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BEGIN:VEVENT
UID:5d421924efb89a82dbb42a03a404bfc1
CATEGORIES:Symmetric Functions & Probability Theory Seminar
CREATED:20250123T202027
SUMMARY:Hyperdeterminantal total positivity
LOCATION:Hill 705
DESCRIPTION:Speaker: Donald Richards (https://science.psu.edu/stat/people/dsr11) (Penn 
 State) \nTitle: Hyperdeterminantal total positivity\nAbstract: This talk wi
 ll cover some recent joint research with K. W. Johnson. \nLet m be a positi
 ve integer, and let K be real-valued function defined on 2m-dimensional Euc
 lidean space.  We define the concept of hyperdeterminantal total positivity
  for the kernel K, thereby generalizing the classical concept of total posi
 tivity. We construct examples of hyperdeterminantal totally positive kernel
 s, extending in particular the fundamental classical example, K(x,y) = exp(
 xy), x,y ∈ ℝ, of a totally positive kernel.  By applying a hyperdeterminant
 al Binet-Cauchy argument, a generalization of Karlin's Basic Composition Fo
 rmula is derived and then applied to construct numerous additional examples
  of hyperdeterminantal totally positive kernels.  Further generalizations o
 f the concept of hyperdeterminantal total positivity by means of the theory
  of finite reflection groups are described, and some open problems are pose
 d.\n
X-ALT-DESC;FMTTYPE=text/html:<p dir="ltr" style="margin: 12px 0px 0px; outline: none; position: relative
 ; color: #212121; font-size: 11pt; font-style: normal; font-weight: 400; fo
 nt-family: Lato, sans-serif; line-height: 1.6667; letter-spacing: normal; o
 rphans: 2; text-align: start; text-indent: 0px; text-transform: none; widow
 s: 2; word-spacing: 0px; white-space: normal;"><span style="font-family: La
 to, Arial; font-variant: normal; font-weight: bold;">Speaker</span><span st
 yle="font-variant: normal;">:&nbsp;</span><a href="https://science.psu.edu/
 stat/people/dsr11" target="_blank" rel="noopener" style="color: inherit; te
 xt-decoration: none;"><span style="color: #006580; font-variant: normal; te
 xt-decoration: underline;">Donald Richards</span></a><span style="font-vari
 ant: normal;">&nbsp;(Penn State)&nbsp;</span></p><p dir="ltr" style="margin
 : 12px 0px 0px; outline: none; position: relative; color: #212121; font-siz
 e: 11pt; font-style: normal; font-weight: 400; font-family: Lato, sans-seri
 f; line-height: 1.6667; letter-spacing: normal; orphans: 2; text-align: sta
 rt; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 0px; w
 hite-space: normal;"><span style="font-family: Lato, Arial; font-variant: n
 ormal; font-weight: bold;">Title</span><span style="font-variant: normal;">
 : Hyperdeterminantal total positivity</span></p><p dir="ltr" style="margin:
  12px 0px 0px; outline: none; position: relative; color: #212121; font-size
 : 11pt; font-style: normal; font-weight: 400; font-family: Lato, sans-serif
 ; line-height: 1.6667; letter-spacing: normal; orphans: 2; text-align: star
 t; text-indent: 0px; text-transform: none; widows: 2; word-spacing: 0px; wh
 ite-space: normal;"><span style="font-family: Lato, Arial; font-variant: no
 rmal; font-weight: bold;">Abstract</span><span style="font-variant: normal;
 ">:&nbsp;</span><span style="color: #242424; font-variant: normal;">This ta
 lk will cover some recent joint research with K. W. Johnson.&nbsp;</span></
 p><p dir="ltr" style="margin: 12px 0px 0px; outline: none; position: relati
 ve; color: #212121; font-size: 11pt; font-style: normal; font-weight: 400; 
 font-family: Lato, sans-serif; line-height: 1.6667; letter-spacing: normal;
  orphans: 2; text-align: start; text-indent: 0px; text-transform: none; wid
 ows: 2; word-spacing: 0px; white-space: normal;"><span style="color: #24242
 4; font-variant: normal;">Let m be a positive integer, and let K be real-va
 lued function defined on 2m-dimensional Euclidean space.&nbsp; We define th
 e concept of&nbsp;</span><span style="color: #242424; font-style: italic; f
 ont-variant: normal;">hyperdeterminantal total positivity</span><span style
 ="color: #242424; font-variant: normal;">&nbsp;for the kernel K, thereby ge
 neralizing the classical concept of total positivity. We construct examples
  of hyperdeterminantal totally positive kernels, extending in particular th
 e fundamental classical example, K(x,y) = exp(xy), x,y ∈ ℝ, of a totally po
 sitive kernel.&nbsp; By applying a hyperdeterminantal Binet-Cauchy argument
 , a generalization of Karlin's Basic Composition Formula is derived and the
 n applied to construct numerous additional examples of hyperdeterminantal t
 otally positive kernels.&nbsp; Further generalizations of the concept of hy
 perdeterminantal total positivity by means of the theory of finite reflecti
 on groups are described, and some open problems are posed.</span><span styl
 e="color: #000000; font-variant: normal;"></span></p>
CONTACT:Donald Richards
DTSTAMP:20260828T140557
DTSTART;TZID=America/New_York:20250129T104500
DTEND;TZID=America/New_York:20250129T234500
SEQUENCE:0
TRANSP:OPAQUE
END:VEVENT
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