On the Homological Theory of Modules over Commutative Rings
This talk will be a partial survey of topics in homological commutative local algebra. In particular, I will discuss results on depth, homological dimensions, and torsion properties of tensor products and powers, as well as characterizations of Gorenstein and regular rings via (co)homology vanishing. I will also mention related conjectures and open problems in this area.
Introductory talk to the online seminar "A study in derived algebraic geometry"
Partial Frobeniuses via black magic
Zoom link: https://eu02web.zoom-x.de/j/66594302263?pwd=6XqRNiAADoXfsLNrwCIji5UZgyh2jG.1
Meeting ID: 665 9430 2263, Passcode: 740104
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Rational points in conic bundles
Asymptotic counting methods will be used to study certain basic questions on the existence of rational points on surfaces. Joint work with Christopher Frei.
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On a question of Douglass and Ono
It is known that the partition function $p(n)$ obeys Benford's law in any integer base $b\ge 2$. A similar result was obtained by Douglass and Ono for the plane partition function ${\text{\rm PL}}(n)$ in a recent paper. In their paper, Douglass and Ono asked for an explicit version of this result. In particular, given an integer base $b\ge 2$ and string $f$ of digits in base $b$ they asked for an explicit value $N(b,f)$ such that there exists $n\le N(b,f)$ with the property that ${\text{\rm PL}}(n)$ starts with the string $f$ when written in base $b$. In my talk, I will present an explicit value for $N(b,f)$ both for the partition function $p(n)$ as well as for the plane partition function ${\text{\rm PL}}(n)$.
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Pointwise bounds for 3-torsion
For $\ell$ an odd prime number and $d$ a squarefree integer, a central question in arithmetic statistics is to give pointwise bounds for the size of the $\ell$-torsion of the class group of $\mathbb{Q}(\sqrt{d})$. This is in general a difficult problem, and unconditional pointwise bounds are only available for $\ell = 3$ due to work of Pierce, Helfgott—Venkatesh and Ellenberg—Venkatesh. The current record is $h_3(d) \ll_\epsilon d^{1/3 + \epsilon}$ due to Ellenberg—Venkatesh. We will discuss how to improve this to $h_3(d) \ll d^{0.32}$. This is joint work with Stephanie Chan.
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Speed talks
Speed talks
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