Abstract:
Let k be an algebraically closed field of characteristic p>2. By a result of Kumar and Thomsen, the standard Frobenius splitting of the affine plane induces a Frobenius splitting of the Hilbert scheme of n points in the plane. In this thesis, we investigate the question, "what is the stratification of the Hilbert scheme of points in the plane by all compatibly Frobenius split subvarieties?" We provide the answer to this question when n is at most 4 and we give a conjectural answer when n=5. We prove that this conjectural answer is correct up to the possible inclusion of one particular one-dimensional subvariety of the Hilbert scheme of 5 points, and we show that this particular one-dimensional subvariety is not compatibly split for at least those primes p between 3 and 23. Next, we restrict the splitting of the Hilbert scheme of n points in the plane (now for arbitrary n) to the affine open patch U_ and describe all compatibly split subvarieties of this patch and their defining ideals. We find degenerations of these subvarieties to Stanley-Reisner schemes, explicitly describe the associated simplicial complexes, and use these complexes to prove that certain compatibly split subvarieties of U_ are Cohen-Macaulay.

Abstract:
We describe a closed immersion from each representation space of a type A quiver with bipartite (i.e., alternating) orientation to a certain opposite Schubert cell of a partial flag variety. This "bipartite Zelevinsky map" restricts to an isomorphism from each orbit closure to a Schubert variety intersected with the above-mentioned opposite Schubert cell. For type A quivers of arbitrary orientation, we give the same result up to some factors of general linear groups. These identifications allow us to recover results of Bobinski and Zwara; namely we see that orbit closures of type A quivers are normal, Cohen-Macaulay, and have rational singularities. We also see that each representation space of a type A quiver admits a Frobenius splitting for which all of its orbit closures are compatibly Frobenius split.

Abstract:
We provide combinatorial formulas for the multidegree and K-polynomial of an (arbitrarily oriented) type A quiver locus embedded inside of its representation space. These formulas are generalizations of three of Knutson-Miller-Shimozono's formulas from the equioriented setting: - The ratio formulas express each K-polynomial as a ratio of specialized double Grothendieck polynomials, and each multidegree as a ratio of specialized double Schubert polynomials. - The pipe formulas express each K-polynomial as an alternating sum over pipe dreams that fit inside of a particular shape, and each multidegree as a positive sum over reduced pipe dreams that fit inside of that same shape. - The component formulas express each K-polynomial as an alternating sum of products of Grothendieck polynomials, and each multidegree as a positive sum of products of Schubert polynomials. The summands are indexed by lacing diagrams associated to the type A quiver locus. The K-polynomial component formula was first conjectured by Buch-Rim\'{a}nyi, and the multidegree component formula was first proved by Buch-Rim\'{a}nyi.

Abstract:
We give an explicit presentation for each lower bound cluster algebra. Using this presentation, we show that each lower bound algebra Grobner degenerates to the Stanley-Reisner scheme of a vertex-decomposable ball or sphere, and is thus Cohen-Macaulay. Finally, we use Stanley-Reisner combinatorics and a result of Knutson-Lam-Speyer to show that all lower bound algebras are normal.

Abstract:
Matrix Schubert varieties are certain varieties in the affine space of square matrices which are determined by specifying rank conditions on submatrices. We study these varieties for generic matrices, symmetric matrices, and upper triangular matrices in view of two applications to algebraic statistics: we observe that special conditional independence models for Gaussian random variables are intersections of matrix Schubert varieties in the symmetric case. Consequently, we obtain a combinatorial primary decomposition algorithm for some conditional independence ideals. We also characterize the vanishing ideals of Gaussian graphical models for generalized Markov chains. In the course of this investigation, we are led to consider three related stratifications, which come from the Schubert stratification of a flag variety. We provide some combinatorial results, including describing the stratifications using the language of rank arrays and enumerating the strata in each case.

Abstract:
We show that locally acyclic cluster algebras have (at worst) canonical singularities. In fact, we prove that locally acyclic cluster algebras of positive characteristic are strongly F-regular. In addition, we show that upper cluster algebras are always Frobenius split by a canonically defined splitting, and that they have a free canonical module of rank one. We also give examples to show that not all upper cluster algebras are F-regular if the local acyclicity is dropped.

Abstract:
The recent explosion of high-throughput technology has been accompanied by a corresponding rapid increase in the number of new statistical methods for developing prognostic and predictive signatures. Three commonly used feature selection techniques for time-to-event data: single gene testing (SGT), Elastic net and the Maximizing R Square Algorithm (MARSA) are evaluated on simulated datasets that vary in the sample size, the number of features and the correlation between features. The results of each method are summarized by reporting the sensitivity and the Area Under the Receiver Operating Characteristic Curve (AUC). The performance of each of these algorithms depends heavily on the sample size while the number of features entered in the analysis has a much more modest impact. The coefficients estimated utilizing SGT are biased towards the null when the genes are uncorrelated and away from the null when the genes are correlated. The Elastic Net algorithms perform better than MARSA and almost as well as the SGT when the features are correlated and about the same as MARSA when the features are uncorrelated.

Abstract:
In the decades, since the advent of shockwave lithotripsy, instrumentation and techniques in both ureteroscopic and percutaneous stone management have improved exponentially, leading to both increased success and lower complication rates. As a result, there have been some controversies revolving around the therapeutic modality of choice for specific stones in terms of their size and location. This review seeks to provide some clarity to the decision-making process with emphasis on patient comfort and choice and due consideration being given to the potential complications associated with the various treatment modalities.

Abstract:
Background: Learning portfolios are increasingly being introduced in higher education including undergraduate and postgraduate medical education. Due to their highly personalized nature, creation of an assessment tool that accurately reflects the value for the learner of the ‘work’ created is challenging, and has prevented a more widespread use of this valuable tool. Innovation & Evaluation: Forty-one physical therapy students were asked to create a learning portfolio as a component of their pathology course. This collection of evidence of learning was evaluated at the midterm and final examination by a synchronous tripod of assessors-the ‘self’, a peer, and the instructor- to provide a formative and summative evaluation. Results: Grades awarded by the three assessors were more similar at the end of the semester when compared with those at the midterm. A quantitative and qualitative satisfaction questionnaire was additionally given to students to determine the usefulness of this educational activity. Though the majority of students responded favourably, with notable self-reported improvements in communication, team-work, and professional growth, primary challenges included negative perceptions related to increased time commitment, student and teacher-related stress, and uncertainty regarding the value and the immediate and long-term relevance of this creative learning activity. Conclusion: Reflection on our study authenticates that the combination of formative and summative evaluations from such tripod assessments of learning portfolios is particularly suited for explicit inclusion in higher educational programs including medicine and allied health professionals. We recommend learning portfolios as a creative learning tool and assessment tool in higher education.

High speed rail systems have blossomed in technological advances since their debut in the 1960’s with the Japanese Shinkansen line. As miles upon miles of tracks increase around the world, bringing added mobility to travelers while decreasing emissions, various technologies are leading the way to a faster tomorrow. This article explores the differences in what countries around the world are using to supply the next generation of travel modes. This paper details the differences in technological implementations from Asia, Europe, and North America. High speed rail systems are far more developed in other countries, especially China, and have required substantial government investment. The United States, with limited HSR development, stands to benefit from the technological advances of others and learn from the economic impacts of HSR in other countries.