The Algebra & Combinatorics Seminar meets (usually) on Fridays from
3–4 pm, in Lecture Hall LH1 of the IISc Mathematics Department.
The organizers are Apoorva Khare and R. Venkatesh.


Abstract.
To any convex integral polygon $N$ is associated a cluster integrable
system that arises from the dimer model on certain bipartite graphs on a
torus. The large scale statistical mechanical properties of the dimer
model are largely determined by an algebraic curve, the spectral curve
$C$ of its Kasteleyn operator $K(x,y)$. The vanishing locus of the
determinant of $K(x,y)$ defines the curve $C$ and coker $K(x,y)$ defines
a line bundle on $C$. We show that this spectral data provides a
birational isomorphism of the dimer integrable system with the Beauville
integrable system related to the toric surface constructed from
$N$.
This is joint work with Alexander Goncharov and Richard Kenyon.




Abstract.
Computing the determinant using the Schur complement of an invertible
minor is wellknown to undergraduates. Perhaps less wellknown is why
this works even when the minor is not invertible. Using this and the
Cayley–Hamilton theorem as illustrative examples, I will gently
explain one "practical" usefulness of Zariski density outside commutative
algebra.




Abstract.
In his seminal paper in 2001, Henri Darmon proposed a systematic
construction of padic points, viz. Stark–Heegner points, on
elliptic curves over the rational numbers. In this talk, I will report on
the construction of padic cohomology classes/cycles in the
Harris–Soudry–Taylor representation associated to a Bianchi
cusp form, building on the ideas of Henri Darmon and Rotger–Seveso.
These local cohomology classes are conjectured to be the restriction of
global cohomology classes in an appropriate Bloch–Kato Selmer group
and have consequences towards the Bloch–Kato–Beilinson
conjecture as well as Gross–Zagier type results. This is based on a
joint work with Chris Williams (Imperial College London).


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2018–19
