**Abstract:** I will describe and study relations between Chern classes and Galois cohomology classes in the Gal(C/R)-equivariant cohomology of real algebraic varieties with no real points. These relations have applications to sums of squares problems, in the spirit of Hilbert's 17th problem. For instance, one can use them to show that a nonnegative real polynomial in R[X_{1},...,X_{n}] of degree at most n-1 is a sum of 2^{n-1} squares in the field R(X_{1},...,X_{n}) of rational functions. This is joint work with Olivier Wittenberg.

**Abstract:** A simple measure of our understanding of the motivic stable homotopy category of a field k is our understanding of the bigraded stable stems in that category. In the case k=C we now understand that the motivic stable stems are intimately tied with the Adams--Novikov spectral sequence and this connection has greatly enhanced our computational understanding of both classical and motivic stems. In this talk I will explain how, for fields containing a square root of -1, there are simple formulas for the p-complete motivic stable stems in terms of Milnor K-theory and stems over C. The proof of this result involves the construction of a category of cellular motives over F_{1} and requires proving many new cases of the Galois reconstruction conjecture. This conjecture asserts that the subcategory of Artin--Tate objects in SH(k) depends only on the absolute Galois group of k and is now proved for fields of small cohomological dimension. This talk represents joint work with Tom Bachmann and Zhouli Xu.

**Abstract:** My talk concerns bordism rings of compact smooth manifolds equipped with a smooth action by a finite group. I will start by recalling classical results on the subject from the 60's and 70's, mostly due to Conner-Floyd, Boardman, Stong and Alexander. Afterwards I will discuss joint work with Stefan Schwede in which we prove an algebraic universal property for the collection of all bordism rings of manifolds with commuting involutions.

**Abstract:** The Grassmann manifold is defined as the space of all k-dimensional linear subspaces of R^{n}, and the oriented Grassmann manifold is the space of all oriented k-dimensional linear subspaces of R^{n}. A covering map exists between these two manifolds, and despite numerous results on the cohomology of Grassmann manifolds, little is known about oriented Grassmannians, even though the first work on this matter dates back to the 19th century.
In this talk, we will focus on the case where k=3 and delve into recent developments in this area. Specifically, we will discuss recent work on mod 2 and integral cohomology algebras, providing insights into approaching these problems.

**Abstract:** As revealed by the Gauss-Bonnet theorem for surfaces and as predicted by the Hopf conjecture in higher dimensions, curvature properties of a Riemannian manifold can determine the sign of the Euler characteristic. One can similarly ask whether the sign of the Euler characteristic of an aspherical space is already determined by the deck transformation groups of finite-sheeted covering spaces. While the answer is "no" in general, we will explain how Galois cohomological methods give an affirmative answer for the important test ground of S-arithmetic quotients of locally symmetric spaces and Bruhat-Tits buildings.

**Abstract:** Given a Banach Algebra A there are two versions of K-theory one might associate with it. Algebraic K-theory and Topological K-theory. Both invariants will give different results in general, but conjecturally they seem to agree when taking finite coefficients. Recent results through condensed mathematics have elucidated this connection. Both algebraic and topological K-theory are contained in a natural condensed structure of the K-theory spectrum of A and they necessarily agree if this structure is discrete. This in turn is equivalent to a generalized version of Gabber Rigidity holding for A. We will show new results that simplify how to prove this type of rigidity in concrete cases.

**Abstract:** In this talk, we discuss equivariant Hermitian K-theory for symplectic groups, Sp_{2} being the most important example. Instead of the classical Grothendieck-Witt ring, we use the Z/2-graded Grothendieck-Witt ring that combines symmetric and symplectic forms, and show how the situation becomes analogous to that for algebraic K-theory and general linear groups. We then establish an Atiyah-Segal completion theorem for the Hermitian K-theory of symplectic groups.

**Abstract:** This talk in the area of low-dimensional topology deals with the problem of determining the topological 4-genus for the special case of 3-braid knots. The 4-genus of a knot is the minimal genus of a "nicely" embedded surface in the 4-dimensional ball with boundary the given knot. Asking whether a knot has 4-genus zero, i.e. whether it bounds a disk in the 4-ball, is a natural generalization in dimension 4 of the question whether it is isotopic to the trivial knot. It is one of the curiosities of low-dimensional topology that constructions such as finding these disks can sometimes be done in the topological category, but fail to work smoothly. The first examples of this phenomenon followed Freedman's famous work on 4-manifolds.
Four decades later, the topological 4-genus of knots - even torus knots - remains difficult to determine. In a joint work with S. Baader, L. Lewark and F. Misev, we classify 3-braid knots whose topological 4-genus is maximal (i.e. equal to their 3-genus). In the talk we will explain the difficulties that arise and draw connections to other problems in low-dimensional topology. We will define all relevant notions and address a broad audience.