Hugh Geller Elevator Pitch

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by Hugh Geller2 views·Updated Sep 13, 2020
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  1. Hi. My name is Hugh Geller, and I am a
  2. graduate student at Clemson University, and I'm going to be
  3. speaking at Champ on September 16th.
  4. So here's a little bit about me, my adviser, Sean
  5. Seder, white staff, and as such, I'm really interested in
  6. communicative and home a logical algebra time speaking at champ.
  7. Um, and in particular, I'm really interested in differential graded
  8. structures. And when I can put these structures on, um,
  9. various minimal resolutions oftentimes simply Shal or cellular.
  10. Um, in particular, I'm looking at minimal free resolutions of
  11. fiber products, and a long term goal is put gg
  12. ultra structures on these resolutions.
  13. But for now, I just trying to get the minimal
  14. free resolution of these fiber products.
  15. Um, so my set up in this case is gonna
  16. have to polynomial rings.
  17. They're gonna have a common base field.
  18. One of them is gonna have X variables.
  19. The other is gonna have y variables.
  20. And I'm going to take quotient rings of these polynomial
  21. rings, and I'm gonna take their fiber product over K.
  22. And I'm gonna view this is being perfecto.
  23. A question ring of the polynomial ring K Adjoined X's
  24. and y's mounted out by a specific ideal.
  25. Now, in this paper, I am actually ableto construct minimal
  26. resolutions for a bunch of different, um, potion rings using
  27. what I just call the star product.
  28. And this allows me Thio reestablish the results of Fisher
  29. Daniel Vischer in in 2006 where he explicitly wrote out
  30. the minimal free resolution for EJ ideals of complete by
  31. partite graphs.
  32. Those actually turned out to be, uh, the rings he's
  33. resolving turned out to be fiber products.
  34. From there, I take that construction and I'm able Thio
  35. build a minimal free resolution for all fiber products.
  36. With Cem, it's a modest restrictions, but if you come
  37. to my talk on the 16th, you'll see that the
  38. restrictions I have are are very modest and pretty standard,
  39. Um, and then having built all these resolutions as as
  40. a corollary, I have explicit formulas for Betty numbers and
  41. incorporates Siris for these fiber products Now, part of why
  42. I want these fiber products is resolutions of these fiber
  43. products is I wanna put differential graded structures on them.
  44. So another project that I'm working on that seems like
  45. it's going to split up into maybe two papers.
  46. Thio length is putting either a DJ algebra structure on
  47. the fiber product or a DJ module structure on the
  48. fiber product and that that's sort of where the splitting
  49. into two papers might be.
  50. Uh, so in this case, I've got some current results
  51. and then some partial results that I'm trying to extend
  52. a little bit more before submitting my paper.
  53. Uh, but taking that that star product from the fiber
  54. product paper under some more and kind of strong, uh,
  55. restrictions, I'm able to put a DJ algebra resolution structure
  56. on that product, which, um, allows me to put the
  57. DJ ultra structure on all of this year's resolutions because
  58. they all satisfy the extra requirements needed.
  59. Uh, but from there, the question becomes, Can I get
  60. a D G eligible structure for the minimal resolution of
  61. thes other fiber products?
  62. The answer, as of right now, is I don't know,
  63. but there are a ton of cases where I can
  64. get a D G module structure instead of algebra structure
  65. on dhe.
  66. This is a partial result because I'm tryingto push the
  67. result to, say, every fiber product, um, they're minimal free
  68. resolution is going to be a D G module on
  69. DDE and all these formulas are given explicitly.
  70. I'll share them in my talk.
  71. Things does lead me to some future projects.
  72. Ah, lot of what I'm doing is I'm trying Thio
  73. prove Cem reprove some tour friendly results on fiber products.
  74. Thes results were originally established by, say and say they're
  75. wag stuff, but they didn't used e g methods while
  76. other causes of rings have similar results that do used
  77. e g methods.
  78. So I'm just seeing if the d G stuff works
  79. here, Um, so a lot of that becomes, you know,
  80. can I get away with using DJ modules instead of
  81. DJ algebra?
  82. Or do I absolutely need DJ algebra?
  83. Xcx? Moreover, other projects include trying Thio get my Star
  84. construction, trying to remove some of the restrictions to make
  85. sure it's a D G algebra.
  86. Now, I also want to look to see if there
  87. are possibly D G applications to math biology, and I
  88. have some ideas in their infancy about how toe possibly
  89. use D G methods when studying neural rings.
  90. Uh huh.
  91. But if you want to hear me talk about any
  92. of the current D G stuff in fiber products stuff.
  93. Here are my upcoming talks, and I hope to see
  94. you at Tramp on 16th.