanujan continued fraction, we prove in the case of $\textrm{A}_{2n}^{(2)}$ that the relevant $q$-series quotients are integral units.

A framework of Rogers-Ramanujan identities and their arithmetic properties

We conjecture that the stable Khovanov homology of torus knots can be described as the Koszul homology of an explicit non-regular sequence of quadratic polynomials. The corresponding Poincare series turns out to be related to the Rogers-Ramanujan identity.

On stable Khovanov homology of torus knots

Arc spaces have been introduced in algebraic geometry as a tool to study singularities but they show strong connections with combinatorics as well. Exploiting these relations we obtain a new approach to the classical Rogers- Ramanujan Identities. The linking object is the Hilbert-Poincar´ e series of the arc space over a point of the base variety. In the case of the double point this is precisely the generating series for the integer partitions without equal or consecutive parts. R´ esum´

http://combinatorics.cis.strath.ac.uk/fpsac2011/proceedings/dmAO0120.pdf

Arc Spaces and Rogers-Ramanujan Identities

Arc spaces have been introduced in algebraic geometry as a tool to study singularities but they show strong connections with combinatorics as well. Exploiting these relations we obtain a new approach to the classical Rogers-Ramanujan Identities. The linking object is the Hilbert-Poincar\'e series of the arc space over a point of the base variety. In the case of the double point this is precisely the generating series for the integer partitions without equal or consecutive parts.

The two Rogers-Ramanujan q-series where σ 0; 1, play many roles in mathematics and physics. By the Rogers- Ramanujan identities, they are essentially modular functions. Their quotient, the Rogers-Ramanujan continued fraction, has the special property that its singular values are algebraic integral units. We find a framework which extends the Rogers- Ramanujan identities to doubly infinite families of q-series identities. If a ∈ (1, 2) and m, n ≥ 1, then we have [infinite product modular function], where the P λ(x, x, ...; q) are Hall-Littlewood polynomials. These q-series are specialized characters of affine Kac-Moody algebras. Generalizing the Rogers- Ramanujan continued fraction, we prove in the case of A that the relevant q-series quotients are integral units.

/pdf/a-framework-of-rogers-ramanujan-identities-and-their-25jn4i4339.pdf

A framework of Rogers–Ramanujan identities and their arithmetic properties

For m 2 N, we determine the irreducible components of the m th Jet Scheme of a complex branch C and we give formulas for their number N(m) and for their codimensions, in terms of m and the generators of the semigroup of C. This structure of the Jet Schemes determines and is determined by the topological type of C.

/pdf/jet-schemes-of-complex-plane-branches-and-equisingularity-3xfhh0dao7.pdf

Jet schemes of complex plane branches and equisingularity

For a plane branch C with g Puiseux pairs, we determine the irreducible components of its jet schemes which correspond to the star (or rupture) and end divisors that appear on the dual graph of the minimal embedded desingularization of C. We exploit these informations to construct a Teissier type resolution of C embedded in $${\mathbb{C}^{g+1}}$$
 , which is special in the sense that its restriction to the strict transform of the plane induces the minimal embedded desingularization of C.

https://webusers.imj-prg.fr/~hussein.mourtada/Jetsanddesing.pdf

Jet schemes and minimal embedded desingularization of plane branches

We prove that for m ∈ N, m big enough, the number of irreducible components of the schemes of m−jets centered at a point which is a double point singularity is independent of m and is equal to the number of exceptional curves on the minimal resolution of the singularity. We also relate some irreducible components of the jet schemes of an E6 singularity to its ”minimal” embedded resolutions of singularities. 2010 Mathematics Subject Classification. Primary 14E18,14B05; Secondary 13D40.

/pdf/jet-schemes-of-rational-double-point-singularities-43dnuid1u8.pdf

Hussein Mourtada

Papers

Arc Spaces and Rogers-Ramanujan Identities

Jet schemes of complex plane branches and equisingularity

Arc Spaces and Rogers-Ramanujan Identities

Jet schemes and minimal embedded desingularization of plane branches

Jet schemes of rational double point singularities