International Workshop on Motives in Tokyo, 2025

Information:

• Date: 17(Mon)-20(Thurs) February, 2025
• Location: Graduate School of Mathematics, University of Tokyo
• Organisers: Thomas Geisser (Rikkyo University), Shane Kelly (University of Tokyo)
• The workshop is supported by JSPS Grant-in-Aid for Scientific Research (B)
  23K25768; 23H01071 (2023) Principal Investigator Thomas Geisser
• Past workshops: 2023 | 2019 | 2018 | 2017 | 2016 | 2014 | 2013 | 2012 | 2011

Speakers:

• Aravind Asok (USC, United States)
• Christian Haesemeyer (Melbourne, Australia)
• Bruno Kahn (FJ-LMI, Japan)
• Wataru Kai (Tohoku, Japan)
• Kenichiro Kimura (Tsukuba, Japan)
• Achim Krause (Oslo, Norway)
• Baptiste Morin (Bordeaux, France)
• Yusuke Nemoto (Chiba, Japan)
• Jinhyun Park (KAIST, Korea)
• Piotr Pstrągowski (RIMS, Japan)
• Fei Ren (Wuppertal, Germany)
• Shuji Saito (Tokyo, Japan)
• Masaya Sato (Tokyo, Japan)
• Vladimir Sosnilo (Regensburg, Germany)
• Yigeng Zhao (Westlake, China)

Recordings:

Photos:

Poster:

Program:

• 10:00-11:00 Saito, Motivic homotopy theory with ramification filtrations [ABSTRACT] [VIDEO]
• 11:15-12:15 Kai, Linear patterns of prime elements in number fields [ABSTRACT] [VIDEO]
• 13:45-14:45 Zhao, A fibration formula for characteristic classes of étale constructible sheaves [ABSTRACT] [VIDEO]
• 15:00-16:00 Pstrągowski, Spectral weight filtrations [ABSTRACT] [VIDEO]

• 10:00-11:00 Haesemeyer, K-theory of singularities, revisited. [ABSTRACT] [VIDEO]
• 11:15-12:15 Park, On the relative Milnor K-theory of some Artin local algebras over a field of any characteristic. [ABSTRACT]
• 13:45-14:45 Morin, Zeta values and the Beilinson fiber square [ABSTRACT] [VIDEO]
• 15:00-16:00 Krause, Prismatic cohomology and K-theory, I [ABSTRACT] [VIDEO]
  
  
• 18:00-20:00 Workshop banquet

• 10:00-11:00 Asok, Algebraic vs analytic vector bundles [ABSTRACT] [VIDEO]
• 11:15-12:15 Sato, Hochschild homology with bounded poles [ABSTRACT] [VIDEO]
• 13:45-14:45 Nemoto, Non-torsion algebraic cycles on the Jacobians of Fermat quotients [ABSTRACT] [VIDEO]
• 15:00-16:00 Ren, Duality for de Rham-Witt sheaves with ramification [ABSTRACT] [VIDEO]

• 10:00-11:00 Kimura, An application of a Hodge realization of Bloch-Kriz mixed Tate motives [ABSTRACT]
• 11:15-12:15 Sosnilo, Equivariant Dundas-Goodwillie-McCarthy theorem. [ABSTRACT] [VIDEO]
• 13:45-14:45 Krause, Prismatic cohomology and K-theory, II [ABSTRACT] [VIDEO]
• 15:00-16:00 Kahn, An \(l\)-adic norm residue epimorphism theorem. [ABSTRACT] [VIDEO]

Abstracts:

• Asok
  Wednesday, 10:00
  Algebraic vs analytic vector bundles
  
  
• Haesemeyer
  Tuesday, 10:00
  K-theory of singularities, revisited.
  Vorst's conjecture relating K-regularity and regularity for affine algebras is now a theorem over perfect fields. However, many questions remain open, for example: Do weaker K-regularity conditions imply algebraic properties weaker than regularity, as is the case for K_0 and K_1-regularity? Is the bound given in the conjecture sharp in any dimension bigger than 1? I will report on recent work with C. Weibel giving one answer to the first question that could have been obtained 50 years ago, and intriguing evidence regarding the second making use of much more recent Hodge-theoretic arguments developed by Mustata, Popa, and other algebraic geometers.
  
  
• Kahn
  Thursday, 15:00
  An \(l\)-adic norm residue epimorphism theorem.
  Let \(X\) be a smooth variety over a finite field. It is known that the continuous étale cohomology groups \(H^i_{\text{cont}}(X,\mathbf{Q}_l(n))\) vanish for \(i < n\) and for \(i>2n+1\). For \(i=n\), one can construct a map \[\mathcal{K}_n^M\otimes \mathbf{Z}_l\to \mathcal{H}^n_{\text{cont}}(\mathbf{Z}_l(n))\] where \(\mathcal{K}_n^M\) is the \(n\)-th Milnor \(K\)-sheaf and the right hand side is the Zariski sheaf associated to the presheaf \(X\supseteq U\mapsto H^n_{\text{cont}}(U,\mathbf{Z}_l(n))\). We show that this map is epi for any \(X\). This is the first unconditional step towards the sheaf-theoretic reformulation of the conjectures of Tate and Beilinson put together.
  
  
• Kai
  Monday, 11:15
  Linear patterns of prime elements in number fields
  Under some linear independence assumption, I establish the asymptotic frequency with which a given finite set of linear polynomials attains simultaneous prime values in a number field. This extends the work of Green, Tao and Ziegler for the case of Q to arbitrary number fields.
  Applications of this result include an even more negative answer to Hilbert's 10th Problem by Peter Koymans and Carlo Pagano, which strengthens the 1970 original negative resolution and its subsequent generalizations.
  
  
• Kimura
  Thursday, 10:00
  An application of a Hodge realization of Bloch-Kriz mixed Tate motives
  Beilinson and Deligne proved a weak version of Zagier's conjucture on special values of Dedekind zeta functions assuming the existence of a category of mixed Tate motives which has certain properties. We show that Bloch-Kriz category of mixed Tate motives together with a Hodge realization which we constructed has the required properties.
  
  
• Krause 🇯🇵 クラウズ
  
  Tuesday, 15:00
  Thursday, 13:45
  Prismatic cohomology and K-theory In these talks I want to give an overview over recent applications of prismatic cohomology to computations of algebraic K-theory. In the first talk, I want to give an overview over prismatic cohomology. In the second, I want to present a recent result with Senger obtaining sharp bounds on the vanishing of even-degree K-groups of Z/p^n, improving on previous results with Antieau and Nikolaus.
  
  
• Morin 🇺🇸 mo-RAN, 🇯🇵 モラン
  
  Tuesday, 13:45
  Zeta values and the Beilinson fiber square
  
  
• Nemoto
  Wednesday, 13:45
  Non-torsion algebraic cycles on the Jacobians of Fermat quotients
  Ceresa cycles are important examples of algebraic cycles that are generically homologically trivial but algebraically non-trivial. However, it is difficult to show the non-triviality or non-torsioness for Ceresa cycles of specific curves. Regarding the algebraic non-triviality or non-torsioness of Ceresa cycles of Fermat curves and their quotients, there are some results by Harris, Bloch, Kimura, Tadokoro and Otsubo. Recently, Eskandari-Murty proved that the Ceresa cycle of the Fermat curve, whose degree is divisible by a prime greater than 7, is non-torsion modulo rational equivalence. In this talk, we prove that the Ceresa cycles of Fermat quotient curves are non-torsion modulo rational equivalence under some assumptions.
  
  
• Park
  Tuesday, 11:15
  On the relative Milnor K-theory of some Artin local algebras over a field of any characteristic.
  When A is a local ring with an ideal I, the n-th relative Milnor K-theory of (A, I) has a good set of generators described by Kato-Saito.
  
  For a k-algebra A with a nilpotent ideal I over a field k with char (k) =0, using the above, we can describe a map called the Bloch map (some call it even Bloch-Artin-Hasse map) from the relative Milnor K-group to a group originating from the absolute Kähler differentials, which is often an isomorphism as observed by some people.
  
  For char (k)=p >0, under the mild assumption ``weakly 5-stable” (due to M. Morrow improving an old notion of W. van der Kallen), together with an outlandishly strong assumption "n! is invertible in k" for the n-th Milnor K-theory, a similar result was obtained, e.g. by Gorchinskiy-Tyurin.
  
  In this talk, in a response to a question posed by M. Morrow years ago, I would like to talk about a new approach to study the relative Milnor K-theory from the geometric perspective of algebraic cycles, and discuss how we may improve the above known results to a field of any characteristic, without imposing an exorbitant invertibility assumption.
  
  
• Pstrągowski 🇺🇸 stra-GOF-ski, 🇯🇵 プストラゴスキ
  
  Monday, 15:00
  Spectral weight filtrations
  In the 1970s, Deligne used the resolution of singularities to define a canonical filtration on rational cohomology of a complex variety, called the weight filtration. Unlike the rational cohomology groups themselves, this filtration is an honest invariant of the algebraic structure, and it depends on more than the homotopy type of a variety. I will talk about joint work with Peter Haine in which we show that the weight filtration has a canonical lift to the level of stable homotopy types.
  
  
• Ren
  Wednesday, 15:00
  Duality for de Rham-Witt sheaves with ramification
  Let X be a smooth proper scheme over a perfect field k of positive characteristic p. For p-primary torsion sheaves on X, there are two duality theories which are fundamentally different: one is the Serre-Grothendieck duality for coherent sheaves on X, which is later generalized by Ekedahl to a duality theory for coherent sheaves on W_nX with the top dRW sheaf being the dualizing sheaf; the other one is the Milnor-Kato-duality, which works for a much bigger class of sheaves, and has the top log dRW sheaf as the dualizing sheaf. The basis of Ekedahl's work is that regular dRW differentials admit a perfect pairing via the wedge product, just as regular Kähler differentials under the Serre-Grothendieck duality. Then it is natural to ask:
  1. what if we allow the differentials to have certain poles along a divisor D, do we still get a good duality theory?
  2. What should be the natural definition for a "dRW sheaf with ramification along D"?
  3. What is the dual of a given ramified dRW sheaf, can it also be defined naturally? Can one say something about its structure?
  In this talk, we will discuss our solutions to these questions, and how our ramified duality theory extends to a ramified version of the Milne-Kato duality. On the side of zeros in terms of Milnor-Kato, we give an explicit description of the structure of our motivic complex.
  Part of this talk is based on joint work with Kay Rülling.
  
  
• Saito (Shuji)
  Monday, 10:00
  Motivic homotopy theory with ramification filtrations
  We connect two important theories apparently unrelated. One is motivic homotopy theory and another is ramification theory. A byproduct is a new motivic interpretation of unramified cohomology. This is a joint work with J. Koizumi and H. Miyazaki.
  
  
• Sato (Masaya)
  Wednesday, 11:15
  Hochschild homology with bounded poles
  There is a map from algebraic K-theory to Hochschild homology, called a trace map. This map developed the study of algebraic K-theory. Algebraic K-theory is A^1-invariant on the category of smooth schemes over a field, so the Voevodsky’s motivic homotopy theory is a nice way to study algebraic K-theory. However, Hochschild homology is not A^1-invariant, so Voevodsky’s theory doesn’t capture it. In this talk, we will extend Hochschild homology of schemes to modulus pairs, and it is representable in the category of motives with modulus defined by Kahn-Miyazaki-Saito-Yamazaki.
  
  
• Sosnilo
  Thursday, 11:15
  Equivariant Dundas-Goodwillie-McCarthy theorem.
  The Goodwillie theorem states that given any Q-algebra R with a nilpotent ideal I, the fiber of the map between the K-theory spaces K(R) and K(R/I) is equivalent to the fiber on the negative cyclic homology. We extend this result to the G-equivariant K-theory and G-equivariant cyclic homology of characteristic 0 rings and affine group schemes G. The technique involves the new abstract notion of a c-category and of nilpotent extensions between them. A general Goodwillie theorem holds for nilpotent extensions of c-categories and the category of perfect complexes over X gives an example of a c-category at least when X is either a qcqs scheme or a quasi-geometric stack that satisfies the resolution property.
  
  
• Zhao
  Monday, 13:45
  A fibration formula for characteristic classes of étale constructible sheaves
  For a constructible étale sheaf, we first review the constructions of cohomological characteristic classes in the classical and relative settings. We then prove a fibration formula to calculate them, which involves a new cohomological class, so-called non-acyclicity classes. This is a joint work with Enlin Yang.