# Colloquia

## Hehui Wu

Vertex Partition with Average Degree Constraint

Date: 5/31/2018

Time: 3:00PM-4:00PM

Place: 315 Armstrong Hall

**Abstract**: A classical result, due to Stiebitz in 1996, states that a graph with minimum degree $s+t+1$ contains a vertex partition $(A, B)$, such that $G[A]$ has minimum degree at least $s$ and $G[B]$ has minimum degree at least $t$. Motivated by this result, it was conjectured that for any non- negative real number s and t, such that if G is a non-null graph with average degree at least $s + t + 2$, then there exist a vertex partition $(A, B)$ such that $G[A]$ has average degree at least $s$ and $G[B]$ has average degree at least $t$. Earlier, we claimed a weaker result of the conjecture that there exist two disjoint vertex set $A$ and $B$ (for which the union may not be all the vertices) that satisfy the required average degree constraints. Very recently, we fully proved the conjecture. This is joint work with Yan Wang at Facebook.

All are welcome.

## Xiaoya Zha

Non-revisiting Paths in Polyhedral Maps on Surfaces

Date: 4/27/2018

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

**Abstract**: The Non-revisiting Path Conjecture (or Wv-path Conjecture) due to Klee and Wolfe states that any two vertices of a simple polytope can be joined by a path that does not revisit any facet. This is equivalent to the well-known Hirsch Conjecture. Klee conjectured even more, namely that the Non-revisiting Path Conjecture is true for all general cell complexes. Klee proved that the Non-revisiting Path Conjecture is true for 3-polytope (3-connected plane graphs). Later, the general Non-revisiting Path Conjecture was varied for polyhedral maps on the projective plane and the torus by Barnette, and on the Klein bottle by Pulapaka and Vince. However, a few years ago, Santos proved that the Hirsch conjecture is false in general.

In this talk, we show that the the non-revisiting path problem is closely related to

(i) the local connectivity $\kappa_G(x; y)$ (i.e. the number of disjoint paths between x and y);

(ii) the number of dfferent homotopy classes of (x; y)-paths;

(iii) the number of (x; y)-paths in each homotopy class.

For a given surface $\Sigma$, we give quantitative conditions for the existence of non-revisiting paths between x and y. We also provide more systematic counterexamples with high number (linear to genus of the surface) of paths between x and y but without any non-revisiting path between them. These results show the importance of topological properties of embeddings of underline graphs for this geometric setting problem.

This is joint work with Michael Plummer and Dong Ye.

This talk will be accessible to graduate students.

All are welcome.

## Xiangwen Li

Every planar graph without adjacent cycles of length at most 8 is 3-choosable.

Date: 4/26/2018

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

**Abstract**: DP-coloring as a generation of list coloring was introduced by Dvovrak and Postle in 2017,

who proved that every planar graph without cycles from 4 to 8 is 3-choosable, which was conjectured by Brodian et al. in 2007. In this paper, we prove that every planar graph without adjacent cycles of length at most 8 is 3-choosable, which extends this result of Dvovrak and Postle.

This talk will be accessible to graduate students.

All are welcome.

## Liming Xiong

Fobidden subgraphs, some propertices: Hamiltonicity and Superieulerian in graphs

Date: 4/26/2018

Time: 4:30PM-5:30PM

Place: 315 Armstrong Hall

**Abstract**: In this talk, we persent some results between forbidden pair of subgraphs and supereuleriancity and hamiltoncity, by using the characterization of supereuleriancity by induced minor, obtained in [Lai, Supereulerian graphs and excluded induced minors, Discrete Math. 146(1995)], we present results on a complete characterizations of the existence of spanning eulerian subgraph, the similar problem for 2-factors is also considered.

This talk will be accessible to graduate students.

All are welcome.

## Robin Baidya

Six ways to compare modules

Date: 4/24/2018

Time: 4:00PM-5:00PM

Place: 315 Armstrong Hall

**Abstract**: How many copies of one object exist in another? How many copies of one object are required to build another? If we can build an object using a particular method, to what extent is that method unique? These questions pervade mathematics, and they must be interpreted diﬀerently in diﬀerent contexts. We will approach these questions as they pertain to modules. To answer the ﬁrst question, we will study how modules embed and factor. For the second question, we will consider the various ways through which a module can be generated and cogenerated. We will investigate the question of uniqueness as it applies to the splitting of modules and to the problem of cancellation. By the end of the talk, we will have covered two ways to interpret each of our three questions, yielding six ways to compare modules.

This talk will be accessible to graduate students.

All are welcome.

## Steve Butler

An introduction to the normalized Laplacian matrix

Date: 4/18/2018

Time: 2:30PM-3:30PM

Place: 315 Armstrong Hall

**Abstract:** Spectral graph theory looks at the interplay between the structure of graphs and the eigenvalues associated with some particular matrix. Different matrices give different information, so it is important to understand how the different matrices behave, and which matrix to use for which types of problems. We will give an introduction to one of the lesser known matrices, the normalized Laplacian matrix, which has ties to the probability transition matrix of a random walk. This matrix is useful in many settings, particularly for graphs which are not regular, but also has some strange quirks.

All are welcome.

## Ilie Ugarcovici

The structure and spectrum of odometer systems

Date: 3/2/2018

Time: 3:00PM-4:00PM

Place: 315 Armstrong Hall

**Abstract:** Odometer systems (or adding machines) are a well studied class of examples in the classical theory of measurable and topological dynamical systems. They can be viewed measure theoretically as cutting and stacking transformations of the unit interval. Alternatively, they can be viewed algebraically as a Z-action on an inverse limit space of increasing quotient groups of Z. Recently, there have been attempts to understand odometer systems over an arbitrary finitely generated and residually finite group. In this talk I will discuss some classification results for odometer systems defined over semidirect product groups, in particular the Heisenberg group. This is joint work with S. Orfanos (DePaul) and A. Sahin (Wright State).

All are welcome.

## Henrik Holm

Prime ideals in commutative and non-commutative rings

Date: 3/9/2018

Time: 4:00PM-5:00PM

Place: 315 Armstrong Hall

Abstract:Prime ideals are important in commutative algebra (e.g. localization and Krull dimension), in algebraic geometry (e.g. affine schemes),

and in number theory (e.g factorization in Dedekind domains). This talk --which is about prime ideals, their generalizations, and their

uses -- has three parts: In the first part, I will talk about certain aspects of prime ideals in commutative rings. In the second part, I will explain

elements of Kanda's recently developed theory of (so-called) atoms. The notion of atoms is a useful and interesting generalization of prime ideals

to non-commutative rings (and to abelian categories). In the third part, I will explain work in progress, joint with R. H. Bak, on how to actually

compute/determine the atoms for certain types of non-commutative rings.

All are welcome.

## Naoki Taniguchi

Almost Gorenstein rings

Date: 3/6/2018

Time: 4:00PM-5:00PM

Place: 315 Armstrong Hall

All are welcome.

## Ihsan Topaloglu

Swarming in domains with boundaries: approximation and regularization by nonlinear diffusion

Date: 2/23/2018

Time: 3:00PM-4:00PM

Place: 422 Armstrong Hall

**Abstract:** In this talk I will consider an aggregation model with nonlinear diffusion in domains with boundaries and present on the zero diffusion limit of its solutions. This model is used in describing phenomena related swarming and social aggregations, such as biological swarms and pattern formation, granular media, and self-assembly of nanoparticles. Using the formulation of the aggregation model as a gradient flow on spaces of probability measures equipped with the Wasserstein metric, I will present the convergence of weak solutions for fixed times, as well as the convergence of energy minimizers in this limit. I will also present numerical simulations that support the analytical results and demonstrates that adding small nonlinear diffusion approximates, as well as regularizes, the plain aggregation model. This is a joint project with Razvan Fetacau and Mitchell Kovacic.

All are welcome.

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