| Sarah Frei | |
Rationality in arithmetic families |
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The rationality problem in algebraic geometry asks whether a given algebraic variety can be parameterized by algebraic functions.
A natural question in the study of rationality is the following: is rationality a deformation invariant in smooth families?
In arithmetic families, this question asks how the rationality of a variety defined over Q interacts with the rationality
of its modulo p reductions for various primes p.
In this talk, I'll discuss work in progress with Asher Auel and Alena Pirutka on algebraic varieties over Q that are not
rational over the complex numbers but are rational modulo p for infinitely many primes p.
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| Joe Kileel | |
Covering numbers and norming sets of real algebraic varieties |
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In this talk I will discuss two different ways to replace a real algebraic variety by "representative" finite
subsets—namely, covering sets and norming sets.
Of prime interest are upper bounds on the size of covering sets and norming sets.
In the former case, we will control the number of \ell_2 balls of radius epsilon needed to cover a real algebraic variety,
image of a polynomial map, or semialgebraic set in Euclidean space, in terms of the degrees of the relevant polynomials and
number of variables.
The bound improves upon the best known general bound, and its proof is much more straightforward.
In the latter case, we will review existing upper bounds based on the Hilbert function of a real algebraic variety.
We will then discuss applications of covering sets and norming sets to applied and computational mathematics.
The talk is based on joint works with Yifan Zhang.
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| Justin Lacini | |
Syzygies and singularities of secant varieties of smooth projective varieties |
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Consider a smooth variety X embedded in projective space.
One may construct a natural sequence of singular varieties, called secant varieties, as follows:
take the closure of the union of all lines connecting two points of X, then take closure of the union of all
planes connecting three points of X and so on.
Secant varieties have generated intense interest across many fields, ranging from classical algebraic geometry to complexity theory.
Recently, Ullery proved that if the embedding of X is sufficiently positive, then the first secant variety is normal.
Ein, Niu and Park generalized this to all secant varieties, in the special case when X is a curve.
In joint work with Choi, Park and Sheridan we undertake a systematic study of the singularities and equations of
secant varieties in all dimensions, and in particular we extend the previous results to the case when X is a surface.
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| Jackson Morrow | |
Arithmetic differential equations and unlikely intersections |
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Unlikely intersection problems occupy a central place in modern arithmetic geometry.
One of the foundational results in this area is Raynaud's theorem resolving the Manin–Mumford conjecture, which
asserts that if a closed subvariety of an abelian variety contains a Zariski dense set of torsion points,
then it is the translate of an abelian subvariety.
In particular, if the subvariety contains no translates of abelian subvarieties, then the intersection of
it with the torsion subgroup is finite, and one may ask for explicit bounds on its size.
In this talk, I will present joint work with Lance Edward Miller in which we establish such a bound when the
subvariety has ample cotangent bundle.
Our approach combines Buium's theory of arithmetic jet spaces with classical intersection theory,
yielding a bound expressed as a sum of cycle classes in the Chow ring of X modulo a prime p of good reduction.
Instead of focusing on technical aspects of our work, I will describe the key features of arithmetic jet spaces
and explain how differential-algebraic techniques can be applied to unlikely intersection problems more broadly.
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| Rachel Webb | |
Twisted weighted stable maps |
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I will present a common generalization of the twisted stable maps of Abramovich-Vistoli and the weighted stable maps
of Alexeev-Guy and Bayer-Manin (building on work of Hassett).
The theory has potential applications to computing Gromov-Witten invariants of Deligne-Mumford stacks with abelian
stabilizer groups.
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