Posts by Collection

The Krein Matrix and an Interlacing Theorem

Published in SIURO, 2014

Consider the linear general eigenvalue problem $Ay = \lambda By$, where $A$ and $B$ are both invertible and Hermitian $N \times N$ matrices. In this paper we construct a set of meromorphic functions, the Krein eigenvalues, whose zeros correspond to the real eigenvalues of the general eigenvalue problem. The Krein eigenvalues are generated by the Krein matrix, which is constructed through projections on the positive and negative eigenspaces of $B$. The number of Krein eigenvalues depends on the number of negative eigenvalues for $B$. These constructions not only allow for us to determine solutions to the general eigenvalue problem, but also to determine the Krein signature for each real eigenvalue. Furthermore, by applying our formulation to the simplest case of the general eigenvalue problem (where $B$ has one negative eigenvalue), we are able to show an interlacing theorem between the eigenvalues for the general problem and the eigenvalues of $A$.

Recommended citation: S. Auyeung, E. Yu (2014). "The Krein Matrix and an Interlacing Theorem." SIURO. Vol. 7. https://www.semanticscholar.org/paper/The-Krein-Matrix-and-an-Interlacing-Theorem-Shamuel-Yu-Kapitula/2cfb79dfc9f546e5d9277cdcaea4976f4c8221bd?p2df

An Algebraic Characterization of Highly Connected $2n$-Manifolds

Published in Rose Hulman Undergraduate Journal, 2016

All surfaces, up to homeomorphism, can be formed by gluing the edges of a polygon. This process is generalized into the idea of a $(n, 2n)$-cell complex: forming a space by attaching a $(2n−1)$-sphere into a wedge sum of n-spheres. In this paper, we classify oriented, $(n-1)$-connected, compact and closed $2n$-manifolds up to homotopy by treating them as $(n, 2n)$-cell complexes. To simplify the calculation, we create a basis called the Hilton basis for the homotopy class of the attaching map of the $(n, 2n)$-cell complex. At the end, we show that two attaching maps give the same, up to homotopy, manifold if and only if their homotopy classes, when written in a Hilton basis, differ only by a change-of-basis matrix that is in the image of a certain map $\Phi$, which we define explicitly in the paper.

Recommended citation: S. Auyeung, J. Ruiter, D. Zhang, (2016). "An Algebraic Characterization of Highly Connected $2n$-Manifolds." Rose Hulman Undergraduate Journal. Vol. 17, Iss. 2. https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1333&context=rhumj

On the algebra generated by $\bar{\mu},\bar{\partial},\partial,\mu$

Published in Complex Manifolds, 2023

In this note, we determine the structure of the associative algebra generated by the differential operators and $\bar{\mu},\bar{\partial},\partial,\mu$ that act on complex-valued differential forms of almost complex manifolds. This is done by showing that it is the universal enveloping algebra of the graded Lie algebra generated by these operators and determining the structure of the corresponding graded Lie algebra. We then determine the cohomology of this graded Lie algebra with respect to its canonical inner differential $[d,−]$, as well as its cohomology with respect to all its inner differentials.

Recommended citation: S. Auyeung, J. Guu, J. Hu, (2023). "On the algebra generated by $\bar{\mu},\bar{\partial},\partial,\mu$." Complex Manifolds. Vol. 10, Iss. 1. https://www.degruyter.com/document/doi/10.1515/coma-2022-0149/html

Local Lagrangian Floer Homology of Quasi-Minimally Degenerate Intersections

Published in Journal of Topology and Analysis, 2023

We define a broad class of local Lagrangian intersections which we call quasi-minimally degenerate (QMD) before developing techniques for studying their local Floer homology. In some cases, one may think of such intersections as modeled on minimally degenerate functions as defined by Kirwan. One major result of this paper is: if $L_0,L_1$ are two Lagrangian submanifolds whose intersection decomposes into QMD sets, there is a spectral sequence converging to their Floer homology $HF_*(L_0,L_1)$ whose $E^1$ page is obtained from local data given by the QMD pieces. The $E^1$ terms are the singular homologies of submanifolds with boundary that come from perturbations of the QMD sets. We then give some applications of these techniques towards studying affine varieties, reproducing some prior results using our more general framework.

Recommended citation: S. Auyeung (2023). "Local Lagrangian Floer Homology of Quasi-Minimally Degenerate Intersections." Journal of Topology and Analysis. https://doi.org/10.1142/S179352532350036X

Adjacent Singularities, TQFTs, and Zariski’s Multiplicity Conjecture

Published in arXiv, 2023

We give a new proof of Zariski’s multiplicity conjecture in the case of isolated hypersurface singularities; this was first proved by de Bobadilla-Pełka. Our proof uses the TQFT structure of fixed-point Floer cohomology and the fact that adjacent singularities produce symplectic cobordisms between the Milnor fibrations of the singularities. The key technical result is to construct a chain map on Floer cochains using the cobordism and as a last step, apply a spectral sequence of McLean. This last step allows us to also recover a theorem of Varchenko.

Recommended citation: S. Auyeung (2023). "Adjacent Singularities, TQFTs, and Zariski's Multiplicity Conjecture." submitted. https://arxiv.org/abs/2308.13925

On Flowers and Fibonacci-Type Sequences

Published in (NA), 2024

This is a note on recreational mathematics I did with friends in which we proved the following statement which arises naturally when studying (sun)flowers: Consider all Fibonacci-recurrence sequences seeded with values strictly smaller than the $n$th Fibonacci number $F_n$. The smallest integer which does not appear in any such sequence is $F_n^2$.

Recommended citation: S. Auyeung, T. Pensyl, J. Shuster, (2024). "On Flowers and Fibonacci-Type Sequences." available upon request.

Stony Brook University

Undergraduate courses, Stony Brook, New York, 2017

I taught a range of undergraduate courses at Stony Brook University; click the link above to see a full list.

I-STEM Math Camp

Math camp, Stony Brook, New York, 2018

I was a teacher for the I-STEM High School Mathematics Program at Stony Brook University. Click the link above to see a list of topics.

Trinity College

Undergraduate courses, Hartford, Connecticut, 2023

I am a Harold L. Dorwart visiting assistant professor at Trinity College. This Fall 2023 semester, I am teaching multivariable and vector calculus where I make use of tools such as Geogebra. Here are some examples: Frenet frame, twisted cubic, level curves, tangents to a surface. For Spring 2024, I will be teaching an introductory statistics course and a differential equations course.