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The podcast in question can be found here. Here is a very weak attempt at a summary. I'd welcome edits to correct or provide more detail. I worked backward from the end and didn't listen earlier than the following -- there might be relevant material earlier.

The context of this interview is the recent publication (2022) in France of Grothendieck's text Récoltes et semailles. In this text, Grothendieck criticized (sometimes severely) several mathematicians and complained about the attitude of the mathematical community in general.

Around 45:30, the journalist claims that topos theory gets very bad press and asks why to Connes, Caramello and Lafforgue. Connes says no, this is a completely external vision of reality. Prompted by the journalist, he explains what is topos theory, where one studies a space not by looking at it directly, but by putting him behind the scenes. One does ordinary mathematics, with a "parameter" which is in the space in question. This leads to a much finer knowledge of the space. He talks about topos theory's relationship to structuralism and category theory.

Around 48:50, Olivia Caramello is asked why topos theory has been rejected by the institutions. She says that she received a lot of opposition since the beginning of her career because she wanted to develop topos theory in a global and systematic way in order to realise Grothendieck's desire of unification. She talks about her own passion for topos theory coming from it providing a place for different mathematical subjects to come into contact, and the incredible efficiency of this theory. She says that the unifying character of topos theory has generated much hostility (she even speaks of ostracism). The reason is not the technicality of topoi as mathematical objects, but rather the global and interdisciplinary character, which disturbs people. Mathematics have become hyper-specialized, the specialist works with his own methods and gets used to think in a certain way. There is a kind of dogmatism in certain mathematical circles, which results in getting used to a certain language and some sort of withdrawing. Around 52:30 she recounts a story of a model theorist spending an afternoon trying to convince her that a theorem of hers was "too general" to be true and trying to produce a counterexample.

Around 53:30, the last question is posed to Laurent Lafforgue, and he speaks about encountering a unique "hostilité" towards topos theory among mathematicians. On the other hand and very surprisingly to him, engineers from Huawei reacted very positively when informed about this theory. Several among them think that Grothendieck topoi could become the mathematics of artificial intelligence. Lafforgue finds it unimaginable that Grothendieck insisted on the importance of topoi already 60 years ago (this can be found in Récoltes et semailles), and it had no effect of the academic environment. Grothendieck does not understand this hostility.

Apparently there will be a second podcast to continue the discussion.

The podcast in question can be found here. Here is a very weak attempt at a summary. I'd welcome edits to correct or provide more detail. I worked backward from the end and didn't listen earlier than the following -- there might be relevant material earlier.

The context of this interview is the recent publication (2022) in France of Grothendieck's text Récoltes et semailles. In this text, Grothendieck criticized (sometimes severely) several mathematicians and complained about the attitude of the mathematical community in general.

Around 45:30, the journalist claims that topos theory gets very bad press and asks why to Connes, Caramello and Lafforgue. Connes says no, this is a completely external vision of reality. Prompted by the journalist, he explains what is topos theory, where one studies a space not by looking at it directly, but by putting him behind the scenes. One does ordinary mathematics, with a "parameter" which is in the space in question. This leads to a much finer knowledge of the space. He talks about topos theory's relationship to structuralism and category theory.

Around 48:50, Olivia Caramello is asked why topos theory has been rejected by the institutions. She says that she received a lot of opposition since the beginning of her career because she wanted to develop topos theory in a global and systematic way in order to realise Grothendieck's desire of unification. She talks about her own passion for topos theory coming from it providing a place for different mathematical subjects to come into contact, and the incredible efficiency of this theory. She says that the unifying character of topos theory has generated much hostility (she even speaks of ostracism). The reason is not the technicality of topoi as mathematical objects, but rather the global and interdisciplinary character, which disturbs people. Mathematics have become hyper-specialized, the specialist works with his own methods and gets used to think in a certain way. There is a kind of dogmatism in certain mathematical circles, which results in getting used to a certain language and some sort of withdrawing. Around 52:30 she recounts a story of a model theorist spending an afternoon trying to convince her that a theorem of hers was "too general" to be true and trying to produce a counterexample.

Around 53:30, the last question is posed to Laurent Lafforgue, and he speaks about encountering a unique "hostilité" towards topos theory among mathematicians. On the other hand and very surprisingly to him, engineers from Huawei reacted very positively when informed about this theory. Several among them think that Grothendieck topoi could become the mathematics of artificial intelligence. Lafforgue finds it unimaginable that Grothendieck insisted on the importance of topoi already 60 years ago, and it had no effect on the academic environment. In the text Récoltes et semailles, Grothendieck does not understand this hostility.

Apparently there will be a second podcast to continue the discussion.

The podcast in question can be found here. Here is a very weak attempt at a summary. I'd welcome edits to correct or provide more detail. I worked backward from the end and didn't listen earlier than the following -- there might be relevant material earlier.

Around 45:30, Alain Connes (I assume) is asked some kind of question which I didn't follow. He brings up topos theory, and gives some kind of explanation of what it is, as "mathematics with parameters" (rough quote). He talks about topos theory's relationship to structuralism and category theory.

Around 48:10, Olivia Caramello is asked some sort of question about the place of topos theory in the mathematical community. She talks about her own passion for topos theory coming from it providing a place for different mathematical subjects to come into contact. She talks about resistance to topos theory (I think I even heard "ostracisme" at some point?) being related to hyper-specialization in mathematics. Around 52:30 she recounts a story of a model theorist spending an afternoon trying to convince her that a theorem of hers was "too general" to be true and trying to produce a counterexample.

Around 53:30, the last question is posed to Laurent Lafforgue, and he speaks about encountering a unique "hostilité" towards topos theory among mathematicians. He mentions talking to engineers who think it could be (rough quote) "the language of artificial intelligence".

The podcast in question can be found here. Here is a very weak attempt at a summary. I'd welcome edits to correct or provide more detail. I worked backward from the end and didn't listen earlier than the following -- there might be relevant material earlier.

The context of this interview is the recent publication (2022) in France of Grothendieck's text Récoltes et semailles. In this text, Grothendieck criticized (sometimes severely) several mathematicians and complained about the attitude of the mathematical community in general.

Around 45:30, the journalist claims that topos theory gets very bad press and asks why to Connes, Caramello and Lafforgue. Connes says no, this is a completely external vision of reality. Prompted by the journalist, he explains what is topos theory, where one studies a space not by looking at it directly, but by putting him behind the scenes. One does ordinary mathematics, with a "parameter" which is in the space in question. This leads to a much finer knowledge of the space. He talks about topos theory's relationship to structuralism and category theory.

Around 48:50, Olivia Caramello is asked why topos theory has been rejected by the institutions. She says that she received a lot of opposition since the beginning of her career because she wanted to develop topos theory in a global and systematic way in order to realise Grothendieck's desire of unification. She talks about her own passion for topos theory coming from it providing a place for different mathematical subjects to come into contact, and the incredible efficiency of this theory. She says that the unifying character of topos theory has generated much hostility (she even speaks of ostracism). The reason is not the technicality of topoi as mathematical objects, but rather the global and interdisciplinary character, which disturbs people. Mathematics have become hyper-specialized, the specialist works with his own methods and gets used to think in a certain way. There is a kind of dogmatism in certain mathematical circles, which results in getting used to a certain language and some sort of withdrawing. Around 52:30 she recounts a story of a model theorist spending an afternoon trying to convince her that a theorem of hers was "too general" to be true and trying to produce a counterexample.

Around 53:30, the last question is posed to Laurent Lafforgue, and he speaks about encountering a unique "hostilité" towards topos theory among mathematicians. On the other hand and very surprisingly to him, engineers from Huawei reacted very positively when informed about this theory. Several among them think that Grothendieck topoi could become the mathematics of artificial intelligence. Lafforgue finds it unimaginable that Grothendieck insisted on the importance of topoi already 60 years ago (this can be found in Récoltes et semailles), and it had no effect of the academic environment. Grothendieck does not understand this hostility.

Apparently there will be a second podcast to continue the discussion.

The podcast in question can be found here. Here is a very weak attempt at a summary. I'd welcome edits to correct or provide more detail. I worked backward from the end and didn't listen earlier than the following -- there might be relevant material earlier.

Around 45:30, Alain Connes (I assume) is asked some kind of question which I didn't follow. He brings up topos theory, and gives some kind of explanation of what it is, as "mathematics with parameters" (rough quote). He talks about topos theory's relationship to structuralism and category theory.

Around 48:10, Olivia Caramello is asked some sort of question about the place of topos theory in the mathematical community. She talks about her own passion for topos theory coming from it providing a place for different mathematical subjects to come into contact. She talks about resistance to topos theory (I think I even heard "ostracisme" at some point?) being related to hypersecialization in mathematics. Around 52:30 she recounts a story of a model theorist spending an afternoon trying to convince her that a theorem of hers was "too general" to be true and trying to produce a counterexample.

Around 53:30, the last question is posed to Laurent Lafforgue, and he speaks about encountering a unique "hostilité" towards topos theory among mathematicians. He mentions talking to engineers who think it could be (rough quote) "the language of artificial intelligence".

The podcast in question can be found here. Here is a very weak attempt at a summary. I'd welcome edits to correct or provide more detail. I worked backward from the end and didn't listen earlier than the following -- there might be relevant material earlier.

Around 45:30, Alain Connes (I assume) is asked some kind of question which I didn't follow. He brings up topos theory, and gives some kind of explanation of what it is, as "mathematics with parameters" (rough quote). He talks about topos theory's relationship to structuralism and category theory.

Around 48:10, Olivia Caramello is asked some sort of question about the place of topos theory in the mathematical community. She talks about her own passion for topos theory coming from it providing a place for different mathematical subjects to come into contact. She talks about resistance to topos theory (I think I even heard "ostracisme" at some point?) being related to hyper-specialization in mathematics. Around 52:30 she recounts a story of a model theorist spending an afternoon trying to convince her that a theorem of hers was "too general" to be true and trying to produce a counterexample.

Around 53:30, the last question is posed to Laurent Lafforgue, and he speaks about encountering a unique "hostilité" towards topos theory among mathematicians. He mentions talking to engineers who think it could be (rough quote) "the language of artificial intelligence".