This course provides an introduction to the interactions between combinatorics and algebraic geometry with focus on combinatorial aspects of Hodge theory. We will be interested in combinatorial description of asymptotics of geometric quantities, such as height, canonical measures, Green functions, etc., associated to algebraic varieties when these varieties vary in family. We will also briefly discuss some applications in physics.
This course will provide an overview of several bijective constructions for planar maps. Planar maps are decompositions of the sphere into polygons. Bijections between planar maps and simpler combinatorial objects (such as trees or lattice paths) have been the key to new discoveries about maps.
There are two big families of bijections: bijections of Schaeffer type, and bijections of Mullin type. The bijections of Schaeffer type (between planar maps and labeled trees) are instrumental in the study of metric properties of maps. The bijections of Mullin type (between decorated planar maps and lattice paths) are instrumental in the study of the scaling limit of statistical models on maps.
I will first give an overview of the bijections of Schaeffer type (using a framework developed with Eric Fusy), and then focus on a particular bijection of Mullin type and its application for relating maps to Liouville quantum gravity and SLE (joint work with Nina Holden and Xin Sun).
Fermions appear naturally in representations of stochastic processes. A prominent example is a family of history dependent stochastic processes, that can be represented in terms of a supersymmetric nonlinear sigma model. I will give an introduction to the problem and some results.
Enumerative geometry is concerned with counting (measuring) some geometric objects, typically surfaces of given topology (genus and number of boundaries), with a certain weight (depending on the target space in which the surfaces are embedded, depending on a metric...). A typical example is the enumeration of maps: counting discrete surfaces made of polygons, possibly carrying colors, with a weight depending on the polygon's sizes and colors.
Many of these enumerative geometry problem share a common feature: once the “disc” enumeration is known (genus 0, 1 boundary, i.e. one marked edge), then all the other topologies can be found by a universal recursion: the “topological recursion”. Beyond its applications as solving many enumerative geometry problems, the topological recursion has beautiful mathematical properties by itself.
In these lectures we shall give some examples of enumerative problems that satisfy the topological recursion, present some general properties of topological recursion, and methods of computation.
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