Ritt Assistant Professor

Department of Mathematics, Columbia

**Office:** Math 427.

**Office Hours:** TBA this semester

**Email:** mmiller@math.columbia.edu Please email me if you have questions about anything!

I received my PhD in Mathematics in June 2019 from UCLA; my advisor was Ciprian Manolescu.

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Teaching

**Fall 2019**

**Calculus III**

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Research

My research is in the field of instanton gauge theory. Using partial differential equations coming from physics, the goal of this field is to assign 'invariants' to 3- and 4-dimensional shapes.

These invariants might be numbers, or they might be groups, or even more complicated objects.

We want these because they help us tell shapes apart, and often contain interesting information about the shapes themselves that was hard to find just from looking at the shape.

In fancier language, I am interested in equivariant instanton homology of 3-manifolds, and its application to topology.

It is my hope and expectation that these new invariants will help unify some disparate ideas in the field.

**Publications**

**Equivariant instanton homology** ~ current as of 8/19/19 ~ comments welcome!

There are some small differences between this version and the arXiv version.

In this paper (which constituted my thesis), I develop some analysis of instanton moduli spaces, and present algebra related to the equivariant (co)homology of differential graded algebras and their modules.

Using these tools, I define four invariants of rational homology spheres that deserve to be called "equivariant instanton homology" and have some formal properties similar to those seen in Heegaard Floer and monopole Floer homology.

Image by Ryan Armand.

Department of Mathematics, Columbia

I received my PhD in Mathematics in June 2019 from UCLA; my advisor was Ciprian Manolescu.

- Tuesday/Thursday 1:10-2:30PM
- Tuesday/Thursday 2:40-4:10PM

**Spring 2020**

**Calculus III**

- Tuesday/Thursday 1:10-2:30PM

**Summer 2020**

**Calculus III**

- Monday-Thursday 6:15-7:50PM

**Fall 2020**

**Calculus IV**

- Tuesday/Thursday 10:10-11:25AM

- Tuesday/Thursday 11:40AM-12:55PM

**Spring 2021**

**Linear Algebra**

- Tuesday/Thursday 11:40AM-12:55PM

These invariants might be numbers, or they might be groups, or even more complicated objects.

We want these because they help us tell shapes apart, and often contain interesting information about the shapes themselves that was hard to find just from looking at the shape.

In fancier language, I am interested in equivariant instanton homology of 3-manifolds, and its application to topology.

It is my hope and expectation that these new invariants will help unify some disparate ideas in the field.

There are some small differences between this version and the arXiv version.

In this paper (which constituted my thesis), I develop some analysis of instanton moduli spaces, and present algebra related to the equivariant (co)homology of differential graded algebras and their modules.

Using these tools, I define four invariants of rational homology spheres that deserve to be called "equivariant instanton homology" and have some formal properties similar to those seen in Heegaard Floer and monopole Floer homology.

Image by Ryan Armand.