Discrete mathematics seminar in UZH - talks given in Fall Semester 2015

• December 15th: Generalization of non-ambiguous trees and differential equations, Adrien Boussicault (Université de Bordeaux). .

In this talk, we present a new definition of non-ambiguous trees as labelled binary trees. We thus get a differential equation whose solution can be described combinatorially. This yield a new formula for the number of non-ambiguous trees. Then, we generalize non ambiguous trees to higher dimension.

• December 7th: On Riemann zeroes, Lognormal Multiplicative Chaos, and Selberg Integral (special session analytic number theory), Dmitry Ostrovsky. .

I will present my conjectures on the relationship of rescaled Mellin transforms of the exponential functional of the Bourgade-Kuan-Rodgers statistic of Riemann zeroes, the multiplicative chaos measure of Mandelbrot-Bacry-Muzy, and the meromorphic extension of the Selberg integral.

• December 1st: Higher Persistence: from Topological Data Analysis to BF theories via Batalin-Vilkovisky algebras, Francesco Vaccarino (Politecnico di Torino). .

We will extend the usual approach of persistent (co-)homology by considering diagrams of incidence algebras of finite posets. Then, we will show that the associated cohomology is a Batalin-Vilkovisky algebra, having as anti-laplacian a suitable dual of the Alain Connes' $$B$$ operator. This technical result pave the way to build on the top of reasonable data (e.g. weighted networks) abstract BF-theories, thus connecting seemingly uncorrelated research areas as topological data analysis and topological quantum field theory.

• November 24th: RNA Bioinformatics and ensemble dynamic programming, Yann Ponty (École polytechnique). .

RiboNucleic Acids (RNAs) are fascinating biomolecules which, similarly to DNA, can be encoded as sequences over a four-letter alphabet, and perform a wide array of biological functions. However, unlike DNA, the precise function of a given RNA depends critically on its structure, adopted as the outcome of a folding process. Luckily, this intricate three- dimensional conformation can be adequately abstracted as a (non-crossing) list of contacts, i.e. a discrete combinatorial object.

In this talk, I will emphasize how, over the past three decades, RNA biology has benefited from a continuous and fruitful cross-talk between discrete mathematicians, computer scientists and biochemists. At the center of this conversation lies the concept of dynamic-programming, an algorithmic design technique which solves a combinatorial optimization problem efficiently by taking advantage of a well-chosen decomposition of its search space. Extensions and optimized instances of this technique now allow to address, at a genomic scale, multiple questions related to the analysis of the Boltzmann ensemble and the sequence-structure(-function) relationship. These developments also raise well-defined open questions, motivating further studies of the underlying discrete structures.

• November 10th: Shifted Schur functions, Valentin Féray (UZH). .

Schur poynomials are classical symmetric polynomials related to representation theory of linear and symmetric groups. They are usually defined either algebraicly, as a quotient of determinant, or combinatorially, as a sum over semi-standard Young tableaux.

We present here another approach, using interpolation symmetric polynomials, due to Knop and Sahi and independently Okounkov and Olshanski. This allows to construct factorial (or shifted) Schur polynomials that generalize Schur polynomials. Most properties of Schur polynomials can be understood from this point of view.

We end this presentation with a new positivity property for these polynomials (joint work with Per Alexandersson).

• October 27th: Simultaneous Core Partitions for Crystallographic Root Systems, Marko Thiel (UZH). .

A simultaneous $$(a,b)$$-core is an integer partition without any hooklengths equal to either $$a$$ or $$b$$. We interpret $$(a,b)$$-cores in terms of the affine symmetric group and thereby generalise them to affine Weyl groups. Using Ehrhart theory, we find simple uniform formulas for the number of $$(a,b)$$-cores and their maximum and average size for simply-laced affine Weyl groups.

Joint work with Nathan Williams.

• October 13th: Sum of matrix entries of representations of the symmetric group and its asymptotics, Dario De Stavola (UZH). .

We consider three quantities related to an irreducible representation matrix $$\pi_\lambda(\sigma)$$ of the symmetric group $$S_n$$, the action described by the so-called seminormal Young representation. These quantities are: the sum of the diagonal elements of the matrix up to a certain index, the sum of all entries of the matrix, and the sum of the entries up to a certain index. We describe them as linear combinations of traces, therefore gathering informations about the asymptotics.

Thank you to all speakers and attendees !

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