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.

`Talks`

## Conferences

- Joint Mathematical Meetings 2022, [website]
*Seattle*, Apr 2022.

- AMS sectional meeting: Model Theory Special session, [website]
*Purdue University*, Mar 2022.

## Seminars

- Fields Institute, University of Toronto,
^{*} *Thematic Program on Tame Geometry, Transseries and Applications to Analysis and Geometry*, Feb 2022.

- Fields Institute, University of Toronto,
- University of Illinois at Chicago,
*Logic Seminar*, Oct 2021.

- University of Leeds and Universidad de los Andes,
^{*} *Topological and Differential Expansions of O-minimal Structures*, Aug 2021.

- University of Leeds and Universidad de los Andes,
- Fields Institute, University of Toronto,
^{*} *Geometry and Model Theory Seminar*, Nov 2020.

- Fields Institute, University of Toronto,
- MSRI, University of California at Berkeley,
^{*} *DDC: Diophantine Problems*, Oct 2020.

- MSRI, University of California at Berkeley,

## Colloquiua

- University of Illinois at Urbana-Champaign,
*AWM Graduate Student Colloquium*, Feb 2022.

- Indraprastha Institute of Information Technology,
*Mathematics Seminar*, May 2019.

`* --> 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
(top 10%) rating.*Outstanding* - On the List of Teachers Ranked as Excellent a total of four times.

- Twice ranked Excellent with
- TA Mentor, Fall 2018
- Advised 5 graduate students though their first semester of teaching.

Co-organized the annual departmental TA orientation.