Indian Institute of Technology Hyderabad (IITH) is a premier institute of science and technology established in 2008. IITH has been consistently ranked in the&nbsp;<a href="https://iith.ac.in/reports/" rel="noopener noreferrer">top 10 institutes</a>&nbsp;in India for Engineering&nbsp;<a href="https://www.nirfindia.org/2021/EngineeringRanking.html" rel="noopener noreferrer" target="_blank">according to NIRF</a>&nbsp;making it one of the most coveted schools for science and technology in the country.

IITH offers&nbsp;<a href="https://iith.ac.in/academics/under-graduate/">undergraduate programs</a>&nbsp;in all the classical engineering disciplines, applied sciences, design, as well as several modern interdisciplinary areas. Students are given a flexibility to explore a broad set of areas, and potentially pursue a minor or double major in a discipline that is not their own. Students who wish to seek a deeper understanding of their own discipline are strongly encouraged to get involved in cutting-edge research with the help of a faculty to mentor them, and earn an Honors in their own field.

The very foundation of IIT Hyderabad is based on&nbsp;<a href="https://iith.ac.in/research/">research and innovation</a>. The vibrant research culture is evident from the number of patents and publications that IITH produces consistently every year. IITH offers&nbsp;<a href="https://iith.ac.in/academics/programmes-offered/">graduate programs</a>&nbsp;at both a masters, and a doctoral level, in several diverse areas. There are separate programs for technology, design, science, and liberal arts. The&nbsp;<a href="https://iith.ac.in/academics/post-graduate/">MTech program</a>&nbsp;is offered in various modes and durations to cater to the ever-growing need of postgraduate level professionals in the country.

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IITH creates a unique holistic ecosystem for education that offers interactive learning, a very flexible academic structure, cutting-edge research, strong industry collaboration, and entrepreneurship. This is an environment which enables students and faculty to translate their dreams into realities.

IIT Hyderabad

We obtain an upper bound on the first Chern class and the Castelnuovo-Mumford regularity of an initialized rank 3 ACM bundle on a general hypersurface in P4. As a corollary, we prove that a general hypersurface in P4 of degree d≥ 4 does not support a rank 3 Ulrich bundle. We also make progress on the base case of a generic version of a conjecture by Buchweitz, Greuel and Schreyer. © 2022, The Author(s), under exclusive licence to Springer Nature B.V.

World University root node

Department of Mathematics

Geometriae Dedicata

On the base case of a conjecture on ACM bundles over hypersurfaces

Fulltext

Journal of Pure and Applied Algebra

Rank 2 Ulrich bundles on general double plane covers

We prove a mild strengthening of a theorem of Česnavičius which gives a criterion for a vector bundle on a smooth complete intersection of dimension at least 3 to split into a sum of line bundles. We also prove an analogous statement for bundles on a general complete intersection surface. ©2021 American Mathematical Society

Proceedings of the American Mathematical Society

Grothendieck-Lefschetz and Noether-Lefschetz for bundles

We prove that a general hypersurface in P5 of degree d≥3 does not support an indecomposable rank 3 arithmetically Cohen-Macaulay (ACM) bundle. This settles the base case of a generic version of a conjecture of Buchweitz, Greuel and Schreyer. © 2019 Elsevier Inc.

Advances in Mathematics

Rank 3 ACM bundles on general hypersurfaces in P5

In terms of the number of generators, one of the simplest non-split rank-3 arithmetically Cohen–Macaulay bundles on a smooth hypersurface in P5 is 6-generated. We prove that a general hypersurface in P5 of degree d≥3 does not support such a bundle. We also prove that a smooth positive dimensional hypersurface in projective space of even degree does not support an Ulrich bundle of odd rank and determinant of the form OX(c) for some integer c. This verifies some cases of conjectures we discuss here. © 2018 Académie des sciences

Comptes Rendus Mathematique

Remarks on higher-rank ACM bundles on hypersurfaces [Remarques sur les fibrés ACM de rang supérieur sur les hypersurfaces]

Let X be a smooth projective hypersurface of dimension ≥5 and let E be an arithmetically Cohen–Macaulay bundle on X of any rank. We prove that E splits as a direct sum of line bundles if and only if H⁎i(X,∧2E)=0 for i=1,2,3,4. As a corollary this result proves a conjecture of Buchweitz, Greuel and Schreyer for the case of rank 3 arithmetically Cohen–Macaulay bundles. © 2017 Elsevier Inc.

Journal of Algebra

Rank 3 arithmetically Cohen–Macaulay bundles over hypersurfaces

Let X be a smooth projective hypersurface. We derive a splitting criterion for arithmetically Cohen–Macaulay bundles over X. As an application, we show that any rank 3 ACM vector bundle over X splits when dim X ≥ 7. We also derive a splitting result for rank 4 arithmetically Cohen–Macaulay bundles. © 2016, Copyright © Taylor & Francis Group, LLC.

Communications in Algebra

Splitting of Low-Rank ACM Bundles on Hypersurfaces of High Dimension

Let k be an algebraically closed field, char k ≠ 2, 3, and let X ⊂ P2 be an elliptic curve with defining polynomial f. We show that any non-trivial torsion point of order r, determines up to equivalence, a unique minimal matrix Φr of size 3r × 3r with linear polynomial entries such that detΦr = fr. We also show that the identity, thought of as the trivial torsion point of order r, determines up to equivalence, a unique minimal matrix Ψr of size (3r - 2) × (3r - 2) with linear and quadratic polynomial entries such that det Ψr = fr. © 2014 World Scientific Publishing Company.

International Journal of Algebra and Computation

Torsion points and matrices defining elliptic curves

Given a smooth, projective variety Y over an algebraically closed field of characteristic zero, and a smooth, ample hyperplane section X ⊂ Y, we study the question of when a bundle E on X, extends to a bundle $\mathcal{E} $ on a Zariski open set U ⊂ Y containing X. The main ingredients used are explicit descriptions of various obstruction classes in the deformation theory of bundles, together with Grothendieck-Lefschetz theory. As a consequence, we prove a Noether-Lefschetz theorem for higher rank bundles, which recovers and unifies the Noether-Lefschetz theorems of Joshi and Ravindra-Srinivas. © 2013 World Scientific Publishing Company.