## Dori Bejleri

Graduate StudentBrown University

### Contact Info

Department of MathematicsBox 1917

Brown University

151 Thayer Street

Providence, RI 02912

dbejleri [at] math [dot] brown [dot] edu

### About me

I'm a fourth year graduate student in the math department at Brown University. My adviser is Dan Abramovich.

### Research

I'm mainly interested in algebraic geometry -- specifically moduli spaces and birational geometry with connections to enumerative geometry, combinatorics and geometric representation theory.

One of my current focuses is on compactifying moduli spaces of surfaces using ideas from the minimal model program. More specifically, I'm studying surfaces that admit a fibration to a smooth curve. Using the theory of twisted stable maps to Deligne-Mumford stacks, one can construct natural degenerations of such fibered surfaces and apply them to compactify moduli spaces by KSBA stable pairs.

My other current focus is on the combinatorial and enumerative geometry of moduli spaces of sheaves, and particularly Hilbert schemes of points. I'm interested in the rich interplay between partition combinatorics and the geometry of Hilbert schemes of points on smooth surfaces and surface orbifolds. I'm also interested in the structure of Hilbert schemes of points on singular curves, espcially generalizing results about planar curves to larger embedding dimension and connections with curve counting on threefolds. I'd also like to understand tropicalizations of these moduli spaces.

You can read more on my research page.

Previously I've worked on foundational questions about algebraic geometry over $\mathbb{F}_1$ with motivation from the Feynman integrals associated to graphs in quantum field theory.

### Papers and preprints

*Motivic Hilbert zeta function of curves are rational.*(With Dhruv Ranganathan & Ravi Vakil).*Moduli of weighted stable elliptic surfaces and invariance of log plurigenera.*(With Kenny Ascher). Submitted.*Moduli of fibered surface pairs from twisted stable maps.*(With Kenny Ascher). Submitted.*Log canonical models of elliptic surfaces.*(With Kenny Ascher).. Volume 320 (2017), 210-243.**Advances in Mathematics***The tangent space of the punctual Hilbert scheme.*(With David Stapleton).Volume 66 (2017), no. 3, 595 - 610.**Mich. Math. J.***The topology of equivariant Hilbert schemes.*(With Gjergji Zaimi).*Quantum field theory over F1.*(With Matilde Marcolli).Volume 69 (2013), 40 – 59.**Journal of Geometry and Physics.**

### Teaching

Instructor for Math 0100: Introductory Calculus II Spring 2017. Section webpage.Teaching assistant for Math 0090: Introductory Calculus I Fall 2015.

### Notes

Notes on $A_{\mathrm{inf}}$ and perfectoid rings from a learning workshop on integral p-adic hodge theory.Notes from a talk given at Brown in February of 2016 on motivic Hilbert zeta functions.

These are notes on the geometry of the Hilbert scheme of points on $\mathbb{A}^2$ for the Brown Graduate Student Seminar. They are aimed at first year graduate students.

Here are some rough notes from my talk on cohomology of equivariant Hilbert schemes. There may be some "typos".

### Some Math Links

### Personal

Previously, I was studying math at Caltech and spending my time in Ricketts House.This is a picture of me and my beautiful wife and our dog Grizzly.

I also like playing music and I'm a big fan of eSports.