ALGANT SELECTION AND PHD CANDIDATES MEETING
REGENSBURG 2015
Schedule
Saturday, February 28th 2015
Presentations by doctoral candidates
Sunday, March 1st 2015
Presentations by doctoral candidates
Algant-DOC meeting
Monday, March 2nd 2015
Selection meeting
Tuesday, March 3rd 2015
Selection meeting
Presentations schedule
Saturday, February 28th 2015
Place: M 104
10.05 – 10.35: Iuliana Ciocanea Teodorescu “Approximating the Jacobson radical of a finite ring”
When trying to answer algorithmic questions about rings and modules, it is often convenient to reduce the problem
at hand to the semisimple case, and then ``lift" the result. However, this approach requires the computation of the
Jacobson radical of the ring, which we cannot efficiently do in general.
Motivated by this, we aim to design a deterministic polynomial time algorithm that, given a finite ring, computes a
good approximation for the Jacobson radical, that is, a two-sided nilpotent ideal that for all practical purposes
behaves like the Jacobson radical and such that when we quotient the ring by it, we are left with a ring that is
``almost" semisimple.
10.40 – 11.10: Nikola Damjanovic “The family of stable curves”
Let f:X \rightarrow Y be a semistable family of curves over a curve Y and smooth over U=Y \backslash S, where S is
a finite set of points. Eckart Viehweg and Kang Zuo proved that for any invertible subsheaf \mathcal{H} of
f_{*}\omega_{X/Y}^{\nu} and every \nu\in \mathbb{N}^{*} we have the inequality:
\deg\mathcal{H}\leq\frac{\nu}{2}\deg(\Omega_{Y}^{1}(\log S)) .
If equality holds for some \mathcal{H}, the family is expected to be very special. Indeed, when \nu=1, Möller proved
that Y is a Teichmüller curve. Our aim is to explore properties of the family when \nu \geq2.
11.15 – 11.45: Francesca Bergamaschi “Stratifications of Hilbert Modular Varieties with
0(p)-Level Structure – Abstract”
Hilbert modular varieties can be seen as a generalization of the modular curve X0(p); they can be roughly described
as spaces parametrizing abelian varieties under the action of a totally real number field. Their relation to number
theory is strong, as Hilbert modular forms can be seen as sections of line bundles over them. In particular, the many
interesting phenomena arising in positive characteristic provide us with sophisticated tools to study their geometry.
In this presentation, we describe the space Mp of Hilbert modular varieties with 0(p)-level structure in
characteristic a prime p. By classifying the p-torsion of points we define stratifications of Mp, that is, we write the
space as a disjoint union of locally closed subspaces. By constructing a local model and by describing the equicharacteristic deformation of points through Zink's theory of displays, we describe the properties of strata.
11.50 – 12.20: Martin Djukanovic “Heights and curves with split Jacobians”
A hyperelliptic curve of genus 2 can have a Jacobian that is isogenous to a product of two elliptic curves. Under
certain conditions, one can relate the Néron–Tate height and the Faltings height of these two abelian surfaces which
allows one to compare the statements of the Lang–Silverman conjecture in these two cases.
LUNCH (M 103)
14.00 – 14.30: Dino Festi “Potential density of rational points on K3 surfaces”
K3 surfaces can be viewed as 2-dimensional analogues of elliptic curves.
Therefore, as for elliptic curves, much interest is devoted to the study of their rational points.
Despite the efforts, little is known in this topic. In 2000, Bogomolov and Tschinkel proved that on many K3 surfaces
over number fields, rational points are potentially dense, that is, there is a number field K such that the set of Kpoints is Zariski dense. In this talk we will see which cases are not covered and we will expose an idea about how to
deal with them.
14.35 – 15.05: Albert Gunawan “Gauss composition on spheres”
Gauss's theorem on the sum of 3 squares relates the number of primitive integer points on the sphere of radius the
square root of n with the class number of some quadratic imaginary order. In 2011, Edixhoven sketched a different
proof of Gauss's theorem by using an approach from arithmetic geometry. He used the action of the special
orthogonal group on the sphere and gives a bijection between the set of SO_3(Z)-orbits of such points, if nonempty, with the set of isomorphism classes of torsors under the stabilizer group. This set of torsors is a group,
isomorphic to the group of isomorphism classes of projective rank one modules over the ring Z[1/2,\sqrt{-n}], that
gives an affine space structure on the set of SO_3(Z)-orbits on the sphere. In this talk, I will give an explicit method
and an example of how this parallelogram law works.
15.10 – 15.40: Guhanvenkat Harikumar “Darmon cycles and overconvergent Shintani lifting”
Darmon cycles are arithmetic objects defined by Victor Rotger and Marco Seveso and can be thought of as a higher
weight analogue of Stark-Heegner points. We consider a p-adic family of half-integral weight modular forms under
the overconvergent Shintani lifting and relate the (derivative of) fourier coefficients of this family to Darmon cycles.
BREAK (M 103)
16.10 – 16.40: Pinar Kilicer “The class number one problem for curves of genus 2 and 3”
The Gauss class number one problem for imaginary quadratic fields is equivalent to finding all elliptic curves over the
rationals with complex multiplication. I will quickly explain the relation between the class number one problem and
the elliptic curves over the rationals. Then I will give the analogue of the class number one problem for curves of
genus 2 and 3 with CM and sketch its solution.
16.45 – 17.15: Djordo Milovic “Divisibility by 16 of class numbers in certain families of quadratic number
fields”
The density of primes p\equiv 3\pmod{4} such that the class number of \mathbb{Q}(\sqrt{-2p}$ is divisible by 2^k is
conjectured to be 2^{-k} for all positive integers k. The conjecture has been resolved for k\leq 3 by the Chebotarev
Density Theorem. We use methods of Friedlander and Iwaniec to prove the conjecture for k = 4.
19.30: Dinner in restaurant „Leerer Beutel“, Bertoldstr. 9
Sunday, March 1st 2015
Place: M 104
9.30 – 10.00: Diego Mirandola “On code products”
Given two (linear) codes C;D with the same length, their product CD is defined as the linear span of all
componentwise products xy, with x 2 C; y 2 D. If C = D, C2 is also called the square of the code C. Our interest in
this notion is motivated by a number of cryptographic applications. We study two different problems.
First, we show that the square of a random code fills the whole space with high probability: more precisely, if n and
k denote the length and the dimension of the code respectively, and the difference k(k+1)=2 n is at least linear in k,
this probability is exponentially close to 1. Combinatorial arguments about bilinear and quadratic forms over finite
fields play an important role. This is a joint work with I. Cascudo, R. Cramer and G. Zemor.
Second, we characterize Product-MDS pairs, i.e. pairs of codes C;D whose product has maximum-possible minimum
distance. We prove in particular, for C = D, that if C2 has minimum distance at least 2, and (C;C) is a Product-MDS
pair, then either C is a Reed-Solomon code, or C is a direct sum of self-dual codes. In passing we establish codingtheory analogues of classical theorems of additive combinatorics. This is a joint work with G. Zemor.
10.05 – 10.35: Maxim Mornev “Tannakian Properties Of Frobenius-Invariant Sheaves”
Let q > 1 be a prime power, Fq a finite field, #Fq = q. Let X be a locally noetherian scheme over Fq, F : X ! X the
absolute q-Frobenius, and _ an Fq-algebra. Let O_X denote the structure sheaf of Spec_ _Fq X. Definition ([2], [1]). A
coherent O_F;X-module is a pair (M; _M), whereMis a co-herent O_X-module, and _M: F_M!Ma morphism of O_Xmodules. A morphism of coherent O_F;X-modules _: (M; _M) ! (N; _N) is a morphism _: M ! N of O_X -modules, such
that _ _ _M = _N _ F__. A coherent O_F;X-module (M; _M) is called unit if _M is an isomorphism. Coherent unit O_F;Xmodules are the Frobenius-invariant sheaves from the title of this talk. They come up naturally in the study of
Drinfeld shtukas [4], and related objects [1], [2], [3]. From the point of view of shtukas an especially interesting case
is when the Krull dimension of _ is positive. In such a situation the category of coherent unit O_F;X-modules has no
chance of being Tannakian since its tensorproduct is not exact. Still, it retains some Tannakian properties. We will
show that if _ is a localization of an Fq-algebra of finite type then the category of coherent unit O_F;X-modules is
abelian, and comes equipped with exact fiber functors, which detect acyclicity of complexes. Furthermore, if the
base scheme X is connected then the fiber functors are faithful, and conservative.
References
[1] G. B ockle, R. Pink. Cohomological Theory of Crystals over Function Fields, EMS Tracts in
Mathematics 9 (2013)
[2] M. Emerton, M. Kisin. The Riemann-Hilbert correspondence for unit F-crystals, Asterisque
293 (2004)
[3] N. Katz, p-adic Properties of Modular Forms, Modular Functions of One Variable, Lecture
Notes in Math. 350, 69{190, Springer, 1973.
[4] L. La_orgue, Chtoucas de Drinfeld et conjecture de Ramanujan
10.40 – 11.10: Carolina Rivera “Canonical height pairings via biextensions”
I explain the construction of a pairing A(K) x B(K) -> Y; where Y is an abelian group, and A and B are abelian varieties
over a global field K, using a biextension of (A;B) by Gm. In particular, we can take B to be the dual of A and the
biextension as the Poincar_e biextension. The problem is now to generalize this construction to a 1-motive
M = [L -> G] over K, which is a complex of group schemes with L a lattice and G a semiabelian variety.
11.15 – 11.45: Ziyang Gao
11.50 – 12.20: Gabriele Spini “Linear Secret Sharing Schemes from Error Correcting Codes and Universal
Hash Functions”
We present a novel method for constructing linear secret sharing schemes (LSSS) from linear error correcting codes
and linear universal hash functions in a blackbox way. The main advantage of this new construction is that the
privacy threshold of the resulting scheme becomes essentially independent of the code we use, only depending on
its rate; this allows us to fully harness the algorithmic properties of recent code constructions such as efficient
encoding and decoding or efficient list decoding. We present two constructions implementing this paradigm: a
linear ramp secret sharing scheme with linear time sharing and reconstructions algorithms, allowing secrets of size
linear in n; and an efficient robust secret sharing scheme for a fraction of dishonest players smaller then 1/2-e
(where e>0 is an arbitrary constant), having optimal share size.
Joint work with R. Cramer, I. Damgaard, S. Fehr and N. M. Doettling.
LUNCH (M 103)
14.00 – 14.30: Qijun Yan
14.35 – 15.05: Chloe Martindale “Isogeny Graphs in Genus 2“
Traditionally, an isogeny graph is a graph whose vertices are elliptic curves and whose edges are isogenies of a given
degree between them. Arranging these graphs in a way that uses the complex multiplication structure of the elliptic
curves (in the cases where this can be done) makes the connected components of the graph into what have been
labelled ‘isogeny volcanoes’. These graphs have nice consequences in number theory and in cryptography, which
has motivated us to analyse the graph structure for curves higher genus. I will talk about the structure of the graph
for genus 2 curves, and about some of the mathematics that goes into understanding this structure, such as
computing models for surfaces coming from Hilbert modular forms.
BREAK (M103)
15.30 – 16.00: Yan Zhao “dg-enhancement of Grothendieck's six functors”
Grothendieck's six functors are fundamental geometric operations between derived categories of sheaves over
topological spaces, schemes, stacks, etc. A full description of derived categories requires enhancements of these
categories over chain complexes. However, it is well known that a derived functor may not admit an enhancement
to a dg-functor and the enhancement, if it exists, may not be unique. In this talk, I will define dg-enhancements of
Grothendieck's six functors for sheaves over ringed spaces. This construction is based on an idea of Valery Lunts and
Olaf Schnürer and is functorial in the dg-category of small dg-categories.
:
After the end of the presentations, at about 16.00 we plan to have a meeting of ALGANT representatives to discuss
the present and the future of the ALGANT-DOC programme.
Selection meeting schedule
Place: M 104, M 201
Monday, March 2nd 2015 EMMC SELECTION MEETING / CONSORTIUM AND ADMINISTRATIVE MEETING
M 104: 9.00 Welcome and consortium meeting (all academic and administrative staff)
10.30 EMMC candidate’s selection (all academic staff and administrative coordinator), list of applications, overview
of assessment results
12.30 LUNCH (MENSA)
14.00 EMMC candidate’s selection (second session) M 104
14.00 Administrative meeting (all administrative staff) M 201
18.00 End of day sessions
20.00 Dinner in restaurant “Bischofshof am Dom”, Krauterermarkt 3
Tuesday, March 3rd 2015 EMJD SELECTION / FINAL MEETING
9.00 EMJD candidate’s selection (all academic staff + administrative coordinator), list of applications, assessment
results
12.30 LUNCH (MENSA)
14.00 EMJD candidate’s selection (second session), absolute ranking list
15.00 Final meeting (all participants), finalization of the selections: approval of the minutes and absolute ranking list
15.30 End of the sessions
INTERNET:
Access to eduroam is available throughout the mathematics building.