January 24th  Robert Hough (Stony Brook) 
The shape of low degree number fields
abstract : In his thesis, M. Bhargava proved parameterizations and identified local conditions which he used to give asymptotic counts for $S_4$ quartic and quintic number fields, ordered by discriminant. This talk will discuss results in an ongoing project to add detail to Bhargava's work by considering in addition to the field discriminant, the lattice shape of the ring of integers in the canonical embedding, and by giving strong rates with lower order terms in the asymptotics. These results build on earlier work of TaniguchiThorne, BhargavaShankarTsimerman and Bhargava Harron.

January 31st  Jacques Tilouine (Paris 13))

Simplicial deformation rings versus Hida theory
abstract : in a joint work in progress with Eric Urban, given a cohomological cuspidal representation $\pi$ on $GL(N)$ over a CM field, we construct a map from the fundamental group of the simplicial deformation ring of $\overline{\rho}_\pi$ to the dual adjoint Selmer group of $\rho_\pi$. In the case of Hida families, these objects still make sense but we conjecture that the corresponding simplicial deformation ring has no higher homotopy groups, so that our map vanish. We investigate evidences and consequences of this conjecture.

February 7th  Spencer Leslie (Duke)

The endoscopic fundamental lemma for unitary FriedbergJacquet
periods
abstract: I will discuss a theory of endoscopy for certain symmetric
spaces associated to unitary groups, the main result being the
fundamental lemma for the "Lie algebra". This is motivated by the study
of certain periods of automorphic forms on unitary groups with
applications to arithmetic. After explaining where the fundamental lemma
fits into this broader picture, I will describe its proof.

February 14th  Congling Qiu (Princeton)

Perfectoid ordinary Siegel spaces and the TateVoloch conjecture
abstract: Tate and Voloch conjectured that, in an abelian variety over a padic field, torsion points outside a closed subvariety
cannot be padically too close to it. There is a wellknown analogy between torsion points in abelian varieties and CM points in Shimura
varieties. In this talk, I will prove the analog of the TateVoloch conjecture for ordinary CM points in a product of Siegel spaces. I will focus on how the theory of perfectoid spaces is applied.

February 28th  Koji Shimizu (IAS)

A padic monodromy theorem for de Rham local systems
abstract: Every smooth proper algebraic variety over a padic field is expected to have semistable model after passing to a finite extension. This conjecture is open in general, but its analogue for Galois representations, the padic monodromy theorem, is known. In this talk, we will explain a generalization of this theorem to etale local systems on a smooth rigid analytic variety.

March 13th  Yifeng Liu (Yale)

Galois deformation, automatic minimality, and application to Selmer groups (via Zoom)
Abstract: In this talk, we will first summarize the final outcome we (= LiuTianXiaoZhangZhu) obtained on bounding Selmer groups of RankinSelberg motives. Then we will explain how Galois deformation can help to give an explicit computation of certain local Galois cohomology, which is a key step toward the Selmer groups. We will also explain a related question, which has been partially answered by us, on automatic minimality of deformations of automorphic Galois representations.

March 27th  Chandrashekah Khare (UCLA)

Wiles defect for Hecke algebras that are not complete intersections (via Zoom)
Abstract: In his work on modularity theorems, Wiles proved a numericalv criterion for a map of rings R>T to be an isomorphism of complete intersections. In addition to proving modularity theorems, this numerical criterion also implies a connection between the order of a certain Selmer group and a special value of an Lfunction. In this talk I will consider the case of a Hecke algebra acting on the cohomology a Shimura curve associated to a quaternion algebra. In this case, one has an analogous map of rings R>T which is known to be an isomorphism, but in many cases the rings R and T fail to be complete intersections. This means that Wiles's numerical criterion will fail to hold. I will describe a method for precisely computing the extent to which the numerical criterion fails (i.e. the 'Wiles defect") at a newform f which gives rise to an augmentation T > Z_p. The defect turns out to be determined entirely by local information at the primes q dividing the discriminant of the quaternion algebra at which the mod p representationarising from f is ``trivial''. (For instance if f corresponds to a semistable elliptic curve, then the local defect at q is related to the ``tame regulator'' of the Tate period of the elliptic curve at q.)
This is joint work with Gebhard Boeckle and Jeffrey Manning.

April 3rd  Mark Kisin (Harvard)

CANCELLED

April 10th  Andreas Mihatsch (Bonn/MIT) Zoom link

Currents on LubinTate space (via Zoom) Slides
Abstract: We will first give an introduction to the formalism of (p,q)forms and currents on nonarchimedean spaces, as defined by ChambertLoirDucros and GublerKünnemann. We explain how a notion of Greens current can be used to reinterpret intersection numbers on padic formal schemes in their generic fiber; at least in some cases. We finally apply these considerations to an intersection problem of quadratic CMcycles on LubinTate space. Here, all expressions in question can be evaluated neatly at infinite level and we recover an intersection number formula from Qirui Li.

April 17th  Brian Smithling (Maryland)

CANCELLED

April 24th  Ashay Burungale (Caltech)

CANCELLED

May 1st at 3pm
 Samuel Mundy (Columbia) Zoom link

Eisenstein series on G_2 and the SkinnerUrban method for
symmetric cube Galois representations (note the Special time 3pm !)
Abstract: Skinner and Urban have been able to constuct nontrivial
elements in the Selmer group for the Galois representation attached to
a certain type of cuspidal automorphic representation of a unitary
group, under the hypothesis that the Lfunction of this representation
vanishes at its central critical value. This is predicted by the
BlochKato conjecture. We will explain an analogous construction for
the symmetric cube of the Galois representation attached to a cuspidal
eigenform, assuming the same kind of vanishing hypothesis. The method
will involve studying Eisenstein series on the exceptional group G_2
and, in the absense of a G_2 Shimura variety, making a careful
analysis of the representation theory of this group at the archimedean
place.

May 8th  Haruzo Hida (UCLA)

CANCELLED
