# 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

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.

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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. document.getElementById('cloakb817493f12c4b984c87ef2ecb5b634c6').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyb817493f12c4b984c87ef2ecb5b634c6 = 'c&#97;r&#97;&#117;j&#111;' + '&#64;'; addyb817493f12c4b984c87ef2ecb5b634c6 = addyb817493f12c4b984c87ef2ecb5b634c6 + '&#105;mp&#97;' + '&#46;' + 'br'; var addy_textb817493f12c4b984c87ef2ecb5b634c6 = 'c&#97;r&#97;&#117;j&#111;' + '&#64;' + '&#105;mp&#97;' + '&#46;' + 'br';document.getElementById('cloakb817493f12c4b984c87ef2ecb5b634c6').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyb817493f12c4b984c87ef2ecb5b634c6 + '\'>'+addy_textb817493f12c4b984c87ef2ecb5b634c6+'<\/a>';

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).

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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. document.getElementById('cloak9e29f6d57970ab3ff4068d0f47809022').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy9e29f6d57970ab3ff4068d0f47809022 = 'ch&#97;ng' + '&#64;'; addy9e29f6d57970ab3ff4068d0f47809022 = addy9e29f6d57970ab3ff4068d0f47809022 + 'm&#97;th' + '&#46;' + 'pr&#105;nc&#101;t&#111;n' + '&#46;' + '&#101;d&#117;'; var addy_text9e29f6d57970ab3ff4068d0f47809022 = 'ch&#97;ng' + '&#64;' + 'm&#97;th' + '&#46;' + 'pr&#105;nc&#101;t&#111;n' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak9e29f6d57970ab3ff4068d0f47809022').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy9e29f6d57970ab3ff4068d0f47809022 + '\'>'+addy_text9e29f6d57970ab3ff4068d0f47809022+'<\/a>';

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. document.getElementById('cloak523d0a493df38d4b4d38b2f2bb647ff7').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy523d0a493df38d4b4d38b2f2bb647ff7 = 'c&#97;.d&#105;' + '&#64;'; addy523d0a493df38d4b4d38b2f2bb647ff7 = addy523d0a493df38d4b4d38b2f2bb647ff7 + '&#117;n&#105;&#97;nd&#101;s' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'c&#111;'; var addy_text523d0a493df38d4b4d38b2f2bb647ff7 = 'c&#97;.d&#105;' + '&#64;' + '&#117;n&#105;&#97;nd&#101;s' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'c&#111;';document.getElementById('cloak523d0a493df38d4b4d38b2f2bb647ff7').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy523d0a493df38d4b4d38b2f2bb647ff7 + '\'>'+addy_text523d0a493df38d4b4d38b2f2bb647ff7+'<\/a>';

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.

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