I am interested in studying convex (and lately, Riemannian) geometric questions from an analytic or probabilistic perspective. My favourite tools come from information theory and mass transport.
Keywords : Convex bodies, entropy, log-concave measures, displacement interpolation, curvature-dimension conditions, heat flow
Publications/preprints:
G. Aishwarya, L. Rotem, and Y. Shenfeld: "Sectional curvature and matrix displacement convexity" 2025 (preprint). [paper]
G. Aishwarya, D. Li, M. Madiman, and M. Meckes: "Metric complexity is a Bryant--Tupper diversity" 2025 (preprint). [paper, related blog post]
G. Aishwarya, D. Li: "Entropic and functional forms of the dimensional Brunn--Minkowski inequality in Gauss space'' To appear Mathematische Annalen 2025. [paper, related talk]
G. Aishwarya, D. Li: "The Kneser--Poulsen phenomena for entropy'' International Mathematics Research Notices, Volume 2025, Issue 12, June 2025, rnaf140. [paper, related talk]
G. Aishwarya, D. Faifman: "Stability in the Banach isometric conjecture and nearly monochromatic Finsler surfaces'' To appear Journal d'Analyse Mathematique 2025. [paper, related talk]
G. Aishwarya, L. Rotem: "New Brunn--Minkowski and functional inequalities via convexity of entropy'' 2023 (preprint). [paper, related talk]
G. Aishwarya, I. Alam, D. Li, S. Myroshnychenko, and O. Zatarain-Vera: "Entropic exercises around the Kneser-Poulsen conjecture''. Mathematika, 69, no. 3, 2023. [paper, related talk]
G. Aishwarya and M. Madiman: "Conditional Rényi entropy and the relationships between Rényi capacities''. Entropy (Special issue on Information Measures with Applications edited by A. Lapidoth), vol. 22, no. 5, 2020. [paper]
G. Aishwarya and M.Madiman: "Remarks on Rényi versions of conditional entropy and mutual information''. Proceedings of the 2019 IEEE International Symposium on Information Theory, Paris, France, July 2019. [conference paper]
In preparation:
G. Aishwarya and M. Madiman: "Magnitude, diversity, and primes: first steps" 2023 (preprint available on request)
Theses/notes:
"Some information-theoretic notions of size in the metric setting" 2021 (PhD thesis, available here)
"Topics in Geometric Measure Theory" 2016 (M.Sc. thesis, available here): Notes on basic Geometric Measure Theory; covers area and coarea formulas, Federer's Tube formula for sets of Postive reach, etc.