Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, October 23, 2018.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, October 23, 2018

1:00 pm in 345 Altgeld Hall,Tuesday, October 23, 2018

#### Expansions of the real field by discrete subgroups of $Gl_n(\mathbb{C})$

###### Erik Walsberg (UIUC)

Abstract: Let $\Gamma$ be an infinite discrete subgroup of $Gl_n(\mathbb{C})$. Then either $(\mathbb{R},<,+,\cdot,\Gamma)$ is interdefinable with $(\mathbb{R},<,+,\cdot, \lambda^{\mathbb{Z}})$ for some $\lambda \in \mathbb{R}$, or $(\mathbb{R},<,+,\cdot,\Gamma)$ defines the set of integers. When $\Gamma$ is not virtually abelian, the second case holds.

2:00 pm in 345 Altgeld Hall,Tuesday, October 23, 2018

#### Tensor algebras of product systems and their C*-envelopes

###### Elias Katsoulis (East Carolina University)

Abstract: Let $(G, P)$ be an abelian, lattice ordered group and let $X$ be a compactly aligned, $\tilde{\phi}$-injective product system over $P$. We show that the C*-envelope of the Nica tensor algebra $\mathcal{N} \mathcal{T} ^+_X$ is the Cuntz-Nica-Pimsner algebra $\mathcal{N} \mathcal{O} _X$ as defined by Sims and Yeend. We give several applications of this result. In particular, we show that the Hao-Ng isomorphism problem for generalized gauge actions of discrete groups on $C^*$-algebras of product systems has an affirmative answer in many cases, generalizing recent results of Bedos, Quigg, Kaliszewski and Robertson and of the second author.

2:00 pm in 243 Altgeld Hall,Tuesday, October 23, 2018

#### Hamiltonian cycles in tough P₂∪P₃-free graphs

###### Songling Shan (Illinois State Math)

Abstract: Let $t>0$ be a real number and $G$ be a graph. We say $G$ is $t$-tough if for every cutset $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. Determining toughness is an NP-hard problem for arbitrary graphs. The Toughness Conjecture of Chvátal, stating that there exists a constant $t_0$ such that every $t_0$-tough graph with at least three vertices is hamiltonian, is still open in general.

A graph is called $P_2\cup P_3$-free if it does not contain any induced subgraph isomorphic to $P_2\cup P_3$, the union of two vertex-disjoint paths of order 2 and 3, respectively. We show that every 15-tough $P_2\cup P_3$-free graph with at least three vertices is hamiltonian.

4:00 pm in 245 Altgeld Hall,Tuesday, October 23, 2018

#### Multi-scale mathematical models of disease

###### Reinhard C. Laubenbacher (University of Connecticut School of Medicine and Jackson Laboratory for Genomic Medicine)

Abstract: Multi-scale mathematical and computational modeling has emerged as a key technology in many areas of engineering, manufacturing, and business. It also holds great promise for biomedicine, with a wide range of potential applications. For instance, well-validated mathematical models could help limit the use of animal experiments for drug discovery or develop better-optimized treatment protocols for patients. Unique challenges arise in this context, however, such as the existence of feedback loops between scales, e.g., the bidirectional interplay between processes at the organism and intracellular levels, or the lack of knowledge about physical or biochemical principles underlying disease mechanisms. In particular, this raises challenging mathematical problems, such as analysis and validation of the dynamics of complex hybrid models. It also complicates the application of optimal control techniques, which is of particular importance, since most problems in biomedicine ultimately are about control. These issues will be illustrated through two ongoing case studies, metabolic drivers of tumor growth and the immune response to respiratory fungal infections.

Dr. Laubenbacher joined the University of Connecticut Health Center in May 2013 as Professor in the Department of Cell Biology and Co-Director of the Center for Quantitative Medicine. Prior to this appointment, he served as a Professor at the Virginia Bioinformatics Institute and a Professor in the Department of Mathematics at Virginia Tech since 2001. He was also an Adjunct Professor in the Department of Cancer Biology at Wake Forest University in Winston-Salem (NC) and Affiliate Faculty in the Virginia Tech Wake Forest University School of Biomedical Engineering and Sciences. In addition, Dr. Laubenbacher was also Professor of Mathematics at New Mexico State University. He has served as Visiting Faculty at Los Alamos National Laboratories, was a member of the Mathematical Science Research Institute at Berkeley in 1998, and was a Visiting Associate Professor at Cornell University in 1990 and 1993. Current interests in Dr. Laubenbacher’s research group include the development of mathematical algorithms and their application to problems in systems biology, in particular the modeling and simulation of molecular networks. An application area of particular interest is cancer systems biology, especially the role of iron metabolism in breast cancer.