This talk will present the ”coreact.wiki” initiative, which aims to develop a novel form of wiki engine that will couple a database of human-readable mathematical knowledge with a database containing machine-readable and -executable representations of this knowledge in proof assistants such as Coq. For the concrete example of analytic combinatorics à la Flajolet and Sedgewick, I will provide...
It is well known that to cover the greatest proportion of the Euclidean plane with identical disks, we have to center these disks in a triangular grid. This problem can be generalized in two directions: in higher dimensions or with different sizes of disks. The first direction has been the most studied (for example, in dimension 3, the Kepler’s conjecture was proved by Hales and Ferguson in...
There exists a way, based on the notion of Quantum Doubles, to introduce analogs of partial derivatives on the so-called Reflection Equation algebras. Analogously to the classical case it is possible to use these ”q-derivatives” for different applications. I plan to explain their utility for constructing q-analogs of the Casimir operators, close to them cut-and-join operators, and the Capelli identity.
The operation of birational rowmotion on a finite poset has been a mainstay in dynamical algebraic combinatorics for the last 8 years.
Since 2015, it is known that for a rectangular poset of the form $[p] \times [q]$, this operation is periodic with period $p + q$. (This result, as has been observed by Max Glick, is equivalent to Zamolodchikov’s periodicity conjecture in type AA, proved by...
We study defining inequalities of string cones via a potential function on a reduced double Bruhat cell. We give a necessary criterion for the potential function to provide a minimal set of inequalities via tropicalization and conjecture an equivalence. This is based on joint work with Gleb Koshevoy.
The work of Conway and Lagarias applying combinatorial group theory to packing problems suggests what we might mean by “domain-wall boundary conditions” for the trimer model on the infinite triangular lattice in which the permitted trimers are triangle trimers and three-in-a-line trimers. Looking at subregions of the lattice with those sorts of boundaries, we find intriguing numerology...
We here introduce some combinatorial and analytic tools, conceived to make possible to perform new expansions in the context of constructive field theory and multiscale analysis. These formulas generalize the idea of performing cluster expansion using a sum indexed by forest to the case of a Taylor expansion of order more than zero. They are expected to help construct new field theories of the...
In this talk, I will use the functor of points approach to Algebraic Geometry to establish that every covariant presheaf X on the category of commutative rings — and in particular every scheme X — comes equipped “above it” with a symmetric monoidal closed category PshModX of presheaves of modules. This category PshModX defines moreover a model of intuitionistic linear logic, whose exponential...
We show how the fundamental statistical properties of quantum fields combined with the superposition principle lead to continuous symmetries including the $SL(2, C)$ group and the internal symmetry groups $SU(2)$ and $SU(3)$. The exact colour symmetry is related to ternary $Z_{3}$-graded generalization of the fermionic commutation relations for quarks. A $Z_{3}$-graded generalization of the...
Structural properties of large random maps and lambda-terms may be gleaned by studying the limit distributions of various parameters of interest. In our work we focus on restricted classes of maps and their counterparts in the lambda-calculus, building on recent bijective connections between these two domains. In such cases, parameters in maps naturally correspond to parameters in lambda-terms...
In a recent paper Don Zagier mentions a mysterious integer sequence $(a_{n})_{n≥0}$ which arises from a solution of a topological ODE discovered by Marco Bertola, Boris Dubrovin and Di Yang. In my talk I show how to conjecture, prove and even quantify that $(a_{n})_{n≥0}$ actually admits an algebraic generating function which is therefore a very particular period. The methods are based on...
I plan to discuss three problems of extremal statistics in which unusual (but related to each other) features arise:
a) statistics of two-dimensional ”stretched” random walks above a semicircle,
b) spectral properties of sparse random matrices,
c) statistics of one-dimensional paths in the Poissonian field of traps. I will pay attention to the relationship of these problems with the...
Wick polynomials are at the foundations of QFT (they encode normal orderings) and probability (they encode chaos decompositions). In this lecture, we survey the construction and properties of noncommutative (or free) analogs using shuffle
Hopf algebra techniques. Based on joint works with K. Ebrahimi-Fard, N. Tapia and L. Zambotti.
I will explain how the computational technique of fibered motives can be used to obtain modularity proofs for certain conifold fibers in Calabi-Yau families (joint with Don Zagier, and with Kilian Bönisch and Albrecht Klemm).
We consider a generalization of Young tableaux in which we allow some consecutive pairs of cells with decreasing labels, conveniently visualized by a ”wall” between the corresponding cells. This leads to new classes of recurrences, and to a surprisingly rich zoo of generating functions (algebraic, hypergeometric, D-finite, differentially-algebraic). Some patterns lead to nice bijections with...
We define Hilbert function of a semiring ideal of tropical polynomials in n variables. For $n = 1$ we prove that it is the sum of a linear function and a periodic function (for sufficiently large values). The leading coefficient of the linear function equals the tropical entropy of the ideal. For an arbitrary n we discuss a conjecture that the tropical Hilbert function of a radical ideal is a...
Convex sets can be defined over ordered fields with a non-archimedean valuation. Then, tropical convex sets arise as images by the valuation of non-archimedean convex sets. The tropicalization of polyhedra and spectrahedra can be described in terms of deterministic and stochastic games with mean payoff, being characterized in terms of sub or super-fixed point sets of Shapley operators, which...
In this talk, basing on the algebraic combinatorics on noncommutative formal power series with holomorphic coefficients and, on the other hand, a Picard-Vessiot theory of noncommutative differential equations, we give a recursive construction of solutions of the Knizhnik-Zamolodchikov equations satisfying asymptotic conditions.
One can always transform a triangulation of a convex polygon into another by performing a sequence of edge flips, which amounts to follow a path in the graph G of the associahedron. The least number of flips required to do so is then a distance in that graph whose estimation is instrumental in a variety of contexts, as for instance in computational biology, in computer science, or in algebraic...
