# Seminar series on basics of (Weil's) Riemann hypothesis, Standard conjectures (including Hodge's conjecture) and Dynamical systems

## This seminar series is co-organised by Fei Hu and Tuyen Trung Truong, in the scope of RCN grant 300814.

The seminar is to provide some basics (including practical tricks and intuitions) on topics around the relations between (Weil's) Riemann hypothesis, Standard conjectures (including Hodge's conjecture) and Dynamical systems.

The current version is for Spring 2021, but can be extended. For Spring 2021, the seminar is on Zoom. We welcome participation or presentation in the seminar. Please write to the organisers for more information.

**Talk 1**: A 10-line proof of Weil's Riemann hypothesis for Abelian varieties, by Tuyen Trung Truong.

Time: 9:30-11:00, Wednesday 3 March 2021.

Abstract: Usual proofs of Weil's Riemann hypothesis for Abelian varieties go through Tate's modules and Rosati's involutions. This talk will present a simpler proof without using these, after some basic background about Standard Conjecture C + intersection theory of algebraic cycles are provided. [Slides of the talk: This is in Notes format] (When you click the link, the file will be downloaded to your computer, and you can open it on a Mac.)

**Talk 2**: Standard conjectures, part 1, by Fei Hu

Time: 9:30-11:00, Wednesday 10 March 2021

Abstract: This talk will provide an up-to-date overview of the Standard Conjectures. [Slides for the talk - This is in pdf format]

**Talk 3**: Standard conjectures, part 2, by Fei Hu

Time: 9:30-11:00, Wednesday 17 March 2021

Abstract: This talk will provide an up-to-date overview of the Standard Conjectures. [Slides - This is in pdf format]

**Talk 4**: Number theory applications of Weil's Rieman hypothesis, part 1, by Tuyen Truong

Time 9:30-11:00, Wednesday 24 March 2021

Abstract: This talk will present some number theory applications of Weil's Riemann hypothesis. Also, it is described an approach by Deninger towards Riemann hypothesis, based on the proof of Weil's Riemann hypothesis for curves. [Slides for the talk: This is in Notes format]

**Talk 5**: Number theory applications of Weil's Rieman hypothesis, part 2, by Tuyen Truong

Time 9:30-11:00, Wednesday 14 April 2021

Abstract: This talk will present some number theory applications of Weil's Riemann hypothesis. Also, it is described an approach by Deninger towards Riemann hypothesis, based on the proof of Weil's Riemann hypothesis for curves. [Slides - This is in Notes format.]

**Talk 6**: Weil's Riemann hypothesis for surfaces, by Fei Hu

Time: 9:30-11:00, Wednesday 28 April 2021

Abstract: This talk will present about the classification of algebraic surfaces and an overview of the proof of Weil's RH for surfaces. [Slides - This is in PDF format]

**Talk 7**: Properties of tropical cohomology and applications: part 1, by Kristin Shaw

Time: 9:30-11:00, Wednesday 5 May 2021

Abstract: This talk will be a survey on a cohomology theory for polyhedral complexes known as tropical cohomology. If a polyhedral complex is obtained by a suitable degeneration of complex projective varieties these cohomology groups encode the Hodge numbers of a generic member of the family, and even the monodromy action can be expressed combinatorially. However there is an even larger class of polyhedral spaces, not obtained from algebraic geometry, for which this cohomology theory has many of the properties of Weil cohomology theories for smooth projective varieties. For example, Poincaré duality, cycle maps, Lefschetz theorems, and Hodge-Riemann bilinear relations. Even the combinatorial monodromy action can be used to formulate a tropical version of the Hodge conjecture.

I will first report on how what is known about this cohomology theory and what questions are still open. I will also explain how the Hard Lefschetz Theorem and the Hodge Riemann bilinear relations have been used to prove open conjectures in discrete mathematics.

**Talk 8**: Properties of tropical cohomology and applications: part 2, by Kristin Shaw

Time: 9:30-11:00, Wednesday 12 May 2021

Abstract: This talk will be a survey on a cohomology theory for polyhedral complexes known as tropical cohomology. If a polyhedral complex is obtained by a suitable degeneration of complex projective varieties these cohomology groups encode the Hodge numbers of a generic member of the family, and even the monodromy action can be expressed combinatorially. However there is an even larger class of polyhedral spaces, not obtained from algebraic geometry, for which this cohomology theory has many of the properties of Weil cohomology theories for smooth projective varieties. For example, Poincaré duality, cycle maps, Lefschetz theorems, and Hodge-Riemann bilinear relations. Even the combinatorial monodromy action can be used to formulate a tropical version of the Hodge conjecture.

I will first report on how what is known about this cohomology theory and what questions are still open. I will also explain how the Hard Lefschetz Theorem and the Hodge Riemann bilinear relations have been used to prove open conjectures in discrete mathematics.

**Talk 9**: Some remarks on the Lefschetz standard conjecture for smooth projective complex varieties: part 1, by Claudio Onorati

Time: 9:30-11:00, Wednesday 26 May 2021

Abstract:

I will try to explain the content of the following two papers

- F. Charles,
*Remarks on the Lefschetz standard conjecture and hyper-Kahler varieties,*Comm. Math. Helv. 88 (2013) - F. Charles, E. Markman,
*The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces,*Comp. Math. 149 (2013)

In the first talk I will focus on the general results on the Lefschetz standard conjecture (LSC) that can be deduced from families of algebraic cycles, giving two simple (and already known) examples. This covers the first parts of the two papers above.

In the second talk I will explain how these ideas can be applied to irreducible holomorphic symplectic manifolds. I will first go through the main results, constructions and examples in the theory, and eventually explain how to apply the machinery developed in the first talk.

**Talk 10**: Some remarks on the Lefschetz standard conjecture for smooth projective complex varieties: part 2, by Claudio Onorati

Time: 9:30-11:00, Wednesday 2 June 2021

Abstract:

I will try to explain the content of the following two papers

- F. Charles,
*Remarks on the Lefschetz standard conjecture and hyper-Kahler varieties,*Comm. Math. Helv. 88 (2013) - F. Charles, E. Markman,
*The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces,*Comp. Math. 149 (2013)

In the first talk I will focus on the general results on the Lefschetz standard conjecture (LSC) that can be deduced from families of algebraic cycles, giving two simple (and already known) examples. This covers the first parts of the two papers above.

In the second talk I will explain how these ideas can be applied to irreducible holomorphic symplectic manifolds. I will first go through the main results, constructions and examples in the theory, and eventually explain how to apply the machinery developed in the first talk.

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Dates for the following talks are not yet scheduled.

**Talk **: Intersection theory of algebraic cycles, motivic and dynamical viewpoints, part 1, by Tuyen Truong

Abstract: This talk will present intersection theory of algebraic cycles from both motivic and dynamical viewpoints.

**Talk **: Intersection theory of algebraic cycles, motivic and dynamical viewpoints, part 2, by Tuyen Truong

Abstract: This talk will present intersection theory of algebraic cycles from both motivic and dynamical viewpoints.