# KPPY 99

Richard Stanley Jang Soo Kim Hongjun Ge Hyungtae Baek Mathias Schacht

Oct 05 2024. 11:30am - 7:00pm at Yeungnam University

Meals will only be provided for registered participants. Registration is free.

Schedule | ||

11:30 - 12:10 |
Richard Stanley MIT | Some combinatorial aspects of cyclotomic polynomials |

12:10 | Lunch | |

1:30 - 2:10 |
Jang Soo Kim SKKU | Lecture hall graphs and the Askey scheme |

2:20 - 3:00 |
Hongjun Ge USTC | A Bose-Laskar-Hoffman theory for $\mu$-bounded graphs with fixed smallest eigenvalue |

3:20 - 4:00 |
Hyungtae Baek KNU | The Anderson rings over a von Neumann regular ring |

4:10 - 4:50 |
Mathias Schacht University of Hamburg | Extremal problems for uniformly dense hypergraphs |

## Abstracts

Richard Stanley

Some combinatorial aspects of cyclotomic polynomials

Some combinatorial aspects of cyclotomic polynomials

Euler showed that the number of partitions of n into distinct parts is
equal to the number of partitions of n into odd parts. MacMahon showed
that the number of partitions of n for which no part occurs exactly
once is equal to the number of partitions of n into parts divisible by
2 or 3. Both these results are instances of a general phenomenon based
on the fact that certain polynomials are the product of cyclotomic
polynomials. After discussing this assertion, we explain how it can be
extended to such topics as counting certain polynomials over finite
fields and obtaining Dirichlet series generating functions for certain
classes of integers.

Jang Soo Kim

Lecture hall graphs and the Askey scheme

Lecture hall graphs and the Askey scheme

We establish, for every family of orthogonal polynomials in the Askey scheme and
the $q$-Askey scheme, a combinatorial model for mixed moments and coefficients in terms of
paths on the lecture hall lattice. This generalizes to all families of orthogonal
polynomials in the Askey scheme previous results of Corteel and Kim for the little $q$-Jacobi
polynomials. This is joint work with Sylvie Corteel, Bhargavi Jonnadula, and Jon Keating.

Hongjun Ge

A Bose-Laskar-Hoffman theory for $\mu$-bounded graphs with fixed smallest eigenvalue

A Bose-Laskar-Hoffman theory for $\mu$-bounded graphs with fixed smallest eigenvalue

In 2018, by using Ramsey and Hoffman theory, Koolen, Yang, and Yang gave a
structural result on graphs with smallest eigenvalue at least $-3$ and large minimum degree.
Without using Ramsey theory, we combine Bose-Laskar type argument and Hoffman theory to show
some structural results about $\mu$-bounded graphs with fixed smallest eigenvalue. A
consequence is that we have a reasonable bound for the minimum degree.
Note that local graphs of distance-regular graphs is $\mu$-bounded. We apply these results
to describe the structure for any local graph
of a distance-regular graph with classical parameters $(D,b,\alpha,\beta)$. As a
consequence, we give a bound on $\alpha$ in terms of $b$. In particular, we show that
$\alpha\leq2$ if $b=2$ and $D\geq 12$. This is joint work with J. Koolen, C. Lv, and Q.
Yang.

Hyungtae Baek

The Anderson rings over a von Neumann regular ring

The Anderson rings over a von Neumann regular ring

Many ring theorists researched various properties of Nagata rings and Serre's
conjecture rings. In this talk, we introduce a subring (refer to the Anderson ring) of both
the Nagata ring and the Serre's conjecture ring (up to isomorphism), and investigate
properties of the Anderson rings. Additionally, we compare the properties of the Anderson
rings with those of Nagata rings and Serre's conjecture rings. More precisely, we examine
the following questions:

- When is the Anderson ring a principal ideal ring?
- When is the Anderson ring a Prüfer-like ring?

Mathias Schacht

Extremal problems for uniformly dense hypergraphs

Extremal problems for uniformly dense hypergraphs

Extremal combinatorics is a central research area in discrete
mathematics. The field can be traced back to the work of Turán and it
was established by Erdős through his fundamental contributions and his
uncounted guiding questions. Since then it has grown into an important
discipline with strong ties to other mathematical areas such as
theoretical computer science, number theory, and ergodic theory.

We focus on extremal problems for hypergraphs, which were introduced by Turán. After solving the analogous question for graphs, Turán asked to determine the maximum cardinality of a set E of 3-element subsets of a given n-element set V such that for any 4 elements of V at least one triple is missing in E. This innocent looking problem is still open and despite a great deal of effort over the last 80 years and our knowledge is still somewhat limited. We consider a variant of the problem by imposing additional restrictions on the distribution of the 3-element subsets in E. These additional assumptions yield a finer control over the corresponding extremal problem. In fact, this leads to many interesting and more manageable subproblems, some of which were already considered by Erdős and Sós in the 1980ies. The additional assumptions on the distribution of the 3-element subsets are closely related to the theory of quasirandom discrete structures, which was pioneered by Szemerédi and became a central theme in the field. In fact, the hypergraph extensions by Gowers and by Rödl et al. of the regularity lemma provide essential tools for this line of research.

We focus on extremal problems for hypergraphs, which were introduced by Turán. After solving the analogous question for graphs, Turán asked to determine the maximum cardinality of a set E of 3-element subsets of a given n-element set V such that for any 4 elements of V at least one triple is missing in E. This innocent looking problem is still open and despite a great deal of effort over the last 80 years and our knowledge is still somewhat limited. We consider a variant of the problem by imposing additional restrictions on the distribution of the 3-element subsets in E. These additional assumptions yield a finer control over the corresponding extremal problem. In fact, this leads to many interesting and more manageable subproblems, some of which were already considered by Erdős and Sós in the 1980ies. The additional assumptions on the distribution of the 3-element subsets are closely related to the theory of quasirandom discrete structures, which was pioneered by Szemerédi and became a central theme in the field. In fact, the hypergraph extensions by Gowers and by Rödl et al. of the regularity lemma provide essential tools for this line of research.