Education
- PhD in Mathematics, Purdue University, August 2017
- B.S. in Mathematics, Oakland University, May 2010
- B.S. in Computer Science, Oakland University, May 2010
Employment
- Assistant Professor at Oakland University, August 2021-Present
- Visiting Assistant Professor at Oakland University, August 2020-July 2021
- Visiting Assistant Professor at Wilkes Honors College, August 2018-June 2020
Selected Publications
See publications.Talks and Colloquiums
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An Example of Nagata and Ribenboim
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Interval Rings
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Interval Rings
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Overrings of 2-Dimensional RLRs
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Overrings of a 2-Dimensional Regular Local Ring
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Convergence of Rees Valuations
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Blowing Up Finitely Supported Ideals of a Regular Local Ring
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Directed Unions of Local Quadratic Transforms
Abstract: Directed Unions of Local Quadratic Transforms
We consider infinite sequences {R_n} of successive local quadratic transforms of a regular local ring. Let S denote the directed union of the sequence of regular local rings R_n and let V denote the unique limit point of the corresponding sequence of order valuation rings. In this talk, I will examine asymptotic properties of this family of order valuations and link this asymptotic behavior to ring-theoretic properties of S, namely whether S is archimedean and whether S is completely integrally closed. I will show how these asymptotic limits give an explicit valuation for the boundary valuation ring V, where the description depends on whether S is archimedean or non-archimedean. This seminar is based on joint work with William Heinzer, Alan Loper, Bruce Olberding, and Hans Schoutens.
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Valuations and Invariants Associated With Directed Unions of Local Quadratic Transforms
Abstract: Valuations and Invariants Associated With Directed Unions of Local Quadratic Transforms
We examine ideal-theoretic properties of the directed union S of an infinite sequence of local quadratic transforms of a regular local ring. We associate a boundary valuation ring V to the sequence and examine its relation to S and to the complete integral closure of S. We define an associated invariant tau and describe how tau determines the structure of V and S, namely the rank of V, whether S is archimedean, and whether S is completely integrally closed.
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Infinite Directed Unions of Local Quadratic Transforms of RLRs
Abstract: Infinite Directed Unions of Local Quadratic Transforms of RLRs
We examine properties of the directed union S of an infinite sequence of local quadratic transforms of a regular local ring. We associate to S a valuation ring V and give an intersection decomposition involving V. I will define an invariant tau and show how tau determines whether S is archimedean or non-archimedean, and in the archimedean case, whether S is completely integrally closed. For each case, we will discuss properties of S and I will give an explicit description for V.
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Ideal Theory of Infinite Directed Unions of Local Quadratic Transforms
Abstract: Ideal Theory of Infinite Directed Unions of Local Quadratic Transforms
We examine ideal-theoretic properties of the directed union S of an infinite sequence of local quadratic transforms of a regular local ring. We associate a boundary valuation ring V to the sequence and examine its relation to S and to the complete integral closure of S. We define an associated invariant tau and describe how tau determines the structure of V and S, namely the rank of V, whether S is archimedean, and whether S is completely integrally closed.
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Blowing Up Finitely Supported Ideals in a Regular Local Ring
Abstract: Blowing Up Finitely Supported Ideals in a Regular Local Ring
We consider singularities of the projective model obtained by blowing up a finitely supported ideal I of a regular local ring. If this blowup is nonsingular, we prove that it is obtained by blowing up the finite set of base points of I. Extending work of Lipman and Huneke-Sally in the 2-dimensional case, we prove that every UFD on the blowup of I that dominates R is regular and infinitely near to R. This is joint work with W. Heinzer and Youngsu Kim.
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Factorizations of Finitely Supported *-Simple Complete Monomials Ideals
Abstract: Factorizations of Finitely Supported *-Simple Complete Monomials Ideals
Zariski proved a unique factorization theorem for the complete ideals in a 2-dimensional regular local ring, and Lipman extended this result by proving in the higher dimensional case that every finitely supported complete ideal has a unique factorization as a *-product of special *-simple complete ideals, with possibly negative exponents. In this talk, for the class of finitely supported complete monomial ideals, we consider (1) the Rees valuations (2) the minimal number of generators (3) when negative exponents arise in the unique factorization as a *-product of special *-simple ideals (4) the order and index with respect to inverse transforms We give examples to illustrate these properties and a non-monomial example where some of these properties do not hold.
Conferences Attended
- MathFest 2019 at Cincinnati, August 2019
- AMS Sectional Meeting at the Ohio State University, March 2018
- Commutative Algebra and Its Interactions With Algebraic Geometry at University of Michigan, July 2016
- Midwest Commutative Algebra and Algebraic Geometry Conference at Notre Dame, May 2016
- Southwest Local Algebra Meeting at Texas State University, February 2016
- AMS Sectional Meeting at Loyola University in Chicago, October 2015
- Midwest Commutative Algebra and Geometry Conference at Purdue University, August 2015
- Midwest Commutative Algebra and Geometry Conference at Purdue University, May 2011
Awards and Grants
- Purdue Research Foundation grant, June 2015 to May 2016
- Ross Fellowship, August 2010 to July 2011
- Louis R. Bragg Graduating Senior Award, Oakland University, May 2010
Programming Language Proficiencies
- Mathematica, Macaulay2, Python, and JavaScript (with Node.js)