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We prove a generalization of the fundamental theorem of algebraic K-theory for Verdier-localizing functors by extending the proof for algebraic K-theory of spaces to the realm of stable $\infty$-categories. The formula behaves much better…

The present work is the author's doctoral thesis, written during his studies at the University of Bonn. Its goal is to establish the foundations of $K$-theory in the context of adic geometry using the formalism of condensed mathematics and…

Jones conjectures the arboreal representation of a degree two rational map will have finite index in the full automorphism group of a binary rooted tree except under certain conditions. We prove a version of Jones' Conjecture for quadratic…

Let $X = \bigcup_k X_k$ be the ind-Grassmannian of codimension $n$ subspaces of an infinite-dimensional torus representation. If $\cE$ is a bundle on $X$, we expect that $\sum_j (-1)^j \Lambda^j(\cE)$ represents the $K$-theoretic…

This paper provides a unified approach to results on representations of affine Hecke algebras, cyclotomic Hecke algebras, affine BMW algebras, cyclotomic BMW algebras, Markov traces, Jacobi-Trudi type identities, dual pairs (Zelevinsky),…

Throughout, let $K$ be an algebraically closed field of characteristic $0$. We provide a generic classification of locally free representations of Geiss-Leclerc-Schr\"oer's algebras $H_K(C,D,\Omega)$ associated to affine Cartan matrices $C$…

Mirkovic-Vilonen (MV) polytopes have proven to be a useful tool in understanding and unifying many constructions of crystals for finite-type Kac-Moody algebras. These polytopes arise naturally in many places, including the affine…

The critical loci of a map $f:X\to Y$ between smooth schemes over a field $k$ are the locally closed subschemes $\Sigma^i(f)\subseteq X$ where the differential of $f$ has constant rank. We prove that if $f : X\to \mathbb A^r$ is the general…

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of $\Lambda$ with fixed…

In order to facilitate the application of standard renormalization techniques, gravitation should be decribed, if possible, in pure connection formalism, as a Yang-Mills theory of a certain spacetime group, say the Poincare or the affine…

Let $F$ be a non-archimedean local field with residue field $\mathbb{F}_q$ and let $G = GL_{2/F}$. Let $\mathbf{q}$ be an indeterminate and let $H^{(1)}(\mathbf{q})$ be the generic pro-p Iwahori-Hecke algebra of the group $G(F)$. Let…

We propose a robust method for location estimation in various matrix manifolds based on the projected Frobenius median, which is closely related to the spatial median. This method applies broadly to matrix manifolds, including Stiefel and…

Let F_0 be a non-archimedean local field of odd residual characteristic and let G be the unramified unitary group U(2,2) defined over F_0. In this paper, we give a classification of the irreducible smooth representations of G of…

The Alt-Caffarelli-Friedman monotonicity formula is a cornerstone in the theory of free boundary problems. In this note we provide a self-contained proof of this result. To prove the main stepping stone, namely the Friedland-Hayman…

We state a precise conjectural isomorphism between localizations of the equivariant quantum K-theory ring of a flag variety and the equivariant K-homology ring of the affine Grassmannian, in particular relating their Schubert bases and…

Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite…

Level-restricted paths play an important role in crystal theory. They correspond to certain highest weight vectors of modules of quantum affine algebras. We show that the recently established bijection between Littlewood--Richardson…

We observe that on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of characteristic $p> h$ (where $h$ is the Coxeter number), with a given (generalized) central character are…

The generalization of the Morse theory presented by Goresky and MacPherson is a landmark that divided completely the topological and geo\-me\-tri\-cal study of singular spaces. Let \{$X_t\}_t$ be a suitable family of germs at $0$ of…

We introduce efficient Markov chain Monte Carlo methods for inference and model determination in multivariate and matrix-variate Gaussian graphical models. Our framework is based on the G-Wishart prior for the precision matrix associated…