| Michael Björklund
(ETH) | Biharmonic functions on groups, measurable cross ratios and central limit theorems.
 (abstract)
The aim of this talk is to discuss a novel approach to central limit theorems for random walks
on groups via biharmonic functions and measurable cross ratios on their associated
Poisson boundaries. The methods are quite general and can be applied quite efficiently to
non-elementary Gromov hyperbolic groups and subdirect products thereof.
|
|
Gautam Chinta (CUNY) | Recent results in multiple Dirichlet series.
(abstract)
I will give a survey of some recent results in the study of Dirichlet series in several complex variables. The focus of the talk will be on applications to number theory.
|
|
Caterina Consani (JHU) | Towards an Archimedean analogue of the Fontaine theory I.
(abstract)
TBA.
|
|
Gunther
Cornelissen (Utrecht) | Curves, dynamical systems, and weighted point counting.
(abstract)
A theorem of Tate and Turner says that two smooth projective algebraic curves over a finite field have the same zeta function if and only if their corresponding Jacobians are isogenous. We prove that an isomorphisms between the Galois groups of their maximal abelian geometric extensions that respects all Dirichlet L-series (rather than just the zeta function) detects the finer property of isomorphism of the curves themselves, up to ``Frobenius twists'' (i.e., morphisms induced by automorphisms of the ground field). This is shown to be equivalent to their respective class field theories being isomorphic as dynamical systems (in a sense that we make precise). The proof uses a variation on the anabelian method of Uchida for function fields, and is much shorter than the corresponding proof for number fields. We also describe an algorithm that in finite time determines an equation for such a curve, taking as consecutive input different L-series of the curve.
|
| Joachim Cuntz (Münster) | C*-algebras associated with endo- and polymorphisms of compact abelian groups (based on a joint article with A.Vershik). (abstract)
A surjective endomorphism or, more generally, a polymorphism, of a compact abelian group H induces a transformation of $L^2(H)$. We study the C*-algebra generated by this operator together with the algebra of continuous functions $C(H)$ which acts as multiplication operators on $L^2(H)$.
Under a natural condition on the endo- or polymorphism, this algebra is simple and can be described by generators and relations. In the case of an endomorphism it is always purely infinite, while for a polymorphism in the class we consider, it is either purely infinite or has a unique trace. We prove a formula allowing to determine the $K$-theory of these
algebras and use it to compute the $K$-groups in a number of interesting examples. To treat the case of a polymorphism we introduce the concept of independence for two commuting endomorphisms. Independence has interesting consequences for instance for the orbit partition associated with a pair of endomorphisms.
|
| Alex Gamburd (CUNY Grad Center) | Expander Graphs, Thin Groups
and Superstrong Approximation.
(abstract)
TBA.
|
| Dorian Goldfeld (Columbia) | Distribution of low-lying zeros for GL(3) families of L-functions.
(abstract)
Katz and Sarnak have conjectured that zeros of families of automorphic L-functions should be distributed like the eigenvalues of random matrices in a classical Lie group which is also called the symmetry type of the family. We shall determine the symmetry types for the following three families: cuspidal GL(3) Maass forms f, the symmetric square family sym^2 f on GL(6), and the adjoint family Ad f on GL(8). We identify the symmetry types as unitary, unitary, symplectic, respectively. This is joint work with Alex Kontorovich.
|
| Alexander Gorokhovsky (Colorado) | Higher analytic indices and symbolic index pairing.
(abstract)
Higher index theory was started in the work of A. Connes and
H. Moscovici on the Novikov conjecture. The goal of my talk is to reinterpret their theorem and to
extend the higher index theory to new
situations. This is joint work with H. Moscovici.
|
| Rostislav Grigorchuk (TAMU) | Schreier Dynamical Systems, totally non-free actions, and self-similar
C*-algebras. (abstract)
I will describe topologies in the space of subgroups of a group and in
the space of Schreier graphs. Then I will discuss totally non free
actions as antipod to the free actions, and Schreier dynamical systems. I will define a concept of IRS (invariant random subgroup), and will describe two examples related to the group of intermediate growth constructed by the speaker in 80th and to famous R.Thompson group F. Self-similar C*-algebras associated with self-similar group actions will be introduced as well as a recurrent trace on them, and it will be explain how they can be used to produce asymptotic expanders and asymptotic Ramanujan graphs.
|
| Piotr M. Hajac
(IMPAN) | Free
actions of compact quantum groups on unital C*-algebras.
(abstract)
Let $F$ be a field, $\Gamma$ a finite group, and $\mathrm{Map}(\Gamma,F)$ the Hopf
algebra of all set-theoretic maps $\Gamma\rightarrow F$. If $E$ is a
finite field extension of $F$ and $\Gamma$ is its Galois group, the
extension is Galois if and only if the canonical map $E\otimes_FE\rightarrow E\otimes_F\mathrm{Map}(\Gamma,F)$ resulting from viewing $E$ as a
$\mathrm{Map}(\Gamma,F)$ - comodule is an isomorphism. Similarly, a finite covering
space is regular if and only if the analogous canonical map is an
isomorphism. In this talk, we extend this point of view to actions of
compact quantum groups on unital C*-algebras. We prove that such an
action is free if and only if the canonical map (obtained using the
underlying Hopf algebra of the compact quantum group) is an isomorphism.
In particular, we are able to express the freeness of a compact Hausdorff
group action on a compact Hausdorff space in algebraic terms. Joint work
with P. F. Baum and K. De Commer.
|
| Masoud Khalkhali
(UWO) | Spectral Zeta Functions and Scalar Curvature for Noncommutative Tori (joint work with Farzad Fathizadeh).
(abstract)
Ideas of spectral geometry can be imported to noncommutative geometry thanks to Connes' notion of spectral triples. In this talk I shall first give a quick review of recent joint work with Farzad Fathizadeh and also the related work by Alain Connes and Henri Moscovici on computing the scalar curvature of noncommutative two tori. The local expression for curvature is computed by evaluating the value of the (analytic continuation of the) spectral zeta function $\zeta_a(s) = {Trace}(a |D|^{-s})$ at $s=0$ as a linear functional in $a \in C^{\infty}({T}_{\theta}^2)$. I shall then report on ongoing work and a formula for the scalar curvature of a family of noncommutative 4-tori, and extensions of the Gauss-Bonnet theorem. Conformal perturbations of higher dimensional noncommutative tori are in general non-Kaehler and this makes the analysis more difficult.
|
| Marcelo Laca (UVIC) | Phase transition for algebraic integers and ideal classes.
(abstract)
We study the equilibrium states of a dynamical system
based on the C*-algebra of the affine semigroup of the ring of integers
in an algebraic number field. Above a certain critical inverse temperature
there is a phase transition associated with ideal classes, which coalesce
at the critical temperature, in a phenomenon that depends on the
asymptotic distribution of the ideal classes of integral ideals. This is
joint work with J. Cuntz and C. Deninger.
|
| Jeff Lagarias
(Michigan) | The Lerch zeta function and noncommutative geometry.
(abstract)
The Lerch zeta function is a three-variable generalization of the Riemann zeta function. We describe algebraic and analytic structures associated to this function. These include a multi-valued analytic continuation of this function in three complex variables, and induced action of these on functional equations and differential equations satisfied by this function. We introduce two-variable Hecke operators for which the Lerch zeta function is an eigenfunction. We note some analogies to topics in non-commutative geometry. (This is joint work with W.-C. Winnie Li).
|
| Giovanni Landi
(Trieste) | On the geometry of quantum projective spaces.
(abstract)
We report on several results on the geometry of quantum projective spaces. In particular: explicit generators for the
K-theory and the K-homology; anti-selfdual connections with computation of corresponding ‘classical’ characteristic
classes (via Fredholm modules), as well as ‘quantum’ characteristic classes via equivariant K-theory and q-indices.
Time allowing we shall also report on complex and holomorphic structures on these spaces.
|
| Hanfeng Li
(Buffalo) | Entropy and $L^2$-torsion
(abstract)
Given any countable discrete group $G$ and any countable left module $M$ of the integral group ring of $G$,
one may consider the natural action of $G$ on the Pontryagin dual of $M$. Under suitable conditions, the entropy of
this action and the $L^2$-torsion of $M$ are defined. I will discuss the relation between the entropy and the $L^2$-torsion,
and indicate how this confirms the conjecture of Wolfgang Luck that any nontrivial amenable group admitting a finite
classifying space has trivial $L^2$-torsion. This is joint work with Andreas Thom.
|
| Francois
Ledrappier (Notre Dame) | Entropy rigidity of non positively curved symmetric spaces.
(abstract)
TBA.
|
| Jianya Liu (Shandong) | The Mobius function and distal flows.
(abstract)
Recently Peter Sarnak proposed a conjecture concerning
the orthogonality of the Mobius function to zero-entropy flows.
Distal flows form a subclass of zero-entropy flows.
In this talk I will report a joint work with Sarnak that the
conjecture is true for various distal flows.
|
| Sheng-Chi Liu (TAMU) | Growth and nonvanishing of restricted Siegel modular forms arising as Saito-Kurokawa lifts.
(abstract)
A Siegel modular form, when restricted to a certain natural submanifold of Siegel's upper half space, is essentially a classical elliptic modular form in each of two variables. In the special case that the Siegel form is a Saito-Kurokawa lift, Ichino gave a formula which explicitly decomposes this restricted Siegel form into elliptic modular forms; the formula involves central values of Rankin-Selberg L-functions on GL3 x GL2. I will talk about some recent results on the average behavior of these L-functions which give some information on how the restricted Siegel form usually behaves. This is joint work with Matt Young.
|
| Oliver Lorscheid (Wuppertal) | Chevalley groups over F1.
(abstract)
One of the main motivations of F1-geometry is to explain the analogy
between Chevalley groups over finite fields and the combinatorial geometry
of their Weyl groups. The idea to explain the combinatorial part as a
geometry over an elusive field, which is nowadays called F1, “the field
with one element”, goes back to a paper of Jaqcues Tits from 1956.
This idea entered the flourishing area of F1-geometry as the formula
G(F1)=W, and many authors contributed to it. However, the viewpoint that
the F1-rational points of a Chevalley group G should equal its Weyl group W
faces a certain functorial problem.
In this talk, I will explain this problem and how to solve by the means of
blueprints. In particular, we will introduce the so-called Tits-category,
which contains models of Chevalley groups over F1 that yield the Weyl
groups in a functorial way.
|
| Stephen Miller
(Rutgers) | Rallis's representations and string theory
(Joint work with Michael Green and Pierre Vanhove).
(abstract)
Steve Rallis and coauthors wrote some pioneering papers on automorphic realizations of "small" infinite dimensional representations of exceptional Lie groups. In some recent papers (arxiv:1004.0163 and arxiv:1111.2983) we showed that some of these appear in a sequence of correction terms to Einstein's 4-graviton scattering amplitude in the low energy limit. As a result, we obtain generalizations to larger automorphic representations, which settles some conjectures of James Arthur.
|
| Hugh Montgomery (Michigan) | Geometric properties of the Riemann zeta function.
(abstract)
We consider the level curves $|\zeta(s)|= c$ and curves of steepest ascent of the zeta function.
For $c < 1$ we find that all connected components are compact. For $c = 1$ there is a parameterized infinite
family of noncompact connected components. For each $c > 1$ there is precisely one noncompact
component, and infinitely many compact components. Parts of this work are joint with
John G. Thompson, while other parts are joint with Steven M. Gonek.
|
|
| Sergey Neshveyev
(Oslo) | Bost-Connes systems associated with function fields.
(abstract)
With a global function field $K$ with constant field $F_q$, a finite
set $S$ of primes in $K$ and an abelian extension
$L$ of $K$, finite or infinite, we associate a
$C^\ast$ - dynamical system. The systems, or at least
their underlying groupoids, defined earlier by Jacob
using the ideal action on Drinfeld modules and by
Consani-Marcolli using commensurability of
$K$-lattices are isomorphic to particular cases of
our construction. We prove a phase transition
theorem for our systems and show that the unique
$KMS_\beta$ - state for every $0<\beta \le 1$ gives
rise to an ITPFI-factor of type $III_{q^{-\beta
n}}$, where $n$ is the degree of the algebraic
closure of $F_q$ in $L$. Therefore for $n=+\infty$
we get a factor of type $III_0$. Its flow of weights
is a scaled suspension flow of the translation by
the Frobenius element on $Gal(\bar F_q/F_q)$. (Joint work with Simen Rustad)
|
| Ryszard Nest
(Copenhagen) | Index theorem and torsion of n-tuples of Toeplitz operators.
(abstract)
TBA.
|
| Hervé Oyono-Oyono (Metz) | Some generalisations of the gap-labeling.
(abstract)
In this talk, we explain how the gap-labeling problem for
quasicrystals, stated by J. Bellissard, can be generalised to provide
topological invariants for pinwheel tilings and Penrose hyperbolic tilings.
|
| Sorin Popa (UCLA) | On the classification of II$_1$ factors arising from free groups acting on spaces. (abstract)
A famous problem of Murray and von Neumann (1943)
asks whether the II$_1$ factors $L(\Bbb F_n)$ associated with free groups with $n$ generators, $\Bbb F_n$,
are non-isomorphic for distinct $n$'s. While this problem is still open,
its ``group measure space'' version, showing that
the II$_1$ factors $L^\infty(X)\rtimes \Bbb F_n$ arising from
free ergodic probability measure preserving actions $\Bbb F_n\curvearrowright X$ are non-isomoprphic
for $n= 2, 3, ...$, independently of the actions, has been recently settled by Stefaan Vaes and myself. I will comment on this result, as
well as on some related problems.
|
| Cristian Popescu
(UCSD) | An equivariant main conjecture in Iwasawa theory and applications.
(abstract)
TBA.
|
| Florin Radulescu
(Roma) | Free groups, $Sl(2,z[1/p])$'s, Countable, measure preserving equivalence
relations and Ramanujan Petersson Conjectures.
(abstract)
TBA.
|
| Bahram Rangipour
(UNB) | Transverse characteristic classes in codimension 2.
(abstract)
TBA.
|
| Marc Rieffel
(Berkeley) | Challenges of Leibniz seminorms.
(abstract)
This is a preliminary report on some investigations I am making
concerning seminorms on C*-algebras that satisfy the
Leibniz inequality. These seminorms are central to the concept of non-commutative (or "C*-" or "quantum") metric spaces. That concept might be of interest for ergodic theory.
I will describe some questions that are of importance for my projects,
and I will present some examples that I have found
that exhibit interesting behavior. The questions I
will discuss involve the relations between a Leibniz
seminorm and the C*-norm, and are already
interesting for finite-dimensional C*-algebras. Thus
I will primarily restrict my attention to that case
so as to avoid the (interesting) analytical
questions of the infinite-dimensional case.
|
| Klaus Schmidt
(Vienna) | Entropy and the growth rate of periodic points for principal algebraic $Z^d$-actions.
(abstract)
Principal algebraic $Z^d$-actions are defined by a Laurent polynomial
$f$ in $d$ commuting variables. Such an action is
expansive precisely when the complex variety of the
polynomial $f$ contains no points whose coordinates
all have absolute value $1$. For such expansive
actions it is known that the limit for the
logarithmic growth rate of periodic points exists
and is equal to the entropy of the action. If the
action is nonexpansive, the connection between
entropy and the growth rate of periodic points is
much more complicated and involves some subtle (and
partly unresolved) diophantine problems concerning
the proximity of torsion points to the variety of
$f$.
This talk is based on joint work with Doug Lind an Evgeny Verbitskiy.
|
| Christian Skau (NTNU) | Toeplitz flows and their K-theory.
(abstract)
A Toeplitz sequence on a finite alphabet is defined in terms of
arithmetic progressions. The resulting shift space
is called a Toeplitz flow. These flows exhibit some
remarkable properties. By considering the C*-crossed
product(which is a non-commutative object)
associated to a Toeplitz flow one is led to an
invariant of K-theoretic nature.This invariant,which
can be computed and interpreted in purely dynamical
terms,is an invariant for (strong) orbit
equivalence. We will show how this K-theoretic
invariant has an attractive presentation, both in
terms of so-called Bratteli diagrams,and also in
terms of the associated dimension groups. We will present several examples to illustrate this.
|
| Ralf Spatzier (Michigan) | Global Rigidity of Higher Rank Actions on Tori and Nilmanifolds.
(abstract)
Consider an action of $Z^k$, $k>1$, on a torus or nilmanifold. Suppose its linearization is irreducible and that the action itself has enough Anosov elements (at least one in each so-called Weyl chamber). Then the action is smoothly conjugate to its linearization.
|
| Nicolas Templier (Princeton) | Harmonic families of automorphic representations.
(abstract)
We consider certain families of automorphic representations of reductive groups over number fields. We establish a general Plancherel equidistribution theorem for the local parameters of these families. The theorem is strong enough to shed new light on the Katz-Sarnak heuristics on zeros of L-functions and eigenvalues of random matrices. In particular we find an interesting criterion for the conjectured symmetry type. This is joint work with Sug-Woo Shin.
|
| Dan Thompson (PSU) | A criterion for topological entropy to decrease under normalised Ricci flow.
(abstract)
TBA.
|
| Jacob Tsimerman (Harvard) | Results of Ax-Lindemann type and the Andre-Oort Conjecture.
(abstract)
Due to recent work of Pila, there has emerged a new approach to the Andre-Oort conjecture that relies on 2 ingredients: obtaining lower bounds for the size of Galois orbits of CM points and a generalization of the Ax-Lindemann theorem to Shimura varieties, coined Ax-Lindemann-Weierstrass. We will explain this picture and present the results thus far, culminating in the unconditional proof of Andre-Oort for $A_{g,1}, g\leq 6.$
This is joint work with J. Pila.
|
| Mariusz Wodzicki
(Berkeley) | Infinitesimal Geometry
and Arithmetic.
(abstract)
TBA.
|
| Bora Yalkinoglu (Max-Planck) | On p-adic Bost-Connes systems and Lubin-Tate theory.
(abstract)
p-adic Bost-Connes systems have been introduced quite recently by Connes and Consani. Their construction is based on the (maximal) unramified
extension of $\mathbb{Q}_p$, and, as for the classical BC-system, the dynamics of the corresponding system is induced by a natural Frobenius action.
In our talk we will explain how to define (complementary) p-adic BC-systems based on the (maximal) ramified extension of $\mathbb{Q}_p$ (here Lubin-Tate theory enters the picture). The first obstacle is that in this case the dynamics induced by the natural Frobenius action is trivial, so that one has to find a replacement for the dynamics. Luckily, there is a natural solution to this problem, going actually back to a construction of Euler, which shows that the natural dynamics is induced by a certain differential operator.
We will show that, when equipped with this new dynamics, p-adic zeta functions appear naturally in form of the partition function of our systems.
Further, we will indicate how our construction (and especially the dynamics) is related to p-adic Hodge and (conjecturally) to p-adic soliton theory.
|
| Yi-Jun Yao (Fudan) | On K-theory of some noncommutative orbifolds.
(abstract)
In this talk(based on our joint work with Xiang Tang), we plan to discuss the computation of K-theory groups
of some crossed-product C*-algebras, by using an equivariant version of Rieffel's
strict deformation.
|
| Matthew Young
(TAMU) | Mass distribution for Hecke eigenforms.
(abstract)
I will discuss joint work with V. Blomer and R. Khan on the $L^4$ norm of Hecke eigenforms of large weight as well as some related restriction estimates for submanifolds. As an application we obtain subconvexity bounds for certain triple product $L$-functions where all three factors vary.
|
| Guoliang Yu (Vanderbilt) | Group actions and operator K-theory.
(abstract)
Crossed product C*-algebras play the role of ``quotient spaces'' in noncommutative geometry.
In this talk, I will discuss a geometric condition which would allow us to compute K-theory of crossed product C*-algebras. This is joint work with Erik Guentner and Rufus Willett.
|
| Steve Zelditch (Northwestern) | Quantum ergodicity on compact hyperbolic surfaces $X$
and distributions on the space $\mathcal{G}(X)$ of its geodesics.
(abstract)
The non-commutative space of this talk is the space of geodesics $\mathcal{G}_{\Gamma}$ of
a compact hyperbolic surface, $D/\Gamma$ i.e. the quotient of the space $\mathcal{G}$ of geodesics of the Poincare
disc by the action of a co-compact discrete group. In quantum ergodicity, one studies
quantum limits of Wigner distributions of eigenfunctions of the Laplacian. I will
discuss two types of Gamma-invariant distributions on
the space $\mathcal{G}$ which are equivalent to the Wigner distributions.
One is the ``Patterson-Sullivan' distributions introduced in a joint work with N. Anantharaman.
The second is new and is a positive measure constructed from the ideal Helgason) boundary
values of the eigenfunctions. I prove a new QE result
which relates quantum limits of automorphic forms where both the weight and eigenvalue vary. The distributions
also raise questions about the ergodic theory of the $\Gamma$ action on $\mathcal{G}$.
|
| Michael Zieve (Michigan) | Factorization of maps between curves, and
noncommutative analogues of Lüroth's theorem.
(abstract)
There is a well-developed theory of factorization
of maps between curves with a totally ramified point, including
results about gcd's and lcm's and (in some situations) even
results describing the extent of nonuniqueness of prime factorization.
After reviewing this, I will present recent work with Brian Wyman
developing analogous results for the functional decomposition of
certain monic polynomials over a noncommutative ring. Although the
same types of conclusions are true in both settings, it is not
known whether there is a common generalization. Further, while
the proofs in the case of curves rely on Lüroth's theorem about
fields between $K$ and $K(x)$, the results in the noncommutative
setting may be viewed as partial analogues of Lüroth's theorem.
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