1/6 Perfectoid Spaces and the WeightMonodromy Conjecture
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Title 
1/6 Perfectoid Spaces and the WeightMonodromy Conjecture

Title of Series  
Part Number 
1

Number of Parts 
6

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CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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Release Date 
2018

Language 
English

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Subject Area  
Abstract 
We will introduce the notion of perfectoid spaces. The theory can be seen as a kind of rigid geometry of infinite type, and the most important feature is that the theories over (deeply ramified extensions of) Q p and over F p((t)) are equivalent, generalizing to the relative situation a theorem of FontaineWintenberger, and also implying a strong form of Faltings's almost purity theorem. This method of changing the characteristic is then applied to deduce many cases of the weightmonodromy conjecture.

00:00
Pairwise comparison
Group action
Manysorted logic
Content (media)
Basis <Mathematik>
Series (mathematics)
Metric system
01:19
Axiom of choice
Geometry
Positional notation
Characteristic polynomial
Routing
Körper <Algebra>
Series (mathematics)
Object (grammar)
Bounded variation
02:52
Point (geometry)
Functional (mathematics)
Perfect group
Multiplication sign
Surface
Characteristic polynomial
Basis <Mathematik>
Chaos (cosmogony)
Canonical ensemble
Limit (category theory)
Equivalence relation
Category of being
Ring (mathematics)
Körper <Algebra>
Object (grammar)
Spacetime
05:54
Group action
Functional (mathematics)
Perfect group
Divisor
Variety (linguistics)
Characteristic polynomial
Analytic set
Algebraic structure
Insertion loss
Lattice (order)
Equivalence relation
Ring (mathematics)
Root
Oval
Right angle
Object (grammar)
Associative property
Condition number
Spacetime
Directed graph
08:39
Root
Variety (linguistics)
Characteristic polynomial
Model theory
Right angle
Lattice (order)
Series (mathematics)
Limit (category theory)
Mereology
Spacetime
Directed graph
11:23
Group action
Military base
Multiplication sign
Limit (category theory)
Theory
Category of being
Ring (mathematics)
Tower
Energy level
Modulform
Körper <Algebra>
Exponential family
Arithmetic progression
Fiber (mathematics)
Spacetime
14:57
Pulse (signal processing)
Manysorted logic
Military base
Series (mathematics)
Spacetime
16:44
Complex (psychology)
Category of being
Functional (mathematics)
Analogy
Körper <Algebra>
Series (mathematics)
Spacetime
19:28
Point (geometry)
Geometry
Group action
Quantum state
Zariski topology
Complete metric space
Theory
Manysorted logic
Ring (mathematics)
Ideal (ethics)
Normal (geometry)
Series (mathematics)
Extension (kinesiology)
Bounded variation
22:36
Point (geometry)
Pairwise comparison
Arithmetic mean
Normal (geometry)
Körper <Algebra>
Series (mathematics)
Game theory
Spacetime
24:27
Group action
Ring (mathematics)
Order (biology)
Ranking
Normal (geometry)
Körper <Algebra>
Bounded variation
25:46
Group action
Equivalence relation
Subgroup
Subset
00:00
when ould so i want to thank the organisers for inviting me to give peace talks so great or to speak.
00:20
here and so i want to talk about something that i called perfect its basis.
00:27
and and. now the first give a sort of some general introduction to the end. what perfected spaces are. so the general aim somehow. to get a comparison.
00:52
between objects. it makes characteristic. syrupy. it was all checks in a characteristic p.. you and. some a. i. the basic serum samoa's so the idea is to get some all in june metric and nice version of the series of winter content of intimacy which says it's the absolute governor groups.
01:33
and yes to govern groups. i. of q p two which you for example there are many other choices you can join all the proceeds of key to the mix characteristic field and have long series field. over fifty are some more think. in some sense of economic the us will think. and and so one see some of that after joining lots of people or routes to mix characteristic saying get something that in some ways looks like and you could characteristic sing and and still you generalize this. to the following we want to have some kind of geometric objects over this is a characteristic fields and some kind of geometric up to them as you could characteristic cute and compare them and so let me just as notation could cases field and. if that this field was some slight variations of cynicism. and then.
02:55
it becomes true that a certain category of socalled perfect art spaces are ok it's canonical equivalent to the category same kind of spaces. over the characteristic field so i do not have flunked function by experts to explain it. he said. and so let me state some general properties of this equivalence the. you. to give you some idea of what's going on so first of all perfect its basis. our local doing to promote your spaces. and so in particular one can talk about the underlying to political space and think that's just the same on both sides. and. and even better not only that a lot of the space of the same but even the top four point i can only be equivalent to some of you just take an axe to be the point spread chaos something and this week sickly reproof someones the serum a fun time into museum. thank you will need at this as an ingredient but some on the surface is a generalization so much as a relative setting of the serum of the content of intimacy. you. and the mother of another property is that this ice and often there between the catch you are some all of a deeply and acknowledge or. in some sense there are some limit procedures involved in size from often and so we have to leave the world of our great geometry. i. instead we have to do some language of.
05:12
richard and to join the tree. i. the judge. he and. and. there's a reason that they're called perfect namely the objects are perfect in a certain sense. so meaning that if iran characteristics p you could disappear. and then all the rings that occur a perfect. he and some mix characteristic.
05:59
we have a similar condition. so guaranteeing some of that they are lots of people route. i. and it. so one thing that we see from this is already that the rings a that we will deal with a highly on the syrian some perfect rings a basic human a syrian. and so. the u.. it may be technically not so easy but i'm. ok. and i want to add some birds about how perfected spaces are related to ride to ease so for example characteristic peace. we have found two. i'm from right he's ok. to protect its base is a look a. i am so what if let them and. so it's not just by x. make maps to explore have an effect in his career that this funk are factors over the kind of your fridge and analytic varieties so.
07:32
four inches over threat so exciting to fix wake. the next to express some of first the associate the identification to eggs and then you make structure she's perfect basically. and so these two are sufficiently close to another for example it's true that. the tall top loss of says a source of the perfect its space is just the same as the total costs. of the associated with genentech variety and in particular the talk of mauritius two objects will agree. and and but there's no such function characteristic zero. you. someone characteristic here there's no canonical way to join people roots in general are you are. nonetheless there some favorable situations where one can still be to give nice constructions have perfected space is also a characteristic zero so let me give some examples of.
08:44
i. perfect or spaces and characteristics year old. and the. and. it's also somewhat gives a flavour of what such a space looks like someone saying we could do this week one takes. the a one over k. and say takes the associated with didn't have the right to own i will take some slightly different sing but think of it as your genetic variety and then. it takes interest limit the was the piece part of it used to coordinate on they want. the u.. so of course i want some a canonical way to join lots of people wrote that's what we do. you are. so more generally. but the for any toward right you can do the same way. and we can take the risk limit or them and five which is on one avoids given by much became a p m l c. which it into a variety of sources to take a. and then and this is the first example. i can't even describe to tilt so i can say this so this i smoke is between the mic equal and makes characteristic and series this is what i call tooting so this case a chilled. but this investment. it is just the same thing constructed uncharacteristic p.. i. but there are some other cases per one model has a canonical way to join something like people roots and say they have a case of you would be a variety.
10:59
i. then you can look at the investment will much became by p.. most associated with genetic variety and again this the perfect to its space. and at least have a us ordinary doctrine. the. i.
11:28
one can describe. to build again as soon as limit will much occasion but he was some broad other righty a primal the is a field where prime mover. they fled you can describe why assertive canonical parameters. described. one sitting. unknown. term trends. the boy. ok. and they're more example so you can take over the ring of introduced in case you can take some people with the group. but. so. the samoan from here on its all in progress and i haven't.
12:36
the checked in which generality expect the i can say will which things but for on a reduction summer. and it's easiest for good or the reduction is now story. i. so they compete with the group. then. you can again take its generic fiber were say it's a form of the repeated was a big group this was supposed to be something that's all you need for you and then again you can take the muslim of the loan modification my people. and then there's also related to some socalled been on this spaces. korean putin coach theory. and also the u.s. bases some of the be right he said he was a big groups have says nice property so if you have. sure righty depending on a level. then you can. takes enormous limit of all levels at p.. it's a corresponding with it and was a friday. or if he can take the next case came running through compact opens up groups have some group fifty p. reporting space. and i can look at the muslim and a bulwark a thick skin. so this means that in this what i mean it's been a problem for a long time if you have say is a movement a tower that you would like to look at its infinite level and now it turns out that its infant level is clearly becomes a space immediate effect on space. it. and so for example once he said. we can take the risk limit of the group photo or. and as we know this can only be a small thing as a perfect to its space was in muslim of the new intake toward.
14:58
so there is also shared mind same working on the snow. the. so as perfect its bases of. the. again. so what. and. the. them. the some of the general introduction. so these perfected spaces are really some originated expenses but have been in for the type so they're not the syrian not anything.
15:54
and in particular for example i don't think that if you want can describe perfected space said it's a culture we have certain formal models what you do as some of the sort of admissible blowups because. these threatening techniques seem to require this year in this light pulses. so we have to have to look for some other approach to which the judge jury and i don't have that it uses a. who were serious attic space as and in the first lecture will basically no explain what an attic space is and compare the series to the existing series in which it and the geometry. i.
16:49
so i wrote that me thinks will the finish is falling. and on a comedian few ok. it is a temple to kill field. i. some who supported the. the. it is induced by. a nontrivial radiation from one. i.
17:27
and. so through all that need fixing on enough uk. the. and. i'm doesn't extend essentially unique. and all mad from. kate to the positive wheels. and you. let's fix one which is not really necessary for hoover syria but for comparing them as other series of my may be helpful. ok so on. some are some basic idea of. which it and the chemistry. some all it should be an analog of syria of complex and his expenses overseen and so that should be a puncture. it will take space is no. i. he said. the founder from the righties over the fields k two of those a category of attic space is. over cake. standing next to fix art. and some on this expensive move should know make sense to talk about the subsidies were function is so mobile and by one or something so. and that of. the following should make sense and four. fall functions on your space. for them. something like this should make sense.
19:28
i. so in particular. one basically see said any point has to give rise to some sort of variation of soon. p.. the. a. the. so it came. so there are no three approaches which have this requirement still some of this approach for didn't have the geometry. take it. you take x. ray which is a set of clothes points of six. i. a. the. for. and so some oh. the effects of so effects. in mexico's point.
20:57
it. and then some of their see. and say this is just fine. but then we get. it was cosponsored some. explore sponsor some. and so. it makes an ideal. so we get a map to almost them which is. some find a few the extension of k.. and the norm one k. extensive any find extension so we. get a natural met to the post of three years and this gives a variation corresponding to this close point. a. in this i wanted to say this but have a good as soon as two hours at all marines a complete. some of the whole series will not change my mom workplaces. the ring but its completion of it takes the freedom to assume that the complete. the. the u.. ok. but in this book which theory when one considers. a certain.
22:39
in all maps from. during our to oppose the fields. so extending. the asian which already have one came. i. so this gives you more points and i will later example describe these points but there's some or something still more general that one can do. the. i. when the lows basically all across to be hiring where you actions. the. the game. the. four. right i will explain the comparison to other series later that again in the situation that i will for perfect its space is i don't know whether this comparison still work and still. let me start by defining what work also asian so it always some ring. the. but. then he. where you asian. on our is a map of them.
24:24
which i mean who writes usually because the norm but i think was to assume a knowledge of calling it the way she was met from all to the union zero.
24:37
some totally order to be a group. so for example could be the positive fields. and. a satisfying the following requirements them. and so. the way of serious years may have one is one way you have expressed why is that most maximum. what's the use of x. and y.. and. so this is now in its multiplicative still there today of them some.
25:24
and some i think it causes wage because one. used to talking about hiring variations but was not used to talk about higher rank norms and room but then came so. the and not ours is a particular ring as it will be in the case is that we're interested in.
25:50
then this members called continuous. the. the fall of the mind i'm a bit. said of all. it's in our city absolutely useless in goma. this is open to. the. the. four. four. for the ride and this is the dumbest not necessary to your group but.
26:46
some will now be fine notion of equivalence which some old does not see where your group so. so much. in this context it makes her the finishes with let them a suit way a subset of gum a p the subgroup.