| Thursday, September 6, 2018 | Organizational Meeting |
| Thursday, September 20, 2018 | Taylor Dupuy, Mochizuki's Inequality and the ABC Conjecture |
| Thursday, October 4, 2018 | Anton Hilado, Elliptic Curves and the abc Conjecture |
| Thursday, October 18, 2018 | Taylor Dupuy, Log Volume Computations |
| Thursday, November 1, 2018 |
Anton Hilado, Anabelian Interpretation of Additive Haar Measure Spaces on p-adic Fields via Local Class Field Theory |
| Thursday, November 15, 2018 |
Taylor Dupuy, More Log Volume Computations |
| Thursday, November 29, 2018 |
Lloyd Simons |
Thursday, September 6, 2018,
12:45-2 p.m. Lord House
Organizational Meeting
Thursday, September 20, 2018,
3-4:30 p.m. Lafayette L307
Taylor Dupuy, Mochizuki's Inequality and the ABC Conjecture
Mochizuki's approach to the ABC conjecture is to 1) prove an inequality
which implies the Szpiro
inequality for elliptic curves under certain technical hypotheses
called "initial theta data" 2) show that these technical restrictions
don't matter. The aim of this talk is to explain exactly what step 1 is
all about. Roughly, for an elliptic curve in "initial theta data",
Mochizuki's inequality says that the size of one region (encoding one
side of Szpiro) is less than the size of a "blurry" region (encoding the
other side of Szpiro). How this blurryness occurs and how to break it
down is what gives rise to an inequality strong enough to imply Szpiro
under the technical hypothesis.
I will explain what these regions are and how they relate to Szpiro
explicitly. In particular we will discuss "indeterminacies", "q-pilots",
"theta-pilots", and "initial theta data". Later in the semester we will
discuss the anabelian
constructions that go into the "blurry construction". This talk is
supposed to set up future talks down the road. Much of this project of
making these inequalities explicit is joint work with Anton Hilado.
Thursday, October 4, 2018,
3-4:30 p.m. Lafayette L307
Anton Hilado, Elliptic Curves and the abc Conjecture
In this talk we state the famous "abc conjecture" of Masser and
Oesterle, and explain how it can be formulated as a statement involving
important quantities related to elliptic curves (Szpiro's conjecture).
We give an introduction to Weierstrass equations, reduction types, and
the conductor and minimal discriminant of an elliptic curve, which are
all needed to state Szpiro's conjecture. We also show how the abc
conjecture is related to Fermat's Last Theorem, and introduce the Frey
curve, which was used to prove the latter, and relate Szpiro's
conjecture to the abc conjecture.
Slides
Thursday, October 18, 2018,
3-4:30 p.m. Lafayette L307
Taylor Dupuy, Log Volume Computations
We are going to continue discussing Mochizuki's inequality. In
particular we will discuss the indeterminacies Ind1,Ind2,Ind3 and start
in on the log-volume computations which give rise to a version of
Szpiro's inequality for elliptic curves sitting in initial theta data.
Thursday, November 1, 2018,
3-4:30 p.m. Lafayette L307
Anton Hilado, Anabelian Interpretation of Additive Haar Measure Spaces on p-adic Fields via Local Class Field Theory
We give a very basic introduction to the ideas of anabelian geometry and give an explicit example of how it works by constructing ("interpreting") the additive Haar measure on a p-adic field K given the absolute Galois group of K.
Using only this group and results from local class field theory, we define a topological abelian group isomorphic to (K,+), with several important quantities associated to it, we define another topological abelian group isomorphic to the real numbers, and construct a set function from the former to the latter satisfying the axioms of a Haar measure with normalization.
Thursday, November 15, 2018,
3-4:30 p.m. Lafayette L307
Taylor Dupuy, More Log Volume Computations
We will perform computations similar to the computations in IUT4 using Mochizuki's Inquality (Corollary 3.12 of IUT3) and the definitions of the indeterminacies therein to give a Szpiro-type inequality for Elliptic Curves in initial theta data (Theorem 1.10 of IUT4).
Thursday, November 29, 2018,
3-4:30 p.m. Lafayette L307
Lloyd Simons, TBA
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