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CATEGORIES:Lecture / Talk / Workshop
DESCRIPTION:Yifeng Huang\, USC \n\n\nTitle: A quadratic form interpolating 
 \texttt{dinv}\n\n\nAbstract:\n\nThe Catalan number C_n=\frac{1}{n+1}\binom{
 2n}{n} is a famous sequence that is the answer to numerous counting problem
 s. Among them is the count of Dyck paths\, namely\, lattice paths above the
  diagonal in an n\times (n+1) rectangle. A celebrated refinement\, called t
 he q\,t-Catalan number\, counts Dyck paths while keeping track of two stati
 stics: \texttt{area} and \texttt{dinv}. While \texttt{area} has a straightf
 orward definition\, \texttt{dinv} is somewhat curious and admits several (n
 ot obviously) equivalent definitions. Why \texttt{dinv} is important owes m
 uch to the symmetry of the q\,t-Catalan numbers\, a remarkable theorem of G
 arsia and Haiman. \n\n\nIn this talk\, we show that \texttt{dinv} has a new
 \, "straightforward" definition using a quadratic form Q\, which initially 
 arises in my joint work on the geometry of singular curves with Jiang and O
 blomkov. More generally\, when evaluated on a pair of Dyck paths\, the symm
 etric bilinear form is nonnegative and has a combinatorial interpretation w
 e call "cross-\texttt{dinv}" (Theorem 1.2). We then use it to prove that Q 
 is positive definite on a certain cone. \n\n\nThe main engine\, Theorem 1.2
 \, is auto-formalized in Lean. This serves as an initial step in a bigger p
 roject with AxiomMath in the intersection of algebraic geometry and q-serie
 s.
DTEND:20260506T210000Z
DTSTAMP:20260511T155036Z
DTSTART:20260506T200000Z
GEO:34.022409;-118.291027
LOCATION:Kaprielian Hall (KAP)\, 265
SEQUENCE:0
SUMMARY:Combinatorics Seminar: A quadratic form interpolating \texttt{dinv}
UID:tag:localist.com\,2008:EventInstance_52763077363478
URL:https://calendar.usc.edu/event/combinatorics-seminar-a-quadratic-form-i
 nterpolating-textttdinv
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