The Steinberg Representation
2026TheorySean Cotner
An introduction to the Steinberg representation of a finite group of Lie type — its alternating-sum construction from parabolic inductions, worked out explicitly for SL_2.
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21 results
CMI 2000, The Millennium Prize Problems
Problem sheet
The Millennium Meeting, Paris, 24 May 2000, at which the seven problems were announced.
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An example of a non-reduced Picard scheme
Theory
An example, due to Serre, of a smooth projective surface in positive characteristic whose Picard scheme fails to be reduced, worked out via the two governing dimension inequalities.
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Finite flat commutative group schemes embed locally into abelian schemes
2026TheorySean Cotner
A proof of Raynaud's theorem: every finite flat commutative group scheme over a locally noetherian base embeds, Zariski-locally, as a closed subgroup of an abelian scheme.
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Hom schemes for algebraic groups
2023PreprintSean Cotner
Accepted for publication in Algebra & Number Theory.
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Quotients of admissible formal schemes and adic spaces by finite groups
2024Journal articleBogdan Zavyalov
Algebra & Number Theory
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A curiosity: “supersmooth” varieties
Theory
A rigidity condition on schemes, strictly stronger than smoothness, introduced with examples and a look at where it fails to be geometric — no known application, just a curiosity.
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On smooth-group actions on reductive groups and spherical buildings
2025PreprintJeffrey D. Adler, Joshua M. Lansky, Loren Spice
With an appendix by Sean Cotner, Joshua M. Lansky, and Loren Spice.
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The torsion component of the Picard scheme
2026OtherBogdan Zavyalov
Every finite flat commutative group scheme over a noetherian local ring is the torsion component of the Picard scheme of a smooth projective scheme with 3-dimensional fibers, built as a quotient of a complete intersection. An application: Hodge numbers that jump in a smooth projective family.
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Some examples of algebraic groups
2026TheorySean Cotner
Two pathological phenomena for algebraic groups over general bases — a group degenerating between the multiplicative and additive group across a DVR, and a non-affine identity component — built as centralizers in SL_n.
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Chow Groups of Abelian Varieties and Beilinson's Conjecture
2016PreprintBogdan Zavyalov
Not intended for publication.
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Topic Modelling for Extracting Behavioral Patterns from Transactions Data
2019Conference paperEvgeny Egorov, Filipp Nikitin, Vasiliy Alekseev, Alexey Goncharov +1
With the increasing popularity of cashless payment methods for everyday, seasonal and special expenses popular banks accumulate huge amount of data about customer operations. In the article, we report a successful application of topic modelling to extract behaviour patterns from the data. The resulting models are built with BigARTM framework: flexible and efficient tool for topic modelling. The framework allows us to experiment with various models including PLSA, LDA and beyond. Results demonstrate ability of the approach to aggregate information about behaviour patterns of different customer groups. The results analysis allows to see the topics of such people clusters varying from travellers to mortgage holders. Moreover, low-dementional embeddings of the customers, which was given with topic model, were studied. We display that the client vector representations store demographic information as well as source data. We also test for a best way of preparing data for the model with metric above in mind.
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