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Papers

(joint with V. Suresh, submitted) Totaro's Question on Zero-Cycles on Torsors (arXiv:1702.00516)

(to appear in *Proceedings of the AMS*) Totaro's Question for Adjoint Groups of Types A_{1} and A_{2n} (arXiv:1701.03124)

(to appear in *Transactions of the AMS*) Totaro's Question for Tori of Low Rank (arXiv:1603.07718)

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Galois Cohomology

For any variety *X* over a field *k*, we can define the arithmetic invariants

ind(*X*) := gcd{[*L*:*k*] : *L/k* is a finite separable field extension such that *X*(*L*) ≠ ∅}

min(*X*) := min{[*L*:*k*] : *L/k* is a finite separable field extension such that *X*(*L*) ≠ ∅}

The index ind(

*X*) divides min(

*X*) by construction.
When are they equal?
The question was originally raised by Serre in the '60s for torsors

*X* under linear algebraic groups

*G* in the ind(

*X*) = 1 case and later generalized by Totaro for the higher index cases.
Affirmative proofs for any groups

*G* are extremely rare.
I settled the conjecture when

*G* is any torus of rank ≤ 2 over an arbitrary field and when

*G* is an absolutely simple classical adjoint group of type A

_{1} or A

_{2n} over a field of characteristic ≠ 2.
Consequently, I proved that any regular variety

*X* containing such a torsor as a dense open subset has a point after a degree ind(

*X*) field extension.
In particular, ind(

*X*) = min(

*X*) when

*X* is a del Pezzo surface of degree 6, a toric variety.

Recently, V. Suresh and I have constructed the first known examples of torsors under linear algebraic groups giving a negative answer to this question, including examples of torsors under tori of all ranks ≥ 3.

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The Brauer Group

In a 2014 paper, Uematsu showed using explicit cocycle computations that Br* X* = Br *k* where *k* is a field of characteristic 0 and *X* is the affine diagonal quadric over *k* determined by

*x*^{2} + by^{2} + cz^{2} + d = 0

for some

*b*,

*c*,

*d* ∈

*k**.
With the guidance of our project leaders Jean-Louis Colliot-Thélène and Yonatan Harpaz at the 2015 Arizona Winter School, four colleagues and I found an alternate, cocycle-free proof of this theorem for

*k* = ℂ(

*b*,

*c*,

*d*) where

*b*,

*c*, and

*d* are algebraically independent indeterminates over ℂ.

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Space-Time Block Coding

Coding theorists and electrical engineers have been using division algebra-based codes in wireless communication since 1998.
Certain invariants arising in algebraic number theory of division algebras and their orders (e.g., maximal orders) directly translate to desirable coding-theoretic properties.
I explored tensor product constructions of maximal order-based codes from existing ones using class field theory.