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.

I am on the job market, seeking positions starting in the latter half of 2022, and happy to share my CV and research statement upon request.

Email: nbhard4 [at] illinois.edu

## Research

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

## Papers

Ax-Kochen-Ersov theory for restricted analytic functions, including the induced structure on residue field and value group,
in preparation.
On the Pila-Wilkie theorem, [pdf, arXiv]
with Lou van den Dries, Expositiones Mathematicae, accepted, pending minor revisions.

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. 86 (2021), no. 4, pp. 1324-1349.
View one of our structures on a mapping of the model theory universe.

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.

## Conferences

• Joint Mathematical Meetings 2022, [website]
Seattle, Apr 2022.
• AMS sectional meeting: Model Theory Special session, [website]
Purdue University, Mar 2022.

## Colloquiua

* --> talks were recorded.

## Teaching

A class picture with my last batch at UIUC!

## Courses

• Math 241, Calculus III (Merit)
Fall 2021.
• Math 241, Calculus III
Fall 2017, Fall 2019, Fall 2020.
• Math 231, Calculus II
Spring 2018, Spring 2019.
• Math 221, Calculus I
Fall 2018.

## Awards

• Twice ranked Excellent with Outstanding (top 10%) rating.
On the List of Teachers Ranked as Excellent a total of four times.
• TA Mentor, Fall 2018