** Persistence modules and barcodes in symplectic geometry and spectral geometry **

**1. Hamiltonian diffeomorphisms **

Arnold:“Symplectic topology has the same relation to ordinary topology as Hamiltonian systems have to general dynamical systems”.

Already surfaces are difficult examples.

Hofer’s length on Hamiltonian diffeomorphism groups . A path is determined by a path of normalized functions . Its length is

This defines a kind of biinvariant Finsler structure on . The corresponding distance is nondegenerate (Hofer, Polterovich, Lalonde-McDuff). It is essentially the only one (Buhovsky-Ostrover). Existence of this metric is remarkable. It is remiscent of commutator norms on finitely generated groups.

Autonomous Hamiltonian diffeos (generated by time-independent Hamiltonians ) correspond to 1-parameter subgroups of . They admit roots of any degree. T-Such flows conserve energy. They are geodesics in Hofer’s metric (but not the only geodesics).

has interesting algebraic properties. It is algebraically simple. Let be the set of Hamiltonian diffeos admitting a root of order . Is metrically dense in ? I.e. is

finite? We conjecture that this is never the case.

Theorem 1 (Polterovich-Shelukin)Let be a closed surface of genus . Let be an arbitrary closed manifold with . Then, for large,

This has been improved since by Jun Zhang, Polterovich-Shelukin-Stojisavljevic.

Our tools are Floer theory and persistence modules and their barcodes.

**2. An example **

In 2 dimensions, autonomous Hamiltonian flows are integrable, i.e. deterministic. Thus we look for chaotic Ham diffeos. Like in an eggbeater, combine two integrable diffeos performing mere shear motions, but on intersecting annuli. As soon as 1992, physicist Franjione-Ottini studied such linked twist maps. A parameter (strength of shear motion) is introduced. As tends to , we show that the distance of resulting diffeo to tends to infinity.

We study periodic orbits in special free homotopy classes. Handles are needed to separate periodic. Our example fails on the 2-sphere, as shown by Khanevsky.

Our invariant survives stabilization by dimension: product with identity does the job.

**3. Motivation **

** 3.1. Dynamics **

In dynamics, it has been known for a long time that vectorfields generate few diffeomorphisms (Palis, Brin 1973). In , non autonomous Ham. diffeos contain a -dense open set (Salamon-Zehnder, Ginzburg-Gurel), for symplectically aspherical manifolds. Our methods upgrade open to Hofer-open, and make it quantitative.

** 3.2. Coarse geometry of **

Polterovich-Rosen: a -generic Hamiltonian generates a nondistorted 1-parameter subgroup, distance to identity grows linearly.

In fact, the only other known behaviour is boundedness.

When I proved that has infinite diameter (for surfaces), Misha Kapovitch asked me wether did lie in a bounded neighborhood of a quasigeodesic. Our main theorem shows that this does not happen for .

** 3.3. Milnor’s constraint **

In 1983, Milnor observed that if a diffeo is a square, , the number of primitive geometrically distinct 2-periodic orbits is even. Indeed, induces a action of such orbits.

Find more restrictions on powers.

**4. Barcodes **

Edelsbrunner, Harer, Carlsson, in the context of topological data analysis. Has developped into a very abstract subject.

A *barcode* is a finite collection of intervals with multplicities . The bottleneck distance is defined as follows. Erase intervals of length and match the remaining intervals up to error . Then infimize .

Given a field , a *persistence module* is a pair of finite dimensional -vectorspaces , , and maps , for . Commuting diagrams. One assumes regularity: for all but a finite number of jump points in , are isomorphisms, together with semicontinuity at jump points.

*Interval module* is the tautological 1-dimensional persistence module supported on an interval.

**Structure theorem**: every persistence module is associated with a unique barcode, as the sum of intervals modules of the intervals of the barcode.

** 4.1. Original example: Morse theory **

closed manifold, Morse function. Persistence module is

Inclusions induce persistence morphisms.

The following statement is a not that easy theorem.

**Robustness**: The map is Lipschitz.

It follows that one can define critical points of merely continuous functions.

** 4.2. Morse homology **

I need be more specific with the homology theory I use.

Let be a Morse function on and a generic Riemannian metric. The Morse complex is spanned by the critical points of with value . The differential counts the number of gradient lines of connecting critical points,

** 4.3. Floer theory **

Born in 1988. In symplectic topology, the role of is played by infinite dimensional manifold of contractible loops . Given a 1-periodic Hamiltonian , define *action functional*

where is an arbitrary disk spanning .

The original action functional has only one critical point of infinite index and coindex. The perturbation is much more interesting, since its critical points correspond to 1-periodic orbits of the Hamiltonian flow generated by .

The solutions of the gradient equation are pseudoholomorphic cylinders. Therefore (Gromov 1985) they constitute a Fredholm problem. Although the gradient equation has no local (in time) solutions, the boundary value problem of gradient connections of critical points is well-posed. Therefore, one gets a well-defined complex, and homology groups , this is *Floer homology*.

Under certain asumptions (asperical, atoroidal,…), the corresponding persistence module depends only on the time 1 map .

One can extend the construction to noncontractible loops.

Theorem 2 (Polterovich-Shelukin)The map

is Lipschitz.

Hence Lipschitz functions on barcodes yield numerical invariants of Ham diffeos. Powers give rise to representations on persistence modules. A Floer-Novikov variant has been developed by Usher-Zhang.

** 4.4. Representations on persistence modules **

Since , acts on the Floer homology of , this is a action . Define be the -eigenspace of . There is a corresponding persistence module. If , then induces action on . Then on it. Thus the multiplicity of each bar in is even. This is reminiscent of Milnor’s constraint.

**Observation**. Distance to full squares is controlled by stable multiplicity, i.e. parity of the dimension of eigenspaces.

**5. Barcodes for eigenfunctions on surfaces **

Joint work with Iosif Polterovich and Vukasin-Stojisavljevic. Elaborates on 2006 work with Misha Sodin.

** 5.1. Oscillation **

**Question**. Given a closed oriented surface, and a smooth Morse function on , how can one define the oscillation of ?

The *Banach indicatrix* is defined as follows. Let denote the number of connected components in . Let

Goes back to Kronrod and Vitushkin in the 1950’s. In the 1980’s, Yomdin rediscovered it, with the idea that if derivatives of are not too large, should be small.

Define the total length of the finite part of the barcode as

The following is an easy fact from surface topology.

Theorem 3 (PPS 2018)

** 5.2. Example **

Consider the Reeb graph of (space of connected components of fibers). descends to a function on it. Then is equal to the total variation of on its Reeb graph. The difference with can be seen on the barcode.

**6. Results on eigenfunctions **

**Main question**. Bound oscillation via analytic properties of functions. Fix a Riemannian metric on surface . Let be the Laplacian. For , consider

The study of the topology of eigenfunctions of the Laplacian has a long history. Richard Courant showed that the number of nodal domains (connected components of ) is for the -eigenfunction.

Theorem 4 (Polterovtch-Sodin 2006)For ,

Corollary 5

** 6.1. Remarks **

1. For Euclidean domains, for Dirichlet boundary values, Alexandrov-Backelman-Pucci-Cabre 1995 show that

2. On the square 2-torus, satisfy , and barcodes have bars of length . This is sharp.

3. Say a critical value of is -significant if it is an endpoint of a bar of length . From our theorem, it follows that

Corollary 6If , the number of -significant critical values of is .

4. Nicolaescu has shown that the expectation of the number of critical points of random linear combinations of eigenfunctions is .

** 6.2. Example **

Let

has a number of critical points that tends to infinity, whereas for every fixed , the number of -significant critical values stays bounded.

**7. Approximation theory **

Let , be smooth functions on . In -norm, what is the best approximation of by functions of the form , ?

Let be the barcode of . Then is Diff-invariant. By robustness theorem,

Thus we get a lower bound from barcodes.

** 7.1. Example **

**Question**. Given a smooth function on the 2-sphere, find optimal approximation of by a function with 2 critical points.

If has 2 index 1 critical values , the lower bound is sharp.

** 7.2. Approximation with eigenfunctions and their images by changes of variables **

From our theorem, it follows that

Corollary 7Let be a Morse function on . Assume that is large compared to . Then

which, in turn, is the half of the average bar length.

**8. Proof **

Our goal is to prove that , imply that .

I do all computations on the square torus, for simplicity. Let denote the Hessian of function .

Equip the unit tangent bundle with the obvious metric . Look at a connected component of a regular fiber , parametrized by arclength. Let be its lift to via its field of normals. Since is not contractible, its length is .

Since

the total length of the lift is , hence

by coarea formula, Cauchy-Schwartz (plus Bochner-Lichnerowicz in the general case).

]]>
** Bandit regret optimization **

Blackboard course. Slides available on the PGMO website. Video available in a few weeks.

**1. Introduction to regret analysis **

** 1.1. Framework **

At each time step , the player selects an action at random, according to a distribution which is independent of the past. Simultaneously, the adversary selects a loss . The loss suffered by the player is . Full information would mean that is observed. Bandit means that only is observed.

The benchmark, to which other strategies will be compared, is

My loss must be compared with the optimal loss ,

The regret is the difference .

The machine learning mindset is that a single picture does not help much, but a large dataset of actions allows for a good strategy. So is a large set, eventually with some structure.

Think of an oblivious adversary, like nature. Adversary merely means that we cannot make any stochastic assumption about it. For instance does not represent what a clever adversary would have done when player applies a constant strategy.

Interpret as the rate at which is identified. How large should one take in order that ?

If there is a good action, we intend to find it. If there is none,…

** 1.2. Some applications **

Add placement. The player runs a website. Action = choose an add. Adversary is a potential customer, who clicks on the add or not. Best

Packet routing. Graph with source and target nodes. Action = choose a path. Huge set of actions, but with a lot of structure (distance between paths).

Hyperparameter optimization.

AI for games, i.e. efficient tree search. In AlphaGo, a determinist game, randomness was introduced to improve exploration of the set of actions.

** 1.3. Overview of results **

\textbf{Full information with finite actions} There,

- for pure noise (case when the past gives no information).
- There exists an algorithm which achieves .

I will give two proofs, an algorithmic one, and a more conceptual one where appears as an entropy.

\textbf{Easy data} I.e. when there is more signal in the data than pure noise. Then .

\textbf{Bandit with finite action} Then . Bad news: in absence of structure, strategy is slow.

\textbf{Large and structured action set} This uses deeper mathematics (Bernoulli convolution, geometry of log-concave measures).

\textbf{Better benchmarks}

**2. Full information **

** 2.1. An algorithm: multiplicative weights **

Rediscovered many times. See Vovk 1990, Freund-Shapira 1996, Littlestone-Warmuth 1994.

Assume oblivious adversary, and .

Idea: keep a weight for each action . Update

Play from .

Key intuition: Say is correct at round , then remains and others shrink, so gets closer to . Otherwise, pays for it a fixed factor .

Theorem 1

Optimizing in yields

**Proof** Liapunov function based argument. Consider . Then

hence

since . Note that . On the other hand,

hence

as announced. End of proof.

** 2.2. Lower bound (pure noise scenario) **

Theorem 2For every algorithm, there exists numbers such that

**Proof**. Probabilistic method. A lower bound on expected regret (with random losses) is given.

Pick at random from a Rademacher distribution ( uniformaly independently). Then ,

according to the central limit theorem. Indeed, the max of independent gaussians has size . End of proof.

** 2.3. A more principled approach **

Based on Abernethy-Warmuth 2008, Bubeck-Dehel-Koren-Peres 2015.

Say players picks and adversary picks .

A deterministic strategy is given by a map .

A bandit is a map .

A randomized strategy is picking at random a deterministic strategy , where is the set of deterministic strategies. Kuhn’s theorem

The minimax regret consists in finding the best strategy, achieving

Sion’s minimax theorem allows to swap inf and sup: let be convex in and concave in , bicontinuous, compact. Then

Now we are in Bayesian setting, with a prior distribution on . In particular, we know the distribution of .

Denote by . By convexity of the simplex, we choose to play , a Doob martingale. In other words, . Let us analyze it.

Each move of a martingale costs entropy. This is encoded in the following folklore inequality,

Assuming this bound, use Cauchy-Schwarz,

which is the optimal bound. End of proof.

The bandit case will be more subtle, since one will not be allowed to condition with respect to .

** 2.4. Proof of the folklore inequality **

Recall the entropy of a finite distribution . The relative entropy is

Given finite random variables and , the entropy of knowing is

The mutual information means how useful is when predicting ,

It is symmetric. Pinsker ‘s inequality states that

**Proof of the folklore inequality**. Apply Pinsker’s inequality,

Since

Hence

**3. Bandit analysis **

Based on Russo-Van Roy 2014.

Back to the instantaneous regret . We used

This is not true any more in the bandit setting, where the filtration is much smaller. Nevertheless, we hope that a weaker inequality would suffice,

The next argument goes back to the very first paper on bandit theory, Thompson Sampling 1933. Bandit is a trade off between exploration and exploitation. For future benefit, it is necessary to explore more.

Denote , . Then

On the other hand,

where the last step is Pinsker’s inequality, plus the fact that . Cauchy-Schwarz inequality gives

Plugging this in the argument of previous section, one gets

Theorem 3In the bandit setting,

The term can be removed.

On the lower bound side, the adversary is picked at random: one is , the others being . One needs samples per arm, whence , regret is .

**4. A primer on bandit linear optimization **

Now , , and the loss of playing is the inner product . Naively, one would assume that regret is .

Theorem 4

This is best possible.

If is infinite but included in the unit ball of for some norm. Then

Theorem 5

where is the cotype of the chosen norm on .

** 4.1. Proof of theorem **

The last application of Cauchy-Schwarz needs be replaced by something smarter. Here

Introduce the (huge) matrix

One needs relate the trace of this matrix to its Frobenius norm. In general,

Thus we win by replacing the size with the rank of matrix . Here where has size , hence has rank .

**5. Mirror descent and online decision making **

We focus on online linear optimization.

Data: a compact body . At each time step , the player picks and the adversary picks (previously, was the simplex) such that for all . My regret is

** 5.1. Stability as a algorithm design principle **

We estimate

The first term is a movement cost. Call the second term the one look ahead regret. The algorithm will find a trade-off between the two terms.

** 5.2. Gradient descent **

This is . In a more invariant manner,

Note the norm is not well suited to our problem, which involves an norm. Assuming that , a third interpretation, in the spirit of regularization, is

We try to avoid overfit, i.e. control entropy. Hence we replace norm with entropy. Also, instead of inner product, duality should be used.

** 5.3. Mirror descent **

Initially due to Nemirovskii, see book, by Nemirovskii and Yudin, 1983.

Le be a convex function. Write

Introduce the Fenchel dual function on the dual space,

Under mild assumptions, .

Given , belongs to the dual space. Move it in the direction of , get . Get back to the primal space by applying , get

Maybe this does not belong to . Thus project it to (in Euclidean distance). In other words,

** 5.4. Analysis of the one look ahead term **

For intuition, I explain the continuous setting. Then , is defined using for all . Hence the one look ahead term becomes

whereas the movement cost becomes . The continuous time mirror descent is

The constrained optimality condition at a point involves covectors in the normal cone

It states that . The gradient of is . Hence the minimizer satisfies

hence the differential inclusion

Theorem 6 (Bubeck-Cohen-Lee…2017)If is continuous, is semicontinuous and is continuous, the differential inclusion has a unique solution.

As a Liapunov function, we use

Let us compute the time derivative

since the Lagrange multiplier is nonpositive on the normal cone. Integrating over time, we get

Lemma 7

Recall the multiplicative weights bound

We have obtained the estimate corresponding to . Next, me move to the other term.

** 5.5. Movement analysis **

Nothing general, here is the simplex equipped with the norm. We hope for an estimate

A stronger estimate would be

Here, . We may choose a suitable convex function . To simplify matters, let us ignore and assume . Then

which is indeed estimated by provided we choose . With this choice, the dynamics becomes

where is such that . This proves the wanted estimate.

** 5.6. Discrete time analysis **

We expect an extra additive error of which, up to a term in , equals

In the last expression, the local norm is defined as follows. The square root of the Hessian of defines a Riemannian metric on (which blows up near the boundary).

Theorem 8Assume that is well-conditionned in the sense that if , then

Then the regret is bounded above by

**Proof**. Recall that and

We must estimate

The last term is by optimality. On the other hand,

Hence, using convexity of ,

By the magic of Fenchel transform,

In turn, this last term is of the order of . Summing up yields the Theorem.

** 5.7. Revisiting multiplicative weights with mirror descent **

Take , whose Hessian is the diagonal quadratic form with eigenvalues . The Riemannian metric is . Then

Hence

Corollary 9If , then well-conditionning holds with .

The Theorem yields

** 5.8. Propensity score estimate for bandits **

Again, we do not observe the full loss function, and need an unbiaised estimator for it. We look for depending on where is pick according to . Start with putting to everything which has not been observed,

Then run mirror descent with instead of , get the Exp3 algorithm of Auer-Cesa-Bianchi-Freund-Schapiro 1995).

Theorem 10Exp3 satisfies

**Proof**.

Note that the variance is finite only beause local norms are used.

** 5.9. More refined regularization **

Change function to make variance smaller. Negentropy gives . Instead, gives . Then

which is slightly better. On the other hand,

Plugging this in the Theorem gives regret bound (Audibert-Bubeck 2009).

In view of easy data bounds, pick (called the log barrier). Then . The local norms cancel, yielding

Plugged in the Theorem, this gives regret bound .

** 5.10. Metrical task systems **

Based on Borodin-Linial-Saks 1982. One moves among states, with a task attached to each state. At each time step, observe . Update state to . Pay movement plus task, .

It is hopeless to try to do as well as the best guy. The ibjective is to do it up to a multiplicative constant. So he regret is replaced with a competitive ratio

where are the optimizers.

If is picked according to distribution , then is Wasserstein distance, and is a one look ahead term.

**Example**. If all distances are equal to 1, is the distance on distributions.

Our convex function should be -Lipschitz with respect to the Wasserstein distance.

Theorem 11 (Bubeck-Cohen-Lee… 2018)The competitive ratio of mirror descent is bounded above by times the terme arising from mouvement cost analysis.

Theorem 12 (Borodin-Linial-Saks 1982)For the trivial distance, the competitive ratio is at least .

**Conjecture**. This is true for any metric.

My intuition is that the metric can only help us. Furthermore, there is an avatar of entropy, called multiscale entropy, associated with any metric. Usual entropy is well suited only to the trivial distance.

]]>** Journee francilienne d’accueil des post-doctorants en mathematiques **

**1. Eleonora di Nezza: Special metrics in Kähler geometry **

I work in Kähler geometry. It has to do with complex manifolds.

Let us start with the uniformization theorem. It implies that compact Riemann surfaces (complex 1-manifolds) admits metrics of constant curvature, the sign of which is determined by the sign of Euler characteristic. This is tremendously useful in the study of the space of Riemann surfaces.

In higher dimensions, do complex manifolds admit special metrics? Kähler metrics are Hermitian metrics which satisfy an extra first order intergability condition: they osculate a flat metric one order more that a general hermitian metric. This is expressed via the imaginary part of the Hermitian metric, which is a differential 2-form : has to be closed. Alternatively, two connections are associated to a Hermitian metrics, the Riemannian Levi-Civita connection, and the Chern connection. The metric is Kähler iff these two connections coincide.

Nevertheless, a given complex manifold admits many Kähler metrics. To single a preferred one, we require that its Ricci curvature be constant. We call them **Kähler-Einstein metrics**, since constancy of Ricci curvature is the content of Einstein’s equations for vacuum with a cosmological consant.

Not every complex manifold admits KE metrics. Indeed, Ricci curvature, viewed a differential 2-form, is a representative of the first Chern class of the (complex) tangent bundle. In the KE case, Ricci curvature is proportional to the Kähler form. For a complex manifold, the fact that the first Chern class can be represented by a closed -form which is positive definite (resp. zero, resp. negative definite) is rather restrictive. Let us assume that this condition is satisfied. Then Calabi showed that KE equations are equivalent to a scalar second order nonlinear PDE of complex Monge-AmpĂ¨re type. The problem splits into 3 different cases. The negative and zero cases were solved in the late 1970’s. The positive case is very recent (Chen-Donaldson-Sun, Tian), after decades of co siderable work by a large community.

The present trend is to extend the theory to singular varieties. Indeed, the classification program (“Minimal Model Program”) requires to handle singular varieties. The unknown in the complex Monge-AmpĂ¨re equation is then a real function defined on the complement of a fixed divisor (i.e. complex mildly singular hypersurface).

Guedj-Zeriahi 2007: existence and uniqueness (up to an additive constant) of a weak solution. With Lu (then in Orsay), I proved in 2014 that their solution is smooth in the complement of the divisor. This is analysis, dealing with a degenerate equation. The french school (Boucksom, Essidieux, Guedj, Zeriahi) has introduced a pluripotential theory which is very efficient for this problem.

**2. Claudio Llosa Isenrich : Kähler groups from branched covers of elliptic curves **

A complex manifold is a real manifold with a preferred atlas ro open subsets of where changes of charts are holomorphic diffeomorphisms. A Kähler manifold is a complex mnifol with a Hermitian metric whose imaginary part, a differential 2-form, is closed.

The most important examples are smooth complex submanifolds of complex projective space (the space of lines in ). They are defined as zero sets of homogeneous polynomials in variables.

The fundamental group of a topological space is the set of homotopy classes of based loops. Loops can be concatenated, which gives rise to a group structure. This is the most ancient, and still the most important, algebraic invariant of topology.

Say a group is Kähler if it is isomorphic to the fundamental group of a compact Kähler manifold. In the 1950’s, Serre raised the question of which groups are Ka\”hler. Notes that every (finitely presented) groups is isomorphic to the fundamental group of a compact complex manifold.

Here are examples. Even rank free abelian groups are Kähler. Surface groups (fundamental groups of compact surfaces) are Kähler. Serre showed that all finite groups are Kähler. The class of Kähler groups is able under taking direct products and finite index subgroups.

Here are restrictions. The first Betti number of a Kähler is even (this rules out even rank free abelian groups, free groups).

My contribution is a construction of new examples of Kähler groups, insped by Dimca-Papadima-Suciu 2009. Let be a complex torus (also called a complex elliptic curve). For , consider branched covers of , , , where is a Riemann surface. For instance, cut open along several slits, take several copies and glue them together along slits in a suitable combinatorial pattern. Since has a commutative group structure, one can take the sum of these maps, and get a map

This is a holomorphic map. A generic fiber is smooth, its fundamental group is Kähler, by construction. It is the kernel of the homomorphism induced by on fundamental groups. I show that if is surjective on fundamental groups, the group enjoys an interesting finiteness property (which I will not define today): it is but not . No previous example was known.

Peng: higher dimensions? Unclear.

**3. Cyril Marzouk: Scaling limits of large random combinatorial structures: Brownian objects **

Motivation from quantum gravity: how to sample a random metric on the 2-sphere? Possibly easier: how to sample a random function?

Approximate answer: look at random walks on . The endpoints obeys a binomial law, which converges to a Gaussian law. In fact (Donsker), the whole path converges in law to Brownian motion, a probability measure on real functions on .

Closer to random metrics: random quadrangulations, also known as planar maps. The graph distance is thought of as a metric on the sphere. The Gromov-Hausdorff distance makes such planar maps into a Polish topological space.

Le Gall and Miermont (independantly): a uniformly random quadrangulation with faces, renormalized by , converges to a random metric space in Gromov-Hausdorff distance. It is homeomorphic to the 2-sphere, but it has Hausdorff dimension 4.

Enrich the model by admitting faces with variable (even) number of edges, and specifiy the number of faces of degree . Assume converges to , converges to , and converges to . Then I show that a uniformly random such map converges to the same random metric space (slightly rescaled).

Le Gall’s method relies on Schaeffer’s encoding of quadrangulations with rooted trees carrying integers at vertices. It was known earlier (Aldous) that the uniformly random tree with vertices, rescaled by , converges to the Brownian tree. I merely explain where the constants come from. A finite tree can be viewed as a walk on . The depth of the tree is the maximal distance reached by the walk. Since random walks are well understood, depths of random trees are known.

**4. Jie Lin: Special values of -functions and a conjecture of Deligne **

Considering sums of inverse squares, cubes,… leads to Riemann’s zeta function. Euler showed that . More generally, is commensurable to . To show this, one extends meromorphically to , and gets a functional equation, which relates to , which is a rational number (one says that is critical, since is holomorphic both at and ).

For odd integers, one must compute residues, which is much harder.

A Dirichlet -function is a weighted inverse power sums, where weights are characters mod . The above theorem extends to -functions. Example:

Adèles. Euler’s product formula for suggests formulae involving all prime numbers, i.e. all absolute values on . The ring of adeles is the (restricted) product of all -adic fields (completion of for the -adic absolute value), and the reals. The idèles are the units of adele ring. The idea of a -function extends to Hecker characters, i.e. continuous characters of .

A Hecke character is a one-dimensional representation of , we must consider more generaly automorphic representations of . Motives are an even wider generalization introduced by Grothendieck. These are geometric objects, like elliptic curves. Analytic continuation of -functions in this generality is a conjecture by Hasse-Weil. In 1979, Deligne conjectured the values of -functions for critical integers.

Langlands’ program relates these 3 types of objects, motives, automorphic representations and Galois representations. This contains the Taniyama-Shimura-Weil conjecture whose solution is used in the solution of Fermat’s last problem.

With coauthors, I have translated Deligne conjecture in automorphic representation terms.

**5. Dena Kazerani : Symmetry, from hyperbolic systems to Green-Naghdi models **

I work in fluid mechanics of incompressible flows. When viscous forces are low, Navier-Stokes equations simplify to Euler equations. A difficulty arises since the domain occupied by the fluid evolves.

I focus today on shallow water, this is the Green-Naghdi model (1976): one assumes that the fluid is irrotational, that the ground is planar and horizontal, vertical velocity depends linearly on the vertical variable, horizontal velocity does not depend on vertical velocity. Then the unknown of the equation, defined on a fixed planar domain, are the horizontal velocity and the water height. One gets a hyperbolic Saint-Venant system with extra dispersive terms.

Hyperbolic systems often have symmetry: Lax-entropy pairs, Godunov structures. It is the case for the Saint-Venant system. The symmetry is expressed by changes of variables involving matrices

We need to generalize matrices to operators on Banach spaces. We give a general definition of Godunov structure, extend the classical results (equivalence with existence of Lax-entropy pairs). We show that Green-Naghdi’s model has this property.

We use this symmetry to establish well-posedness of the equations: global existence, asymptotic stability.

Di Nezza: the definitions make sense only for smooth solutions, is this a problem? Yes, singular solutions behave differently (shocks).

]]>

** Existence of conic Kaehler-Einstein metrics **

Joint work with Feng Wang, Zhejiang university.

A log-Fano manifold is the date of a compact Kaehler manifold , a divisor with normal crossings such that the line bundle

is positive.

A metric is a conic Kaehler-Einstein metric if it is smooth Kaehler in and for every point where is defined by in some coordinates, is equivalent (between two multiplicative constants) to the model cone metric

Say that is conic Kaehler-Einstein if

**1. Necessary conditions **

Berman 2016: If admits a conic KE metric with , then is log-K-stable.

Log-K-stability is defined as follows.

A special degeneration of is a 1-parameter family of log-pairs, consisting of

- A normal log-pair with a -equivariant map ,
- is an equivariant -ample -line bundle.
- is isomorphic to for every .

There is a natural compactification of that maps to . Defined number

If the central fiber is a log-Fano variety embedded in by , then can be interpreted as a Futaki invariant.

Say that is log-K-semistable if for any special degeneration has . Say that is log-K-stable if for any special degeneration has and equality holds only for the trivial degeneration .

**2. The result **

Theorem 1If is log-K-stable, the there exists a conic KE metric with .

Many special cases were known, as consequences of existence of KE metrics on smooth closed manifolds. For instance when is a multiple of .

**3. Motivation **

We are interested in -Fano varieties . Assume admits a resolution such that , . For small enough , define

If there exists a KE metric on , then is a degenerate conic KE metric on with conic angles along . We expect that there exist conic KE metrics on with , which Gromov-Hausdorff converge to as .

We think that we are now able to prove the following. *If is a K-stable -Fano variety. Then it admits a generalized KE metric in the above sense*.

**4. Proof **

Many steps are similar to the smooth case. Pick a large integer such that has a smooth divisor . We use a continuity method, solving

. The set of such that a solution exists is easily shown to be non-empty (it contains 0) and open. Is it closed? The key point is a estimate. It follows from a “partial -estimate” and log-K-stability. In turn, this follows from an -estimate and compactness a la Cheeger-Colding-Tian.

** 4.1. Smoothing conical KE metrics **

Say that has a K-approximation if there exist Kaehler metrics in the same cohomology class such that

- uniformly on and smoothly outside ,
- ,
- in Gromov-Hausdorff topology.

We show that if and if for all ,

for some , then has a K-approximation where .

We solve a modified equation with an extra term involving ‘s. For this, we use the variational approach by Boucksom-Eyssidieux-Guedj-Zeriahi and results of Darwan-Robinstein, Guenancia-Paun.

** 4.2. Extend B. Wang-Tian’s results to conic case **

**5. Work in progress **

To handle -Fano varieties, we need to extend Cheeger-Colding to conic cases.

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** Geometry and analysis of waves in manifolds with boundary **

The wave-front is a subset of the cotangent bundle, whose projection is the singular support. In all dimensions, in Euclidean space, it travels at constant speed along straight lines (Fermat,…, Hormander).

In general Riemannian manifolds without boundary, it travels along geodesics as long as time stays less than the injectivity radius (Duistermaat-Hormander).

We impose Dirichlet boundary conditions. Then transverse waves reflect according to Snell’s law of reflection (Chazarain). What about tangencies? Assume obstacle is convex. Do waves propagate in the shadow?

Melrose-Taylor 1975: if the boundary is , no smooth singularities in the shadow region. However, analytic singularities occur.

Inside strictly convex domains, waves reflect a large number of times. The wave shrinks in size between two reflections, it refocusses, therefore its maximum increases. Caustics appear, together with swallowtail and cusp singularities.

In the non-convex case, especially if infinite order tangencies occur, one does not even know what the continuation of a ray should be (Taylor 1976).

**1. Dispersive estimates **

It is a measurement of the decay of amplitude of waves due to spreading out while energy is conserved.

In , after a high frequency cut-off around frequency , the maximum amplitude decays like . Indeed, the wave is concentrated in an annulus of width . The same holds in Riemannian manifolds without boundary.

In the presence of boundary, propagation of singularities has brought results in the 1980’s. Later on, people have tried a reduction to the boundary-less case with a Lipschitz metric: this requires no assumptions on the boundary, but ignores reflection and its refocussing effect.

** 1.1. Within convex domains **

Theorem 1 (Ivanovici-Lascar-Lebeau-Planchon 2017)For strictly convex domains, dispersion is in

This follows from a detailed description of the wave-front, including swallow-tails. It takes into account infinitely many reflections. It is sharp.

** 1.2. Outside convex obstacles **

The Poisson spot. This is a place where diffracted light waves interfere. It is in the shadow area, but much more light concentrates there. This was confirmed experimentally by Arago, following a debate launched by Fresnel who did not believe in the wave description of light. It should exist if one believes in Fermat’s principle that light rays follow geodesics, including those which creep along the boundary surface (Keller’s conjecture). In 1994, HargĂ© and Lebeau proved that, when light creeps along the bounday, it decays like .

Theorem 2 (Ivanovici-Lebeau 2017)For strictly convex obstacles,

- if , dispersion estimates hold like in ,
- if , they fail at the Poisson spot.

The reason is that a -dimensional surface lits the Poisson point.

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** Asymptotic expansions of holonomy **

Joint with Pierre Pansu.

**1. Motivation **

Given a connection on a principal bundle , holonomy along a based loop of is an element of resulting from lifting horizontally to . We look for an expression such that is a good approximation of holonomy when is short,

We want that be simpler to compute than holonomy, and be related to curvature.

Hatton-Choset: motion of a snake with two joins. , . Experimentalists have been led to choose the Coulomb gauge, and for the integral over a disk spanning of curvature expressed in Coulomb gauge.

In this practical example, motions are tangent to a sub-bundle of the tangent bundle of . Hence our interest in expansions which are particularly efficient on such curves. We call this setting *sub-Riemannian*.

Sub-Riemannian curvature is not easy to define. The obvious approach of using adapted connections on the tangent bundle is not illuminating.

**2. Results **

- Asymptotic, gauge-free formula in Euclidean space.
- Riemannian case not that different.
- Sub-Riemannian case suggests a notion of curvature.
- For certain sub-Riemannian structures,

** 2.1. Euclidean case **

Dilations define radial fillings of loops. Use radial gauge (frame is parallel along rays through the origin). They turn out to be optimal. Using radial gauge, integrate curvature over radial filling. This defines

Say a differential form has weight if dilates are . Use radial gauge to define weight of forms on .

Theorem 1If the curvature has weight , thenFurthermore, one can expand in termes of Taylor’s expansion of curvature.

Since curvature has weight at least 2, one gets a 4-th order approximation.

** 2.2. Sub-Riemannian case **

The flat sub-Riemannian case corresponds to Carnot groups, i.e. a Lie group whose Lie algebra has a gradation

and is generated by . **Example**: Heisenberg group.

Fix a norm on . Left translates of define a sub-Riemannian metric, for which dilations on are homothetic.

According to Le Donne, sub-Riemannian Carnot groups are characterized by being the only locally compact homogeneous geodesic metric spaces with homothetic homeos.

Carnot groups come with a left-invariant horizontal basis, we pick a connection on the tangent bundle which makes it parallel. It has torsion. We combine it with the principal bundle connection to define iterated covariant derivatives of curvature. We organize them according to weights adapted to the Lie algebra grading. The above theorem extends.

** 2.3. Horizontal holonomy **

Since we are interested only in holonomy along horizontal loops, we have the freedon to change the connection outside the horizontal subbundle.

Chitour-Grong-Jean-Kokkonen: using this freedom, there are choices which minimize the curvature in the sense that as many components as possible vanish identically. This tends to increase the weight of curvature.

**Example**: on 3-dimensional Heisenberg group, the preferred connection has curvature which vanishes on the horizontal distribution, hence has weight instead of 2. Above Theorem provides a 6-th order expansion, whose terms can be computed algebraically.

More generally, on free -step nilpotent Lie groups, the curvature of a preferred connection has order at least , whence a -th order expansion whose terms are linear in curvature (in fact, in the preferred curvature).

We expect to use it to refine the Euclidean expansion.

**3. Question **

What does this give in case of the two-joints snake? Requires to push computations further.

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** The borders of Outer Space **

Joint work with Kai-Uwe Bux and Peter Smillie.

**1. Duality groups **

I am interested in Poincare duality. For a group, assume is a smooth -manifold, then

Bieri-Eckmann observed that is suffices that acts freely cocompactly on a contractible space whose compactly supported cohomology vanishes in all degrees but , and is torsion free. Then is a duality group.

If the action is merely proper and cocompact, is a virtual duality group. Borel-Serre used this for lattices. Bestvina-Feighn used this to show that $latex {Out(F_n) is a virtual duality group. Mapping class groups also act on a contractible space.

To achieve cocompactness, Borel-Serre added to the symmetric space copies of Euclidean space forming a rational Euclidean building. The resulting bordification of the quotient is a manifold, with boundary homotopy equivalent to a wedge of spheres (Solomon-Tits). Instead, Grayson constructed an invariant cocompact subset of symmetric space.

Grayson’s work was used by Bartels-Lueck-Reich-Ruping to prove Farrel-Jones for }&fg=000000$Sl(n,{\mathbb Z})$latex {.

For }&fg=000000$Out(F_n)J_n\subset X_n$latex {. It is much easier and gives more information on the boundary.

\section{Construction}

}&fg=000000$X_nJ_n$ is obtained by chopping off some of their corners.

A core graph is a subgraph such that, when one shrinks it, one gets out of Outer Space. Only corner facing core graphs need be chopped off.

The boundary appears as a union of contractible walls (every intersection of walls is contractible).

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** Topology of ends of nonpositively curved manifolds **

Joint work with T. Nguyen Pham.

I am interested in complete Riemannian manifolds with curvature in , and finite volume.

**Example**. Product of two hyperbolic surfaces. The end is homeomorphic to , with some extra structure: is made of two pieces.

More generally, for locally symmetric spaces of noncompact type, lifts of ends are homeomorphic to , with a wedge of spheres. This description goes back to Borel-Serre.

**1. Thick-thin decomposition **

Gromov-Schroeder: assume there are no arbitrarily small geodesic loops. Then the thin part is homeomorphic to , with a closed manifold.

The condition is necessary. Gromov gives an example of a nonpositively curved infinite type graph manifold of finite volume.

Theorem 1 (Avramidi-Nguyen Pham)Under the same assumptions, any map of a polyhedron to the thin part of the universal cover can be homotoped within the thin part into a map to an -dimensional complex, .

**Consequences**:

- If , each component of the thin part is aspherical and has locally free fundamental group.
- for all .
- .

**2. Proof **

Maximizing the angle under which two visual boundary points are seen gives Tits distance, and the corresponding path metric .

In the universal cover, the thin part is the set of points moved less than away by some deck transformation . Isometries are either hyperbolic (minimal displacement is achieved) or parabolic (infimal displacement is 0). Parabolic isometries have a nonempty fix-point set at infinity. At each point , the subgroup generated by isometries moving no more than is virtually nilpotent, hence virtually has a common fixed point at infinity. This allows to define a discontinuous projection to infinity. The point is to show that the image has dimension .

** 2.1. Busemann simplices **

If and are Busmeann functions, need not be a Busemann function again, but on each sphere, there is a unique point where it achieves its minimum, and tis point depends in a Lipschitz manner on . This defines an arc in Tits boundary, hence simplices . We claim that

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** Free-by-cyclic groups and trees **

Joint work with S. Dowdall and I. Kapovich.

The Bieri-Neumann-Strebel invariant is an open subset of , it is the set of such that is surjective on . Here, is the torsion free abelian cover of and is an equivariant map representing .

If is free-by-cyclic, one can refine

Geoghegan-Mihalik-Sapir-Wise show that for every , is locally free and there exists an outer automorphism and a finitely generated subgroup such that . In particular, if , then one can take .

From now on, we assume that is atoroidal and fully irreducible. Then is hyperbolic, and there exists an expanding irreducible train track representative (Bestvina-Handel). Let be the mapping torus. It carries the suspension of , which is a one-sided flow (action of semi-group ). The representative of integral cohomology class factors to a map . Let be the subset of cohomology classes such that the representative can be chosen to be increasing along the flow. Then

Theorem 1

- is a component of . It is a rational polyhedral cone.
- For , inverse images of points are cross-sections of the flow. The first return map is an expanding irreducible train track representative of , with .

Stretch factors form a nice function on .

Theorem 2 (Algom-Kfir-Hironaka-Rafi)There exists an -analytic, convex function such that for all such that for al and ,

- .
- .
- If , then .

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** Action dimension and Cohomology **

Joint work with Giang Le and Mike Davis.

**1. Action dimension **

This is the minimal dimension of contractible manifolds which admit a proper -action. The geometric dimension replaces manifolds with complexes.

** 1.1. Examples **

If is of type , then . This comes from embedding complexes into . would be easy. is Stallings’ theorem, using a suitable model of .

Bestvina-Feighn: For lattices in semi-simle Lie groups, is the dimension of the symmetric space.

Desputovic: Teichmuller space.

** 1.2. Our favourite examples **

Today, we focus on graph products of fundamental groups of closed aspherical manifolds and complements of hyperplane arrangements. We are concerned with lower bounds: when can one reduce from the obvious dimension?

The first class (circles) includes RAAG, covered by Avramidi-Davis-Okun-Shreve.

** 1.3. Motivation from -cohomology **

Let denote the -Betti numbers of the universal covering.

**Singer conjecture**: If is a closed aspherical manifold of dimension , then vanish if .

This suggests

**Action dimension conjecture**. If , then .

Okun and I have shown that both conjectures are in fact equivalent.

**2. Graph products **

Let be a flag complex with vertex set . The graph product of a family of groups over is the quotient of the free product of by the normal subgroup generated by , when is an edge of .

**Examples**. If all , we get RAAG. If all are finite cyclic, we get RACG.

Theorem 1Let be a -dimensional flag complex, let be the corresponding graph product of fundamental groups of closed aspherical -manifolds. Then

- If , then .
- If , then .

** 2.1. Constructing aspherical manifolds **

The only way to make new aspherical manifolds is to glue aspherical manifolds with boundary along codimension 0 submanifolds of their boundaries. For instance, Salvetti complexes, made of tori, do not work. We replace tori with tori interval.

In general, we glue together products of , which is dimensional, which is sharp in some cases, as we show next. The fact that has vanishing homology allows to decrease dimension.

** 2.2. Obstructions to actions **

Bestvina-Kapovitch-Kleiner coarsify van Kampen’s obstruction to embedding complexes into . This lives in (configuration of pairs of points).

Theorem 2 (Bestvina-Kapovitch-Kleiner)Let be or hyperbolic, let with . Then

**Example**. If , contains , hence (in fact, ).

For graph products of closed aspherical manifolds, we construct a complex, denoted by , in . It is a join of -spheres based on .

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