Igor Krylov: Local inequalities for cA_n-singularities and applications to birational rigidity
Abstract:
I will discuss the idea of proof of birational rigidity of threefolds and the
importance of local inequalities for their proof. Then I will discuss the birational
rigidity results that follow from local inequalities for cA_n points, in particular
I will talk about birational rigidity of sextic double solids with cA_n-singularities.
At the end of the talk I will talk a bit about the differences in approach that allow
us to get inequalities extended from cA_1-points to cA_n-points.
Yongqi Liang: Hasse principle and Weil restriction
Abstract: We consider the Hasse principle for existence of rational points on
algebraic varieties defined over number fields. There are many possible cohomological
obstructions to the Hasse principle. We compare the etale-Baruer-Manin obstruction
on Weil restrictions of a variety with respect to extensions of number fields, and
prove that they can be naturally identified to each other.
This is a joint work with Yang Cao.
Jinhyung Park: Singularities and syzygies of secant varieties of smooth projective varieties
Abstract: The k-th secant variety of a smooth projective variety embedded in
projective space is the Zariski closure of the union of the planes spanned by
k+1 distinct points. Suppose that the embedding is given by the complete linear
system of a sufficiently positive line bundle. About 10 years ago, Ullery and
Chou-Song proved that the first secant variety has normal Du Bois singularities.
About 5 years ago, in joint work with Ein and Niu, we generalized this result to
higher secant varieties of curves, and showed that the k-th secant varieties of
curves satisfy N_{k+2,p}-property meaning that the minimal free resolution of the
section ring is as simple as possible until the p-th step. In this talk, I report
recent joint work with Choi, Lacini, and Sheridan. We undertake a systematic study of
secant varieties in all dimensions based on geometry of Hilbert schemes of points.
In particular, we determine exactly when secant varieties have extra singularities
not lying in the previous secant varieties, and we extend the previous results on
singularities and syzygies of secant varieties in the case when the Hilbert scheme
of k+1 points is smooth.
Guo Chuan Thiang: Exact fractional quantization
Abstract: I will give an overview on how large-scale index theory appears
in real physical systems. Specifically, the index problem for elliptic
operators on noncompact manifolds can be quantified using renormalized
traces, which are proved to be integer quantized in a universal way.
These trace formulae have precise physical meaning as "exact quantization",
as manifested in quantum Hall effect experiments. We also discovered hidden
rationally quantized trace formulae, whose geometric meaning remains mysterious.
Zhiyu Tian: Kato homology of rationally connected fibrations
Abstract: Kato homology is defined as the homology of the Gersten complex of certain
etale cohomology groups. In this talk I will explain some conjectures about Kato
homology of rationally connected fibrations, which generalize many conjectures by
Voisin and Schreieder, bear a link to some Floer-theoretic conjectures of
Cohen-Jones-Segal, and are central to the understanding of some arithmetic
properties of rationally connected varieties over global fields. I will also
present some new evidences, which are mostly about quotient singularities and
Severi-Brauer schemes.
Jorge Gigante Valcarcel: Algebraic classification of the gravitational field in general metric-affine geometries
Abstract: We present the full algebraic classification of the gravitational field in four-dimensional
general metric-affine geometries. As an immediate application, we determine the algebraic types
of the broadest family of static and spherically symmetric black hole solutions with spin,
dilation and shear charges in Metric-Affine Gravity.
Joonyeong Won: Cylinders in varieties embedded in weighted projective spaces
Abstract: For a projective Fano variety X , an -K-polar cylinder in X is an open ruled affine subset
whose complement is a support of an effective Q-divisor Q-rationally equivalent to anticanonical divisor.
This notion links together affine, birational and Kaehler geometries. We discuss the existence and
non-existence of –K-polar cylinders in varieties embedded in weighted projective spaces.
Bin Zhang: Drinfled associator from locality algebraic geometry
Abstract: Drinfeld associator is an interesting and important object in mathematical physics.
In this talk, we will briefly introduce how it emerges in locality algebraic geometry.
Locality algebraic geometry is developed by combining the locality principle in quantum
field theory with algebraic geometry, with the aim of preparing suitable mathematical
tools for research of mathematical physics. The locality principle is a fundamental
principle in quantum field theory, which we introduce the locality structure to express.
Locality algebraic structures, especially locality commutative algebras, naturally appear
in many mathematical directions. Locality algebraic geometry is a geometric study of
local commutative algebras and we are now studying a rather special locality commutative algebra,
that is, the locality commutative algebra generated by Chen fractions, which provide a geometric
frame work to study Drinfled associator. This talk is based on the joint work with L. Guo and S. Paycha.