AGNT Ical Atom

I will begin with an introduction to algebraic K-theory, highlighting its connections to number theory and algebraic geometry, with a focus on descent phenomena. Next, I will provide an overview of chromatic homotopy theory and its relationship to K-theory. I will then discuss the redshift conjecture and several significant advances from recent years. Finally, I will present joint work with Carmeli, Schlank and Yanovski on the compatibility of K-theory with chromatic analogues of cyclotomic extensions. I will explain how this result contributed to the recent refutation of Ravenel’s long-standing telescope conjecture.

I will begin with a tour of classical ideas in geometric representation theory including Kazhdan-Lusztig polynomials and Beilinson-Bernstein localization. Then I will delve into a logical continuation of these ideas in the work of Bezrukavnikov on exotic t-structures. Finally I will explain an application of these ideas to finding explicit character formulas for affine Lie algebras in our joint work with Andrei Ionov and Vasily Krylov.

I will describe various results, some old and some new, on the structure of the groups of points of an abelian variety over a finite field. The talk will focus on the case of varieties of large dimension over a fixed finite field. In this regime, the Weil bounds allow for the possibility of the exponent of the group staying bounded as the dimension grows. I will explain that at least in the case of Jacobians this cannot be the case. Part of the talk is based on recent joint work with Daniel Keliher.

In the talk, I will discuss the Turner-Turaev formalism for unoriented Topological Quantum Field Theory (TQFT). Building upon this formalism, I will introduce an analogous version for (d+1)-dimensional TQFT for pro-p Poincare duality groups. In the case of d = 1, this enables us to study cobordisms and TQFTs for both the maximal pro-p quotient of absolute Galois groups of p-adic fields and pi_1(X)^p, the pro-p completions of fundamental groups of surfaces. This generalisation gives a framework for arithmetic TQFTs and strengthens the analogies within arithmetic topology, which relates p-adic fields to surfaces (oriented mod p^r). I will explain the classification of TQFTS for the (1+1)-dimensional case, in terms of Frobenius algebras with some extra structure.

If time permits, I will explain how we define a Dijkgraaf Witten like theory, to get formulas for counting G-covers of X, where X is either a surface, or a p adic field, and G is a p-group (these formulas are similar to the ones given by Mednykh for surfaces using TQFTs, and by Masakazu Yamagishi using a more algebraic approach). I will also try to outline how we plan to also get similar formulas for Hom(\pi_1(X)^p,G), where G=GL_n(k) for k=F_{p^r} or Z/p^rZ.

The talk is based on joint work with Oren Ben-Bassat.

For a number field $K$ admitting an embedding into the field of real numbers $\mathbb{R}$, it is impossible to construct a functorial in $G$ group structure in the Galois cohomology pointed set $H^1(K,G)$ for all connected reductive $K$-groups $G$. However, over an arbitrary number field $K$, we define a diamond (or power) operation of raising to power $n$

We show that this operation has many functorial properties. When $G$ is a torus, the set $H^1(K,G)$ has a natural group structure, and $x^{\Diamond n}$ coincides with the $n$-th power of $x$ in this group.

For a cohomology class $x \in H^1(K,G)$, we define the period $\operatorname{per}(x)$ to be the greatest common divisor of $n>0$ such that $x^{\Diamond n}=1$, and the index $\operatorname{ind}(x)$ to be the greatest common divisor of the degrees $[L:K]$ of finite separable extensions $L/K$ splitting $x$. These period and index generalize the period and index of a central simple algebra over $K$ (in the special case where $G$ is the projective linear group $\operatorname{PGL}_n$, the elements of $H^1(K, G)$ can be represented by central simple algebras). For an arbitrary reductive group $G$ defined over a local or global field $K$, we show that $\operatorname{per}(x)$ divides $\operatorname{ind}(x)$, that $\operatorname{per}(x)$ and $\operatorname{ind}(x)$ have the same prime factors, but the equality $\operatorname{per}(x) = \operatorname{ind}(x)$ may not hold.

The talk is based on joint work with Zinovy Reichstein. All necessary definitions will be given, including the definition of the Galois cohomology set $H^1(K,G)$.

I will describe various results, some old and some new, on the structure of the groups of points of an abelian variety over a finite field. The talk will focus on the case of varieties of large dimension over a fixed finite field. In this regime, the Weil bounds allow for the possibility of the exponent of the group staying bounded as the dimension grows. I will explain that at least in the case of Jacobians this cannot be the case. Part of the talk is based on recent joint work with Daniel Keliher.

Let X be an integral projective variety defined over Q of degree at least 2 and B > 0 an integer. The (uniform) dimension growth conjecture, now proven in almost all cases following works of Browning, Heath-Brown and Salberger, provides a uniform upper bound on the number of rational points of height at most B lying on X, where the bound depends only on the degree of X, the dimension of its ambient space, and on B.

In this talk, I will report on current developments which go beyond classical uniform dimension growth bounds, focusing on an affine variant (which implies the projective one).

This is based on joint work with Cluckers, Dèbes, Nguyen and Vermeulen.