Discrete quantum systems from topological field theory

ArXiv.org · 2025-06-05 · 1 citations

We introduce a technique to construct gapped lattice models using defects in topological field theory. We illustrate with 2+1 dimensional models, for example Chern-Simons theories. These models are local, though the state space is not necessarily a tensor product of vector spaces over the complex numbers. The Hamiltonian is a sum of commuting projections. We also give a topological field theory construction of Levin-Wen models.

On the Whitehead theorem for nilpotent motivic spaces

Annals of K-Theory · 2025-11-28

Algebraic vector bundles and $p$-local $𝔸^1$-homotopy theory

Annales Scientifiques de l École Normale Supérieure · 2025-12-18

Unstable motivic and real-étale homotopy theory

ArXiv.org · 2025-01-26

We prove that for any base scheme $S$, real étale motivic (unstable) homotopy theory over $S$ coincides with unstable semialgebraic topology over $S$ (that is, sheaves of spaces on the real spectrum of $S$). Moreover we show that for pointed connected motivic spaces over $S$, the real étale motivic localization is given by smashing with the telescope of the map $ρ: S^0 \to {\mathbb G}_m$.

The Odd Fermion

ArXiv.org · 2024-01-08 · 2 citations

In this short note we use the geometric approach to (topological) field theory to address the question: Does an odd number of quantum mechanical fermions make sense?

On P^1-stabilization in unstable motivic homotopy theory

arXiv (Cornell University) · 2023-06-07 · 1 citations

We analyze stabilization with respect to ${\mathbb P}^1$ in the Morel--Voevodsky unstable motivic homotopy theory. We introduce a refined notion of cellularity (a.k.a., biconnectivity) in various motivic homotopy categories taking into account both the simplicial and Tate circles. Under suitable cellularity hypotheses, we refine the Whitehead theorem by showing that a map of nilpotent motivic spaces can be seen to be an equivalence if it so after taking (Voevodsky) motives. We then establish a version of the Freudenthal suspension theorem for ${\mathbb P}^1$-suspension, again under suitable cellularity hypotheses. As applications, we resolve Murthy's conjecture on splitting of corank $1$ vector bundles on smooth affine algebras over algebraically closed fields having characteristic $0$ and compute new unstable motivic homotopy of motivic spheres.

On the Whitehead theorem for nilpotent motivic spaces

arXiv (Cornell University) · 2022-10-12

We improve some foundational connectivity results and the relative Hurewicz theorem in motivic homotopy theory, study functorial central series in motivic local group theory, establish the existence of functorial Moore--Postnikov factorizations for nilpotent morphisms of motivic spaces under a mild technical hypothesis and establish an analog of the Whitehead theorem for nilpotent motivic spaces. As an application, we deduce a surprising unstable motivic periodicity result.

Intersection forms of spin 4-manifolds and the pin(2)-equivariant Mahowald invariant

Communications of the American Mathematical Society · 2022-02-23 · 12 citations

In studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P i n left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Pin</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Pin}(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -equivariant stable maps between certain representation spheres. A precise answer of Furuta’s problem was later conjectured by Jones. In this paper, we completely resolve Jones conjecture by analyzing the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P i n left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Pin</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Pin}(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -equivariant Mahowald invariants. As a geometric application of our result, we prove a “10/8+4”-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B upper P i n left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mi>Pin</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">B\operatorname {Pin}(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j"> <mml:semantics> <mml:mi>j</mml:mi> <mml:annotation encoding="application/x-tex">j</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -based Atiyah–Hirzebruch spectral sequence.

Corporate Social Responsibility, Sustainability and Leadership

2022-10-17 · 4 citations

In Chapter 8, Professor Michael Hopkins, Director of the Institute for Responsible Leadership and former Senior Economist at the United Nations International Labour Organization, persuasively argues that corporate social responsibility (CSR) and sustainability can bring broader benefit in the private, as well as the public, sector. Having defined CSR primarily in terms of treating key stakeholders responsibly and outlining the conditions for sustainability, he claims that benefit especially arises because it brings financial rewards for the companies concerned – along with other vital non-monetary gains in the context of the United Nations Sustainable Development Goals. Further encouragement is given to following this path in relation to the global Covid pandemic.

Localization and nilpotent spaces in -homotopy theory

Compositio Mathematica · 2022-03-01 · 5 citations

Abstract For a subring $R$ of the rational numbers, we study $R$ -localization functors in the local homotopy theory of simplicial presheaves on a small site and then in ${\mathbb {A}}^1$ -homotopy theory. To this end, we introduce and analyze two notions of nilpotence for spaces in ${\mathbb {A}}^1$ -homotopy theory, paying attention to future applications for vector bundles. We show that $R$ -localization behaves in a controlled fashion for the nilpotent spaces we consider. We show that the classifying space $BGL_n$ is ${\mathbb {A}}^1$ -nilpotent when $n$ is odd, and analyze the (more complicated) situation where $n$ is even as well. We establish analogs of various classical results about rationalization in the context of ${\mathbb {A}}^1$ -homotopy theory: if $-1$ is a sum of squares in the base field, ${\mathbb {A}}^n \,{\setminus}\, 0$ is rationally equivalent to a suitable motivic Eilenberg–Mac Lane space, and the special linear group decomposes as a product of motivic spheres.