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have now added speakers, titles and abstracts for our first little workshop (public on zoom) which might be of interest to some readers here:
Homotopical Perspectives on TDA
Speakers: Chacholski, Ginot, Jardine & Zhou
Wasn’t there a mission statement some place? Was it the article, Sati and Schreiber 2021: ’Topological and Quantum Systems’, mentioned at computational trilogy?
Besides whats in the nLab entry now, there is the following paragraph on the center landing page here (the domain nyuad.nyu.edu
seems to have a hiccup right now, does it work for you?):
The Center for Quantum and Topological Systems serves as a nucleation point for cross-disciplinary expertise in theory and application of Quantum Topological Systems in general, with an emphasis towards the unifying goal of robust Quantum Computation in particular — combining all questions from theoretical foundations (quantum error-correction) over hardware (topological quantum materials and novel quantum chips), architecture (parameterized quantum circuits) and software (quantum programming languages and hardware-aware software optimizations) to applications (quantum machine learning and quantum cryptography)
have added pointer to the upcoming external talk:
15 Sep 2022 at PlanQC 2022
U.S. on joint work with Hisham Sati:
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Also, I am eager to add the list of names of our group of postdoc researchers who are arriving in the next weeks (or have already), but don’t want to say anything that may not be fully official yet. What I can safely say is that among them are are Adrian Clough, David Jaz Myers and Mitchell Riley.
Great team! Let me know when you need a philosopher.
with the summer break over, we are resuming next week. In the GTP-seminar wel’ll have:
13 Sep 2023
Mayuko Yamashita (Kyoto University, Japan):
Topological Modular Forms and Heterotic String Theory
In this talk, I will explain my works with Y. Tachikawa to study anomalies in heterotic string theory via homotopy theory, especially the theory of Topological Modular Forms (TMF). TMF is an E-infinity ring spectrum which is conjectured by Stolz-Teichner to classify two-dimensional supersymmetric quantum field theories in physics. In the previous work [arXiv:2108.13542], we proved the vanishing of anomalies in heterotic string theory mathematically using TMF. Additionally, we have a recent update on the previous work [arXiv:2305.06196]. Due to the vanishing result, we can consider a secondary transformation of spectra, which coincides with the Anderson self-duality morphism of TMF. This allows us to detect subtle torsion phenomena in TMF by differential-geometric methods.
As before, I’ll add (here) resources as they become available
Looks like a great conference.
Your abstract had me remember those ideas on jet $\infty$-toposes approximating twisted cohomology:
Higher-order approximations should involve a notion of higher-order forms of the tangent (∞,1)-topos, in parallel with the relationship between the jet bundles and tangent bundle of a manifold. It is clear that whatever we may say in detail about the $k$th-jet (∞,1)-topos $J^k\mathbf{H}$, its intrinsic cohomology is a version of twisted cohomology which is in between nonabelian cohomology and stable i.e. generalized (Eilenberg-Steenrod) cohomology.
It seems that a layered analysis of nonabelian cohomology this way in higher homotopy theory should eventually be rather important, even if it hasn’t received any attention at all yet. It seems plausible that a generalization of Chern-Weil theory which approximates classes of principal infinity-bundles not just by universal characteristic classes in ordinary cohomology and hence in stable cohomology, but that one wants to consider the whole Goodwillie Taylor tower of approximations to it.
If the first approximation prompts extension of HoTT to linear HoTT, is there some ’twisted HoTT’ out there?
The twisting of cohomology seen in an $\infty$-topos just corresponds to dependency of the types in the corresponding internal logic. In this sense plain (L)HoTT is already “twisted”.
In the paragraph you quote it is not the twisting that makes the difference, but the “degree of linearity” of what is being twisted.
Ah, OK. So what makes the first degree of linearity stand out as so central to quantum physics?
What would, say, second degree linearity relate to?
The degree $k$ in $J^k \mathbf{H}$ refers to Goodwillie polynomials of degree $k$.
By all that is known, coherent quantum processes are linear maps, not more general polynomial maps or worse.
(This reminds me of how Maxim Kontsevich trolled the 2015 Breakthrough Prize ceremony by declaring that the universe must be a simulation because it is “impossible” that quantum physics is really about linear spaces instead of curved manifolds like the rest of physics — around 19:30.)
Funny. Fine to present it as a source of bafflement, but the step to simulation seems bizarre.
So what is the more precise statement about linearity here? Would it be “coherent quantum processes observable to an observer are linear maps”? This reminds me of the recent series of papers by Witten (e.g. this) where one describes an algebra of observables (hence linear) only in relation to an observer (there described by a timelike curve).
Not sure what you are after here. Coherent quantum processes are unitary linear operators between Hilbert spaces. That’s essentially one of the axioms of QM.
Maybe I’m misunderstanding what you are referring to as quantum physics, since what I mentioned is supposed to be an observation of quantum gravity. I guess the question is, doesn’t quantum gravity (understood as a generalization of qft, not as a class of qft’s) take those higher degrees $k$ into account? In the sense you mention an axiom of QM is restricting to order $k=1$, isn’t the point of QG that one of the to-be axioms is the incorporation of those higher degrees?
No, quantum gravity is not meant to change the rules of quantum physics. Those $C^\ast$-algebras of quantum observables that you point to are still (embedded into) algebras of linear operators.
video recordings of all talks at Running HoTT 2024 now available: find the link below the corresponding speaker/title in the list here
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