Events

Title: Linear arboricity of sparse multigraphs via orientations

Abstract:

The linear arboricity $la(G)$ of a loopless multigraph $G$ is the minimum number of colors required to edge-color $G$ into linear forests, that is, forests whose components are all paths. The Linear Arboricity Conjecture of Akiyama, Exoo, and Harary asserts that the linear arboricity of a simple graph $G$ is at most $\lceil (\Delta(G)+1)/2 \rceil$.

Title: Generalized Subspace Subcode with Application in Cryptology

Abstract:

Most codes with an algebraic decoding algorithm are derived from Reed-Solomon codes. They are obtained by taking equivalent codes, for example Generalized Reed-Solomon codes, or by using the so-called subfield subcode method, which leads to Alternant codes over the underlying prime field, or over some intermediate subfield. The main advantage of these constructions is to preserve both the minimum distance and the decoding algorithm of the underlying Reed-Solomon code.

Title: Virtual characters of permutation statistics

Abstract:

Functions of permutations are studied in a wide variety of fields including probability, statistics and theoretical computer science. I will introduce a method for studying such functions using representation theory and symmetric functions. As a consequence, one can extract detailed information about asymptotic behavior of many permutation statistics with respect to non-uniform measures that are invariant under conjugation. The key new tool is a combinatorial formula called the path Murnaghan-Nakayama rule that gives the Schur expansion of a novel basis of the ring of symmetric functions. This is joint work with Brendon Rhoades.

Title: Algebraic Graph Theory

Abstract:

A graph is distance-regular if we can write the distance adjacency matrices as polynomials in the adjacency matrix. Distance-regular graphs are a class of graphs of significant interest to algebraic graph theorists for their structural and algebraic properties. The notion of distance-regularity can be weakened to a local property on vertices, but when every vertex in the graph is locally distance-regular, the graph will either be distance-regular or in the closely related class of distance-biregular graphs.

Title: Combinatorial atlas for log-concave inequalities

Abstract:

The study of log-concave inequalities for combinatorial objects have seen much progress in recent years. One such progress is the solution to the strongest form of Mason’s conjecture (independently by Anari et. al. and Brándën-Huh).

Title: An Introduction to Nonnegativity and Polynomial Optimization

Abstract:

In science and engineering, we regularly face polynomial optimization problems, that is: minimize a real, multivariate polynomial under polynomial constraints. Solving these problems is essentially equivalent to certifying of nonnegativity of real polynomials -- a key problem in real algebraic geometry since the 19th century.

Title: An into introduction to the chromatic number of digraph

Abstract: A proper $k$-coloring of a digraph $D$ is a coloring of the vertices such that every color class is acyclic, and the dichromatic number of a digraph $D$ is the minimum number $k$ such that there is a proper $k$-coloring of $D$. Many questions about the chromatic number can be asked about the dichromatic number, but as one will quickly observe, unsuspected complications arise when dealing with digraphs.

Title: On the Security of the NIST lightweight Finalist Ascon

Abstract: 

The ongoing NIST lightweight cryptographic standardization project for the selection of ciphers which are suitable for constrained environments is in the final stage. The authenticated encryption algorithm Ascon, designed by Dobrauing et al., is one out of the 10 finalists. Ascon is also one of the winners of the CAESAR competition in the lightweight applications category.

Title: A Brief Introduction to World of Erd\H{o}s-Ko-Rado Theorems

Abstract:   The Erd\H{o}s-Ko-Rado (EKR) theorem is a famous result that is one of the cornerstones of extremal set theory. This theorem answers the question "What is the largest family of intersecting sets, of a fixed size, from a base set?"

Title: Determinantal formulas with major indices

Abstract: Krattenthaler and Thibon discovered a beautiful formula for the determinant of the matrix indexed by permutations whose entries are q^maj( u*v^{-1} ), where “maj” is the major index. Previous proofs of this identity have applied the theory of nonsymmetric functions or the representation theory of the Tits algebra to determine the eigenvalues of the matrix.