#ExplainingMyResearch
In order to say something, you first need to have the words for it, and so it is with #physics and #mathematics.
(I know that many pure #mathematicians view their work very differently, but this is how I approach it.)

#ExplainingMyResearch
Very early in my studies, one of the things that gripped me most was how on its face very #abstract #mathematics serves as a language to describe physical phenomena. I got into #math and in particular #geometry and #topology more and more because it is the #language that physics is written in.
The research that I do day-to-day now is far removed from physics, but this is still how I view my work: I develop this language, the language of the universe, if you want.

#ExplainingMyResearch
(Here, I am not going to go into how this can, on occasion, make physicists arrogant, insufferable and very wrong. As a physicist, you always have to be careful to not view the entire world as nails just because you have this really neat hammer.)
https://xkcd.com/793/

#ExplainingMyResearch
Something different today: Since I've already turned this hashtag into a birds-eye view of my research, today I want to talk about how I understand my work in a broader context.
I started studying #physics at uni, and I am still fascinated by it: Of course the boundaries between different scientific disciplines are and have always been permeable - whether something is physics, #biology or #chemistry is often a question of interpretation and tools, rather than content.

Since I turned #ExplainingMyResearch into a fairly general rumination on my larger #research area, I'm starting another more specialist hashtag for what I'm actually doing on a more day-to-day basis: #TodaysMath
(I will not post for it every day, nor will I abandon the more general one.)
#math #geometry

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To start, I am working on 2-dimensional "universes" like the disc with boundary, which are not strictly speaking GC, but similar enough to be useful.
My #preprint
https://arxiv.org/abs/2207.06894
describes how to define #Floer #cohomology for so-called log-#symplectic surfaces. I am currently finishing off the full description of the category of branes in this setting, so keep your eyes open! 🙂

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Now I mentioned before that I am chiefly interested in #GeneralizedComplex geometry underlying all this: All universes that exhibit #MirrorSymmetry are GC, and #branes are GC objects.
However, there is currently no general version for a "GC category of branes". Part of my current work is to define this in certain cases.

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The extremely simple example I showed earlier with the lines inside a circle is one extremely simple version of a #category of #branes: The disc with the circular boundary is the "universe" and the lines are the branes. The interaction between the branes is given by the cohomology associated to the intersection points. Because of its low dimension, this example is much less complex for higher dimensional universes like our own (which in string theory is 10-dim!).

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This is Kontsevich's famous Homological Mirror Symmetry #Conjecture. It has been demonstrated in examples, but it is certainly not fully understood fully yet (and it is also not exactly what physicists understand mirror symmetry to be).

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It turns out that #MirrorSymmetry can be described in terms of #branes, which I have described before: For each of the two different mirror geometries, we can define a #category of branes (again, this is very simplified and not exactly right), which is a mathematical structure encoding not only the branes themselves, but also in some sense their interactions. Two universes are mirror if the categories of branes are equivalent in a certain way.

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Today: A short explanation of connections between the different concepts I have talked about:
#MirrorSymmetry is originally an observation from #StringTheory, which can behave the same in different universes with different geometric structures - they are mirror partners. 🧵
#math #physics

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Now, in this particular setting, this does not seem very useful - you can tell what the minimal number of intersection points is by just looking at the picture.
But this is actually just an extremely simple example of a complicated invariant called #Floer #cohomology, which is central to the #math description of #MirrorSymmetry!

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HOWEVER, there is a #cohomology associated to this setting which essentially counts the *minimal number of intersection points* given a particular arrangement of endpoints, no matter what the actual lines look like.
Computing it for a particular arrangement of lines involves counting the number of intersection points that is actually present and substracting a certain number of them again (those that are "exact", a distinct property).

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Clearly, we cannot tell from the endpoints alone: If we keep the endpoints fixed, but stretch and wiggle the middle of one line, we can produce more intersection points - the intersection points do not constitute an invariant!

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Here is an example that is actually pretty close to an aspect of my current big research project (fusegc.kirchhofflukat.de):
Take two (or more) lines inside a circle that begin and end at the edge and are not allowed to cross themselves, only each other. How many intersection points do they have, based on where their endpoints are?

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They are called "invariants" because they stay the same under many superficial transformations; they will only change if we modify the object in some fundamental way.

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An invariant is some simple algebraic object - e.g. just a number - computed from the more complicated thing. If the numbers are different, the two complicated objects cannot be equivalent!

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The principle is always as follows: You want to understand some kind of complicated mathematical object, for example a knot or a manifold (see earlier: a potentially very complicated #geometric shape of any dimension). In particular, this involves being able to tell different such objects apart: Are they actually fundamentally different, or equivalent in some way?
In order to do this, mathematicians use #invariants, of which cohomologies are powerful examples.

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Today's thread is about #cohomology. Almost no matter which area of pure #maths, theoretical #physics and also many more #applied fields you work in, you will encounter some form of cohomology. While these are defined in very different ways depending on context, they sometimes turn out to still compute the same thing; and their core properties are always the same.

Back to #ExplainingMyResearch! I did 2 big threads last week; you can click the hashtag to find the earlier ones!
In today's 🧵 I am once again trying to be very general and explain a very fundamental #math concept. (Sorry, math expert followers; I'll eventually get to more specialist stuff.)