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.

Useful links

• Logic arXiv
• Map of the model theoretic universe
• A collective blog on model theory
• Model theory wiki

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

Advised 5 graduate students though their first semester of teaching.
Co-organized the annual departmental TA orientation.