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6 results on '"Philbin, McCleary"'

1. Matrix Formulae and Skein Relations for Quasi-cluster Algebras

Author

Gilbert, Cody, Philbin, McCleary, and Wright, Kayla

Subjects
Mathematics - Combinatorics, Mathematics - Geometric Topology, Mathematics - Rings and Algebras
Abstract

In this paper, we give matrix formulae for non-orientable surfaces that provide the Laurent expansion for quasi-cluster variables, generalizing the orientable surface matrix formulae by Musiker-Williams. We additionally use our matrix formulas to prove the skein relations for the elements in the quasi-cluster algebra associated to curves on the non-orientable surface.

Published
2023

2. A Workshop to Build Community and Broaden Participation in Mathematics: Reflections on the Mathematics Project at Minnesota

Author

Banaian, Esther, Brauner, Sarah, Chandramouli, Harini, Klinger-Logan, Kim, Nadeau, Alice, and Philbin, McCleary

Abstract

We detail our experience running an annual four-day workshop at the University of Minnesota, called the Mathematics Project at Minnesota (MPM). The workshop is for undergraduates who come from groups underrepresented in mathematics and aims to increase the participation and success of such groups in the mathematics major at the University. In this paper, we explain how MPM is organized, discuss its objectives, and highlight some of the sessions that we feel are emblematic of the program's success. The paper concludes with an analysis of achievements and obstacles in the programs' first three years. [Funding for the Mathematics Project at Minnesota was provided by the National Science Foundation.]

Published
2022
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3. Splines over integer quotient rings

Author

Philbin, McCleary, Swift, Lindsay, Tammaro, Alison, and Williams, Danielle

Subjects
Mathematics - Combinatorics
Abstract

Given a graph with edges labeled by elements in $\mathbb{Z}/m\mathbb{Z}$, a generalized spline is a labeling of each vertex by an integer $\mod m$ such that the labels of adjacent vertices agree modulo the label associated to the edge connecting them. These generalize the classical splines that arise in analysis as well as in a construction of equivariant cohomology often referred to as GKM-theory. We give an algorithm to produce minimum generating sets for the $\mathbb{Z}$-module of splines on connected graphs over $\mathbb{Z}/m \mathbb{Z}$. As an application, we give a quick heuristic to determine the minimum number of generators of the module of splines over $\mathbb{Z}/m\mathbb{Z}$. We also completely determine the ring of splines over $\mathbb{Z}/p^k\mathbb{Z}$ by providing explicit multiplication tables with respect to the elements of our minimum generating set. Our final result extends some of these results to splines over $\mathbb{Z}$.

Published
2017

4. A Workshop to Build Community and Broaden Participation in Mathematics: Reflections on the Mathematics Project at Minnesota.

Author

Banaian, Esther, Brauner, Sarah, Chandramouli, Harini, Klinger-Logan, Kim, Nadeau, Alice, and Philbin, McCleary

Subjects
COMMUNITY involvement, COLLEGE majors
Abstract

We detail our experience running an annual four-day workshop at the University of Minnesota, called the Mathematics Project at Minnesota (MPM). The workshop is for undergraduates who come from groups underrepresented in mathematics and aims to increase the participation and success of such groups in the mathematics major at the University. In this paper, we explain how MPM is organized, discuss its objectives, and highlight some of the sessions that we feel are emblematic of the program's success. The paper concludes with an analysis of achievements and obstacles in the programs' first three years. [ABSTRACT FROM AUTHOR]

Published
2022
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6. On the Algebraic de Rham Homology of Quasi-projective Schemes

Author

Philbin, McCleary

Abstract

Let Y be a nonempty, connected, quasi-projective scheme over a field k of characteristic 0, embedded in Pnk. We show that the object En,n1 on the E1 page of the Hodge-de Rham spectral sequence abutting to the algebraic de Rham homology of Y is a one dimensional k-space when Y is complete of dimension at least one, an infinite dimensional k-space when Y is zero dimensional, and zero when Y is not complete. Further, we show that when Y is of dimension at least one, En,n1 is a stable object, and when Y is of dimension zero, En,n2 is a stable object. In the latter case, the spectral sequence degenerates on the E2 page. Together these results allow us to compute the zero-th algebraic de Rham homology of Y .

Published
2023

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