I am a mathematics doctoral candidate at UIUC. My research interests lie primarily in model theory and its applications. I am part of the Logic group, and my advisor is Lou van den Dries.

Before coming over to Illinois, I received my undergraduate education at IIT Kanpur.

Email: nbhard4 [at] illinois.edu

## Research

My recent work is with the model theory of valued fields, and o-minimality and its applications.

## Papers

On the Pila-Wilkie theorem. [pdf, arXiv]
with Lou van den Dries, Submitted.

This expository paper gives an account of the Pila-Wilkie counting theorem and some of its extensions and generalizations. We use semialgebraic cell decomposition to simplify part of the original proof. Included are complete treatments of a result due to Pila and Bombieri and of the o-minimal Yomdin-Gromov theorem that are used in this proof.
The additive groups of $ℤ$ and $ℚ$ with predicates for being square-free. [pdf, arXiv, DOI]
with Minh Chieu Tran, The Journal of Symbolic Logic, vol. 85 (2020).

We consider the four structures $(ℤ;\mbox{Sqf}^ℤ)$, $(ℤ;<,\mbox{Sqf}^ℤ)$, $(ℚ;\mbox{Sqf}^ℚ)$, and $(ℚ;<,\mbox{Sqf}^ℚ)$ where $ℤ$ is the additive group of integers, $\mbox{Sqf}^ℤ$ is the set of $a\in ℤ$ such that $v_p(a)<2$ for every prime $p$ and corresponding $p$-adic valuation $v_p$, $ℚ$ and $\mbox{Sqf}^ℚ$ are defined likewise for rational numbers, and $<$ denotes the natural ordering on each of these domains. We prove that the second structure is model-theoretically wild while the other three structures are model-theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences.

View one of our structures in the model theory universe.

## Talks

These talks were recorded.

Fields Institute, University of Toronto,
Geometry and Model Theory Seminar, Nov 2020.
MSRI, University of California at Berkeley,
DDC: Diophantine Problems, Oct 2020.

## Teaching

A brief profile of my teaching at UIUC.

## Courses

• Math 241, Calculus III
Fall 2017, Fall 2019, Fall 2020.
• Math 231, Calculus II
Spring 2018, Spring 2019.
• Math 221, Calculus I
Fall 2018.

On the List of Teachers Ranked as Excellent four times.
Twice ranked Excellent with Outstanding rating - (top 10% of instructors university-wide).
TA Mentor, Fall 2018