# Invited talks

These lectures will be accessible live only to registered participants via Zoom. You can find the Zoom link to join on our forum.

Monday, July 19, 13:30 ~ 14:30 UTC-3

## Solutions of the divergence, decomposition of $L^p$-functions, and applications.

### Ricardo Durán

#### University of Buenos Aires and CONICET, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

The variational analysis of the classical equations of mechanics is strongly based on some inequalities involving a function and its derivatives (for example, Poincare and Korn type inequalities). Many of these results can be obtained from the so-called Lions lemma .

The Lions lemma is equivalent to the existence of appropriate solutions of the equation $div\,u=f$. In this talk we recall how solutions of the divergence equation can be constructed by elementary arguments in a very general class of bounded domains. Then, we show how these solutions can be used to obtain a decomposition of a function of vanishing integral in a domain into a sum of locally supported functions with the same property.

Finally, we show how such a decomposition can be used to prove a local version of the classic result of Fefferman and Stein on the boundedness of the sharp maximal function. We apply these results to obtain weighted a priori estimates for elliptic problems and give some applications to the analysis of finite element approximations of elliptic problems with singular data.

Monday, July 19, 13:30 ~ 14:30 UTC-3

## Higher Fano Manifolds

### Carolina Araujo

#### IMPA, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

Fano manifolds are complex projective manifolds having positive first Chern class. The positivity condition on the first Chern class has far reaching geometric and arithmetic implications, making Fano manifolds a central subject in modern algebraic geometry. For instance, Fano manifolds are covered by rational curves, and families of Fano manifolds over one dimensional bases always admit holomorphic sections. In recent years, there has been great effort towards defining suitable higher analogues of the Fano condition. Higher Fano manifolds are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. For instance, they should be covered by higher dimensional rational varieties, and families of higher Fano manifolds over higher dimensional bases should admit meromorphic sections (modulo Brauer obstruction). In this talk, I will discuss a possible notion of higher Fano manifolds in terms of positivity of higher Chern characters, and discuss special geometric features of these manifolds.

Joint work with Roya Beheshti (Washington University, USA), Ana-Maria Castravet (University of Versailles, France), Kelly Jabbusch (University of Michigan-Dearborn, USA), Svetlana Makarova (University of Pennsylvania, USA), Enrica Mazzon (Max Planck Institute, Germany), Libby Taylor (Stanford University, USA) and Nivedita Viswanathan (The University of Edinburgh, UK).

Tuesday, July 20, 12:15 ~ 13:15 UTC-3

## Compactness of conformally compact Einstein manifolds

### Sun-Yung Alice Chang

#### Princeton University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

Given a manifold $ (M^n; [h])$, when is it the boundary of a conformally compact Einstein manifold $(X^{n+1}; g+)$ with $r^2 g+ |_{M} = h$ for some defining function $r$ on $X^{n+1}$? This problem of finding "conformal filling in" is motivated by problems in the AdS/CFT correspondence in quantum gravity (proposed by Maldacena in 1998) and from the geometric considerations to study the structure of non-compact asymptotically hyperbolic Einstein manifolds.

In this talk, instead of addressing the existence problem of conformal filling in, we will discuss the compactness problem. That is, given a sequence of conformally compact Einstein manifold with boundary, we are interested to study the compactness of the sequence class under some local and non-local boundary constraints. I will survey some recent development in this research area. As an application, I will report some recent work on the global uniqueness of a conformally compact Einstein metric defined on the (n+1)-dimensional ball with its boundary metric with its boundary metric infinity a metric near the canonical metric on the n-dimensional sphere when $n \geq 3$, where the existence of conformally filling in metric was constructed in the earlier work of Graham-Lee.

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Tuesday, July 20, 14:45 ~ 15:45 UTC-3

## Topological Ramsey spaces: a space of partitions

### Carlos Di Prisco

#### Universidad de Los Andes, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.

F. P. Ramsey published in 1930 an article on the decidability of a fragment of first order logic. A central element of the proof is a combinatorial theorem of independent interest that originated a whole branch of combinatorial theory of infinite sets, known today as Ramsey Theory. This theory has been developed substantially during the last fifty years with applications to mathematical analysis and other areas of mathematics.

Carlson and Simpson initiated the study of a dual Ramsey theory centered on partitions of the set of natural numbers, instead of sets of natural numbers. They also proposed a general framework to study similar combinatorial properties shared by a diversity of spaces, collected under the name of Ramsey spaces. The theory of Ramsey spaces was reformulated and expanded by Todorcevic and continued by other researchers.

In this talk we will present several results about the Ramsey space of infinite partitions of the set of natural numbers. In particular, we propose a definition for coideals of this space and present several of their main properties complementing research initiated by Matet and Halbeisen.

Bibliography.

Carlson, T. J. and S. G. Simpson, A dual form of Ramsey's theorem. Adv. in Math. 53 (1984), no. 3, 265–290.

Matet, P., Partitions and filters. Journal of Symbolic Logic 51 (1986) 12-21.

Halbeisen, L., Ramseyan ultrafilters. Fundamenta Mathematicae 169 (2001) 233-248.

Todorcevic, S., Introduction to Ramsey spaces. Princeton University Press, Princeton, New Jersey, 2010.

Tuesday, July 20, 14:45 ~ 15:45 UTC-3

## Subset sums, completeness and colorings

### Jacob Fox

#### Stanford University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we discuss novel techniques which allow us to prove a diverse range of results on representing integers as subset sums, including solutions to several long-standing open problems in the area. These include: solutions to the three problems of Burr and Erdős on Ramsey complete sequences, for which Erdős later offered a combined total of \$350 for their solution; analogous results for the new notion of density complete sequences; the solution to a conjecture of Alon and Erdős on the minimum number of colors needed to color the positive integers less than $n$ so that $n$ cannot be written as a monochromatic sum; the exact determination of an extremal function introduced by Erdős and Graham on sets of integers avoiding a given subset sum; and, answering a question reiterated by several authors, a homogeneous strengthening of a seminal result of Szemerédi and Vu on long arithmetic progressions in subset sums.

Joint work with David Conlon (Caltech, USA) and Huy Tuan Pham (Stanford University, USA).

Tuesday, July 20, 14:45 ~ 15:45 UTC-3

## The Mathematics of Interacting Particle Systems by Boltzmann Type flows

### Irene Matínez Gamba

#### The University of Texas at Austin, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

The dynamics of Multilinear Interactive particle systems are statistically described by kinetic collisional modeling. Introduced in the last quarter of the nineteenth century by L. Boltzmann and J.C. Maxwell, independently, gave birth to the area of Mathematical Statistical Mechanics and Thermodynamics. These evolution models concern a class of non-local, and non-linear integro-differential operator equations of Boltzmann type, whose solutions are probability densities. These are dissipative systems. Their rigorous mathematical treatment and approximations are still emerging in comparison to classical non-linear PDE theory. All these models share a unified functional analysis framework identifying the role of scattering and partition functions as integrable forms over compact manifolds, endowed by a priori estimates generating coercivity as well as uniform bounds. A functional ODE in Banach spaces framework yields well-posedness in the Banach space in probability densities with moments, for well prepared weights, that induces dissipative effects and propagate entropy monotonicity. From there, it follows the propagation of $L^p_{2k}(\mathbb{R}^d)$-norms, $1\leq p \leq \infty$ and $k>k_*\ge 1$, where $k_*$ depends on the model's scattering mechanism, exponential moments, as well as higher Sobolev regularity theory. Their bounds can be traced back to the coercive constants and a priori moments upper bounds estimates. Applications range from the modeling of classical elastic billiard of single species of a monoatomic gas, to polyatomic gases interchanges, to system of particle species such as in mixtures with disparate masses, to low temperature regimes for quantum interactions, collisional plasmas or electron transport in nano structures, and to self-organized or social interacting dynamics.

Joint work with The most recent part of this work has been developed with Ricardo J. Alonso (Texas A&M - Qatar), Erica de la Canal (The University of Texas at Austin, USA), Milana Pavić-Čolić (Univerity of Novi Sad, Serbia; and RWTH Aachen University, Germany) and Maja Tascović (Emory University, USA).

Wednesday, July 21, 12:15 ~ 13:15 UTC-3

## On the Geometry of a Homological Invariant of a Plane Curve Singularity

### Xavier Gomez-Mont

#### Centro de Investigación en Matemáticas, México - This email address is being protected from spambots. You need JavaScript enabled to view it.

A first invariant of an oriented manifold $M$ consists of its (co)homology groups (over $\mathbb{Z},\mathbb{Q}$), that comes provided with a bilinear form $<\ ,\ >$ induced from the intersection of cycles of complementary dimension. If part of $M$ contracts to a point, we obtain an isolated singularity. In case of a hypersurface $\{f=0\}$ in $\mathbb{C}^n$, going around the critical value gives rise to an automorphism $h$ of $M$ called the geometric monodromy, and its action on (co)homology is the algebraic monodromy $h_*$. The algebraic monodromy has a periodic and a non-periodic part. The non-periodic part is codified in $N$: the logarithm of the unipotent part of $h_*$. Hodge Theory distinguishes the bilinear forms $\langle N^j\,,\, \rangle$. The problem we address is to find the geometric content of these bilinear forms in terms of the geometric monodromy $h$.

We will describe for plane curves singularities the geometric content of the symmetric bilinear form $\langle N\ ,\ \rangle$ in terms of Dehn twists of $h$. The Theory of Resolution of Plane Curve Singularities provides a finite number of closed curves $ \{\gamma_j\}$ in the smooth oriented compact surface with boundary $S :=f^{-1}(\varepsilon)$ and positive integers $m$ and $m_j$, attached to each $\gamma_j$, so that $h^m$ is the identity outside of tubular neighbourhoods of $\gamma_j$ and it is a Dehn twist by $m_j$ on a tubular neigbourhood of $\gamma_j$ (an explicit Thurston-Nielsen representative of $h$). We show $$\langle N\alpha,\beta \rangle =\frac{1}{m} \sum_jm_j \langle \alpha ,\gamma_j \rangle \langle \beta ,\gamma_j \rangle \hskip 1cm,\hskip 1cm \alpha,\beta \in H_1(S,\mathbb{Z}).$$ The bilinear form is the sum with weights of the product of the common times that the closed curves $\alpha$ and $\beta$ intersect the curves $\gamma_j$, everything done in 1-homology $H_1(S,\mathbb{Z})$. This bilinear form induces a non-degenerate bilinear form on $W_1:=\frac{H_1(S,\mathbb{Z})}{Ker(N)}$, and the above expression shows that it is positive definite (i.e. Riemann-Hodge positivity). We identify $W_1$ as the 1-homology group of a graph $\tilde \Gamma$ (the graph of the semistable reduction of the singularity), providing explicit generators of the non-periodic part $W_1$ of $H_1(S,\mathbb{Z})$, since in $\tilde \Gamma$ there are no 2-chains (arXiv:2011.12332).

Joint work with Lily Alanis (Universidad Aut\'onoma de Nuevo Le\'on, M\'exico), Enrique Artal (Universidad de Zaragoza, Espa\~na), Christian Bonatti (Universit\'e de Bourgogne, Dijon, France), Manuel Gonz\'alez-Villa (CIMAT, M\'exico) and Pablo Portilla (CIMAT, M\'exico).

Wednesday, July 21, 12:15 ~ 13:15 UTC-3

## Potential Theory: new results

### San Martin Jaime

#### Universidad de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

We will present new results on (probabilistic and classical) Potential Theory with impact on Markov Processes, Linear Algebra and the maximum principle. On the one hand, we will give significant advances in the so-called inverse M-matrix problem, and on the other hand we will show how these results give us new ways to construct Markov processes, even in the Brownian context.

Wednesday, July 21, 14:45 ~ 15:45 UTC-3

## The equivalence problem in analytic dynamics

### Christiane Rousseau

#### University of Montreal, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

A central problem in local dynamics is the equivalence problem: when are two analytic systems locally equivalent under an analytic change of coordinates? In the neighbourhood of a singular point, representatives of equivalence classes could be given by normal forms. But, most often, the changes of coordinates to normal form diverge. Why? What does it mean? Unfolding the singularities reveals the geometric obstructions to the convergence to normal form. In this talk, we discuss a class of singularities and their unfoldings for which we can provide moduli spaces for the equivalence problems. We explain the common geometric features of these singularities, and how the study of the unfolding of these singularities allows understanding both the singularities themselves, and the obstructions to the existence of analytic changes of coordinates to normal form.

Wednesday, July 21, 14:45 ~ 15:45 UTC-3

## On the stability of homogeneous Einstein manifolds

### Jorge Lauret

#### Universidad Nacional de Córdoba and CIEM (CONICET), Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $M$ be a compact differentiable manifold and let $\mathcal{M}_1$ denote the space of all unit volume Riemannian metrics on $M$. The critical points of the simplest curvature functional, given by the total scalar curvature $ Sc:\mathcal{M}_1\rightarrow {\Bbb R}, $ are called Einstein metrics and play a fundamental role in Differential Geometry and Physics. Among Einstein metrics with positive scalar curvature, those which are stable as critical points of Sc (i.e., negative definite Hessian) on the subspace $\mathcal{C}_1$ of unit volume constant scalar curvature metrics on $M$, and in particular local maxima of $Sc|_{\mathcal{C}_1}$, seems to be extremely rare.

In this talk, after some general preliminaries, we will focus on the case when the metrics and the variations are considered to be $G$-invariant for some compact Lie group $G$ acting transitively on $M$. This takes us to work on the overwhelming and sophisticated class of all compact homogeneous spaces.

Joint work with Emilio Lauret (Universidad Nacional del Sur and INMABB (CONICET), Argentina) and Cynthia Will (Universidad Nacional de Córdoba and CIEM (CONICET), Argentina).

Wednesday, July 21, 14:45 ~ 15:45 UTC-3

## Quantum Toric Geometry

### Ernesto Lupercio

#### Center for Research and Advanced Studies IPN Mexico (Cinvestav), Mexico - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk I will introduce a (non-commutative) generalization to the field of toric geometry that we have developed in collaboration with Katzarkov, Meersseman and Verjovsky. Just as classical toric manifolds are made up of real and complex tori, Quantum Toric Manifolds are made up of quantum tori. The talk will be a friendly tour of the field at a colloquium level. We add some final remarks relating this to complex systems and tropical geometry. This is mainly based on "Quantum (Non-commutative) Toric Geometry: Foundations" (arxiv:2002.03876).

Joint work with Ludmil Katzarkov (University of Miami), Laurent Meersseman (University of Angers, France) and Alberto Verjovsky (UNAM).

Thursday, July 22, 12:15 ~ 13:15 UTC-3

## The Waist Inequality and Positive Scalar Curvature

### Davi Maximo

#### University of Pennsylvania, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

The topology of three-manifolds with positive scalar curvature has been (mostly) known since the solution of the Poincare conjecture by Perelman. Indeed, they consist of connected sums of spherical space forms and S^2 x S^1's. In spite of this, their "shape" remains unknown and mysterious. Since a lower bound of scalar curvature can be preserved by a codimension two surgery, one may wonder about a description of the shape of such manifolds based on a codimension two data (in this case, 1-dimensional manifolds). In this talk, I will show results elucidating this question for closed three-manifolds.

Thursday, July 22, 12:15 ~ 13:15 UTC-3

## Multinomial random tilings

### Richard Kenyon

#### Yale University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

Random tiling models are an important subarea of combinatorics and statistical mechanics, but tend to be quite difficult to analyze. We define a variant of the random tiling model, the ``multinomial random tiling model" which is not just computationally effective but leads to explicit and diverse long-range behavior including phase transitions, crystallization and quasicrystallization, and conformal invariance. This is joint work with Cosmin Pohoata.

Joint work with Cosmin Pohoata (Yale University, USA).

Thursday, July 22, 14:45 ~ 15:45 UTC-3

## Observable events and typical trajectories in dynamical systems

### Lai-Sang Young

#### New York University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

I will present ideas related to ``typical solutions" for finite and infinite dimensional dynamical systems, deterministic or stochastic. In finite dimensions, one often equates observable events with positive Lebesgue measure sets, and view invariant measures that reflect large-time behaviors of positive Lebesgue measure sets of initial conditions as physically relevant. Accepting these ideas, there is a simple and very nice picture that one might hope to be true. Reality is messier, unfortunately, at least for deterministic systems. I will argue that the addition of a small amount of random noise will bring this picture about. As for infinite dimensional systems, such as those defined by semi-flows generated by evolutionary PDEs, a different notion of observability is needed. I will finish with some results that suggest a notion of "typical solutions" for certain kinds of infinite dimensional systems.

Joint work with Alex Blumenthal (Georgia Institute of Technology) and others.

Thursday, July 22, 14:45 ~ 15:45 UTC-3

## Asymptotic Mean Value Properties for Non Linear Partial Differential Equations

### Julio Daniel Rossi

#### Buenos Aires Univ., Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In recent years there has been an increasing interest in whether a mean value property, known to characterize harmonic functions, can be extended in some weak form to solutions of nonlinear equations. This question has been partially motivated by the surprising connection between Random Tug-of-War games and the normalized $p-$Laplacian discovered some years ago, where a nonlinear asymptotic mean value property for solutions of a PDE is related to a dynamic programming principle for an appropriate game. Our goal in this talk is to show that an asymptotic nonlinear mean value formula characterizes weak solutions (in the viscosity sense) for the wide class of partial differential equations, including eigenvalues of the Hessian and the classical Monge-Amp\`ere equation.

References

P. Blanc -- J. D. Rossi. Game Theory and Partial Differential Equations. De Gruyter Series in Nonlinear Analysis and Applications Vol. 31. 2019.

P. Blanc -- F. Charro -- J. J. Manfredi -- J. D. Rossi. A nonlinear mean value property for Monge-Ampere. To appear in Journal of Convex Analysis.

P. Blanc -- C. Esteve -- J. D. Rossi. The evolution problem associated with eigenvalues of the Hessian. Journal of the London Mathematical Society. Vol. 102(3), 1293--1317, (2020).

P. Blanc -- J. D. Rossi. Games for eigenvalues of the Hessian and concave/convex envelopes. Journal de Math\'ematiques Pures et Appliquees. Vol. 127, 192--215, (2019).

Joint work with P. Blanc (Jyvaskyla, Finland), F. Charro (Detroit, USA), C. Esteve (Bilbao, Spain) and J.J. Manfredi (Pittsburgh, USA)..

Thursday, July 22, 14:45 ~ 15:45 UTC-3

## Canonical Heights on Shimura Varieties and The Andre-Oort conjecture

### Jacob Tsimerman

#### University of Toronto, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

The Andre-Oort conjecture describes how CM points lie in algebraic subvarieties of Shimura varieties. Through a series of works by many authors, the conjecture had been reduced to establishing an upper bound on the height of CM points, which has been shown in the case of the Siegel modular variety $\mathcal{A}_g$ . We explain how to deduce this height bound in general by reducing to the case of $\mathcal{A}_g$ using a classical idea of Deligne. To carry out this idea we construct a variant of the Faltings height for arbitrary Shimura varieties, so that we may compare heights between distinct Shimura Varieties. We do this in the case of proper Shimura varieties using recent ideas in relative p-adic hodge theory.

Joint work with Ananth Shankar(University of Wisconsin) and Jonathan Pila(Oxford University).

Friday, July 23, 12:15 ~ 13:15 UTC-3

## Symbolic systems, automorphism groups, and invariant measures

### Bryna Kra

#### Northwestern University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

The automorphism group of a symbolic system capture its symmetries, reflecting the dynamical behavior and the complexity of the system. For example, for a topologically mixing shift of finite type, the automorphism group contains isomorphic copies of all finite groups and the free group on two generators and such behavior is common for shifts of high complexity. In the opposite setting of low complexity, there are numerous restrictions on the automorphism group, and for many classes of shift systems, it is known to be virtually abelian. I will give an overview of relations among dynamical properties of the system, algebraic properties of the automorphism groups, and measurable properties of the system and of its automorphism group, all of which quickly lead to open questions.

Friday, July 23, 12:15 ~ 13:15 UTC-3

## Hardy fields and Transseries

### Lou van den Dries

#### University of Illinois at Urbana-Champaign, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

This concerns joint work with Matthias Aschenbrenner and Joris van der Hoeven. The focus of our book ``Asymptotic Differential Algebra and Model Theory of Transseries '' (Annals of Mathematics Studies 195, 2017, Princeton University Press) is on understanding the differential field of transseries. To deal with Hardy fields we recently refined the general theory from that book considerably. This has benefits beyond Hardy fields. I will start from scratch by introducing Hardy fields and state our main result about Hardy fields, which basically amounts to a complete theory about extending Hardy fields. I will include some material on transseries and on how they relate to Hardy fields.

Joint work with Matthias Aschenbrenner (University of Vienna, Vienna, Austria) and Joris van der Hoeven (Ecole Polytechnique, Paris, France).

Friday, July 23, 14:45 ~ 15:45 UTC-3

## Partially hyperbolic dynamics in hyperbolic 3-manifolds

### Rafael Potrie

#### Universidad de la República, Uruguay - This email address is being protected from spambots. You need JavaScript enabled to view it.

Partially hyperbolic dynamics arise from the need to weaken the notion of uniform hyperbolicity in order to understand larger classes of dynamics. These systems can be detected with finitely many iterates and recently many results have been obtained in the direction of a general topological classification of these systems, at least in dimension 3. I will then try to present motivations and tools used to understand the dynamics and topology of partially hyperbolic systems in 3-manifolds, specially hyperbolic 3-manifolds. This includes joint works with a lot of people including T.Barthelme,C. Bonatti, S. Fenley, S. Frankel, A. Gogolev, A. Hammerlindl, K. Parwani and M. Shannon.

Friday, July 23, 14:45 ~ 15:45 UTC-3

## Counting distances and directions in fractals

### Pablo Shmerkin

#### University of British Columbia and T. Di Tella University, Canada and Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

Counting patterns such as distances or directions in finite sets is a classical problem in geometric combinatorics. Starting with work of Falconer in the 1980s, there has been a lot of interest in studying geometric patterns in "large" subsets of Euclidean space (fractals). I will introduce this general class of problems, and describe recent progress on two specific instances - the Falconer distance set problem and the radial projection problem. Based partly on joint work with T. Keleti and with H. Wang. No specialized background will be assumed.

Friday, July 23, 14:45 ~ 15:45 UTC-3

## Characterizations of Nonnegative polynomials on some varieties

### Mauricio Velasco

#### Universidad de los Andes, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk I will describe some recent results on the characterization of those polynomials that are nonnegative on a variety $X$ in $\mathbb{R}^n$. In the first part of the talk I will explain why this is an interesting problem: it turns out to have a wealth of applications ranging from nonconvex optimization to stochastic control. In the second part of the talk I will explain how this problem can be approached on algebraic curves and surfaces.

Joint work with G. Blekherman (GA Tech), R. Sinn (U. Lepizig) and G.G. Smith (Queen's U).