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MSRI-UP 2017 Colloquia

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Dr. Anastasia Chavez, University of California, Berkeley

The Dehn-Sommerville Relations and the Catalan Matroid

The f-vector of a d-dimensional polytope P stores the number of faces of each dimension. When P is simplicial the Dehn--Sommerville relations condense the f-vector into the g-vector, which has length ⌈d+12⌉. Thus, to determine the f-vector of P, we only need to know approximately half of its entries. This raises the question: Which (⌈d+12⌉)-subsets of the f-vector of a general simplicial polytope are sufficient to determine the whole f-vector? We prove that the answer is given by the bases of the Catalan matroid.

Dr. Erika Camacho, Arizona State University

Mathematical Models of the Retina and In Silico Experiments: Shedding Light on Vision Loss

Mathematical modeling has been used to study diverse biological topics ranging from protein folding to cell interactions to interacting populations of humans but has only recently been used to study the physiology of the eye.  In recent years, computer (in silico) experiments have given researchers invaluable insights and in some cases have re-directed experimental research and theory.  In this talk I will give a brief overview of the relevant physiology of the eye as it pertains to Retinitis pigmentosa (RP), a group of inherited degenerative eye diseases that characterized by the premature death of both rod and cone photoreceptors often resulting in total blindness. With mathematics and in silico experiments, we explore the experimentally observed results highlighting the delicate balance between the availability of nutrients and the rates of shedding and renewal of photoreceptors needed for a normal functioning retina.  This work provides a framework for future physiological investigations potentially leading to long-term targeted multi-faceted interventions and therapies dependent on the particular stage and subtype of RP under consideration.  The mathematics presented will be accessible to an undergraduate math audience and the biology will be at the level of a novice (and with a little help from Dr. Seuss).

Dr. Maria Mercedes Franco, Queensborough Community College of The City University of New York

Impact of Undergraduate Research on Student Learning at a Community College

Undergraduate research (UR) has been identified as a High Impact Practice (HIP), that is, as an effective pedagogical practice proven to promote student engagement, satisfaction, acquisition of desired knowledge, skills and competencies, persistence, and attainment of educational goals. Moreover, education research to assess the benefits of UR has shown a strong positive correlation between undergraduate research opportunities and persistence in STEM (Science, Technology, Engineering, and Mathematics) and STEM-related careers. These benefits of UR are reported by students in both dedicated research experiences and research-like courses from, almost entirely, baccalaureate-granting institutions. Can we replicate these findings at the community college level? If so, community college students have much to gain, particularly CUNY community college students who fit the description of those who are more likely to benefit from UR and other HIPs:  low income, first-generation students who come from educationally disadvantaged backgrounds (Kuh, 2008).

In this talk, I will present an overview of the UR program at Queensborough and share preliminary results about the impact of both dedicated research experiences and research-like courses on student learning.  

Dr. Alexander Diaz-Lopez, Villanova University

How to be "successful" in the mathematics world?

In this talk, I will share three pieces of advice regarding how to be "successful" in mathematics (think about the meaning of the word "success"). I will then discuss the effect MSRI-UP has had in my career. We will then discuss some open problems regarding roots of "peak polynomials".    

Dr. Michael Young, Iowa State University

Exponential Domination in Grids

Domination in graphs has been an important and active topic in graph theory for over 40 years. It has immediate applications in visibility and controllability. In this talk we will discuss a generalization of domination called exponential dom- ination. A vertex v in an exponential dominating set assigns weight 21−dist(v,u) to vertex u. An exponential dominating set of a graph G is a subset of V (G) such that every vertex in V (G) has been assigned a sum weight of at least 1. We will specifically look at grid graphs and graphs on the torus.