I am currently working on my Ph.D. in mathematics at Kansas State University.

This website is currently under construction.

My research involves transferring theorems, ideas, and intuition from algebraic geometry into noncommutative algebraic geometry,
stable homotopy
theory, and geometry over the field with one element.

The program of my research can be thought of as "affine categorical
geometry". The objects of study of affine categorical geometry are categories of modules over some algebraic structure. Relevant
examples of such
categories would be a braided monoidal category of modules over a Hopf
algebra, modules over a monoid in a closed symmetric monoidal category,
\(U(\mathfrak{g})-\) mod and \(U_q(\mathfrak{g}) - \) mod for a semisimple Lie algebra \(\mathfrak{g}\), and the model
category of modules over a ring spectrum. One of the goals of affine categorical geometry is to use tools developed by my advisor
Alex Rosenberg in
[1], [2], [3] to define an appropriate notion of the spectrum of such categories
described above. That is if \( \mathcal{A}\) is a category as above we wish to construct a
pair \( ( {\bf Spec}( \mathcal{A}), \mathcal{O}_{\mathcal{A}} ) \), where \( {\bf Spec}(\mathcal{A}) \) is a topological space, and
\[ \mathcal{O}_{\mathcal{A}} : Open({\bf Spec}(\mathcal{A}))^{op} \rightarrow Cat \]
is a presheaf on \( {\bf Spec}(\mathcal{A}) \) with values in small categories. Such a spectrum would be analagous to the set of prime ideals of a commutative ring.
In fact, if \( R \) is a commutative ring, \( {\bf Spec}(R - mod) \) as defined in [1] is
homeomorphic to the prime spectrum \( {\bf Spec}(R) \) through the map
\[ Spec(R) \rightarrow {\bf Spec}(R - mod) \]
\[ p \mapsto ker(R_p \otimes_R - ). \]
The presheaf \( \mathcal{O}_{R-mod} \) is as follows
\[ R-mod :Open( {\bf Spec} (R - mod))^{op} \rightarrow Cat \]
\[ U(S) \rightarrow S^{-1}R - mod \]
where \( S \subseteq R \) is a saturated multiplicative set in \( R \), and \( U(S) \) is the open
set of \( {\bf Spec} (R - mod) \) whose points \( T \in {\bf Spec} (R - mod) \) such that the stalk
of \( \mathcal{O}_{R-mod} \) at \( T \) is nonzero.

Although it may seem like the end goal of defining \( {\bf Spec }(\mathcal{A}) \) for various
categories \( \mathcal{A} \) is algebraic geometry, it is actually representation theory. If
\( \mathcal{A} \) is an algebraic structure (e.g., one of the ones mentioned above), then
the topological space \( {\bf Spec}(\mathcal{A} - mod) \) can be thought of as parameterizing
the category \( \mathcal{A} - mod \); that is, representations of \( \mathcal{A} \). Furthermore, this
parameterization is coherent in a way that allows one to reconstruct the
category \( \mathcal{A} - mod \) if one knows the pair \( ({\bf Spec}(\mathcal{A} - mod), \mathcal{O}_{\mathcal{A}-mod} ) \).

[1] A. Rosenberg, *Noncommutative Algebraic Geometry and Representations of Quantized Algebras. *
Kluwer Academic Publishers Group, Dordecht, 1995.

[2] A. Rosenberg, * Spectra Related with Localizations. * Max-Planck-Institut für Mathematik
Preprint Series 110, 2003.

[3] A. Rosenberg, * Geometry of Noncommutative `Spaces' and Schemes. * Max-Planck-Institut für Mathematik
Preprint Series 68, 2011.

eric.a.bunch@gmail.com

@ericbunch0

Cardwell Hall 127

Kansas State University

Mathematics Department

Manhattan, KS 66506