# Mathematics Colloquia Archives - Fall 2014 and Spring 2015

**Colloquia - Fall 2014 & Spring 2015**

Speaker: | Christine Laurent-Thiebaut (Institut Fourier, Fance) | Aug. 28, 2014 |

Title: |
On the Hartogs phenomenon and global extension of CR functions | |

Abstract: | It is well known that any domain in the complex plane is the domain of definition of a holomorphic function (i.e. given a domain there exists a holomorphic fonction defined on it, which cannot be extended holomorphically to a larger domain). This is no more the case in several complex variables, a new phenomenon appears, which was discovered by Hartogs in 1905 : any holomorphic function defined outside a compact subset of a domain extends holomorphically to the whole domain.We will discuss this extension phenomenon and present some generalizations in different settings. in different settings. | |

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Speaker: | Sunil Chebolu (Illinois State University) | Sept. 4, 2014 |

Title: | Mersenne primes, Fermat primes, and the Catalan's Conjecture | |

Abstract: | We investigate the following natural problem. When is the multiplicative group of a field indecomposable? We solve this problem for finite fields and fields of characteristic not equal to 2. The answer involves Mersenne and Fermat primes, and one solution of this problem involves the famous Catalan's conjecture. We also have some partial results when the characteristic of the field is 2. I will discuss these results which are in joint work (arXiv:1407.3481) with Keir Lockridge. | |

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Speaker: | Li Li (Oakland University) | Oct. 23, 2014 |

Title: | Bases of Cluster algebras and Geometry | |

Abstract: | Cluster algebras were introduced by Fomin and Zelevinsky in trying to understand canonical bases in algebraic groups. A lot of activity in the theory of cluster algebras has been directed towards various constructions of natural bases in them. In this talk, I will discuss some recent advances in the study of cluster algebras, in particular, various natural bases of the cluster algebras and their connection to two classes of special varieties, namely the quiver Grassmannians and Nakajima's graded quiver varieties. | |

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Speaker: | Ben Schmidt (Michigan State University) | Nov. 6, 2014 |

Title: | Illuminating the shape of space with light | |

Abstract: | To what extent do the focal properties of light rays (geodesics) in a space determine the space's global shape? In spaces with large symmetry groups, light rays tend to focus in a highly predictable fashion. Conversely, a rigidity theme asserts that highly symmetric spaces are characterized by special focusing properties of light. I'll present a number of theorems and open problems in this direction. One such problem, in my opinion, is perhaps the most scandalously unsolved problem in all of mathematics. | |

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Speaker: | S. Ejaz Ahmed (Brock University) | Nov. 7, 2014 |

Title: | Big Data Big Bias Big Surprise? | |

Abstract: | In high-dimensional statistics settings where number of variables is greater than observations, or when number of variables are increasing with the sample size, many penalized regularization strategies were studied for simultaneous variable selection and post-estimation. However, a model may have sparse signals as well as with number predictors with weak signals. In this scenario variable selection methods may not distinguish predictors with weak signals and sparse signals. The prediction based on a selected submodel may not be preferable in such cases. For this reason, we propose a high-dimensional shrinkage estimation strategy to improve the prediction performance of a submodel. Such a high-dimensional shrinkage estimator (HDSE) is constructed by shrinking a full model ridge estimator in the direction of a candidate submodel. We demonstrate that the proposed HDSE performs uniformly better than the ridge estimator. Interestingly, it improves the prediction performance of given candidate submodel generated from existing variable selection methods. The relative performance of the proposed HDSE strategy is appraised by both simulation studies and the real data analysis. | |

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Speaker: | Jung-Chen Liu (National Taiwan Normal University) | Nov. 10, 2014 |

Title: | Asymptotic behavior of index of reducibility | |

Abstract: | In a Noetherian ring, every ideal can be factored as an intersection of irreducible ideals. Moreover, the number of irredundant irreducible factors is unique and is called the index of reducibility of this ideal. In this talk, I will present some interesting behavior of index of reducibility while we raise the powers of the generators of the ideal. I will also mention some results related to the asymptotic behavior of the index of reducibility of parameter ideals in a Noetherian local ring. | |

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Speaker: | Yunus Zeytuncu (University of Michigan Dearborn) | Nov. 13, 2014 |

Title: | Mapping Properties of the Bergman Projection and Geometry of Domains in $\mathbb{C}^n$ | |

Abstract: | On a bounded domain in $\mathbb{C}^n$, the Bergman projection is the orthogonal projection operator from the space of square integrable functions onto the space of square integrable holomorphic functions. Mapping properties of this operator depend on geometric properties of the underlying domain. In particular, boundedness on $L^p$ spaces can be related to smoothness of the boundary of the domain. In this talk, we will talk about this relation and present some new results. | |

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Speaker: | Gabriel Huerta (University of New Mexico) | Nov. 14, 2014 |

Title: | Spatial modeling and trend analysis for block maxima extremes | |

Abstract: | We consider some novel approaches for extremes value analysis that rely on spatial statistical methods. Through a block-maxima approach based on the Generalized Extreme Value (GEV) distribution, we characterize extreme precipitation from a regional climate model via a hierarchical structure that assigns a latent spatial process to its location and scale parameters defined through a Gauss Markov Random Field (GMRF) . This model also includes an annual shift in the location parameter that may be able to predict trends over time. In addition, I will discuss how block maxima of extremes of ozone levels can be analysed with related and simpler models that are based on regression trends on the location parameter of the GEV distribution and where the regression parameters are random effect parameters arising from a spatial distribution where correlations are modeled as function of distance between stations on a monitoring network. All these approaches depend on customized Markov Chain Monte Carlo (MCMC) methods to approximate the posterior distribution of observable and un-observable quantities underlying extremal events. | |

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Speaker: | Isola Ajiferuke (University of Western Ontario) | Nov. 18, 2014 |

Title: | Applicability of Count Regression Models in Informetric Studies | |

Abstract: | The purpose of the study is to investigate the applicability of count regression models in informetric studies. Identified count response variables in informetric studies include the number of authors, the number of references, the number of views, the number of downloads, and the number of citations received by an article. Also of a count nature are the number of links from and to a website. Data were collected from the United States Patent and Trademark Office (www.uspto.gov), an open access journal (www.informationr.net/ir/), and Web of Science. The datasets were then used to compare the performance of linear regression model with Poisson, negative binomial, and generalized Poisson regression models. The regression analyses showed that linear regression model performed slightly better than the appropriate count model in two datasets, predicted some negative values for the response variable in three datasets, and performed much worse than the appropriate count regression model in two other datasets. In addition, it was found that due to over-dispersion in most response variables, the negative binomial regression model seems to be more appropriate for informetric datasets. | |

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Speaker: | Amit Savkar (University of Connecticut) | Jan. 15, 2015 |

Title: | Placement, knowledge gap, and assessment: Common challenges for University educators | |

Abstract: | This talk will look at the current challenges faced at the university level in terms of placing incoming freshmen students in correct first-year mathematics courses. In particular, I will highlight some of the steps taken to place students in pre-calculus, differential calculus, and integral calculus using ALEKS. We will look at the potential effect of student placement on DFW rates. Finally, I will introduce a way for data integration and assessment of data for understanding student knowledge gap and future projects to bridge this gap through systemic changes using pedagogy through technology. The concept mapping through the use of "tagging" will be to pin point knowledge gaps and areas of potential interventions. | |

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Speaker: | Kasso Okoudjou (University of Maryland) | Jan. 22, 2015 |

Title: | Preconditioning of frames | |

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Speaker: | Jungho Yoon (Ewha Womens University, South Korea) | Feb. 5, 2015 |

Title: | Data approximation methods and their applications | |

Abstract: | In this talk, we discuss (linear and non-linear) data approximation methods and their applications to CAGD and image processing. Some approximation theories and algorithms such as moving least squares methods and subdivisions are presented. We first discuss advantages and limitations of linear methods. Then, on the purpose of overcoming their drawbacks, we introduce adapted methods using `non-local' penalty functions and regularizations. The suggested methods are based on the moving least squares projection technique and total variation minimizations, but introduce a fundamental modification such that the weights are determined based on local data similarities. | |

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Speaker: | Peter Tingley (Loyola University) | Feb, 19, 2015 |

Title: | Crystals: a combinatorial way to study representations of Lie groups. | |

Abstract: | Group theory is about understanding symmetries of (mathematical) objects. But sometimes it is useful to turn this around: instead of starting with an object and trying to understand its symmetries, one starts with a group and tries to understand all objects or spaces that have those symmetries. At this point, one is doing representation theory. I will discuss the representation theory of various Lie groups, mostly the group SL(n) of n by n matrices with determinant 1, and how it can be studied combinatorially using Kashiwara's theory of crystals. I will then briefly discuss my own research, which is largely concerned with understanding this relationship between combinatorics and representations theory in the case of affine Lie groups. | |

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Speaker: | Zalman Usiskin (University of Chicago) | March 3, 2015 |

Title: | Teachers' Mathematics: A Branch of Applied Mathematics | |

Abstract: | Teachers' mathematics" refers to that mathematics that is helpful to a teacher of the subject but may not be of particular interest to people who apply mathematics in other domains. This mathematics arises from classroom situations in much the same way that statistics arises from real data. It includes pedagogical content knowledge, but also the analysis of mathematical concepts, Polya-style problem analysis, and connections among mathematical topics. Examples of each of these types of teachers' mathematics related to standard high school mathematics content will be shown. | |

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Speaker: | Shyamal Peddada (NIH) | March 5, 2015 |

Title: | Some challenges in the analysis of microbiome data: Old wine in a new bottle with a twist! | |

Abstract: | Understanding differences in the microbial composition and abundance of taxa in different groups of study subjects (e.g. gut microbiome in pre-term and full term babies) is of great interest. Some of the existing "off the shelf" computational tools, which are based on linear regression, t-test, ANOVA etc., as well as some of the recently developed models/methods such as ZIG, Dirichlet-Multinomial models etc., are flawed for these data because they either ignore the underlying structure in the microbiome data or use inappropriate models. As a result they are subject to inflated false discovery rate (FDR) while also losing power. Consequently, these methods; (a) may result in wasted resources in following up "leads" that cannot be replicated because they are false, (b) may result in missing important findings that should have been discovered, and most importantly (c) misinterpretation of the underlying biology. In this talk we introduce a novel methodology called Analysis of Composition of Microbiomes (ANCOM) that accounts for the underlying compositional structure in the data, and controls the FDR at a nominal level (less than 5%) while being more powerful than the existing methods. More importantly, it enables a better biological interpretation of the data. NOTE: This talk is designed to be conceptual (with minimal mathematical details) and should be broadly accessible. | |

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Speaker: | Tracy Payne (Idaho State University) | March 17, 2015 |

Title: | Evolutionary Game Theory and Applications to Geometry | |

Abstract: | Evolutionary game theory is a continuous model for interactions with frequency dependent selection. For example, the classical Lotka-Volterra predator-prey system can be modeled this way. Another example is an infinite population of people playing the game rock-paper-scissors. (Obviously, if almost everyone else is playing rock, you will play paper.) Asymptotic behavior, Nash equilibria and stability of fixed points are studied. This framework may also be used to analyze the evolution of geometric structures; in fact, in some cases, the Ricci flow is exactly a “replicator equation” for evolutionary game theory. We will describe one such geometric evolution equation-the combinatorial Ricci flow- and how methods of evolutionary game theory may be applied to it. | |

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Speaker: | Robert delMas (University of Minnesota) | March 19, 2015 |

Title: | Trends in students’ conceptual understanding of statistics | |

Abstract: | The talk will begin with a brief presentation on best practices in assessment followed by a more detailed presentation on results from the ARTIST CAOS assessment. The ARTIST project has collected data since 2005 on students' understanding of statistics through the administration of the Comprehensive Assessment of Outcomes in Statistics (CAOS) instrument. The CAOS test consists of 40 multiple-choice items that cover six topics: data collection and design, graphical representations, variability, sampling variability, tests of significance, and bivariate data. From 2005 through 2014, over 23,000 tertiary level students in the United States of America who were enrolled in a college-level first course in statistics completed the CAOS test at the end of their respective courses. Information on the psychometric properties of the CAOS assessment will be presented followed by a look at trends in students' understanding of the six statistical topics across the 9-year period. | |

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Speaker: | William Fulton (University of Michigan) | April 2, 2015 |

Title: | On the Origins of Intersection Theory in Algebraic Geometry | |

Abstract: | In the middle of the 19th century, two independent but related questions led to the development of intersection theory as we know it today. One was enumerative geometry — counting geometric figures satisfying a collection of conditions — which is thriving today largely because ideas from physics. The other is the study of degeneracy loci — typified by spaces of matrices of at most a given rank. In this talk I will describe the early history of these subjects. | |

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Speaker: | Bella Bollobas (University of Memphis) | April 9, 2015 |

Title: | Bootstrap Percolation | |

Abstract: | Professor Bollobas will present some basic facts about bootstrap percolation. He will go on to describe some important theorems proved by Aizenman, Lebowitz, Cerf, Manzo, Cirillo and Holroyd, culminating in some substantial results he has proved with Balogh, Duminil-Copin and Morris. | |

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Speaker: | Bella Bollobas (University of Memphis) | April 10, 2015 |

Title: | Universality for Monotone Cellular Automata | |

Abstract: | Recently, with Smith and Uzzell, Dr. Bollobas initiated the study of a far-reaching generalization of bootstrap percolation on lattices and lattice-like finite graphs. The only assumptions made about such a process is that it is local, homogeneous and monotone. Surprisingly, much can be proved about these very general processes; in particular, they can be classified into three classes, telling us much about the critical probability. In two dimensions such a classification was given by Smith, Uzzell, Balister, Przykucki and Dr. Bollobas. Very recently, Duminil-Coplin, Morris, Smith and Dr. Bollobas have gone much further: they have proved fairly precise results about the process in all dimensions, indicating that a rich theory of "universality theory' is waiting to be discovered. | |

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Speaker: | Maria Tjani (University of Arkansas) | April 16, 2015 |

Title: | Closed range composition operators on the Bergman space | |

Abstract: | Let $\phi$ be a holomorphic self map of the unit disk $\mathbb D$. Associate to $\phi$ the composition operator $C_\phi$ defined by $C_\phi f=f\circ\phi$. If $\phi$ is non-constant then $C_\phi$ is a one-to-one linear operator that is bounded on many spaces of analytic functions. We will discuss $C_\phi$ on the Bergman space $A^2$. We give necessary and sufficient conditions for the composition operator to have closed range on $A^2$. The pull-back measure of area measure on $\mathbb D$ plays an important role. This talk is based on joint work with Pratibha Ghatage. | |