### Fall 2018 Mathematics Colloquium

###### Time and Location:

Minard 214 at 3:00 PM (refreshments at 2:30 PM in the Math Conference Room (Minard 404))

Special Colloquia or Tri-College Colloquia venues and times may vary, please consult the individual listing.

#### Tuesday, September 25 Jeremiah Bartz (UND)

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Multinets in the complex projective plane

Abstract: Multinets are certain configurations of lines and points with multiplicities in the complex projective plane. They appear in the study of resonance and characteristic varieties of complex hyperplane arrangement complements and cohomology of Milnor fibers. Although the notion of multinets has been around since 2006, very few examples or general properties of multinets with non-trivial multiplicities were known until much more recently.

In this talk, we will describe the motivation behind studying multinets, give numerous examples with pictures and recipes for constructing them, and discuss recent advances along with open problems in this area.

#### Tuesday, October 9 Kaisa Taipale (UMN)

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The shape of data: applied topology

Abstract: Topological data analysis looks at the shape of data -- the underlying structure -- by adapting traditional methods from topology to modern computation and applications. Using simplicial complexes and homological algebra, we can analyze structures that persist, visualize the topological structures of datasets, and approach questions in manifold learning. Anyone can carry out data analysis with Python and R, and you might be surprised what insights mathematicians can bring to the world of applications. We'll look at real-life problems from health to finance to politics, and I hope to challenge your idea of what "applied mathematics" can mean.

#### Tuesday, October 16 Tri-College Colloquium

Doğan Çömez at Concordia

Quantization for probability measures on Cantor Dust

TIME and ROOM : Integrated Science Center (ISC) 301 at 3:00 Pm. Refreshments at 2:30 IN ISC 354

Abstract: Quantization for a probability measure is the process of estimating it by a discrete probability that assumes only a finite number of levels in its support. The focus of this talk is the quantization of continuous singular self similar probability measures on Cantor Dusts. For such a probability measure, we determine the optimal sets of points that ensures optimality and their quantization errors. In addition, we will show that the quantization dimension of the underlying measure exists, whereas its quantization coefficient, which determines the efficient rate of convergence of errors, does not exist. However, the interval in which accumulation points of the quantization coefficient lie is determined.

#### Tuesday, October 23 Semyon Litvinov (Penn State-Hazelton)

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On convergence of ergodic averages for Dunford-Schwartz operators

in fully symmetric function spaces.

Abstract: Let $(\Omega,\mu)$ be a sigma-finite measure space, and let $X\subset L^1(\Omega)+L^\infty(\Omega)$ be a fully symmetric space. If $\mu(\Omega)=\infty$, we give necessary and sufficient conditions for almost uniform convergence in $X$ (in Egorov's sense) of Cesàro averages $M_n(T)(f)=\frac{1}{n}\sum_{k=0}^{n-1}T^k(F)$ for all Dunford-Schwartz operators $T$ in $L^1(\Omega)+L^\infty(\Omega)$ and any $f\in X$. Besides, if $(\Omega,\mu)$ is quasi-non-atomic, we show that the averages $M_n(T)$ converge strongly in $X$ for each Dunford-Schwartz operator $T$ in $L^1(\Omega)+L^\infty(\Omega)$ if and only if $X$ has order continuous norm and $L^1(\Omega)$ is not contained in $X$.

#### Tuesday, October 30 William Martin (NDSU)

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PRIUM: Promoting Reasoning in Undergraduate Mathematics

Abstract:

PRIUM is a collaborative NSF-funded project (NSF DUE 1624906) involving mathematics department leaders and research mathematicians at North Dakota State University and Georgia State University. Our cyclic, iterative assessment model seeks insights about the development of undergraduates’ abilities to read, interpret, critique and write proofs. Research-based approaches to teach mathematical proof or assessing student comprehension of proof have been scantly documented in the literature in spite of its attention in mathematics education research for over 30 years (Mejia-Ramos, Fuller, Weber, Rhoads, & Samkoff, 2012). PRIUM is designed to support mathematics educators and mathematicians to work closely to utilize insights gained from research-based proof assessments. This presentation will include a brief overview of the project and preliminary results from the assessments implemented during 2016-18. We will present several proof assessments implemented at the two universities and will engage with participants about how these assessments differ from the traditional “memorize and reproduce” techniques of proof assessment.

References

Mejia-Ramos, J.P., Fuller, E., Weber, K., Rhoads, K., & Samkoff, A. (2012). An assessment model for proof comprehension in undergraduate mathematics. *Educational Studies in Mathematics*, 79: 3–18.

#### Tuesday, November 6 Dan Florentin (Kent State)

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Caustic duality in Minkowski billiards

TIME CHANGE: The talk starts at 3:15 pm.

Abstract: Mathematical billiards are a classical and well-studied class of dynamical systems,"a mathematician’s playground". Convex caustics, which are curves to which billiard trajectories remain forever tangent, play an important role in the study of billiards. In this talk we will discuss convex caustic in Minkowski billiards, which is the generalization of classical billiards in non-Euclidean normed planes. In this case a natural duality arises from, roughly speaking, interchanging the roles of the billiard table and the unit ball of the (dual) norm. This leads to duality of caustics in Minkowski billiards. Such a pair of caustics is dual in a strong sense, and in particular they have equal perimeters and other classical parameters. We will show that, whenthe norm is Euclidean, every caustic possesses a dual caustic, but in general this phenomenon fails. Based on joint work with S. Artstein-Avidan, Y. Ostrover, and D. Rosen.

#### Tuesday, November 13 Ivan Yegorov (NDSU)

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Optimal feedback strategies for bacterial growth.

Abstract: Mechanisms of bacterial adaptation to environmental changes are of great interest for both fundamental biology and engineering applications. This talk presents a continuous-time dynamic problem of resource allocation between metabolic and gene expression machineries for a self-replicating bacterial population. In compliance with evolutionary principles, the criterion is to maximize the accumulated structural biomass. In the model, we include both the degradation of proteins into amino acids and the recycling of the latter (i.e., using as precursors again). On the basis of the analytical investigation of our problem by Pontryagin's principle, we develop a numerical method to approximate the switching curve of the optimal feedback control strategy. The obtained field of extremal state trajectories consists of chattering arcs and 1 steady-state singular arc. The constructed feedback control law can serve as a benchmark for comparing actual bacterial strategies of resource allocation. We also study the influence of temperature, whose increase intensifies protein degradation. While the growth rate suddenly decreases with the increase in temperature in a certain range, the optimal control synthesis appears to be essentially less sensitive. Besides, we present an extended model that shows the potential of optimal control frameworks for better understanding and improving biotechnological production processes.

#### Tuesday, November 27 Tri-College Colloquium

Doug Anderson at NDSU

Equilibrium Stability vs. Hyers-Ulam Stability

Abstract: We compare the equilibrium stability of a simple first-order differential equation with its Hyers-Ulam stability, appropriately defined. Then we analyze these stabilities when the problem is discretized, either with difference equations or with more general time scales. Whether a given equation has Hyers-Ulam stability, and if so, whether there is a minimum Hyers-Ulam constant and unique accompanying solution is also discussed.

#### Tuesday, December 4 Mohamed Baghzali (NDSU)

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The Effectiveness of the NDSU Math Emporium

Abstract: In the Fall 2015 semester, the NDSU Math Department created a learning center and has continuously tried to achieve three overarching goals:

1.) challenge students and assist them to become active learners;

2.) provide multiple ways to accommodate student learning; and

3.) provide appropriate and adequate resources.

Six semesters later, the true effectiveness of the NDSU Math Emporium courses redesign in introductory mathematics courses (College Algebra and Trigonometry) was measured and the results as well as the logistics will be presented during this presentation.