Audio Transcript Auto-generated
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Hi. My name is Tyler, anyway, and I'm a graduate
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student at U.
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T. A.
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I work under the direction of Dr David Jorgensen, and
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my primary research interests are in communicative and home, a
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logical algebra.
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Okay, so I'd like to give an overview of my
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thesis topics.
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There are two main areas of study, one being the
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classification of totally a cyclic complexes over certain rings and
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these other the study of triangle resolutions and their properties.
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So throughout the talk, I use ah, couple of terms
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that I would like to talk about real quick and
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the first being totally basically complexes.
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So these are the primary object of study in my
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research and in some sense, there a doubly infinite analog
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of re resolutions.
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Um, if you take the collection of all totally cyclic
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complexes and their home on top of equivalent classes of
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chain maps, you get the category of K tech and
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we do our primary work in K tech.
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The next being triangle resolutions and triangle resolutions are the
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triangulated analog of of resolutions.
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Um, these consists of complexes and chain maps between them
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and their built upon the idea of approximations and the
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axioms of a triangulated category, and we'll get into more
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of that later.
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So approximations originally defined by Auslander and Small Oh, and
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independently Knox in 1918 1981.
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Um, they're often referred to in literature as a cover
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or envelope or pre cover pre envelope, and they're a
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categorical notion of kind of similarity between objects.
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Um, so I'd like to do a deeper dive into
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my classification topic.
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And, um, in order to facilitate this classification scheme, I
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have defined a notion of Arnold pupils which apply for
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Total basically complexes over and selling and see em local
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rings. And these conditions are to make sure that my
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classifications is well defined and I can actually do them.
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So once I have this classification scheme, what I what
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I do is I focus in on rings of finite
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cm type where this classifications most appropriate.
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I use building blocks I use in decompose herbal, totally
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sick complexes as building blocks.
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And so, if you only have finally many, um, it
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makes your job a lot easier.
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One thing I do is I actually show that, um,
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some analogous properties of finite cm type in the module
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case extend nicely to the category of totally cycle complexes
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things like air quivers.
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So I'd like to take this classifications and use it
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to analyze the relationship between modules, where if you take
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the module and you find it's totally basically complex and
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those have totally different or very similar Arnold couples, what
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can you say?
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I'd also like to understand Arnold couples of mapping cones
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and what information you can get.
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Um, from that next, I'd really like to jump into
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triangle resolutions and triangle resolutions, as mentioned previously, our Anna
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logs of resolutions in the module case.
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Except now we are using the axioms and categorical methods
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to build them.
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So instead of taking kernels, we now apply mapping cones,
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and the way this is actually done is very interesting.
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You look at ah, totally basically complex, and if you
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approximate it, you get a map, and then you take
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the mapping cone of that and you can actually build
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a triangle out of that.
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And then, if you kind of do some, use the
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triangle at the triangulated category axioms.
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You can actually start to kind of repeat that process
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and build your resolution well, one important, um, one important.
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A property of your resolution, even in the module case,
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is minimal ity.
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Right? And so we try to extend this idea of
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minimal, minimal iti to the triangulated resolutions case.
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And so we have to define what it means to
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be a minimal triangle resolution.
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And under what conditions do we actually get a minimal
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triangle resolution?
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Well, once we have the idea of minimum ality, we
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can use that to kind of broaden our definitions on
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these resolutions.
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So one thing I'd like to do is use that
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use a minimal triangle resolution to define a Betty number.
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And once you have a betting number, Well, what does
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it mean?
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What does the complexity of this resolution And can I
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use that complexity to attempt to prove certain conjectures that
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have been imposed by my advisor?
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Well, that's all I have for today.
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Thank you for your time.
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And if you want to hear more, I'll be talking
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on the 16th in champs and I appreciate your time