We explain a classical formula of Waldspurger and its generalization to
higher rank unitary groups. We will also discuss the relative trace formula
approach of Jacquet--Rallis.

We explain our attempt to formulate a refinement of Manin's conjecture using
birational geometry, e.g., the minimal model program and the boundedness of log
Fano varieties. This is joint work with Brian Lehmann and Yuri Tschinkel.

Joint work with Ronald Cramer, Chris Peikert and Oded Regev

A handful of recent cryptographic proposals rely on the conjectured hardness of
the following problem in the ring of integers of a cyclotomic number field:
given a basis of an ideal that is guaranteed to have a ``rather short''
generator, find such a generator. Recently, Bernstein and
Campbell-Groves-Shepherd have sketched potential attacks against this problem.
Most notably, the latter authors claimed a *quantum polynomial-time* algorithm.
(Alternatively, replacing the quantum component with an algorithm of Biasse and
Fieker would yield a classical subexponential-time algorithm.) A key claim of
Campbell etal. is that one step of their algorithm---namely, decoding the
*log-unit* lattice of the ring to recover a short generator from an arbitrary
one---is efficient (whereas the standard approach on general lattices takes
exponential time). However, very few convincing details were provided to
substantiate this claim.

In this work, we clarify the situation by giving a rigorous proof that the
log-unit lattice is indeed efficiently decodable, in cyclotomics of
prime-power index. The proof consists of two main technical contributions: the
first is a geometrical analysis, using tools from analytic number theory, of the
standard generators of the group of cyclotomic units. The second shows that for
a wide class of typical distributions of the short generator, a standard
lattice-decoding algorithm can recover it, given any generator.

To any knot one can assign an algebraic curve together with a distinguished
point. Roughly speaking it parametrizes the representations of the fundamental
group into SL(2). I will explain how to find the curves in two different ways
and how to extract interesting numbers from it. For example the figure eight
knot gives rise to 15A8 in the Cremona table.

We show how the Arthur-Selberg trace formula can be used to study the Newton stratification of
Shimura varieties. In the first half of the talk we will recall the definition of the Newton
stratification, and give some examples. In the second half of the talk we explain the (idea of)
the Kottwitz point counting formula, and how it can be compared with the Arthur-Selberg trace
formula. This comparison can then be used to answer geometric questions, such as non-emptiness
and dimensions of Newton strata.

Elliptic curves over fields have been studied in various guises for thousands of
years. Around 1965, André Néron studied how 1-parameter families of elliptic
curves degenerate, and constructed canonical (or Néron) models for them. These
models have many wonderful properties, some of which we will outline in the
first part of the talk.

We will then investigate what happens in families with more parameters (more
formally, over base-schemes of higher dimensions). In this situation things
become considerably more complicated. We will give a reasonably complete
classification of when Néron models do and do not exist, and describe various
applications.

As the culmination of work of many mathematicians, Yuan has obtained a very
general equidistribution result for small points in arithmetic varieties.
Roughly speaking Yuan's theorem states that given a « very » small generic
sequence of points, with respect to a positive hermitian line bundle, the
associated sequence of measures converges weakly to the measure associated
to the hermitian line bundle. Here very small means that the height of the
points converges to the lower bound of the essential minimum given by Zhang
inequalities. The existence of a very small generic sequence is a strong
condition on the arithmetic variety because it implies that the essential
minimum attains its lower bound. We will say that a sequence is small if the
height of the points converges to the essential minimum. By definition every
arithmetic variety contains small generic sequences.

We show that for toric line bundles on toric varieties arithmetic Yuan's
theorem can be splitted in two parts.

A) Given a small generic sequence of points, with respect to a positive
hermitian line bundle, the associated sequence of measures converges weakly
to a measure.

B) If the sequence is very small, the limit measure agrees with the measure
associated to the hermitian line bundle.

In this talk we discuss a bound for the height of the pre-image of a special
point on a Shimura variety in a fundamental set of the associated Hermitian
symmetric domain. This bound is polynomial in terms of standard invariants
associated with the corresponding Mumford-Tate torus and generalises an earlier
result of Pila and Tsimerman for the moduli space of principally polarised
abelian varieties of dimension g. The result constitutes the final ingredient
needed to complete a new proof of the André-Oort conjecture under the
Generalised Riemann Hypothesis, using a strategy of Pila and Zannier. This is
joint work with Martin Orr (University College London).

Severi published in 1921 a result on nodal curves on the projective plane. He
showed that the set of nodal curves of degree d with exactly k nodes on the
projective plane has an obvious embedding in a projective space. He also stated
that the projective scheme obtained by taking the Zariski closure is non-empty
and of codimension k. His proofs of the first and third statement were correct,
but it was not until 1985 that a correct proof for the irreducibility was found.

The result on the dimension shows that given enough points on a surface, the
number of nodal curves through these points does not depend on the choice of
points. These number, depending on d and k, were computed recursively by
Caporaso and Harris in 1998. Eventually it were Fomin and Mikhalkin in 2009 who
showed that these Severi degrees are polynomials in d, for k small in comparison
to d.

Even before this proof, mathematicians have been trying to generalize these
results and conjectures to arbitrary smooth projective surfaces. One will need
to replace degree d curves by a line bundle and in this way one can still define
the Severi variety and degree. Göttsche conjectured in 1997 that under an
assumption on this line bundle, generalizing ampleness, the Severi degree should
be a polynomial in topological invariants of the surface and the line bundle.
Tzeng was able to prove this theorem by using algebraic cobordism: an
equivalence relation on classes of surfaces with a line bundle. The crux of the
proof lies in the fact that any equivalence class of algebraic cobordism is
uniquely determined by the topological invariants, occurring in Göttsche's
conjecture.

A popular subject in Diophantine geometry is the existence of rational
points in varieties, a subject which includes topics such as the Hasse
principle and the Brauer-Manin obstruction. On the opposite end of the
spectrum there are varieties whose rational points are Zariski dense and
where one is interested in more refined properties of their distribution.
Manin's conjecture attempts to describe the distribution of points in
specific subvarieties via the use of heights. It has never been established
for a single smooth cubic surface although it is known for particular del
Pezzo surfaces of any degree d>3. In this talk we will prove the one sided
estimate predicted by Manin's conjecture for a family of smooth cubic
surfaces containing a rational line.

If an elliptic curve over a number field admits a rational p-isogeny over every
completion, does it necessarily admit one over the number field itself? As
Sutherland has shown, the answer is 'No' in general. I will discuss joint work
with John Cremona going further into this question, giving some interesting
counterexamples. There'll also be an (ultimately failed though nevertheless
worthwhile) attempt, with Alex Bartel, at proving isomorphisms between Jacobians
of modular curves of genus 3 and level 13.

Let R be a local ring which is complete for a discrete valuation, with fraction
field K and algebraically closed residue field k. Let X be a smooth, proper
K-variety. Nicaise conjectured that the rational volume of X is equal to the
trace of the tame monodromy operator on the l-adic cohomology of X, if X
satisfies a cohomological tameness condition. We prove his conjecture for a
large class of varieties satisfying this condition: those with logarithmic good
reduction.

Let π be a cuspidal automorphic representation of GL(n) over a number field F.
Let p be a rational prime, and let X be the set of all finite-order Hecke
characters of F obtained by composition with the norm homomorphism from F to ℚ
with some Dirichlet character of p-power conductor. I will explain why one
expects (and sometimes knows) that if π is cohomological, then for all but
finitely many characters ξ in the set X, the central value L(1/2,π×ξ) does not
vanish. I will also present some preliminary results, as well as explain how the
general conjecture can be reduced to certain estimates with hyper-Kloosterman
sums of p-power modulus.

The problem of lifting Galois covers of curves to characteristic zero asks
roughly which Galois covers of smooth projective curves defined over a field of
positive characteristic arise as reduction of Galois covers of the same group
in characteristic zero. This question has been studied in several fashions
since the '60s and is still a source of open problems and ongoing research.
The last big breakthrough was the proof of the so-called "Oort conjecture",
achieved in 2012 by Pop, and built on the work of Obus-Wewers. The aim of this
talk is to explain the techniques used in proving main results on the topic, to
discuss new open problems and to introduce a new viewpoint on the subject,
related to Berkovich theory of non-Archimedean analytic spaces.

In 1979, G.Belyi proved his theorem characterizing, analytically, all compact
Riemann surfaces which can be defined over a number field. Using that as a
springboard, A.Grothendieck defined dessins d'enfants (children's drawings) as
certain graphs embedded on Riemann surfaces which essentially are a
combinatorial description of Belyi's characterization. As such they admit an
action of the absolute Galois group ie they are an instance of the so called
geometric Galois actions. Dessins were used to study Riemann surfaces and to
provide combinatorial invariants of the Galois action. In this talk we will
outline Belyi's thoerem, introduce dessins and some of their properties along
with some connections with different areas.

We first briefly discuss the standard method for solving
polynomial systems, using Groebner bases. Then we will discuss the
first fall degree conjecture, which predicts the complexity of such
computations. We present some evidence in favor and some evidence
against this conjecture. We will propose a new concept, the last fall
degree, which can be used to give complexity of Groebner basis
computations as well, without any heuristics. Finally, we discuss how
one can prove that HFE (hidden field equations) is insecure by
bounding the complexity of Groebner basis algorithms using the last
fall degree.

Up to isomorphism, there are approximately $2 q^3$ curves of genus 2 over the
finite field ${\bf F}_q$. How many of these curves have nonsimple Jacobian
varieties? Jeff Achter (Colorado State University) and I have recently shown
that up to explicit powers of $\log q$, the number of genus-2 curves with
nonsimple Jacobians is $q^{2.5}$, so that roughly speaking there is one chance
out of $\sqrt q$ that a randomly-chosen curve has nonsimple Jacobian. I will
outline part of the proof of this result, and say something about the problem
that motivated Achter to ask about this question.

The André-Oort Conjecture (AOC) states that the irreducible components of the
Zariski closure of a set of special points in a Shimura variety are special
subvarieties. Here, a special variety means an irreducible component of the
image of a sub-Shimura variety by a Hecke correspondence. The AOC is an analogue
of the classical Manin-Mumford conjecture on the distribution of torsion points
in abelian varieties. In fact, both conjectures are special instances of the
far-reaching Zilber-Pink conjecture(s).

I will present a rarely known approach to the AOC that goes back to Yves André
himself: Before the recent model-theoretic proofs of the AOC in certain cases by
the Pila-Wilkie-Zannier approach, André presented in 1998 the first
unconditional (i.e., not depending on GRH) proof of the AOC in a non-trivial
case, namely, a product of two modular curves. In my talk, I discuss several
results in the style of André's method, allowing to compute all special points
on a non-special curve in a product of two modular curves.

These results are effective - in stark contrast to those obtainable by the
Pila-Wilkie-Zannier approach - and have sometimes the additional advantage of
being uniform both in the degrees of the curve and its definition field. For
example, this allows to show that there are actually no two singular moduli
(= j-invariants associated with complex CM-elliptic curves) x and y satisfying
x+y=1.

It is a fundamental problem in representation theory to classify the
finitely generated modules for a given finite dimensional algebra
over a field. One approach to this problem is to give conditions for
one module to be "simpler" than another in some sense. For example,
because of the Jordan-Hölder Theorem, one can take the
semi-simplification of a module. In this talk I will introduce several
partial orders that can be put on the isomorphism classes of finitely
generated modules. I will then concentrate on one particular partial
order coming from taking closures of orbits under the action of an
algebraic group. I will discuss two different settings in which this occurs,
one affine and the other projective. At the end I will raise some questions
and give some answers in a low-dimensional setting. This is joint work
with T. Chinburg and B. Huisgen-Zimmermann.

The classical case of Iwasawa theory has to do
with how quickly the p-parts of the ideal class groups of
number fields grow in certain towers of number fields.
I will discuss the connection of these growth rates to Chern
classes. By the end of the talk I'll describe some joint work
over imaginary quadratic fields with F. Bleher, R. Greenberg, M. Kakde,
G. Pappas, R Sharifi and M. Taylor.

Let F be a non-discreet non-Archimedean local field. In this talk we
will be concerned with irreducible smooth representations of GL_4(O_F)
which determine the inertial support of irreducible smooth
representations of GL_4(F). There representations are called typical
representations. We will obtain a complete classification of typical
representations in terms of Bushnell-Kutzko theory of types for the
case where the cardinality of the residue field is greater than 3. The
case of GL_4(F) will serve as an example for showing techniques used
for the general case.

I will discuss some work in progress, with Rob Kurinczuk, attempting
to understand the category of smooth l-modular representations of
p-adic classical groups, in particular towards a classification of
cuspidal/supercuspidal representations, the notion of "endo-class" and
implications for decompositions of the category.

This is a report on joint work in progress with C. J. Bushnell. Let F
be a non-Archimedean locally compact field, F^sep a separable
algebraic closure of, G_F its Galois goup over F. The Langlands
correspondence for GL(n,F) associates to a smooth irreducible
representation r of G_F, of dimension n, a smooth irreducible cuspidal
representation pi(r) of GL(n,F). When n=1 that is given by class field
theory, and the higher ramification theorem of class field theory says
that for two characters r_1 and r_2 of G_F, the characters pi(r_1) and
pi(r_2) of GL(1,F)=F* have the same restriction to the higher unit
subgroup U_F^i of F* if and only if r_1 and r_2 have the same
restriction to the higher ramification subgroup G_F^i of G_F. We
determine what happens if n>1; that introduces Herbrand functions of a
new type.

At the beginning of the XX century Borel introduced the notion of normal number
and showed that almost every real number is normal. Later he conjectured that
every irrational algebraic number is normal, but this result is still out of
reach. After giving the necessary background we will use the p-adic Subspace
Theorem to prove a transcendence criterion for numbers with a special kind of
b-ary expansion. Time permitting, we will then use this criterion to show that
the complexity function of the b-ary expansion of every irrational algebraic
number grows more than linearly, and that every irrational automatic number is
transcendental.

This work is basically the analysis of the problem of under which conditions
pullbacks commute with colimits, but from an homotopical point of view. Hence
the role of ordinary pullback and colimit functor in the classical theory is
played by homotopy pullback and homotopy colimit functors. The framework is that
of a general category of simplicial presheaves, i.e. one of the form sSet^I, for
some small category I.

Arakelov intersection theory arises as a way to generalize the intersection
theory on smooth projective surfaces over a field to an intersection theory on
arithmetic surfaces. In 1987 Paul Hriljac gave, for each element in the
Tate-Shafarevich group of a semi-stable elliptic curve over a number field, an
upper bound for the discriminant of a splitting field of this element in terms
of its order. We will look at Hriljac's proof for this upper bound using
Arakelov intersection theory, and generalize the result of Hriljac to include
all torsors of this elliptic curve.