Abstracts
Sir Michael Atiyah
Geometry and Arithmetic
Geometry and Arithmetic have been at the heart of Mathematics from at least
the time of Pythagoras. We can learn lessons from the past, from the great
Euler. I will discuss the present and speculate on the future.
Valentin Blomer
Eigenfunctions on Riemannian spaces
It is a problem in classical analysis to determine the asymptotic behaviour of
Laplacian eigenfunctions on Riemannian manifolds. Number theory enters the
scene if the manifold carries additional arithmetic structure such as an algebra
of Hecke operators. I will present methods from number theory, Lie groups
and automorphic forms to obtain information on the mass distribution of such
eigenfunction on various arithmetic spaces, and discuss some applications.
Jean-Benoı̂t Bost
Theta series and the analogy between number fields and function fields
In the classical analogy between number fields and function fields, an Euclidean
lattice E := (E, k.k) — defined by a free Z-module of finite rank E and some
Euclidean norm k.k on the R-vector space ER := E ⊗Z R — appears as the
counterpart of a vector bundle V on a smooth projective curve C over some
field k.
In this analogy, the arithmetic counterpart of the dimension
h0 (C, V ) := dimk Γ(C, V )
of the space of sections of V is the non-negative real number, defined by means
of the theta function of the Euclidean lattice E:
X
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h0θ (E) := log
e−πkvk .
v∈E
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This originates in Riemann’s “second proof” of the meromorphic extension and
functional equation for the Riemann zeta function, and to its extension by Hecke
to the Dedekind zeta functions of general number fields.
This talk will be devoted to diverse aspects of this analogy, notably to generalizations of the invariant h0θ attached to some infinite dimensional avatars
of Euclidean lattices, and to their applications to transcendence theory and to
algebraization theorems in Diophantine geometry.
Jan Bruinier
Kudla-Rapoport divisors on arithmetic ball quotients
We report on joint work with B. Howard, S. Kudla, M. Rapoport, and T. Yang.
A celebrated result of Hirzebruch and Zagier states that the generating series of
Hirzebruch-Zagier divisors on a Hilbert modular surface is an elliptic modular form
with values in the cohomology. The goal of this talk is to prove an analogue for
special divisors on integral models of ball quotients. In this setting the generating
series takes values in an arithmetic Chow group. We discuss some applications
to arithmetic theta lifts and the Colmez conjecture.
Bianca Dittrich
Quantum space time engineering
Following the lead that space time is geometry I explore notions of quantum space
time and quantum geometry. I discuss new realizations of quantum geometry
that have been recently constructed using methods from extended topological
field theories. I lay out a program to understand the dynamics of quantum gravity
based on such notions of quantum geometry.
Dennis Gaitsgory
Langlands duality for quantum groups
The geometric Satake isomorphism says that the category of perverse sheaves on
G(O)\G(K)/G(O) is canonically equivalent to Rep(GL ) as a monoidal category.
Now, the category Rep(GL ) admits a canonical deformation–into the category of
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representations of the quantum group. On the automorphic side this corresponds
to the metaplectic twist.
The talk is meant to explain how exactly that happens.
Robert Haslhofer
The moduli space of 2-convex embedded spheres
We investigate the topology of the space of smoothly embedded n-spheres in
Rn+1 , i.e. the quotient space Mn := Emb(S n , Rn+1 )/Diff(S n ). By Hatcher’s
proof of the Smale conjecture, M2 is contractible. This is a highly nontrivial
theorem generalizing in particular the Schoenflies theorem and Cerf’s theorem.
In this talk, I will explain how geometric analysis can be used to study the
topology of Mn respectively some of its variants. I will start by sketching a proof
of Smale’s theorem that M1 is contractible. By a beautiful theorem of Grayson,
the curve shortening flow deforms any closed embedded curve in the plane to
a round circle, and thus gives a geometric analytic proof of the fact that M1
is path-connected. By flowing, roughly speaking, all curves simultaneously, one
can improve path-connectedness to contractibility. In the second half of my talk,
I’ll describe recent work on space of smoothly embedded spheres in the 2-convex
case, i.e. when the sum of the two smallest principal curvatures is positive.
Our main theorem (joint with Buzano and Hershkovits) proves that this space is
path-connected, for every n. The proof uses mean curvature flow with surgery.
Hans-Joachim Hein
Calabi-Yau manifolds with conical singularities
Yau’s solution to the Calabi conjecture provided the first known examples of
compact Riemannian manifolds with zero Ricci curvature that are not flat, i.e.
not isometric to a quotient of Euclidean space by a discrete group of Euclidean
motions. The underlying topological manifolds of Yau’s examples are actually
complex algebraic. A typical example would be a smooth complex hypersurface
of degree n+2 in CP n+1 . We prove an extension of Yau’s theorem that produces
compact Riemannian spaces with zero Ricci curvature and with isolated conical
singularities. For example, the underlying algebraic space could now be a singular
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degree n + 2 hypersurface of CP n+1 with at worst ordinary double points. Joint
work with Song Sun.
Nigel Higson
On Weyl’s spectral decomposition theorem
Early in his career, Hermann Weyl examined and solved the problem of decomposing a function on a half-line as a continuous combination of the eigenfunctions
of a Sturm-Liouville operator with asymptotically constant coefficients. Weyl’s
theorem served as inspiration for Harish-Chandra in his pursuit of the Plancherel
formula for semisimple groups, and for this and other reasons it continues to be
of interest. I’ll try to explain how Weyl’s theorem arises again in efforts to view
Harish-Chandra’s work from the perspective of noncommutative geometry, and
I’ll describe a new, geometric, proof of Weyl’s theorem that seems to fit better
with representation theory. This is joint work with Tyrone Crisp and Qijun Tan.
Renate Loll
Riemann meets Quantum at the Planck scale
We expect that the - still elusive - quantum gravity theory underlying General
Relativity will give us a quantitative description of the structure and dynamics of
”quantum spacetime”. It is a fascinating question which concepts from classical
geometry can still be (made) meaningful at the ultrashort Planck scale and in
the presence of large quantum fluctuations. Causal Dynamical Triangulations
(CDT) is a candidate theory for nonperturbative quantum gravity, based on a
regularized path integral that (after ”Wick rotation”) is defined on quantum
ensembles of piecewise flat Riemannian geometries. I will describe which generic
and specific insights have been gained on the nature of ”quantum geometry”,
whose properties are not captured by smooth metrics, but by more elementary
notions of distance and volume. I will give intriguing examples of unexpected
quantum behaviour of familiar quantities like the dimensionality and curvature.
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Kobi Peterzil
Complex geometry in tame model theoretic settings: definability, analyticity and algebraicity
Let X be a complex analytic subset of C n . An old result of Fortuna and Lojasiewicz (1981) says: If X is a semialgebraic set (when one identifies the complex
numbers with the real plane) then X is a complex algebraic variety. Formulated
in a model theoretic language- If X above is definable in the real field then it is
an algebraic variety.
In this talk I will describe how various model theoretic assumptions affect the
possible definable complex analytic sets in a structure. I will mainly focus on
the question: Under which assumptions such sets are necessarily algebraic or
constructible? Chow’s classical theorem on the algebraicity of projective analytic
sets also fits into this framework.
Michael Rapoport
p-adic uniformization of Shimura curves
Shimura curves are algebraic curves that arise through complex uniformization by
an arithmetic group acting on the complex half plane. Forty years ago, Cherednik
observed that under suitable assumptions, these curves can also be uniformized
by the Drinfeld p-adic half plane. Now we are close to a reasonable proof (of
a variant) of this statement. I will report on joint work with S. Kudla and Th.
Zink, and related work of P. Scholze.
Claude Sabbah
Exponential-Hodge theory
Starting from the Riemann - resp. Birkhoff - existence theorem for linear differential equations of one complex variable, I will motivate on the example of hypergeometric - resp. confluent hypergeometric - equations the variant of Hodge
theory, called exponential-Hodge theory, originally introduced by Deligne. I will
also explain the interest of this theory in relation with mirror symmetry of Fano
manifolds.
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Norbert Schappacher
Bernhard Riemann as a puzzle for the historian
Considering Riemann’s importance for the recent history of mathematics and
physics, and looking at the wealth of literature that has accumulated around
him, one would think that it should be a fairly straightforward affair to present
a historical account of his life and work. The aim of the talk is to explain why
this is not so, and what would have to be done to improve the situation. To be
sure, facts that are known and that one ought to know about Riemann will be
duly recalled in the course of the lecture.
Ulrike Tillmann
Riemann moduli space and operads
Riemann’s moduli spaces are at the hard of much modern mathematics. In this
lecture we will explore their properties as an operad.
Operads were introduced in the 1970 in homotopy theory to study loop spaces.
Infinite loop spaces are of particular interest as they give rise to generalised homology theories. In the 1990’s operads had a renaissance with much interest
stimulated from mathematical physics. Thus Segal’s axiomatic approach to conformal field theory defines an operad of Riemann surfaces. We will show that
this is an example of a new generation of operads detecting infinite loop spaces.
The talk will introduce the main concepts and is addressed to a general mathematical audience.
Todor Tsankov
On metrizable universal minimal flows
It is an old theorem in topological dynamics that to every topological group, one
can associate a unique universal minimal flow (UMF): a flow that maps onto
every minimal flow of the group. For some groups (for example, the locally
compact ones), this flow is not metrizable and does not admit a concrete description. However, somewhat surprisingly, for many ”large” Polish groups, the
UMF is metrizable, can be computed, and carries interesting combinatorial information. Examples of extremely amenable groups (groups for which the UMF
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is a point) have been known since the 70s but more recently, Kechris, Pestov,
and Todorcevic, inspired by previous results of Pestov and Glasner and Weiss,
developed a systematic approach connecting metrizable UMFs with structural
Ramsey theory and many new examples were found. I will describe the state
of the art in the area and also discuss some new results that give a complete
characterization of metrizable UMFs for Polish groups. The new results are jonit
with I. Ben Yaacov, J. Melleray and L. Nguyen Van Thé.
Stefaan Vaes
Classification of von Neumann algebras
The theme of this talk is the dichotomy between amenability and non-amenability.
Because the group of motions of the three-dimensional Euclidean space is nonamenable (as a group with the discrete topology), we have the Banach-Tarski
paradox. In dimension two, the group of motions is amenable and there is therefore no paradoxical decomposition of the disk. This dichotomy is most apparent
in the theory of von Neumann algebras: the amenable ones are completely classified by the work of Connes and Haagerup, while the non-amenable ones give rise
to amazing rigidity theorems, especially within Sorin Popa’s deformation/rigidity
theory. I will illustrate the gap between amenability and non-amenability for von
Neumann algebras associated with countable groups and with group actions on
probability spaces, as well as for the type III factors arising in Voiculescu’s free
probability theory.
Rainer Weissauer
Groups and the geometry of Riemann theta divisors
In general the theta divisor of a polarized abelian variety is a highly singular
variety. That the singularities often have an intrinsic meaning most prominently is
demonstrated by the Riemann singularity theorem that describes the singularities
of theta divisors of Jacobians. In fact, many approaches to the Schottky problem
- i.e. to the question how to characterize Jacobians in terms of their theta
divisors - subsequently were based in one or the other way on refinements of
Riemann’s theorem. In this context we present a new invariant that attaches a
reductive group to a theta divisor, a group that is closely related to the nature of
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the singularities, and we discuss some of the consequences and open questions
arising from the study of the underlying structures.
Anna Wienhard
Positivity and higher Teichmüller theory
Classical Teichmüller space describes the space of conformal structures on a given
topological surface S. It plays an important role in several areas of mathematics
as well as in theoretical physics. Higher Teichmüller theory generalizes several
aspects of classical Teichmüller theory to the context of Lie groups of higher
rank, such as the symplectic group PSp(2n; R) or the special linear group PSL(n;
R). So far, two families of higher Teichmüller spaces are known. The Hitchin
component, which is defined when the Lie group is a split real forms, and the
space of maximal representations, which is defined for Lie groups of Hermitian
type. Interestingly, both families are linked with various notions of positivity in
Lie groups.
In this talk I will give an introduction to higher Teichmüller theory, introduce
new positive structures on Lie groups and discuss the (partly conjectural) relation
between the two.
Guoliang Yu
Secondary invariants of elliptic operators and applications
I will introduce certain secondary invariants of elliptic operator defined in the
context of C ∗ -algebra K-theory by Higson and Roe and discuss their applications
to Riemannian geometry and topology. This is joint work with Zhizhang Xie and
Shmuel Weinberger.
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