The TRUTH about tensors (Part 1 ???) K-Theory 2021-12-12 | Following Michael Penn's recent and wonderful video, some people had questions about what this has to do with tensors in physics. In this off-the-cuff video, I outline some of what goes into connecting these ideas. Michael's video: youtube.com/watch?v=K7f2pCQ3p3U&t=205s&ab_channel=MichaelPennA classic intuitive explanation for physics: youtube.com/watch?v=f5liqUk0ZTw
Who Gives a Sheaf? Part 15: A concrete model for (co)limits K-Theory 2024-09-30 | We finally come back down to earth and provide an explicit model for (co)limits in certain scenarios. Examples of stalks as limits are given.
Who Gives a Sheaf? Part 14: Know your limits!! K-Theory 2024-09-19 | In this video we define limits as adjoints of the diagonal functor.
Who Gives a Sheaf? Part 13: Adjunction Addendum K-Theory 2024-09-18 | Talking about some more abstract nonsense about adjunctions. In particular, we explain how the unit of an adjunction can be used to define the counit.
Who Gives a Sheaf? Part 12: Pass the adjoint! K-Theory 2024-09-17 | We use our previous examples to motivate the definition of adjoint functors.
Who Gives a Sheaf? [With typos! (see comments)] Part 11: A universal property K-Theory 2024-09-16 | TYPO: Everywhere I say and write "product of stalks" I mean "disjoint union of stalks". We talk about the universal property of sheafification and probably go into too many details.
Who Gives a Sheaf? [With typos! (see comments)] Part 10: Forget everything! K-Theory 2024-09-15 | TYPO: Everywhere I write and say "product of stalks" I mean "disjoint union of stalks". (Thanks to Xander Lewis)In this video we talk about sheafification as a functor and its special friend the forgetful functor.(0:00) Sheafification as functor(10:55) Adjoints (but you don't know that yet)
Math Talk! Professor Wolfgang Soergel, Koszul duality K-Theory 2024-07-04 | A lovely chat with professor Wolfgang Soergel about life and math, under a tree.
Introduction to Lie algebras, Episode 5: Complexification K-Theory 2024-06-23 | In this video we talk about "complexifying" a real Lie algebra, and the notion of compact forms of complex reductive Lie algebras. Google site: sites.google.com/view/kristapsjohnbalodis/aboutLink Tree: https://linktr.ee/kristapsjohnbalodis(0:00) Introduction(1:10) Forgetting and complexifying vector spaces(8:30) Digression on abstract nonsense(11:00) Complexification of Lie algebras(17:00) Compact forms(23:15) Cartan subalgebras
Math Talk! Dr. Mishty Ray, local Langlands program K-Theory 2024-06-20 | Chatting with my good friend about the Ph.D. experience.
Math Talk! Dr. Adam Clay, Orderable Groups & Topology K-Theory 2024-06-01 | Better mics! Worse sound quality! A good time was had by all.
Introduction to Lie algebras, Episode 4: General themes, and representations K-Theory 2024-05-16 | In this video we discuss some ideas that arise in the classification of low-dimensional Lie algbras, write down some important examples, and discuss the definition of representations.https://linktr.ee/kristapsjohnbalodis
Math Talk! Dr. Anwesh Ray, Iwasawa theory & arithmetic statistics K-Theory 2024-05-12 | In this video we chat about mathematics, the mathematical process, and even the socio-political aspects of the field!Arizona Winter School: swc-math.github.iohttps://linktr.ee/kristapsjohnbalodis
Introduction to Lie algebras, Episode 3: Remarks on general structure and themes. K-Theory 2024-05-07 | In this video we define subalgebras, ideals, quotients, the derived subalgebra, direct sums, and the adjoint representation of a Lie algebra.Here's a PDF I found which offers a more hands-on approach to the classification of low-dimensional Lie algebras which appears in Chapter 10 of Fulton-Harris:https://mathweb.ucsd.edu/~abowers/downloads/survey/3d_Lie_alg_classify.pdfhttps://linktr.ee/kristapsjohnbalodis
Representations of GL(n, Qp), Epsiode 7: The Zelevinksy classification K-Theory 2024-05-05 | In this video we present the Zelevinksy classification of irreducible representations of p-adic GL(n, Qp).https://linktr.ee/kristapsjohnbalodis
Representations of GL(n, Qp), Episode 6: Zelevinsky Segments K-Theory 2024-05-04 | In this video we introduce Zelevinksy's notion of multisegments.https://linktr.ee/kristapsjohnbalodis
Representations of GL(n, Qp), Episode 5: Parabolic Induction and the Jacquet Functor K-Theory 2024-05-01 | It's all about adjunctions babyyyyyyyyyyyyyyy!
Introduction to Lie algebras, Episode 2: Its an Abelian world K-Theory 2024-05-01 | We define Abelian Lie algebras!https://linktr.ee/kristapsjohnbalodis
Representation of GL(n, Qp), Episode 4: A non-answer about smoothness K-Theory 2024-05-01 | In this video, I don't answer a question about smooth functions. My Stack Exchange question:math.stackexchange.com/questions/4233814/why-is-our-definition-of-smooth-function-on-a-totally-disconnected-space-theThe other Stack Exchange Question:mathoverflow.net/questions/64239/locally-constant-functions-with-compact-support-smoothFiona Murnaghan's Notes:math.utoronto.ca/murnaghan/courses/mat1196/rnotes.pdfhttps://linktr.ee/kristapsjohnbalodis
Lie Algebras, Episode 1: Introduction K-Theory 2024-04-29 | We look at the definition and some basic examples of Lie algebras.https://linktr.ee/kristapsjohnbalodis
Representation Theory of GL(n, Qp), Episode 3: Induction Introduction K-Theory 2024-04-29 | We take a first pass at induction for p-adic groups.
Representation Theory of GL(n, Qp), Episode 2: Finite Dimensional Irreducible Representations K-Theory 2024-04-23 | In this video we classify all the finite-dimensional irreducible representations, which is pretty easy since they're all 1-dimensional! Sans proof.
Representation theory of GL(n, Qp), Episode 1: The Basics K-Theory 2024-04-22 | The first in a series of videos about the representation theory of GL(n, Qp).https://linktr.ee/kristapsjohnbalodis
Who Gives a Sheaf? Part 9: Some operations K-Theory 2024-01-22 | A very casual and short video about some basic sheaf concepts.
Who Gives a Sheaf? Part 8: Babys first sheafification K-Theory 2024-01-21 | In this video we sheafify the constant presheaf.
Math Talk! David Jaramillo, Arithmetic Dynamics K-Theory 2023-11-18 | Finally back with more math talk! In this video we talk about life, math, and the arithmetic dynamics of elliptic curves.
Should you learn category theory? VVLOG: Oct 4th K-Theory 2023-10-04 | Just pop'n off about categories.
You want this book! Whats on My Shelf? Episode 6 K-Theory 2023-08-04 | In this video we talk about a couple of absolutely legendary books!Introduction (0:00)Functions of One Complex Variable (0:36)Riemann Surfaces (4:28)Gamm (8:40)Algebraic Number Theory (13:10)
Whats on my Shelf? Epsiode 5 K-Theory 2023-07-28 | Today we talk about why Hardy was #sorrynotsorry, THE textbook on commutative algebra, and some stuff I don't really understand.Introduction (0:00)A Mathematician's Apology (0:45)Complex Analysis (3:49)Lectures on Morse Homology (6:50)Introduction to Commutative Algebra (8:43)A Brief Introduction to Theta Functions (13:56)Analysis, Manifolds and Physics (16:30)
Underrated Number Theory, Whats on my shelf? Episode 4. K-Theory 2023-07-21 | In this video I discuss a few different texts, including the book I first learned number theory from, which is highly underrated.Free English translation of the first few hundred pages of Recoltes et Semailles:https://web.ma.utexas.edu/users/slaoui/notes/recoltes_et_semailles.pdfMy elementary number theory videos:youtube.com/playlist?list=PLnNqTHlK5sGKRS54aVKU6jxWsvYHwuAYGMy series on SL2(Z):youtube.com/playlist?list=PLnNqTHlK5sGJ7JOdRY7Cx8d2lnNlyLCYSFollow me on Twitter:twitter.com/KristapsBalodi3Introduction (0:00)Linear Algebraic Groups (0:30)Recoltes et Semailles (2:04)SL2(R) (9:00)Lectures on Rings and Modules (12:35)Automorphic Functions (15:38)
The Greatest Book Ever Written. Whats on my Shelf? Episode 3. K-Theory 2023-07-16 | Today we talk about a great triple of books spanning many different topics, but we start off with one of the greatest books ever printed. Infinite Series videos on ordinal numbers and Goodstein sequences. Part 1: youtube.com/watch?v=uWwUpEY4c8o&t=10s&ab_channel=PBSInfiniteSeriesPart 2: youtube.com/watch?v=oBOZ2WroiVY&t=572s&ab_channel=PBSInfiniteSeriesIntroduction (0:00)The Greatest Book (0:45)A Concise Course in Algebraic Topology (9:15)Introduction to Set Theory (13:12)
Whats on my shelf? Episode 2 K-Theory 2023-07-10 | In this video, I continue talking about random books on my floor, including one of the most underrated differential geometry texts of all time.Introduction (0:00)Algebraic Geometry (0:38)Automorphic Forms (3:57)An Introduction to the Langlands Program (6:15)Admissible Dual of GL(n) via Compact Open Subgroups (8:45)Analyse Harmonique dans les Systèmes de Tits Bornologiques de Type Affine (10:08)I Want to be a Mathematician (11:08)Differential Analysis on Complex Manifolds (12:48)
Who gives a sheaf? Part 7: Sheafification K-Theory 2023-07-09 | In this video, we begin to discuss the sheafification of a presheaf. Stack exchange thread:math.stackexchange.com/questions/146996/sheafification-of-a-presheaf-through-the-etale-spaceA video on vector bundles:youtube.com/watch?v=RqQ-Jm2wtAI&list=PLnNqTHlK5sGL8I-RH5dzU6s4ZnZdeTdUt&index=9&ab_channel=K-TheoryOverview (0:00)(re)Constructing sheaves (7:38)We love Abelian (19:25)Recap and looking ahead (25:30)
Whats on my shelf? Episode 1 K-Theory 2023-07-07 | Talking about some of the math books on my shelf. Geoff's thesis: arxiv.org/abs/2110.01130Konno's article: semanticscholar.org/paper/A-NOTE-ON-THE-LANGLANDS-CLASSIFICATION-AND-OF-OF-Konno/c6e856eb9f0682a705dd6c7b964b27ca97baf451Introduction (0:00)Linear Algebraic Groups (1:00)Automorphic Forms and Representations (3:45)Representations of Reductive Groups (8:15)Perverse Sheaves and Applications to Representation Theory (9:58)Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups (14:40)The Shape of Books to Come (17:30)
A random digression on the Iwahori-Hecke algebra of a reductive p-adic group. K-Theory 2023-07-06 | A continuation of a conversation on Overflow:mathoverflow.net/questions/450110/parabolic-induction-and-tensoring-iwahori-affine-hecke-algebras/450123?noredirect=1#comment1163520_450123
Analysis With Friends, Part 3: Sums of limits K-Theory 2023-07-01 | In this video we start to explore sums of limits, and ask ourselves what else might be possible with limits.
Analysis With Friends, Part 2: Limits K-Theory 2023-06-16 | In this video, we arrive at the formal definition of a limit, and prove some basic results.
Analysis With Friends, Part 1: Sequences K-Theory 2023-06-08 | In this video, we review the concept of sequences and connect our intuitive understanding of them to something more formal. We also begin a discovery-based approach to finding the "right" definition of the limit.Textbook: Introduction to Analysis by Edward GaughanSome set theory series:youtube.com/watch?v=rjV4AO0iIVY&list=PLjJhPCaCziSQyON7NLc8Ac8ibdm6_iDQf&ab_channel=AntonioMontalbanZF - axioms described by a Fields medalist:youtube.com/watch?v=R7pMHqIIdbc&list=PL8yHsr3EFj52EKVgPi-p50fRP2_SbG2oi&ab_channel=RichardE.BORCHERDSMy own series on sets:youtube.com/playlist?list=PLnNqTHlK5sGLD2zRusrX7q4uBshw9PNuDIntroduction & Overview (0:00)0.999... (6:20)Intuition (10:40)Definition (14:33)Exploring Limits (27:28)Homework (50:25)
Who Gives a Sheaf? Part 6: Natural Transformations K-Theory 2023-05-31 | In this video, I correct an error from the previous one and introduce more category theoretic terminology via sheaves.
Who gives a sheaf? Part 5: Some Categories K-Theory 2023-05-28 | In this video we introduce the notion of categories through some examples.Tom Leinster's "Basic Category Theory":arxiv.org/pdf/1612.09375.pdf
Who gives a sheaf? Part 4: Stalks K-Theory 2023-05-17 | In this video, we discuss a more classical definition of sheaves and germs.Introduction (0:00)Zooming in (1:04)Stalk definition (12:12)Tangent space (14:00)Helping out with morphisms (17:21)
Math Talk! Dr. Emily Riehl, to infinity categories and beyond. K-Theory 2023-02-09 | In this video I have a lovely discussion with Dr. Emily Riehl about math, HoTT, infinity categories, and more! Dr. Riehl's site, with links to publications: emilyriehl.github.ioDr. Riehl's band, Unstraight: unstraightmusic.comSpectra: http://lgbtmath.org
The TRUTH about TENSORS, Part 10: Frames K-Theory 2023-01-22 | What do the octonions have to do with spheres? Skip to the end of the video to find out!
Math Talk! Heejong Lee, Ph.D. Langlands Program K-Theory 2023-01-15 | This is the first official in-person Math Talk! the episode, hopefully with many more to come!
Who Gives a Sheaf? Part 3: Mighty Morphn Morphisms K-Theory 2023-01-13 | In this video we discuss the definition of a morphism of sheaves.
The TRUTH about TENSORS, Part 9: Vector Bundles K-Theory 2023-01-13 | In this video we define vector bundles in full abstraction, of which tangent bundles are a special case.
Who Gives a Sheaf? Part 2: A non-example K-Theory 2023-01-11 | In this video we compare two pre-sheaves, one which is a sheaf, and one which is not.
Who Gives a Sheaf? Part 1: A First Example K-Theory 2023-01-10 | We take a first look at (pre-)sheaves, as being inspired from first year calculus.
HoTT Boi Summer K-Theory 2023-01-05 | Interviews with famous people, introductions to type theory and Coq, cringy editing and puns, this video has it all babyyyyyyyyyy! It only took me half a year to compile this video about my summer school experience in Cortona Italy.The HoTT Book: https://www.cs.uoregon.edu/research/summerschool/summer14/rwh_notes/hott-book.pdfEgbert Rijke's book on HoTT: arxiv.org/abs/2212.11082Introduction (0:00)Exploring Italy (2:08)Intro to Type Theory (3:48)Intro to Coq (8:24) Spritz! (18:27)Homotopy Type Theory (18:46)0 =/= 1 (34:28)The Final Days (46:36)Interviews 1 (47:46)QED (53:52)Interviews 2 (54:39)Italian Pizza (1:05:29)$h!t Posting (1:06:16)
Math Talk! Dr. Andrej Bauer on proof assistants, constructive mathematics, philosophy, and more. K-Theory 2022-12-07 | In this wonderful discussion with Dr. Andrej Bauer we discuss a whole host of topics centering around constructive mathematics, and proof assistants.Support Ukraine through Shtab Dobra:Instagram: instagram.com/shtab.dobraFacebook: facebook.com/shtab.dobraPayPal: bf.shtabdobra@gmail.cometransfer: pochtanasha@gmail.comThe 5 stages of accepting constructivism:ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdfDrawing + proof assistants:homotopy.ioeff-lang.orgIs Dr. Bauer a constructivist?math.andrej.com/2012/10/03/am-i-a-constructive-mathematicianPropositions as [Types]:academic.oup.com/logcom/article-abstract/14/4/447/933532?redirectedFrom=fulltextSupporting Ukraine (0:00)Introduction & discovering mathematics (1:52)HoTT & Voevodsky (11:10)AI (17:50)Constructivism & plurality (23:35)Proof assistants (32:50)Folk knowledge (41:49)The future of proof & implicit knowledge (47:50)Philosophy (1:02:40)
Fundamental Groups & Local Systems (an Equivariant Ending) K-Theory 2022-12-04 | In this off-the-cuff video, I give a rambly bird's eye view of the equivalence of categories between local systems and representations of the fundamental group of a sufficiently nice topological space. I also discuss the corresponding story for the equivariant etale fundamental group of a variety.Introduction (0:00)Fundamental Groups (1:04)Local Systems (4:40)The Main Theorem (9:50)Getting Etale (20:40)Equivariance (25:50)
