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15

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Introduction

**Skills and Expertise**

## Publications

Publications (15)

We show that the sheaf of $\mathbb A^1$-connected components of a reductive algebraic group over a perfect field is strongly $\mathbb A^1$-invariant. As a consequence, torsors under such groups give rise to $\mathbb A^1$-fiber sequences. We also show that sections of $\mathbb A^1$-connected components of anisotropic, semisimple, simply connected al...

The definition of Milnor-Witt cycle modules in \cite{Feld} can easily be adapted over general regular base schemes. However, there are simple examples to show that Gersten complex fails to be exact for cycle modules in general if the base is not a field. The goal of this article is to show that, for a restricted class of Milnor-Witt cycle modules o...

We explicitly describe the $\mathbb{A}^1$-chain homotopy classes of morphisms from a smooth henselian local scheme into a smooth projective surface, which is birationally ruled over a curve of genus $> 0$. We consequently determine the sheaf of naive $\mathbb{A}^1$-connected components of such a surface and show that it does not agree with the shea...

We show that the sheaf of $\mathbb A^1$-connected components of a Nisnevich sheaf of sets and its universal $\mathbb A^1$-invariant quotient (obtained by iterating the $\mathbb A^1$-chain connected components construction and taking the direct limit) agree on field-valued points. Given any natural number $n$, we construct an $\mathbb A^1$-connected...

We show that $\mathbb A^1$-connectedness of a large class of varieties over a field $k$ can be characterized as the condition that their generic point can be connected to a $k$-rational point using (not necessarily naive) $\mathbb A^1$-homotopies. We also show that symmetric powers of $\mathbb A^1$-connected varieties (over an arbitrary field), as...

We explicitly describe the $\mathbb A^1$-chain homotopy classes of morphisms from a smooth henselian local scheme into a smooth projective surface, which is birationally ruled over a curve of genus $> 0$. We consequently determine the sheaf of naive $\mathbb A^1$-connected components of such a surface and show that it does not agree with the sheaf...

A conjecture of Morel asserts that the sheaf of $\mathbb A^1$-connected components of a space is $\mathbb A^1$-invariant. Using purely algebro-geometric methods, we determine the sheaf of $\mathbb A^1$-connected components of a smooth projective surface, which is birationally ruled over a curve of genus $>0$. As a consequence, we show that Morel's...

Using sheaves of A^1-connected components, we prove that the Morel-Voevodsky singular construction on a reductive algebraic group fails to be A^1-local if the group does not satisfy suitable isotropy hypotheses. As a consequence, we show the failure of A^1-invariance of torsors for such groups on smooth affine schemes over infinite perfect fields....

We show that if G is an anisotropic, semisimple, absolutely almost simple, simply connected group over a field k, then two elements of G over any field extension of k are R-equivalent if and only if they are A(1)-equivalent. This has two important consequences. First, the A(1)-singular construction Sing(*)(G) cannot be A(1)-local for such groups. S...

A conjecture of Morel asserts that the sheaf of A^1-connected components of a
simplicial sheaf X is A^1-invariant. A conjecture of Asok-Morel asserts that
A^1-connected components of smooth k-schemes coincide with their
A^1-chain-connected components and are birational invariants of smooth proper
schemes. In this article, we exhibit examples of sch...

We define a notion of p-adic measure on Artin n-stacks that are of strongly finite type over the ring of p-adic integers. p-adic measure on schemes can be evaluated by counting points on the reduction of the scheme modulo p(n). We show that an analogous construction works in the case of Artin stacks as well if we count the points using the counting...

It is shown that it is impossible to construct a free theory of fermions on infinite hypercubic Euclidean lattice in even number of dimensions that: (a) is ultralocal, (b) respects the symmetries of hypercubic lattice, (c) chirally nonsymmetric part of its propagator is local, and (d) describes single species of massless Dirac fermions in the conti...

Geometric motivic integration on Artin N-stacks