Piotr Achinger

About me

Before coming to Berkeley, I studied mathematics and computer science at the University of Warsaw and wrote a Master's thesis in mathematics under supervision of Adrian Langer. I also spent a year as a PhD student at the Hausdorff Center for Mathematics in Bonn under supervision of Nicolas Perrin. I am supported by the Fulbright Science and Technology Fellowship.

I passed my qualifying examination on May 17, 2012. I am now a student of Arthur Ogus.

Interests

• algebraic geometry in characteristic p
• Frobenius splittings
• toric varieties
• representability of matroids

Publications

• Frobenius Push-Forwards on Quadrics, Communications in Algebra (arXiv) (pdf)
• A characterization of toric varieties in positive characteristic (pdf)

Other projects

• Translation of Serre's FAC (pdf) (source)

Seminars

Spring 2013 - Student Algebraic Geometry Seminar (with Jason Ferguson)

Fall 2012 - Abelian varieties (with Shelly Manber)

Fall 2012 - Student Algebraic Geometry Seminar (with Jason Ferguson)

Spring 2012 - Algebraic geometry in positive characteristic (with Arthur Ogus)

• Preliminaries on the Frobenius morphism (Doosung Park, 31.01.2012)
• Cartier isomorphism (Andrew Niles, 07.02.2012)
• Cartier descent and p-curvature (Yuhao Huang, 14.02.2012)
• Lifting to characteristic zero (Peter Mannisto, 21.02.2012)
• The Deligne-Illusie theorem (P.A., 28.02.2012)
• Counterexamples in characteristic p (George Melvin, 06.03.2012)
• Mori's bend and break (P.A., 13.03.2012)
• (PD-)Differential operators (James McIvor, 20.03.2012)
• The Grothendieck-Katz p-curvature conjecture (Katrina Honigs, 03.04.2012)
• Non-abelian Hodge theory in characteristic p (Yuhao Huang, 10.04.2012)
• The algebraic fundamental group (Jason Ferguson, 17.04.2012)
• Supersingular elliptic curves and Elkies' theorem (Shelly Manber, 24.04.2012)
• Arithmetic of the category of modules with integrable connection in characteristic p (Martin Olsson, 02.05.2012)

Spring 2012 - Crystalline cohomology reading group (with Peter Mannisto)

Teaching

Fall 2012 - Math H1B - Honors Calculus with Prof. Bergman

Short course on homological algebra

• Course webpage

Conference notes

Geometrie Algebrique en Liberte XX, June 2012, Grenoble

• Curves, abelian varieties and the Schottky problem (I) - S. Grushevsky (Stony Brook)
• Curves, abelian varieties and the Schottky problem (II) - S. Grushevsky (Stony Brook)
• Curves, abelian varieties and the Schottky problem (III) - S. Grushevsky (Stony Brook)
• Introduction to derived algebraic geometry (I) - B. Toën (Montpellier)
• Introduction to derived algebraic geometry (II) - B. Toën (Montpellier)
• Introduction to derived algebraic geometry (III) - B. Toën (Montpellier)
• Introduction to derived algebraic geometry (IV) - B. Toën (Montpellier)
• Moduli spaces of K3 surfaces and irreducible holomorphic symplectic manifolds (I) - K. Hulek (Hannover)
• Moduli spaces of K3 surfaces and irreducible holomorphic symplectic manifolds (II) - K. Hulek (Hannover)
• Moduli spaces of K3 surfaces and irreducible holomorphic symplectic manifolds (III) - K. Hulek (Hannover)
• Partial compactification of the zero section of the universal abelian variety - D. Zakharov (Stony Brook)
• Białynicki-Birula decomposition of Deligne-Mumford stacks - J. Skowera (Zurich)
• A ratio of integration relating GIT quotients by abelian and non-abelian groups - Z. Maddock (Columbia)
• Derived categories and deformations of smooth rational surfaces - A. Bodzenta-Skibińska (Warsaw)
• Chow groups and stable maps - M. Kemeny (Bonn)
• The Automorphisms group of \overline{M}_{g,n} - A. Massarenti (SISSA)
• Effective Iitaka fibration - G. di Cerbo (Princeton)
• Bogomolov–Sommese vanishing on log canonical pairs - P. Graf (Freiburg)
• On symplectic automorphisms of holomorphic symplectic fourfolds - G. Mongardi (Roma Tre)

char-p & p-adic geometry, June 2012, Mainz

• Rational and Witt-rational singularities (I) - A. Chatzistamatiou, K. Rülling (Essen)
• Rational and Witt-rational singularities (II) - A. Chatzistamatiou, K. Rülling (Essen)
• Rational and Witt-rational singularities (III) - A. Chatzistamatiou, K. Rülling (Essen)
• p-adic Simpson correspondence (I) - G. Faltings (MPI Bonn)
• p-adic Simpson correspondence (II) - G. Faltings (MPI Bonn)
• The Hochschild-Witt complex (I) - D. Kaledin (Steklov)
• The Hochschild-Witt complex (II) - D. Kaledin (Steklov)
• The Hochschild-Witt complex (III) - D. Kaledin (Steklov)
• char p version of Manin's theorem of the kernel - D. Rössler (Toulouse)
• Nef line bundles over finite fields - A. Langer (Warsaw)
• Geometric deformations of vector bundles and their torsors - H. Brenner (Osnabrück)
• The Fundamental Group Scheme of a Family of Varieties over W - V. Mehta (Tata)
• Comparison theorems in p-adic Hodge theory - B. Bhatt (Michigan)
• On the center of the ring of differential operators on a smooth variety over Z/p^nZ - V. Vologodsky (Oregon)
• Frobenius maps as source of cohomology of reductive groups - W. van der Kallen (Utrecht)
• Tropical geometry and metrics on p-adic analytic curves - S. Payne (Yale)
• On the F-purity of log canonical singularities - S. Takagi (Tokyo)
• A Frobenius variant of Seshadri constants - K. Schwede (Penn State)
• Wild quotient singularities - S. Schröer (Düsseldorf)
• On the birational nature of lifting - C. Liedtke (Bonn)
• A global division theorem for test ideals - K. Tucker (Princeton)
• Asymptotic base loci in positive characteristic - M. Mustata (Michigan)

Arithmetic, Motives and Moduli Spaces, January 2012, Institut Henri Poincare, Paris

• Notes
• Conference webpage

Lecture notes

Math 274 - Log geometry (A. Ogus; Fall 2011)

• Lecture 1 - Introduction (26.08.2011)
• Lecture 2 - Limits of monoids (29.08.2011)
• Lecture 3 - Ideals and geometry (31.08.2011)
• Lecture 4 - Finiteness (02.09.2011)
• Lecture 5 - Exactness (07.09.2011)
• Lecture 6 - Duality (09.09.2011)
• Lecture 7 - Cones (12.09.2011)
• Lecture 8 - Duality for cones (14.09.2011)
• Lecture 9 - Valuative monoids (16.09.2011)
• Lecture 10 - Monoid schemes (19.09.2011)
• Lecture 11 - Toric log structures (21.09.2011)
• Lecture 12 - Local and logarithmic homomorphisms (23.09.2011)
• Lecture 13 - Exact and Kummer homomorphisms (26.09.2011)
• Lecture 14 - Flat actions (28.09.2011)
• Lecture 15 - Regular actions (30.09.2011)
• Lecture 16 - Integrality (03.10.2011)
• Lecture 17 - Integrality and local exactness (05.10.2011)
• Lecture 18 - Topology of cones (07.10.2011)
• Lecture 19 - Sheaves of monoids (10.10.2011)
• Lecture 20 - (12.10.2011)
• Lecture 21 - (14.10.2011)
• Lecture 22 - (17.10.2011)
• Lecture 23 - (19.10.2011)
• Lecture 24 - (21.10.2011)
• Lecture 25 - (24.10.2011)
• Lecture 26 - (26.10.2011)
• Lecture 27 - (28.10.2011)
• Lecture 28 - (31.10.2011)
• Lecture 29 - (02.11.2011)
• Lecture 34 - (14.11.2011)
• Lecture 35 - (16.11.2011)
• Lecture 36 - (18.11.2011)
• Lecture 37 - (23.11.2011)
• Lecture 38 - (28.11.2011)
• Lecture 39 - (30.11.2011)
• Lecture 41 - (05.12.2011)

Seminar notes

Number Theory Seminar (K. Ribet; Fall 2011)

• Perfectoid Spaces and Applications I (P. Scholze, 07.09.2011)
• Perfectoid Spaces and Applications II (P. Scholze, 09.09.2011)
• Perfectoid Spaces and Applications III (P. Scholze, 12.09.2011)
• Perfectoid Spaces and Applications IV (P. Scholze, 14.09.2011)
• Perfectoid Spaces and Applications V (P. Scholze, 16.09.2011)

Student Arithmetic Geometry Seminar (M. Olsson)

• Organizational meeting (M. Olsson, 02.09.2011)
• Schemes as big Zariski sheaves (A. Niles, 09.09.2011)
• Grothendieck topologies (S. Stich, 16.09.2011)
• Fibered categories (A. Mazel-Gee, 23.09.2011)
• Descent (A. Dudzik, 30.09.2011)
• Algebraic spaces (Y. Huang, 07.10.2011)
• Classical invariant theory (S. Manber, 14.10.2011)
• Quasicoherent sheaves on algebraic spaces (J. Ferguson, 21.10.2011)
• Algebraic stacks (D. Park, 28.10.2011)

• 1-Motives I (P. Mannisto, 03.02.2012)
• 1-Motives II (P. Mannisto, 10.02.2012)
• K3 surfaces in positive characteristic I (P. A., 17.02.2012)
• K3 surfaces in positive characteristic II (P. A., 24.02.2012)

Student Algebraic Geometry Seminar (A. Critch, C. Crissman; Fall 2011)

• Organizational meeting (02.09.2011)
• Birational geometry of curves (N. Ilten, 09.09.2011)
• Moduli of elliptic curves (A. Niles, 16.09.2011)
• Rational points on elliptic curves (S. Manber, 23.09.2011)
• Picard schemes (P. Mannisto, 14.10.2011)
• Representations of finite algebraic groups (G. Melvin, 28.10.2011)

Formal and Rigid Geometry (M. Baker; Fall 2011)

• Organizational meeting (M. Baker, 06.09.2011)
• Tate algebras (D. Park, 13.09.2011)
• Ideals in Tate algebras (A. Dudzik, 20.09.2011)
• Affinoid algebras (R. Morrison, 27.09.2011)
• Affinoid subdomains (K. Nguyen, 04.10.2011)
• Tate's acyclicity (S. Backman, 11.10.2011)

Perfectoid Seminar (D. Mondragon; Fall 2011)

• Motivation and Organization (D. Mondragon, 19.09.2011)
• Rigid Analytic Spaces (K. Nguyen, 22.09.2011)
• Adic Spaces I (D. Mondragon, 26.09.2011)
• Adic Spaces II (D. Mondragon, 29.09.2011)
• Almost Ring Theory I (V. Murali, 03.10.2011)
• Almost Ring Theory II (V. Murali, 06.10.2011)
• p-adic Hodge Theory (K. Nguyen, 10.10.2011)
• Perfectoid rings (D. Mondragon, 13.10.2011)
• Tilting and Galois groups (A. Dudzik, 24.10.2011)

Various talks - 2011/2012

• Perfectoid Spaces -- Mathematics Department Colloquium (Peter Scholze, 08.09.2011)
• p-adic Local Langlands Correspondence -- Mathematics Department Colloquium (Pierre Colmez, 22.09.2011)
• Tropical hyperelliptic curves -- Combinatorics Seminar (Melody Chan, 27.09.2011)
• Singularities of Cox rings -- Algebraic Geometry and Commutative Algebra Seminar (Morgan V. Brown, 04.10.2011)
• Introduction to Gross-Zagier formula -- Mathematics Department Colloquium (Xinyi Yuan, 19.01.2012)
• The Stacks Project + Derived Categories of Stacks -- (Aise Johan de Jong, 24.01.2012)
• Local models of Shimura varieties -- Number Theory Seminar (Xinwen Zhu, 01.02.2012)
• Integral Hodge conjecture for 3-folds -- Number Theory Seminar (Burt Totaro, 15.02.2012)

Various talks - Fall 2012

• Arithmetic of Hyperelliptic Curves Lecture 1 Lecture 2 Lecture 3 (Benedict Gross, Bowen Lectures 2012)
• Various stuff (heights and L-functions) (Xinyi Yuan, Number Theory Seminar 9/26/2012)
• Consequences of the Theorem of the Cube (Will Johnson, Abelian Varieties Seminar 9/25/2012)
• Theorem of the Cube (Abelian Varieties Seminar)
• Multigraded Hilbert functions (Diane MacLagan. MSRI/Evans Talk 9/24/2012)
• Torus quotients of binomial D-modules (Christine Berkesch, Eisenbud Seminar 9/11/2012)
• Koszul algebras and their syzygies (Aldo Conca, Eisenbud Seminar 9/18/2012)
• Toric ideals oh hypergraphs (Elizabeth Gross, Eisenbud Seminar 9/18/2012)
• Cohomology of A_g (Samuel Grushevsky, UC Berkeley Oct 2012)
• A_{g,d} is a DM stack (I) (Katrina Honigs, Olsson Seminar 28/09/2012)
• A_{g,d} is a DM stack (II) (Katrina Honigs, Olsson Seminar 10/05/2012)
• The Mordell-Weil Theorem (Khoa Nguyen, Abelian Varieties Seminar 10/16/2012)
• F-pure threshold of CY hypersurfaces (Anurag Singh, Eisenbud Seminar 10/16/2012)

Various talks - Spring 2013

• Rational curves on symplectic varieties (Brendan Hassett, Berkeley Colloquium 2/21/2013)
• p-adic Hodge Theory - statements (Peter Mannisto, Arithmetic Geometry Seminar 2/22/2013)
• Overview of the "Hassett-Keel program" (LMMP for \bar M_g) (Noah Giansiracusa, Moduli Seminar 2/22/2013)
• Introduction to Hodge Theory (Penghui Li, SAGS 2/22/2013)
• Koszul Algebras and the Frobenius endomorphism (Thanh Vu, Eisenbud Seminar 4/16/2013)