Motivated by joint work with Joachim Kock (UAB Barcelona & U Copenhagen), I plan to present an introduction to the powerful machinery of decomposition spaces (also known as 2-Segal spaces) from the viewpoint of its applications to combinatorics and its computational properties.
We consider positive integer sequences $\rho(n)$, $n=0,1 \cdots$, expressible through the ratios of products of factorials, or of ratios of products of factorials along with Gamma functions. Admitting certain forms of these ratios, the generating functions (gf) of $\rho(n)$'s become generalized hypergeometric functions (gf), which turn out to be algebraic. Detailed conditions for the...
The behavior of multiplicity automata is computable by means of the star of a matrix with noncommutative coefficients taken within a semiring (commutative or noncommutative). Our purpose here is to review applications of this unifying concept (Sweedler's duals, Topological algebras, Infinite iterated integrals). In passing, we indicate how to extend holomorphic-valued shuffle characters as,...
A partition of a positive integer n is a non-increasing sequence of positive integers, called parts, whose sum is n. A partition identity is a theorem stating that for all n, the number of partitions of n satisfying some conditions (often congruence conditions on the parts) equals the number of partitions of n satisfying some other conditions (often difference conditions between the parts)....
I plan to compare the roles of Reflection Equation Algebras and Quantum Groups in different problems of Combinatorics and Mathematical Physics. A special attention will be paid to the Quantum version of the Capelli identity.
Kronecker moduli are algebraic varieties parametrizing linear algebra data up to base change. We consider generating series of their Euler characteristic and/or Betti numbers, and discuss their algebraicity and more general functional/q-difference equations defining them.
We will show that HOMFLY knot invariant can be used to study representation theory of quantum groups and Hecke algebras and other combinatorial problems. The work is an interpretation of the talks by D.Gurevich at this conference.
Last year at CAP2023 I gave a talk about a class of matrices filled by variables, whose eigenvalues are linear forms in the variables. The construction was based on finite posets. I'll speak about a generalization to convex sets in general reflection groups, and even to the case where there is no group at all.
We define higher d-operads, d ≥ 1. For d = 1, 1-operads are nc-operads.
We show that higher Bruhat orders on the discrete Grassmannians $\left({[n] \atop d}\right)$, n ≥ d, form a d-operad.
This is joint work with Vadim Schechtman.
The Nambu-determinant Poisson brackets on $\mathbb{R}^d$ are expressed by the formula
$ \{f,g\}_d (\mathbf{x}) = \varrho(\mathbf{x}) \cdot \det\bigl( \partial(f,g,a_1,...a_{d-2}) / \partial(x^1,...,x^d) \bigr), $
where $a_1$, $\ldots$, $a_{d-2}$ are smooth functions and $x^1$, $\ldots$, $x^d$ are global coordinates (e.g., Cartesian), so that...
For a monoidal category ${\mathcal T}$, if there exists a "real commuting family $(C_i,R_{C_i},\phi_i)_{i\in I}$", we can define a localization $\widetilde {{\mathcal T}}$ of ${\mathcal T}$ by $(C_i,R_{C_i},\phi_i)_{i\in I}$.
Let $R=R({\mathfrak g})$ be the quiver Hecke algebra(=KLR algebra) associated with a symmetrizable Kac-Moody Lie algebra ${\mathfrak g}$ and ${\mathscr C}_w$ the...
The character table of the symmetric group $S_n$, of permutations of n objects, is of fundamental interest in theoretical physics, combinatorics as well as computational complexity theory. We investigate the implications of an identity, which has a geometrical interpretation in combinatorial topological field theories, relating the column sum of normalised central characters of $S_n$, to a sum...
The injective norm is a natural generalization to tensors of the operator norm of a matrix. In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states, where it is known as the geometric entanglement. In this paper, we give a high-probability upper bound on the injective norm of real and complex Gaussian random tensors,...
This talk is dedicated to the survey of some of our results related to q-deformations of the Fock spaces and related to q-convolutions for probability measures on the real line R. The main idea is done by the combinatorics of moments of the measures and related q-cumulants of different types. The main and interesting q-convolutions are related to classical continuous (discrete) q-Hermite...
Crystal bases B(∞), B(λ) are powerful tools to study representations of Lie algebras and quantum groups. We can get several essential information of integrable highest weight representations or Verma modules from B(λ) or B(∞). To obtain such information from crystal bases, we need to describe them by combinatorial objects. The polyhedral realizations invented by Nakashima-Zelevinsky are...
The Weyl algebra (or Heisenberg-Weyl algebra) is the free algebra with two generators $D$ and $U$ and single relation $DU - U D = 1$. As a consequence of this relation, certain monomials are equal, such as $DU U \, D$ and $U \, DDU$. We characterize all such equalities over a field of characteristic 0, describing them in several ways: operational (by a combinatorial equivalence relation...
The kinematic spaces modeling massless particle scattering can be parametrized in different ways such as via the spinor helicity formalism or using momentum twistor coordinates. Either case yields an algebraic variety that we call the spinor helicity variety, respectively the momentum twistor variety. In a general set up (not assuming neither planarity nor dual conformal symmetry in the model)...
Large Language Models are neural networks which are trained to produce a probability distribution on the possible next words to given texts in a corpus, in such a way that the most likely word predicted, is the actual word in the training text.
We will explain what is the mathematical structure defined by such conditional probability distributions of text extensions.
Changing the viewpoint...
Spin glass models have been a interesting research subject due to the multiple valuable insights it brought to various fields such as statistical physics and machine learning. The spherical p-spin glass model in particular has been proven an excellent candidate to investigate the landscape of such models. On the other side, theoretical tools for the study of random tensors have been developed...
