
Kurinczuk, Robert; Matringe, Nadir

Blasiak, Jonah; Fomin, Sergey
We review and further develop a general approach to Schur positivity of
symmetric functions based on the machinery of noncommutative Schur functions.
This approach unifies ideas of Assaf, Lam, and Greene and the second author.

Campbell, Justin
We give a new geometric construction of the big projective module in the
principal block of the BGG category O, or rather the corresponding Dmodule on
the flag variety. Namely, given a oneparameter family of nondegenerate additive
characters of the unipotent radical of a Borel subgroup which degenerate to the trivial
character, there is a corresponding oneparameter family of Whittaker sheaves. We
show that the unipotent nearby cycles functor applied to this family yields the big
projective Dmodule.

Rostami, Sean
This paper proves that the nearby cycles complexes on a certain family of
PEL local models are central with respect to the convolution product of sheaves on
the corresponding affine flag varieties. As a corollary, the semisimple trace functions
defined using the action of Frobenius on those nearby cycles complexes are, via the
sheaffunction dictionary, in the centers of the corresponding Iwahori–Hecke algebras.
This is commonly referred to as Kottwitz’s conjecture. The reductive groups associated
with the PEL local models under consideration are unramified unitary similitude
groups with even dimension. The proof follows the method of Haines and Ngô (Compos
Math 133:117–150, 2002). Upon completion of the first...

Ritter, A.; Smith, Ivan

Nadler, David
The aim of this paper is to apply ideas from the study of Legendrian singularities
to a specific example of interest within mirror symmetry. We calculate the
Landau–Ginzburg Amodel with M = C3, W = z1z2z3 in its guise as microlocal
sheaves along the natural singular Lagrangian thimble L = Cone(T 2) ⊂ M. The
description we obtain is immediately equivalent to the Bmodel of the pairofpants
P1\{0, 1,∞} as predicted by mirror symmetry.

Su, Changjian
We give restriction formula for stable basis of the Springer resolution and
generalize it to cotangent bundles of partial flag varieties. By a limiting process, we
get the restriction formula of Schubert varieties.

Jan Nowak, Krzysztof

Mészáros, Karola; Morales, Alejandro H.; Rhoades, Brendon
We introduce the Tesler polytope Tesn(a), whose integer points are the
Tesler matrices of size n with hook sums a1, a2,..., an ∈ Z≥0. We show that Tesn(a)
is a flow polytope and therefore the number of Tesler matrices is counted by the type
An Kostant partition function evaluated at (a1, a2,..., an, −n
i=1 ai). We describe
the faces of this polytope in terms of “Tesler tableaux” and characterize when the
polytope is simple. We prove that the hvector of Tesn(a) when all ai > 0 is given by
the Mahonian numbers and calculate the volume of Tesn(1, 1,..., 1) to be a product
of consecutive Catalan...

Kuznetsov, Alexander; Perry, Alexander
Abstract Given a variety Y with a rectangular Lefschetz decomposition of its derived
category, we consider a degree n cyclic cover X → Y ramified over a divisor Z ⊂ Y .
We construct semiorthogonal decompositions of Db(X) and Db(Z) with distinguished
components AX and AZ and prove the equivariant category of AX (with respect
to an action of the nth roots of unity) admits a semiorthogonal decomposition into
n − 1 copies of AZ . As examples, we consider quartic double solids, Gushel–Mukai
varieties, and cyclic cubic hypersurfaces.

Farah, Ilijas; Hart, Bradd; Rordam, Mikael; Tikuisis, Aaron
The relative commutant A ∩ AU of a strongly selfabsorbing algebra A is
indistinguishable from its ultrapower AU. This applies both to the case when A is the
hyperfinite II1 factor and to the case when it is a strongly selfabsorbing C∗algebra.
In the latter case, we prove analogous results for ∞(A)/c0(A) and reduced powers
corresponding to other filters on N. Examples of algebras with approximately inner flip
and approximately inner halfflip are provided, showing the optimality of our results.
We also prove that strongly selfabsorbing algebras are smoothly classifiable, unlike
the algebras with approximately inner halfflip.

Ayala, David; Francis, John; Tanaka, Hiroyuki
This work forms a foundational study of factorization homology, or topological
chiral homology, at the generality of stratified spaces with tangential structures.
Examples of such factorization homology theories include intersection homology,
compactly supported stratified mapping spaces, and Hochschild homology with
coefficients. Our main theorem characterizes factorization homology theories by a
generalization of the Eilenberg–Steenrod axioms; it can also be viewed as an analogue
of the Baez–Dolan cobordism hypothesis formulated for the observables, rather
than state spaces, of a topological quantum field theory. Using these axioms, we
extend the nonabelian Poincaré duality of Salvatore and Lurie to the setting of strati
fied spaces—this is a nonabelian version of the Poincaré...

Zhang, Ruixiang
We prove the discrete analogue of Kakeya conjecture over Rn. This result
suggests that a (hypothetically) lowdimensional Kakeya set cannot be constructed
directly from discrete configurations. We also prove a generalization which completely
solves the discrete analogue of the Furstenberg set problem in all dimensions. The
main tool of the proof is a theorem of Wongkew (Pac J Math 159:177–184, 2003),
which states that a lowdegree polynomial cannot have its zero set being too dense
inside the unit cube, coupled with Dvirtype polynomial arguments (Dvir in J Am
Math Soc 22(4):1093–1097, 2009). From the viewpoint of the proofs, we also state a
conjecture that is stronger than and...

Zhu, Xinwen
We show that the irregular connection on Gm constructed by Frenkel and
Gross (Ann Math 170–173:1469–1512, 2009) and the one constructed by Heinloth
et al. (Ann Math 177–181:241–310, 2013) are the same, which confirms Conjecture
2.16 of Heinloth et al. (Ann Math 177–181:241–310, 2013).

Achar, Pramod N.; Henderson, Anthony; Juteau, Daniel; Riche, Simon
We study some aspects of modular generalized Springer theory for a complex
reductive group G with coefficients in a field k under the assumption that the
characteristic of k is rather good for G, i.e. is good and does not divide the order
of the component group of the centre of G. We prove a comparison theorem relating
the characteristic generalized Springer correspondence to the characteristic0 version.
We also consider Mautner’s characteristic ‘cleanness conjecture’; we prove
it in some cases; and we deduce several consequences, including a classification of
supercuspidal sheaves and an orthogonal decomposition of the equivariant derived
category of the nilpotent cone.
P.

Proudfoot, Nicholas; Schedler, Travis
Abstract We prove a conjecture of Etingof and the second author for hypertoric
varieties that the Poisson–de Rham homology of a unimodular hypertoric cone is isomorphic
to the de Rham cohomology of its hypertoric resolution. More generally, we
prove that this conjecture holds for an arbitrary conical variety admitting a symplectic
resolution if and only if it holds in degree zero for all normal slices to symplectic
leaves. The Poisson–de Rham homology of a Poisson cone inherits a second grading. In
the hypertoric case, we compute the resulting 2variable Poisson–de Rham–Poincaré
polynomial and prove that it is equal to a specialization of an enrichment of the Tutte
polynomial...

Kapranov, Mikhail; Schiffmann, Olivier; Vasserot, Eric
Let X be a smooth projective curve over a finite field. We describe H,
the full Hall algebra of vector bundles on X, as a Feigin–Odesskii shuffle algebra.
This shuffle algebra corresponds to the scheme S of all cusp eigenforms and to the
rational function of two variables on S coming from the Rankin–Selberg Lfunctions.
This means that the zeroes of these Lfunctions control all the relations in H. The
scheme S is a disjoint union of countably many Gmorbits. In the case when X has a
thetacharacteristic defined over the base field, we embed H into the space of regular
functions on the symmetric powers of...

MacQuarrie, J. W.; Zalesskii, Pavel A.
A second countable virtually free prop group all of whose torsion elements
have finite centralizer is the free prop product of finite pgroups and a free prop
factor. The proof explores a connection between padic representations of finite pgroups
and virtually free prop groups. In order to utilize this connection, we first
prove a version of a remarkable theorem of A. Weiss for infinitely generated profinite
modules that allows us to detect freeness of profinite modules. The proof now proceeds
using techniques developed in the combinatorial theory of profinite groups. Using an
HNNextension, we embed our group into a semidirect product F K of a free prop
group...

Furusho, Hidekazu; Komori, Yasushi; Matsumoto, Kohji; Tsumura, Hirofumi
We construct padic multiple Lfunctions in several variables, which are
generalizations of the classical Kubota–Leopoldt padic Lfunctions, by using a specific
padic measure. Our construction is from the padic analytic side of view, and
we establish various fundamental properties of these functions. (a) Evaluation at nonpositive
integers: We establish their intimate connection with the complex multiple
zetafunctions by showing that the special values of the padic multiple Lfunctions at
nonpositive integers are expressed by the twisted multiple Bernoulli numbers, which
are the special values of the complex multiple zetafunctions at nonpositive integers.
(b) Multiple Kummer congruences: We extend Kummer congruences for Bernoulli
numbers to congruences for the twisted multiple...

Ross, Dustin
We prove the crepant resolution conjecture for Donaldson–Thomas invariants
of toric Calabi–Yau 3orbifolds with transverse Asingularities.