28 files changed, 343 insertions, 0 deletions

diff --git a/src/talks/cube-2min-slides.pdf b/src/talks/cube-2min-slides.pdf
new file mode 100644
index 0000000..f332eb8
--- /dev/null
+++ b/src/talks/cube-2min-slides.pdf
diff --git a/src/talks/cubo-cagliari.pdf b/src/talks/cubo-cagliari.pdf
new file mode 100644
index 0000000..2efdee3
--- /dev/null
+++ b/src/talks/cubo-cagliari.pdf
diff --git a/src/talks/division-groningen.pdf b/src/talks/division-groningen.pdf
new file mode 100644
index 0000000..c2ad7f6
--- /dev/null
+++ b/src/talks/division-groningen.pdf
diff --git a/src/talks/division-leiden.pdf b/src/talks/division-leiden.pdf
new file mode 100644
index 0000000..a4566c2
--- /dev/null
+++ b/src/talks/division-leiden.pdf
diff --git a/src/talks/group-cube.pdf b/src/talks/group-cube.pdf
new file mode 100644
index 0000000..191b9db
--- /dev/null
+++ b/src/talks/group-cube.pdf
diff --git a/src/talks/kummer-tarrach.pdf b/src/talks/kummer-tarrach.pdf
new file mode 100644
index 0000000..7dbe643
--- /dev/null
+++ b/src/talks/kummer-tarrach.pdf
diff --git a/src/talks/linuxday2019.pdf b/src/talks/linuxday2019.pdf
new file mode 100644
index 0000000..7a18aa1
--- /dev/null
+++ b/src/talks/linuxday2019.pdf
diff --git a/src/talks/notes-bilbao.pdf b/src/talks/notes-bilbao.pdf
new file mode 100644
index 0000000..5c0500c
--- /dev/null
+++ b/src/talks/notes-bilbao.pdf
diff --git a/src/talks/notes-division-modules.pdf b/src/talks/notes-division-modules.pdf
new file mode 100644
index 0000000..0883299
--- /dev/null
+++ b/src/talks/notes-division-modules.pdf
diff --git a/src/talks/notes-etale.pdf b/src/talks/notes-etale.pdf
new file mode 100644
index 0000000..6b31e10
--- /dev/null
+++ b/src/talks/notes-etale.pdf
diff --git a/src/talks/notes-injectivity.pdf b/src/talks/notes-injectivity.pdf
new file mode 100644
index 0000000..7dd3da8
--- /dev/null
+++ b/src/talks/notes-injectivity.pdf
diff --git a/src/talks/notes-katz.pdf b/src/talks/notes-katz.pdf
new file mode 100644
index 0000000..c248b71
--- /dev/null
+++ b/src/talks/notes-katz.pdf
diff --git a/src/talks/notes-kummerec.pdf b/src/talks/notes-kummerec.pdf
new file mode 100644
index 0000000..9a91d78
--- /dev/null
+++ b/src/talks/notes-kummerec.pdf
diff --git a/src/talks/script-kummer-kwg.txt b/src/talks/script-kummer-kwg.txt
new file mode 100644
index 0000000..8759ee7
--- /dev/null
+++ b/src/talks/script-kummer-kwg.txt
@@ -0,0 +1,224 @@
+--- Kummer theory ---
+0:50
+
+Kummer theory is the study of field extensions generated by the n-th roots
+of elements of a base field, such as the rational numbers.  Taking all
+n-th roots ensures that such an extension is Galois and that it contains
+the cyclotomic field: indeed, one can write the n-th roots of unity as
+ratios of different n-th roots of the same element.
+
+To study these extensions it is convenient to start not just with a
+set of elements, but with a multiplicative subgroup of the base field.
+For example, one can take the group generated by one element.  This does
+not change anything on the field-theoretic side: the extensions we are
+considering do not change. But it makes the "purely algebraic" side of
+things more convenient: the group sqrt[n]{A} now is group which contains
+A and the roots of unity.
+
+
+--- Kummer theory for algebraic groups ---
+1:00
+
+"Kummer theory for algebraic groups" is a similar, more general
+problem. If we take G to be a commutative algebraic group over a number
+field K, we can take a subgroup of the K-rational points of G and you
+consider the n-division points of this group, which is an analogue of
+the group of n-th roots in the previous case.
+
+If you add the coordinates of these points to your base field you obtain
+what a field extension which has properties remarkably similar to the
+classical Kummer extensions: it is Galois over K and it contains the
+n-torsion field of G, an analogue of the cyclotomic field generated by
+the torsion points of G.
+
+This is a generalization of the classical case, because if you take G
+to be the multiplicative group you obtain exactly that case.
+
+These field extensions are the kind of objects that I am studying.
+
+
+--- Results for elliptic curves ---
+1:50
+
+So, what do we want to know about these kind of field extensions?  One of
+the things we care about is estimating, or computing, their degrees.
+This is because they have applications in other areas of number theory:
+when studying problems related to Artin's primitive root conjecture it
+can happen that the density of certain set of primes can be expressed
+in terms of the degrees of Kummer extensions.
+
+The degree of a Kummer extension over the torsion field is always
+between a certain power of n and the same power of n times a constant.
+This power of n is for example 2 in the case of elliptic curves and
+groups of points generated by one non-torsion element. So they cannot
+be much smaller than the maximum.
+
+In recent years there has been effort in making these results effective.
+In a recent work with Lombardo we were able to quantify this constant, or
+a possible value for it, in terms of computable properties of the curve
+and of the chosen point, for curves without complex multiplication.  In
+particular, one of these properties are the p-adic Galois representation
+associated with the curve. Over Q we even have an explicit and uniform
+estimate for such a constant.
+
+In our work we had problems when the curve had non-trivial endomorphisms
+defined over the base field, so CM curves.  However Abtien Javan Peykar, a
+student of Lenstra, managed to get similar results for CM curves only. So,
+how did he manage?
+
+
+--- Endomorphism rings ---
+0:55
+
+The problem Lombardo and I had was caused by considering the A and its
+division groups as abelian groups. Instead, Javan Peykar decided to
+take them as modules over the endomorphism ring of the curve, an order
+in a quadratic imaginary field - and even with some extra technical
+limitations, such as considering only maximal orders.
+
+So my idea was: if we can do the same over a *general* ring, regardless
+of it being Z or an order in a number field or anything else, maybe we
+can build a general framework to study these division groups, or rather
+division modules, and then apply all of this to do Kummer theory over
+other classes of algebraic groups.
+
+So this is what I did.
+
+
+--- Division modules ---
+1:00
+
+The first part is understanding what "division in modules is", starting
+from the "denominator": what do we divide by?
+
+As is often the case in commutative algebra, it is convenient to use,
+instead of the elements of the ring, ideals of the ring. So for M
+contained in N we define the I-division module of M inside N to be the
+set of elements of N that multiplied by I end up inside M. This is a
+classical definition that is found in some commutative algebra books.
+
+We also want to consider infinite unions of such division modules. For
+example we might want to work with the set of all division points of
+our subgroup of rational points. In our first example, all n-th roots
+of a certain number.
+
+To do this, we introduce the concept of "ideal filter", which like a
+fiter in set theory but for ideals.
+
+
+--- Ideal filters ---
+0:20
+
+In practice I always want to divide by one of these two families of
+ideals: either the one generated by all positive integers or the one
+generated by powers of a given prime.
+
+These are the main example of ideal filters that we should keep in mind,
+but I will develop my theory in general.
+
+
+--- J-injectivity ---
+1:25
+
+The set of all division points is a divisible abelian group. But over
+a general ring this divisibility property can be awkward to work with,
+and we prefer to use injectivity, which is equivalent to divisibility
+over the ring Z.
+
+A module is called injective when maps to it can be lifted along
+injective morphisms. We call it instead J-injective when maps to it can
+be lifted along certain injective morphisms, namely those such that the
+codomain coincides with the module of J-division points of the image.
+This definition captures the concept of "dividing only by J".
+
+This a nice and simple generalization of a classical concepts, but it has
+some noteworthy properties. First of all it is a true generalization:
+taking J to be the set of all right ideals of R it becomes equivalent
+to injectivity. And for example one can use it to extend the definition
+of p-divisible abelian group: over Z p-divisibility is equivalent to
+p^\infty-injectivity, where p^\infty is the ideal filter I introduced
+in the previous slide.
+
+One might say that this definition highlights the connection between
+injectivity and divisibility better than the classical one does.
+
+
+--- (J,T)-extensions ---
+1:20
+
+The last ingredient to complete our algebraic theory is the torsion. We
+are building all this theory of division modules abstractly, in a way
+independent of the agebraic group G that we started with.  But when we do
+this and we consider division modules, there is no way for this objects
+to know that they are supposed to live in some elliptic curve rather than
+in the multiplicative group or in some higher-dimensional abelian variety.
+
+We need some extra structure. We need to fix a torsion and J-injective
+module T that plays the role of the torsion subgroup of G.
+
+Then we consider only those extensions of a base module M that consist
+of division points and whose torsion embeds into T.
+
+These objects form a category with many nice properties, that strongly
+resembles the category of field extensions of a fixed field.
+
+We also have an analogue of an algebraic closure, that plays the role
+of the "set of all division points" that we have mentioned. This can be
+constructed as a "J-hull", which is the analogue of the injective hull,
+or injective envelope, for our generalization of injectivity.
+
+With this category we can establish many properties of division modules,
+and study their automorphisms.
+
+
+--- Galois representations ---
+0:55
+
+And finally, how do I use this whole theory to study my number theoretical
+problems?  I can consider the Galois group of my Kummer extension,
+say the one generated by all division points, and it embeds into the
+automorphism group of this maximal (J,T)-extension.
+
+This automorphism group fits into a short exact sequence.
+
+Then the standard short exact sequence of Galois theory embeds into this
+and we obtain this commutative diagram of groups, with exact rows.  This
+sequence is the main tool to study Kummer theory for algebraic groups.
+
+Finding an explicit lower bound for the degrees I talked about amounts
+to proving an explicit open image theorem for this "representation"
+on the left-hand side.
+
+In short, this diagram is the key to derive number-theoretic results
+from certain key properties of the group.
+
+
+--- New results ---
+0:50
+
+So, what kind of new results were we able to obtain with this technical
+tools in our hands?
+
+First of all, it was easy to unify the CM and non-CM cases and
+show that one does not need to separate the two cases, except for
+studying some specific properties of the curves related to their Galois
+representations. Once you have have them, you can plug in any elliptic
+curve with any endomorphism ring into our general framework and you
+obtain the (already known) results. This also completes the CM case,
+that had some missing pieces due to technical difficulties.
+
+More importantly, in my opinion, we have now a better understanding of
+these objects.
+
+You see, when studying a problem cases by case is like you are trying
+to find your way in a forest step by step. With this general framework
+we have a way-better overview of the landscape we are moving in.
+
+Lastly, the generality of this theory allows one to obtain some results
+for higher-dimensional abelian varieties. This is work in progress,
+but we already have results for some classes of varieties. There some
+technical things to work out related to understanding the torsion subgroup
+as a module over the endomorphism ring, but I am optimistic that we will
+work this out.
+
+
diff --git a/src/talks/slides-alggroupsfieldext.pdf b/src/talks/slides-alggroupsfieldext.pdf
new file mode 100644
index 0000000..2b48adb
--- /dev/null
+++ b/src/talks/slides-alggroupsfieldext.pdf
diff --git a/src/talks/slides-ecm.pdf b/src/talks/slides-ecm.pdf
new file mode 100644
index 0000000..9305984
--- /dev/null
+++ b/src/talks/slides-ecm.pdf
diff --git a/src/talks/slides-fieldextec.pdf b/src/talks/slides-fieldextec.pdf
new file mode 100644
index 0000000..6b54ff7
--- /dev/null
+++ b/src/talks/slides-fieldextec.pdf
diff --git a/src/talks/slides-groupcohomec.pdf b/src/talks/slides-groupcohomec.pdf
new file mode 100644
index 0000000..72a2a3c
--- /dev/null
+++ b/src/talks/slides-groupcohomec.pdf
diff --git a/src/talks/slides-journey-kummer.pdf b/src/talks/slides-journey-kummer.pdf
new file mode 100644
index 0000000..9edf804
--- /dev/null
+++ b/src/talks/slides-journey-kummer.pdf
diff --git a/src/talks/slides-kummec-barcelona.pdf b/src/talks/slides-kummec-barcelona.pdf
new file mode 100644
index 0000000..7af439d
--- /dev/null
+++ b/src/talks/slides-kummec-barcelona.pdf
diff --git a/src/talks/slides-kummec-diamant.pdf b/src/talks/slides-kummec-diamant.pdf
new file mode 100644
index 0000000..d66f002
--- /dev/null
+++ b/src/talks/slides-kummec-diamant.pdf
diff --git a/src/talks/slides-kummer-diamant.pdf b/src/talks/slides-kummer-diamant.pdf
new file mode 100644
index 0000000..2160eac
--- /dev/null
+++ b/src/talks/slides-kummer-diamant.pdf
diff --git a/src/talks/slides-kummer-final.pdf b/src/talks/slides-kummer-final.pdf
new file mode 100644
index 0000000..e24e01a
--- /dev/null
+++ b/src/talks/slides-kummer-final.pdf
diff --git a/src/talks/slides-kummer-kwg.pdf b/src/talks/slides-kummer-kwg.pdf
new file mode 100644
index 0000000..3940f4a
--- /dev/null
+++ b/src/talks/slides-kummer-kwg.pdf
diff --git a/src/talks/slides-kummerdegrees.pdf b/src/talks/slides-kummerdegrees.pdf
new file mode 100644
index 0000000..2b13012
--- /dev/null
+++ b/src/talks/slides-kummerdegrees.pdf
diff --git a/src/talks/slides-local-global.pdf b/src/talks/slides-local-global.pdf
new file mode 100644
index 0000000..32a84fd
--- /dev/null
+++ b/src/talks/slides-local-global.pdf
diff --git a/src/talks/slides-tronto-bristol.pdf b/src/talks/slides-tronto-bristol.pdf
new file mode 100644
index 0000000..4ece0b7
--- /dev/null
+++ b/src/talks/slides-tronto-bristol.pdf
diff --git a/src/talks/talks.md b/src/talks/talks.md
new file mode 100644
index 0000000..b7cea5b
--- /dev/null
+++ b/src/talks/talks.md
@@ -0,0 +1,119 @@
+# Talks
+
+## Cubing
+
+* *Cubo di Rubik e Speedcubing (ITA)*.
+  Invited talk at Università di Cagliari, January 2020.
+  [Slides: [pdf, 2.0Mb](cubo-cagliari.pdf)]
+
+* *The Rubik's cube group structure in 2 minutes*.
+  Speed talk at the AAN Symposium, May 2019.
+  [Slides: [pdf, 1.2Mb](cube-2min-slides.pdf)]
+
+## Math
+
+* *Kummer theory for elliptic curves*.
+  Short talk for Tarrach Prize 2023, March 2023.
+  [Slides: [pdf, 288Kb](kummer-tarrach.pdf)]
+
+* *Kummer theory for commutative algebraic groups*.
+  Seminar at the university of Leiden, September 2022.
+  [Slides: [pdf, 1.2Mb](slides-kummer-final.pdf)]
+
+* *My journey in Kummer theory*.
+  Leiden-Luxembourg Number Theory PhD Days 2022, July 5-6.
+  [Slides: [pdf, 521Kb](slides-journey-kummer.pdf)] 
+
+* *A tour of group theory with our companion cube*.
+  University of Luxembourg PhD seminar, May 2022.
+  [Slides: [pdf, 1.1Mb](group-cube.pdf)]
+
+* *Kummer theory via (J,T)-extensions*.
+  University of Leiden algebra seminar, May 2022;
+  similar to *A category of division modules*, see below.
+
+* *Kummer theory for algebraic groups*.
+  Nederlands Matematisch Congres, April 2022;
+  repeated at the DIAMANT symposium, April 2022.
+  [Notes (NMC): [pdf, 215Kb](slides-kummer-kwg.pdf);
+   Script (NMC): [txt, 9Kb](script-kummer-kwg.txt);
+   Notes (DIAMANT, longer version): [pdf, 223Kb](slides-kummer-diamant.pdf)]
+
+* *Division in modules*.
+  University of Leiden algebra seminar, April 2022;
+  similar to *A generalization of injective modules* and
+  *Division in modules and Kummer theory*, see below.
+  [Notes: [pdf, 245Kb](division-leiden.pdf)]
+
+* *Division in modules and Kummer theory*.
+  Invited talk at the University of Groningen, February 2022.
+  [Slides: [pdf, 249Kb](division-groningen.pdf)]
+
+* *A category of division modules*.
+  University of Luxembourg Number Theory seminar, November 2021.
+  [Notes: [pdf, 270Kb](notes-division-modules.pdf)]
+
+* *Introduction to étale cohomology*.
+  University of Luxembourg seminar series on perverse sheaves, October 2021.
+  [Notes: [pdf, 237Kb](notes-etale.pdf)]
+
+* *A generalization of injective modules*.
+  University of Luxembourg Number Theory seminar, October 2021.
+  [Notes: [pdf, 259Kb](notes-injectivity.pdf)]
+
+* *Integer factorization and elliptic curves*.
+  University of Leiden Bachelor seminar (guest talk), April 2021.
+  [Slides: [pdf, 393Kb](slides-ecm.pdf)]
+
+* *Group cohomology and elliptic curves*.
+  Invited talk at Number Theory Online conference, February 2021.
+  [Slides: [pdf, 210Kb](slides-groupcohomec.pdf);
+   [Video](https://vimeo.com/526814236/1a640285c0?embedded=true&source=video_title&owner=47245911)]
+
+* *Kummer theory for algebraic groups*.
+  Invited talk at the University of Bristol (online), October 2020.
+  [Slides: [pdf, 239Kb](slides-tronto-bristol.pdf)]
+
+* *Algebraic groups and field extensions*.
+  University of Luxembourg PhD seminar (online), April 2020.
+  [Slides: [pdf, 280Kb](slides-alggroupsfieldext.pdf)]
+
+* *Field extensions and elliptic curves*.
+  University of Luxembourg PhD Day, October 2019.
+  [Slides: [pdf, 6.0Mb](slides-fieldextec.pdf)]
+
+* *Kummer theory for elliptic curves*.
+  University of Leiden algebra seminar, September 2019;
+  repeated with slides at the DIAMANT symposium, November 2019 and
+  at Seminari de Teoria de Nombres de Barcelona, February 2020.
+  [Notes: [pdf, 181Kb](notes-kummerec.pdf);
+   Slides DIAMANT: [pdf, 271Kb](slides-kummec-diamant.pdf);
+   Slides Barcelona: [pdf, 290Kb](slides-kummec-barcelona.pdf)].
+
+* *Divisibility of points in algebraic groups*.
+  Invited talk at the University of the Basque Country, May 2019.
+  [Notes: [pdf, 231Kb](notes-bilbao.pdf)]
+
+* *Kummer theory for number fields*.
+  Fifth symposium of the Roman Number Theory association, April 2019;
+  repeated at the Algant Alumni symposium in Benasque, May 2019.
+  [Slides: [pdf, 291Kb](slides-kummerdegrees.pdf)]
+
+* *The local-global principle*.
+  University of Luxembourg interdisciplinary seminar, March 2019.
+  [Slides: [pdf, 598Kb](slides-local-global.pdf)]
+
+* *Local-global principle for torsion*.
+  University of Luxembourg Number Theory seminar, November 2018.
+  [Notes: [pdf, 208Kb](notes-katz.pdf)]
+
+## Software
+
+* *L'importanza del software libero nella ricerca (ITA)*.
+  Linux day in Feltre, October 2019.
+  [Slides: [pdf, 804Kb](linuxday2019.pdf);
+   [Video](https://www.youtube.com/watch?v=FB0YIYNp3rw)]
+
+* *Linux from scratch (ITA)*
+  Linux day in Sedico, October 2012.
+  [Video](https://www.youtube.com/watch?v=h7xW7iq-ApA).