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.