Aspects of symmetry, topology and anomalies in quantum matter by Wang, Juven Chun-Fan

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Lower central series of a free associative algebra over the integers and finite fields by Bhupatiraju, Surya, Jordan, David, Kuszmaul, William, Li, Jason, Etingof, Pavel I.

“…We describe the torsion in the reduced quotient [bar over B][subscript 1] and B[subscript 2] geometrically in terms of the De Rham cohomology of Z[superscript n]. As a corollary we obtain a complete description of [bar over B][subscript 1](A[subscript n](Z)) and [bar over B][subscript 1](A[subscript n](F[subscript p])), as well as of B[subscript 2](A[subscript n](Z[1/2])) and B[subscript 2](A[subscript n](F[subscript p])), p > 2. …”
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Calabi-Yau manifolds, discrete symmetries and string theory by Mishra, C

“…These were later verified by independent cohomology computations. We go on to develop the machinery to understand the geometry of CY manifolds embedded as hypersurfaces in a product of del Pezzo surfaces. …”

On certain problems in homotopy theory by Stasheff, J

“…</p> <p>The second direction is guided by William Massey's definition of cohomology products. The essential properties of the singular chain complex on an A_n-space are embodied in the concept of an A(n)-algebra. …”

Non-Abelian string and particle braiding in topological order: Modular SL(3,Z) representation and (3 + 1)-dimensional twisted gauge theory by Wen, Xiao-Gang, Wang, Juven

“…String and particle braiding statistics are examined in a class of topological orders described by discrete gauge theories with a gauge group G and a 4-cocycle twist ω[subscript 4] of G's cohomology group H[superscript 4](G,R/Z) in three-dimensional space and one-dimensional time (3 + 1D). …”
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Bosonic anomalies, induced fractional quantum numbers, and degenerate zero modes: The anomalous edge physics of symmetry-protected topological states by Santos, Luiz H., Wen, Xiao-Gang, Wang, Chun-Fan Juven

“…We provide a manifest correspondence from the physical phenomena, the induced fractional quantum number, and the zero energy mode degeneracy to the mathematical concept of cocycles that appears in the group cohomology classification of SPTs, thus achieving a concrete physical materialization of the cocycles. …”
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Duan's fixed point theorem: proof and generalization

“…We then use rational homotopy to generalize to spaces <mml:math alttext="$X$"> <mml:mi>X</mml:mi> </mml:math> whose rational cohomology is the tensor product of an exterior algebra on odd dimensional generators with the tensor product of truncated polynomial algebras on even dimensional generators. …”
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Superstrings by Greene, B, Greene, Brian

“…Phenomenological considerations place severe constraints on the dimensions and transformation properties of certain cohomology groups and thereby lead to a highly restricted class of acceptable models.…”

Cosystolic Expansion of Sheaves on Posets with Applications to Good 2-Query Locally Testable Codes and Lifted Codes by First, Uriya A., Kaufman, Tali

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Commutative K-theory by Gritschacher, S

“…If <em>G</em> = <em>U;O</em> (the infinite unitary / orthogonal groups) then <em>B</em>_com<em>U</em> and <em>B</em>_com<em>O</em> are E_∞-ring spaces. The corresponding cohomology theory is called commutative <em>K</em>-theory.…”

Theory and applications of algebraic topology by Sutherland, W

“…The first stage uses the cohomology theories dealing with stable vector bundles over a space, while the criteria used in the second stage arise mainly from work of M.A. …”

Harmonic maps and associated energy functionals by Tošić, O

“…This is an algebraic conjecture stating that, given a finite cover of closed surfaces p : S'→S and a cohomology class χ ∈ H1(˜S', Z) \ {0}, the orbit of χ under the group of mapping classes on S that lift via p to S' is infinite. …”

Higher dimensional discrete Cheeger inequalities by Anna Gundert, May Szedlák

“…</span></pre><pre style="-qt-block-indent: 0; text-indent: 0px; margin: 0px;"><span style="color: #000000;">Recently, a topologically motivated notion analogous to edge expansion that is based on </span><span style="color: #008000;">$\mathbb{Z}_2$</span><span style="color: #000000;">-</span><span style="text-decoration: underline; color: #000000;">cohomology</span><span style="color: #000000;"> was introduced by </span><span style="text-decoration: underline; color: #000000;">Gromov</span><span style="color: #000000;"> and independently by Linial, Meshulam and </span><span style="text-decoration: underline; color: #000000;">Wallach</span><span style="color: #000000;">. …”
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