The direct PDF links are often more updated than the arXiv versions.
Papers/Preprints

Stickelberger's discriminant theorem for algebras
(with
Owen Biesel and John Voight), preprint
arXiv:2208.06138 (15 pp.)
Stickelberger proved that the discriminant of a number field is congruent to 0 or 1 modulo 4. We generalize this to an arbitrary (not necessarily commutative) ring of finite rank over the integers using techniques from linear algebra. Our proof, which relies only on elementary matrix identities, is new even in the classical case.

Maximal BrillNoether loci via K3 surfaces
(with
Richard
Haburcak), preprint arXiv:2206.04610 (29 pp.)
The BrillNoether loci M^{r}_{g,d} parameterize curves of genus g admitting a linear system of rank r and degree d$; when the BrillNoether number is negative, they sit as proper subvarieties of the moduli space of genus g curves. We explain a strategy for distinguishing BrillNoether loci by studying the lifting of linear systems on curves in polarized K3 surfaces, which motivates a conjecture identifying the maximal BrillNoether loci. Via an analysis of the stability of LazarsfeldMukai bundles, we obtain new lifting results for line bundles of type g^{3}_{d} which suffice to prove the maximal BrillNoether loci conjecture in genus 919, 22, and 23.

Thinking about abstracts
Notices of the American Mathematical Society 68, no. 10 (November 2021), pp. 17551756.
A column in the November 2021 Early Career section about the
history of abstracts in scientific publishing, and how this history
can provide some strategies for thinking about writing abstracts.

Explicit descent on elliptic curves and splitting Brauer classes
(with Benjamin Antieau)
preprint arXiv:2106.04291, submitted (31 pp.)
We prove new results on splitting Brauer classes by genus one curves,
settling in particular the case of index 7 classes over global
fields. Though our method is cohomological in nature, and proceeds
by considering the more difficult problem of splitting
μ_{N}gerbes, we use crucial input from the
arithmetic of modular curves and explicit Ndescent on
elliptic curves.

Unramified Brauer groups of conic bundles threefolds in characteristic two
(with
Alessandro Bigazzi, Christian
Boehning, and HansChristian
Graf von Bothmer)
American Journal of Mathematics 143 (2021), pp. 16011631.
We establish a formula for computing the unramified Brauer group of tame conic bundle threefolds in characteristic 2. The formula depends on the arrangement and residue double covers of the discriminant components, the latter being governed by ArtinSchreier theory (instead of Kummer theory in characteristic not 2). We use this to give new examples of threefold conic bundles defined over the integers that are not stably rational over the complex numbers.

BrillNoether special cubic fourfolds of discriminant 14
Facets of Algebraic Geometry: A Volume in Honour of William Fulton's 80th Birthday, Volume 1 (editors Paolo Aluffi, David Anderson, Milena Hering, Mircea Mustață, Sam Payne), London Mathematical Society Lecture Note Series 472 (2022), Cambridge University Press, pp. 2953.
We study the Brill–Noether theory of curves on K3 surfaces that
are Hodge theoretically associated to cubic fourfolds of
discriminant 14. We prove that any smooth curve in the polarization
class has maximal Clifford index and deduce that a cubic fourfold
contains disjoint planes if and only if it admits a Brill–Noether
special associated K3 surface of degree 14. As an application, we
the complement of the pfaffian locus, inside the
Noether–Lefschetz divisor of discriminant 14 in the moduli space of
cubic fourfolds, is shown to be contained in the irreducible locus of cubic fourfolds containing two disjoint planes.

Conic bundle fourfolds with nontrivial unramified Brauer group
(with
Christian
Boehning, HansChristian Graf von Bothmer, and Alena Pirutka)
Journal of Algebraic Geometry 29 (2020), pp. 285327.
We derive a formula for the unramified Brauer group of a general class
of rationally connected fourfolds birational to conic bundles over
smooth threefolds. We produce new examples of conic bundles over
P^{3} where this formula applies and which have
nontrivial unramified Brauer group. The construction uses the theory of contact surfaces
and, at least implicitly, matrix factorizations and symmetric
arithmetic CohenMacaulay sheaves, as well as the geometry of special
arrangements of rational curves in P^{2}. We also prove the
existence of universally CH_{0}trivial resolutions for the general
class of conic bundle fourfolds we consider. Using the degeneration
method, we thus produce new families of rationally connected fourfolds
whose very general member is not stably rational.

Universal triviality of the Chow group of 0cycles and the Brauer group
(with
Alessandro Bigazzi, Christian
Boehning, and HansChristian
Graf von Bothmer)
International Mathematics Research Notices, rnz171 (2019)
We prove that a smooth proper universally CH_{0}trivial variety X over a field k has universally trivial Brauer group. This fills a gap in the literature concerning the ptorsion of the Brauer group when k has characteristic p.

The mathematics of Grace Murray Hopper
Notice of the American Mathematical Society 66 (March
2019), no. 3, pp. 330340.
Grace Murray Hopper (19061992) is well known as a pioneering
computer scientist and decorated Naval officer. However, what is
often overlooked in accounts of Hopper's life and work is her
mathematical legacy. For the first
time, using archival material from Yale University's collections,
this article illuminates Hopper's foundational
mathematical training as well as the specific contributions of her
1934 Yale Ph.D. thesis advised by Øystein Ore. Her results
concern necessary conditions for the irreducibility of polynomials with
rational coefficients based on a novel construction, called here the
NewtonHopper polygon, which is related to the modern
notion of archimedean Newton polygon. The bounds that Hopper obtains
on the distance between the NewtonHopper polygon of a product of
polynomials and the Minkowski sum of their respective NewtonHopper
polygons, are explained and clarified.

Failure of the localglobal principle for isotropy of quadratic forms over rational function fields
(with
V. Suresh)
preprint arXiv:1709.03707 (7 pp.)
We prove the failure of the localglobal principle, with respect to all discrete valuations, for isotropy of quadratic forms over a rational function field of transcendence degree at least 2 over the complex numbers. Our construction involves the generalized Kummer varieties considered by Borcea and CynkHulek.

Periodindex bounds for arithmetic threefolds
(with
Benjamin Antieau, Colin Ingalls, Daniel Krashen, Max Lieblich)
Inventiones Mathematicae 26 (2019), no. 2, pp. 301335.
The standard periodindex conjecture for the Brauer group of a field
of transcendence degree 2 over a padic field predicts that
the index divides the cube of the period. Using Gabber's theory of
primetol alterations and the deformation theory of twisted
sheaves, we prove that the index divides the fourth power of the
period for every Brauer class whose period is prime to 6p,
giving the first uniform periodindex bounds over such fields. For
classes whose period is prime to p (and not necessarily prime
to 6), we prove that the index divides the fifth power of the period.

Azumaya algebras without involution
(with
Uriya First and Ben Williams)
Journal of the European Mathematical Society 21 (2019), no. 3, pp. 897–921.
Generalizing a theorem of Albert, Saltman showed that an Azumaya
algebra A over a ring represents a 2torsion class in the Brauer group
if and only if there is an algebra A' in the Brauer class of A
admitting an involution of the first kind. Knus, Parimala, and
Srinivas later showed that one can choose A' with deg A' = 2 deg A. We
show that 2 deg A is the lowest degree one can expect in
general. Specifically, we construct an Azumaya algebra A of degree 4
and period 2 such that the degree of any algebra A in the Brauer
class of A admitting an involution is divisible by 8. Separately, we
provide examples of Azumaya algebras of degree 2
admitting symplectic involutions, but no orthogonal involutions. These
stand in contrast to the case of central simple algebras of even
degree over fields, where the presence of a symplectic involution
implies the existence of an orthogonal involution and vice versa.

Stable rationality of quadric and cubic surface bundle fourfolds
(with
Christian
Boehning and Alena Pirutka)
European
Journal of Mathematics 4 (2018), no. 3, pp. 732760.
Our main result is that a very general
hypersurface X of bidegree (2,3) in P^{2} x
P^{3} is not stably rational, which completes the
stable rationality analysis of such bidegree (2,n) fourfold hypersurfaces. Via
projections onto the two factors, X is a cubic surface bundle over
P^{2} and a conic bundle over P^{3},
and we analyze the stable rationality problem from both these points
of view. We also show that this case provides another example of a
smooth family of rationally connected fourfolds with rational and
nonrational fibers. Along the way, we introduce new quadric surface bundle fourfolds over P^{2} with discriminant curve of any even degree at least 8, having nontrivial unramified Brauer group and admitting a universally CH_{0}trivial resolution.

Some nonspecial cubic fourfolds
(with
Nick Addington)
Documenta Mathematica 23 (2018), pp. 637651.
Ranestad and Voisin have showed, quite surprisingly, that the divisor
in the moduli space of cubic fourfolds consisting of cubics "apolar
to a Veronese surface" is not a NoetherLefschetz divisor. We give
an independent proof of this by exhibiting an explicit cubic
fourfold X in the divisor and using point counting methods
over finite fields to show X is NoetherLefschetz general. We
also show that two other divisors considered by Ranestad and Voisin are not NoetherLefschetz divisors.

Cycles, derived categories, and rationality
(with
Marcello
Bernardara)
in Surveys on Recent Developments in Algebraic Geometry, Proceedings of Symposia in Pure Mathematics, vol. 95, Amer. Math. Soc., Providence, RI, 2017, pp. 199266.
We survey recent developments in the (stable) rationality problem from
the point of view of unramified cohomology and 0cycles, as
well as derived categories and semiorthogonal decompositions, and
how these perspectives intertwine and reflect each other. For smooth
projective surfaces, we show how the existence of phantom
subcategories of the derived category can be viewed as a stronger
measure of rationality than the universal triviality of the Chow group
of 0cycles, equivalently, the existence of a decomposition
of the diagonal.

Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields
(with
Marcello
Bernardara)
Proceedings of the London Mathematical Society
117 (2018), no. 1, pp. 164
We study the birational properties of geometrically rational surfaces
from a derived categorical perspective. In particular, we give a
criterion for the rationality of a del Pezzo surface
S over an arbitrary field, namely, that its derived category
decomposes into zerodimensional components. For deg S ≥ 5, we
construct explicit semiorthogonal decompositions by subcategories of
modules over semisimple algebras arising as endomorphism algebras of
vector bundles and we show how to retrieve information about the index of
S from Brauer classes and Chern classes associated to these vector
bundles.

Surjectivity of the total Clifford invariant and Brauer dimension
Journal of Algebra 443 (2015), 395421
This article addresses previous attempts to generalized
Merkurjev's theoremthat every 2torsion Brauer class is represented
by the Clifford algebra of a quadratic formto base rings and
schemes other than fields. Parimala, Scharlau, and Sridharan found
smooth complete padic curves for which Merkurjev's theorem is
equivalent to the existence of a rational theta characteristic. The
main result of this work is that over a smooth proper padic curve
X, replacing the Witt group by the total GrothendieckWitt group of
line bundlevalued quadratic forms, these obstructions vanish; every
2torsion Brauer class on X is represented by the Clifford invariant
of some line bundlevalued quadratic form. The key new ingrediant is
the actual construction of the Clifford invariant of a line
bundlevalued quadratic form, when the value line bundle is
nontrivial.

Universal unramified cohomology of cubic fourfolds containing a plane
(with JeanLouis
ColliotThélène, R. Parimala)
in Brauer Groups and Obstruction Problems: Moduli Spaces and
Arithmetic, Progress in Mathematics, vol. 320, Birkhäuser Basel, 2017,
pp. 2956.
We prove that a general complex cubic fourfold containing a plane has
constant unramified cohomology in degree three over any field
extension of the complex numbers. The proof uses results on the
unramified cohomology of quadrics due to Kahn, Rost, and Sujatha.
This has bearing on the rationality question for cubic fourfolds.
Along the way, we develop some of the foundations of the universal
triviality of the Chow group of 0cycles with applications to the
rationality problem in algebraic geometry.

Quadric surface bundles over surfaces
(with
R. Parimala and
V. Suresh)
Documenta Mathematica, Extra Volume: Alexander S. Merkurjev's Sixtieth Birthday (2015), pp. 3170.
Let f : T > S be flat morphism of degree 2 of regular integral
schemes of dimension at most 2 with regular branch divisor D.
We establish a bijection between Azumaya quaternion algebras on
T and quadric surface bundles on S with simple
degeneration along D.
This is a manifestation of the exceptional isomorphism
^{2}A_{1}=D_{2} degenerating to the exceptional
isomorphism A_{1}=B_{1}=C_{1}. Along the
way, we study the orthogonal group schemes, which are smooth yet
nonreductive, of quadratic forms with simple degeneration.

Cubic fourfolds containing a plane and a quintic del Pezzo surface
(with
Marcello Bernardara,
Michele
Bolognesi, and
Anthony VárillyAlvarado),
Compositio Algebraic Geometry 1 (2014), issue 2, pp. 181193.
We isolate a class of smooth rational cubic fourfolds X
containing a plane whose associated quadric surface bundle does not
have a rational section. This is equivalent to the nontriviality of
the Brauer class of the even Clifford algebra over the K3 surface
S of degree 2 associated to X. Specifically, we show
that in the moduli space of cubic fourfolds, the intersection of the
divisors C_{8} and C_{14} has five
irreducible components. In the component corresponding to the
existence of a tangent conic, we prove that the general member is both
pfaffian and has nontrivial Brauer class. This answers a question of
B. Hassett. Such cubic fourfolds also provide twisted derived
equivalences between K3 surfaces of degree 2 and 14, hence further
corroboration of Kuznetsov’s derived categorical conjecture on the
rationality of cubic fourfolds.

Fibrations in complete intersections of quadrics, Clifford
algebras, derived categories, and rationality problems
(with
Marcello Bernardara and
Michele
Bolognesi)
Journal de mathématiques pures et appliquées 102 (2014), issue
1, pp. 249291.
We develop the categorical and algebraic tools of relative
homological projective duality and the Moritainvariance of the even Clifford
algebra under quadric reduction by hyperbolic splitting, to study
semiorthogonal decompositions of the bounded derived category of
fibrations in intersections of quadrics. Together
with input from the theory of quadratic forms, we apply these tools in the case
of fibrations in intersections of two quadrics X > Y having relative dimension 1, 2, or 3, in which case the
fibers are elliptic curves, Del Pezzo surfaces of degree 4, or Fano
threefolds, respectively. In the latter two cases, if Y is the projective line
over an algebraically closed field of characteristic zero, we relate
rationality questions to categorical representability of X.

Failure of the localglobal principle for isotropy
of quadratic forms over surfaces
Quadratic Forms and Linear Algebraic Groups, Oberwolfach Reports
10 (2013), issue 2, report 31, pp. 18231825.
We construct counterexamples to the localglobal principle
for isotropy of quadratic forms over the function field of any surface
over an algebraically closed field of characteristic not 2.

The transcendental lattice of the sextic Fermat surface
(with
Christian
Boehning and HansChristian Graf von Bothmer)
Mathematical Research Letters 20 (2013), no. 6, pp. 10171031.
We prove that the integral polarized Hodge structure on the
transcendental lattice of a sextic Fermat surface is
decomposable. This disproves a conjecture of Kulikov related to a
Hodge theoretic approach to proving the irrationality of the very
general cubic fourfold.

Exceptional collections of line bundles on projective homogeneous varieties
(with
Alexey Ananyevskiy,
Skip Garibaldi,
and
Kirill
Zainoulline)
Advances in Mathematics 236 (2013), pp. 111130.
We construct new examples of exceptional collections of line bundles
on the variety of Borel subgroups of a split semisimple linear
algebraic group G of rank 2 over a field. We exhibit exceptional
collections of the expected length for types A_{2} and B_{2}=C_{2} and prove
that no such collection exists for type G_{2}. This settles the question
of the existence of full exceptional collections of line bundles on
projective homogeneous Gvarieties for split linear algebraic groups G
of rank at most 2.

Vector bundles of rank 4 and A_{3}=D_{3}
International Mathematical Research Notices 2013 (2013), no. 15, pp. 34503476.
This article shows how a version of
"Pascal's rule" for vector bundles gives an explicit isomorphism between
the moduli functors represented by projective homogeneous bundles for
reductive group schemes of type A_{3} and D_{3}.
This is exploited to show that a vector bundle V of rank
4 has a sub or quotient invertible sheaf if and only if the canonical
symmetric bilinear form on its exterior square has a lagrangian subspace.
Under certain hypotheses on the base scheme,
this is equivalent to the vanishing of the Witt theoretic Euler class
e(V) in W^{4}(X, det(V)).

Remarks on the Milnor conjecture over schemes
GaloisTeichmüller Theory and Arithmetic Geometry (Kyoto, 2010), H. Nakamura, F. Pop, L. Schneps, A. Tamagawa eds.,
Advanced Studies in Pure Mathematics 63, 2012, pp. 130
This article mostly consists of a survey of results
concerning the attempts to generalize the Milnor conjecturesboth
the Ktheoretic and quadratic form aspectsto base rings and
schemes other than fields. Included is a nontrivial example of a
padic curve (with bad reduction) that does not satisfy the classical
generalization of the Milnor conjectures for quadratic forms. By
explicitly computing its GrothendieckWitt groups, Brauer group, and
Clifford invariant map, it is shown how the author's previous results
on the Clifford invariants of line bundlevalued quadratic forms
recover the Milnor conjecture for this curve.

Clifford invariants of line bundlevalued quadratic forms
MPIM preprint
series 33 (2011), 55 pp.
This article concerns an extension of the Clifford invariant (or 2nd
StiefelWhitney class) to similarity classes of line bundlevalued
quadratic forms of even rank and fixed discriminant on a scheme (where
2 is invertible). This invariant resides in degree two étale
cohomology with coefficients in a twisted group scheme of order four
and "interpolates" between the classical Clifford invariant and the
1st Chern class (modulo 2) of the value line bundle. This invariant
is further related to classes in the "involutive" Brauer group
associated to generalized Clifford algebras. In rank at most six,
this invariant is related to the classification of line bundle
valuedforms.

Open problems on central simple algebras
(with Eric Brussel,
Skip Garibaldi, and Uzi Vishne)
Transformation Groups 16, Issue 1 (2011), pp. 219264
A survey of past research on central simple algebras and the Brauer
group of a general field, organized as a list of currently open
problems. The open problems surveyed include those concerning
noncyclic algebras in prime degree, crossed products, generation by
cyclic algebras, periodindex, rationality of the center of generic
matrices, essential dimension, involutions (the Springer problem),
surfaces over finite fields (ArtinTate conjecture), tensor product
decompositions, and the birational geometry of SeveriBrauer varieties
(Amitsur conjecture). Where appropriate, results intended to fill
gaps in the literature are also proved.
Books

Brauer Groups and Obstruction Problems: Moduli Spaces and Arithmetic
(with B. Hassett, A. VárillyAlvarado, and B. Viray)
Progress in Mathematics, Vol. 320, Birkhäuser Basel, 2017.
Brauer groups and related cohomological invariants have long been important tools for analyzing arithmetic questions on algebraic varieties. Two central examples are the BrauerManin obstruction to the existence of rational points and the application of unramified cohomology to rationality questions. Recent advances in derived categories, derived equivalences of varieties, and moduli of twisted sheaves have provided new geometric tools for interpreting elements of the Brauer group. The contributions of this volume explore the implications of these ideas for arithmetic geometry, and hone geometric tools with a view towards future arithmetic applications.
Theses

Cohomological invariants of line bundlevalued symmetric bilinear
forms
(submitted version, or the better single spaced version)
Ph.D. thesis at the University of Pennsylvania, May 2009 (advisor Ted Chinburg).

Une démonstration d'un
théorème de Bernstein sur les représentations de
quasicarréintégrable de GL_{n}(F)
où F est un corps local non archimédien
(submitted [PDF])
(A proof of Bernstein's theorem concerning quasisquareintegrable
representation of GL_{n}(F) where
F is a nonarchimedian local field.)
Diplôme d'Études Approfondies (DEA) Mathématiques
Pures thesis at, Université ParisSud XI Orsay, France, July
2004 (advisor Guy Henniart).

Volumes of integer polynomials over local fields
(submitted [PDF], with
hiresolution images)
Undergraduate senior thesis at Reed College, May 2003, (advisor Joe P. Buhler).
Notes
Slides from selected talks

Noether's problem
Dartmouth Colloquium, Hanover, NH, May 2018

Platonic solids
Synapse Resonance, Outreach program of the Yale Scientific Magazine,
New Haven, CT, December 2017

Geometry via point counting
Center for Communications Research (CCR), Princeton, NJ, March 2017

Platonic solids and symmetry
Math Mornings, Yale
University, New Haven, CT, October 2016
Publicity

Stable rationality of quadric bundles
Geometry over nonclosed fields, Simons Symposium, Schloss Elmau,
Bavaria, Germany, April 2016

BrillNoether special cubic fourfolds
AMS Summer Research Institute in Algebraic Geometry, Salt Lake City,
Utah, July 2015

Pick’s Theorem
Pathways to Science Café, New Haven, CT, October 2014

Quadric surface bundles and quaternion algebras
AMS/MAA Joint Meetings, Special Session on Number Theory and Geometry,
San Diego, CA, January 2013

Azumaya algebras without involution
AMS Special Session on Galois Cohomology and the Brauer Group,
Knoxville, TN, March 2012

Points, lines, and conics
Yale University, Yale Undergraduate Math Society (YUMS), April 2014

Points and conics
University of Connecticut, Math Club, March 2014

Fröhlich twisting via orthogonal motives
AMS Special Session on Hopf Algebras and Galois Module Theory, Tampa,
FL, March 2012

Quaternions and conic sections: from algebra to geometry
Mathematics colloquium, Wake Forest University, WinstonSalem, NC,
November 2009

Clifford sequences in the theory of line bundlevalued quadratic forms
over arithmetic schemes
AMS Special Session on Arithmetic Geometry, Boca Raton, FL, October 2009

Selfdual Galois representations
Center for Communications Research (CCR) West, La Jolla, CA, November 2008

padic polynomial volumes
Graduate student pizza seminar, University of Pennsylvania, October 2004

Which numbers have square roots?
Special thesis talk for trustees, Reed College, Portland, OR, April 2003
