Danilo Lewański

Research interests:

I am interested in the interaction between Algebraic Geometry, Mathematical Physics and Theoretical Physics. In particular, I study the connections between the cohomology of moduli spaces of curves in Algebraic Geometry, integrable systems and integrable hierarchies in Mathematical Physics,  Physics models arising from (topological) string theory and Statistical Physics models arising from random matrix integrals. My research employs the innovative methods of Topological Recursion (TR) theory. 

Promoted to a mathematical theory in 2007 by Eynard and Orantin, Topological Recursion has proved to be a universal and unifying tool across different fields, leading to significant progress in enumerative algebraic geometry, integrable systems and hierarchies, random matrix models, Gromov-Witten theory, Hurwitz theory, enumeration of maps on surfaces, topological string theory, moduli spaces of curves, Hitchin systems, knot theory theory, resurgence, and more. 

Here you can find the video of two of my recent talks at the Topological Recursion seminar.

FIGURE 1: A schematic drawing on the interaction between Theoretical Physics, Mathematical Physics, and Algebraic Geometry; the hidden underlying instanton corrections from Resurgence; Topological Recursion as a tool to build a solid understanding of the theory.

A little bit about topological recursion:

Random matrix models carry a fundamental object called spectral curve, and TR generates the asymptotic expansion of the correlation functions of the model from this spectral curve. In different words, TR can be thought as an algorithm, concretely  implementable, which takes as input a spectral curve and produces as output the infinite list of numbers. These numbers are solutions of some enumerative problem --- matching enumerative problems  and their corresponding spectral curves is one direction of research for TR. TR has three very valuable features:

1. The algorithm only needs the spectral curve to work, whereas the matrix model can be non-existent or extremely hard to find. Therefore TR promotes a handy shift of focus for what concerns the initial data of the recursion.

2. The algorithm is universal: it is the same recursive formula for very different enumerative problems. This is possible because the recursion “takes place” over different spectral curves.

3. A cohomological description (often a cohomological field theory, CohFT for short) is provided directly from the spectral curve, and the output numbers are its integrals over the moduli spaces of stable curves of fixed genus and number of marked points. This feature has been used several times to prove or re-prove involved ELSV-type formulae.

In a simplified setting, a spectral curve can be thought of the tuple  (C, x(z), y(z), B(z_1, z_2)), where:

•  C is (possibly open in) a Torelli marked Riemann surface, 

• the functions x,y are meromorphic on C, 

• x(z) has only simple branch points (which moreover do not intersect the branch points of y(z)), and defines C as a cover over the Riemann sphere.

• B is a meromorphic differential defined on the topological product CxC with a double pole on the diagonal with bi-residue one. The canonical B is (dz_1 dz_2)/(z_1 - z_2)^2, but it can have a meaningful addition of a holomorphic part.

The initial data of the recursion is given by ω_{0,1}(z) = y(z)dx(z) and ω_{0,2}(z_1, z_2) = B(z_1, z_2).

FIGURE 2. A schematic representation of Topological Recursion (TR). It can be thought as an implementable algorithm which takes a spectral curve as input and spits as output an infinite set of numbers {N_{g,μ}} indexed by a natural number g and a partition μ of positive length n. These numbers arise from the series expansion of meromorphic n-multidifferentials ω_{g,n}(z_1, . . . , z_n) defined on the topological product of n copies of the curve. Each ω_{g,n} is computed recursively in terms of all other ω_{g ′ ,n ′} with 2g′ −2+n′ < 2g−2+n, and this is why it is called a recursion. The recursion is performed layer by layer, each layer corresponding to a fixed Euler characteristic 2g - 2 + n, and that motivates the adjective topological. Let us make a few examples. The input spectral curve already produces ω_{0,1} and ω_{0,2}.  So the first layers read:

Layer 2g - 2 + n = 1: computation of ω_{1,1}, ω_{0,3}

Layer 2g - 2 + n = 2: computation of ω_{1,2}, ω_{0,4}

Layer 2g - 2 + n = 3: computation of ω_{2,1}, ω_{1,3}, ω_{0,5}

Layer 2g - 2 + n = 4: computation of ω_{2,2}, ω_{1,4}, ω_{0,6}

Layer 2g - 2 + n = 5: computation of ω_{3,1}, ω_{2,3}, ω_{1,5}, ω_{0,7}

Layer 2g - 2 + n = 6: computation of ω_{3,2}, ω_{2,4}, ω_{1,6}, ω_{0,8}

Layer 2g - 2 + n = 7: computation of ω_{4,1}, ω_{3,3}, ω_{2,5}, ω_{1,7}, ω_{0,9}

Layer 2g - 2 + n = 8: computation of ω_{4,2}, ω_{3,4}, ω_{2,6}, ω_{1,8}, ω_{0,10}

.......

and so on. Each layer uses all preceding layers in combination with ω_{0,1} and ω_{0,2} and a certain recursion kernel.

 Let us mention a few examples of enumerative geometric problems computed by TR (we are omitting important details, normalisations, and citations, to keep the reading light. For more details we refer to Eynard's beautiful book "Counting Surfaces") : 

1. The simplest possible spectral curve (the parabola x(z) = z^2) produces the simplest possible CohFT: the Witten-Kontsevich case, i.e. the identity CohFT 1. 

2. A slightly more complicated curve, x(z) as the sine function, produces as CohFT the exponent of the first Mumford kappa class. Its integrals are the volumes of moduli spaces of hyperbolic structures, computed recursively in the beautiful work of M. Mirzakhani. TR provided a new proof of the two results. 

3. Recent works of Stanford-Witten (2019) and of Norbury (2000) show that x(z) as the cosine function produces after integration the volumes of moduli spaces of hyperbolic structures of super Riemann surfaces. 

4. Bouchard-Mariño conjecture (now theorem) states that the curve given by x(z) = - z + log(z) generates Hurwitz numbers (H). Hurwitz numbers enumerate conencted coverings of the Riemann sphere of fixed genus and fixed ramification profile over infinity, with all other ramifications being simple (only two sheets branching). The cohomology class produced by TR is the total Chern class of the Hodge bundle of holomorphic differentials.

5. The curve given by x(z) = - z^r + log(z), y(z) = z^q, generates by Zvonkine Generalised conjecture (now theorem) certain Gromov-Witten (GW) invariants of equivariant P^1. By Okounkov-Pandharipande GW/H correspondence, these invariants are q-orbifold r-spin Hurwitz numbers. The cohomology class produced by TR in this case plays a role in very many enumerative problems. Its Chern characters were computed by Chiodo.

6. The curve x(z) = z^2, y(z) = z, with canonical B shifted by the Riemann-Hurwitz zeta function generates the Masur-Veech volumes of the principal stratum of the moduli space of quadratic differentials of unit area. However, the spectral curve x(z) = -z - log(z), y(z) = z^2, and canonical B also produces the same numbers. In fact the two spectral curves generate two distinct families of polynomials P_{g,n}: the constant terms of both families coincide and are the Masur-Veech volumes, whereas the other numbers are in general different.

All examples above are computed using the same unifying universal formula. Many more examples have already been found and worked out, despite the methods being so recent. Several important restatements and generalisations of TR have been developed.

More to be written (and much more to be discovered) ....