Subjects: Fano manifolds and their mirror dual pencils, maximally mutable Laurent polynomials, the first and second structure connections, gamma conjectures and the gamma class, drops in the images of monodromies in Landau-Ginzburg models, congruence Galois reps and congruence sheaves, Bloch-Kato classes and numerator/denominator of L-values, GSp_4 - Galois representations and paramodular non-lifts, hypergeometric families and weighted projective spaces, hypergeometric motives and their L-functions, extensions of Picard-Fuchs equations and subcritical L-values.

Talks on Saturday and Sunday (exact times tba.):

* A. Corti: Mirror symmetry for orbifold del Pezzo surfaces

* T. Coates: An update on classification of Fano 3-folds and 4-folds

* A. Kasprzyk: Maximal mutable Laurent polynomials.

The aim would be to explain the idea of maximal mutable Laurent polynomials in two dimensions, where everything is reasonably clear, and then report on results attempting to generalise this to higher dimensions (in particular dimension three, and a sampling in dimension four).

* Andew Strangeway: Fano bundles and Apéry-like numbers.

A Fano bundle is a vector bundle whose projectivisation is a Fano manifold. I will discuss a technique to calculate the quantum cohomology of the projectivisation of such a bundle on projective space. In certain examples the J-function (a way of packaging the information of quantum cohomology) is observed to contain Apéry-like numbers (sequences that rapidly converge to a 'numerically interesting’ limit) . I will propose a geometric method to obtain high order (>5) differential operators that conjecturally have Apéry-like solutions. The relation, if any, of these high order operators to lower order operators is unclear, and the limits of some of these sequences are as yet undetermined. This talk will contain work in progress and comments are welcome.

* H. Iritani: tba.

* Mark Watkins: Hypergeometric motives, particularly at t=1 A family of hypergeometric motives can be determined by a coprime pair of products of cyclotomic polynomials, each product having the same degree $d$. For each $t$ not 0, 1, or infinity, one can then (following Katz) associate a motive $M_t$ to this data. At each prime $p$, one obtains an Euler factor by considering the monodromy action (over $Q_p$) on the solution space of the associated hypergeometric differential equation. For good primes, this can be explicitly calculated by a trace formula (again see Katz). There are three kinds of bad primes. The wild primes are those which divide one of the indices of the cyclotomic polynomials. The tame primes are with $v_p(t)$ nonzero, and the "multiplicative" primes are those with $v_p(t-1)$ nonzero. These last are the easiest to understand, in the complex case they correspond to the fact that there are $(d-1)$ independent holomorphic solutions about $t=1$. Indeed, one can consider the formal motive $M_1$ at $t=1$ -- as an example, in the case of $\Phi_5$ and $\Phi_1^{^4}$, one obtains the weight 4 modular form of level 25.

We report specifically on a "database" of $t=1$ degenerations, including all choices (about 1500 total) of cyclotomic data up through degree 6. In each case, we are able to numerically determine the wild Euler factors and verify the functional equation of the L-function to (say) 15 digits of precision. Of additional interest is the fact that approximately half the relevant (odd motivic weight) examples of even parity appear to have analytic rank 2.

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