BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Caroline Turnage-Butterbaugh (Carleton College)
DTSTART;VALUE=DATE-TIME:20220922T160000Z
DTEND;VALUE=DATE-TIME:20220922T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/1
DESCRIPTION:Title: Moments of Dirichlet L-functions\nby Caroline Turnage-Butterbau
gh (Carleton College) as part of FRG Grad Seminar (Averages of of L-functi
ons and Arithmetic Stratification)\n\n\nAbstract\nIn recent decades there
has been much interest and measured progress in the study of moments of L-
functions. Despite a great deal of effort spanning over a century\, asympt
otic formulas for moments of L-functions remain stubbornly out of reach in
all but a few cases. I will begin this talk by reviewing what is known fo
r moments of the Riemann zeta-function on the critical line\, and we will
then consider the problem for the family of all Dirichlet L-functions of e
ven primitive characters of bounded conductor. A heuristic of Conrey\, Far
mer\, Keating\, Rubenstein\, and Snaith gives a precise prediction for the
asymptotic formula for the general 2kth moment of this family. I will out
line how to harness the asymptotic large sieve to prove an asymptotic form
ula for the general 2kth moment of approximations of this family. The resu
lt\, which assumes the generalized Lindelöf hypothesis for large values o
f k\, agrees with the prediction of CFKRS. Moreover\, it provides the firs
t rigorous evidence beyond the so-called “diagonal terms” in their con
jectured asymptotic formula for this family of L-functions. This is joint
work with Siegfred Baluyot.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hua Lin (UC Irvine)
DTSTART;VALUE=DATE-TIME:20220926T160000Z
DTEND;VALUE=DATE-TIME:20220926T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/3
DESCRIPTION:Title: One-level density of zeros of Dirichlet L-function over function fi
elds\nby Hua Lin (UC Irvine) as part of FRG Grad Seminar (Averages of
of L-functions and Arithmetic Stratification)\n\n\nAbstract\nFor this talk
\, we compute the one-level density of zeros of cubic and quartic Dirichle
t $L$-functions over function fields $\\mathbb{F}_q[t]$ in the Kummer sett
ing ($q\\equiv1\\pmod{\\ell}$) and for order $\\ell=3\,4\,6$ in the non-Ku
mmer setting ($q\\not\\equiv1\\pmod{\\ell}$). In each case\, we obtain a m
ain term predicted by Random Matrix Theory (RMT) and a lower order term no
t predicted by RMT. We also confirm the symmetry type of the family is uni
tary\, supporting the Katz and Sarnak philosophy. I will first talk about
some history and background on the subject\, make the analogy and describe
the primitive characters over function fields in each setting\, and then
show the computation in more detail.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Louis Gaudet (Rutgers University)
DTSTART;VALUE=DATE-TIME:20221003T160000Z
DTEND;VALUE=DATE-TIME:20221003T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/4
DESCRIPTION:Title: The least Euler prime via a sieve approach\nby Louis Gaudet (Ru
tgers University) as part of FRG Grad Seminar (Averages of of L-functions
and Arithmetic Stratification)\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/frggradseminar/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lasse Grimmelt (University of Oxford)
DTSTART;VALUE=DATE-TIME:20221010T160000Z
DTEND;VALUE=DATE-TIME:20221010T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/5
DESCRIPTION:Title: Primes in large arithmetic progressions and applications to additiv
e problems\nby Lasse Grimmelt (University of Oxford) as part of FRG Gr
ad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\
nAbstract\nResults on the distribution of primes up to $X$ in an arithmeti
c progression with modulus $q$ fall\, depending on the relative size of $q
$ and $X$\, roughly speaking into three categories. For small $q$ (say up
to a power of $\\log X$)\, multiplicative analytic methods in the form of
Dirichlet L-functions are used\, in the medium range ($q < N^{1/2-\\epsilo
n}$) the large sieve gives us the Bombieri-Vinogradov Theorem\, and finall
y one can handle slightly larger $q$ by bounds for sums of Kloosterman sum
s. In this talk I will give a background about these results and highlight
some recent progress in the third category. I will also explain how this
progress can be applied to additive problems involving (subsets of) the pr
imes.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Goldston (San José State University)
DTSTART;VALUE=DATE-TIME:20221017T160000Z
DTEND;VALUE=DATE-TIME:20221017T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/6
DESCRIPTION:Title: Small Gaps and Spacings between Riemann zeta-function zeros\nby
Dan Goldston (San José State University) as part of FRG Grad Seminar (Av
erages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nI w
ill discuss joint work with Hung Bui\, Micah Milinovich\, and Hugh Montgom
ery on differences between consecutive zeros of the Riemann zeta-function
that are smaller than the average spacing between zeros. We assume the Rie
mann Hypothesis. One result is that by using the pair correlation method o
ne can prove there is a positive proportion of consecutive zeros closer th
an 0.6039 times the average spacing. One limitation of this method is that
these close pairs of zeros could all be multiple zeros\, and thus the met
hod may not be finding any small gaps between zeros at all - here we requi
re a gap between two numbers to have non-zero length because that is what
a gap is. We refer to differences between consecutive zeros including diff
erences equal to zero as “spacings”. There are three methods known to
deal with close zeros\, and all three actually produce small spacings betw
een zeros rather than small gaps. (One method is unconditional\, the other
two assume RH.) For small gaps\, or differences between distinct zeros\,
the three methods only produce gaps larger than the average spacing. Our s
econd result is based on a new fourth method that on RH proves there are s
mall gaps between zeros closer than 0.991 times the average spacing betwee
n zeros. The method however does not produce a positive proportion of such
gaps\, and I believe proving this on RH for a positive proportion is a di
fficult problem.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:George Dickinson (University of Manchester)
DTSTART;VALUE=DATE-TIME:20221219T170000Z
DTEND;VALUE=DATE-TIME:20221219T180000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/7
DESCRIPTION:Title: Second moments of Dirichlet L-functions\nby George Dickinson (U
niversity of Manchester) as part of FRG Grad Seminar (Averages of of L-fun
ctions and Arithmetic Stratification)\n\n\nAbstract\nThe asymptotic formul
ae for moments of L-functions are well studied objects in analytic number
theory as they are useful tools when investigating the L-functions themsel
ves. Often especially useful are the moments that have been twisted by a D
irichlet polynomial\, and the longer the twist the better. However\, findi
ng formulae gets more difficult as the length increases. In this talk\, we
will compare methods for finding different types of twisted second moment
s of Dirichlet L-functions\, as well as looking at some of their applicati
ons.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emma Bailey (CUNY)
DTSTART;VALUE=DATE-TIME:20221031T160000Z
DTEND;VALUE=DATE-TIME:20221031T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/8
DESCRIPTION:Title: Large values of $\\zeta$ on the critical line\nby Emma Bailey (
CUNY) as part of FRG Grad Seminar (Averages of of L-functions and Arithmet
ic Stratification)\n\n\nAbstract\nSelberg’s central limit theorem tells
us that typically $|\\zeta(1/2 + it)|$ is of size $\\exp(\\sqrt{\\log \\lo
g T})$ for $t\\in [T\, 2T]$. One can ask about /atypical/ values\, or abou
t large deviations to Selberg’s central limit theorem. By exploring a co
nnection between $\\zeta$ and branching random walks\, we are able to show
that the Gaussian tail extends to the right\, on the scale of the varianc
e. In this talk I will focus on the connection to branching random walks a
nd show how this probabilistic interpretation allows us to understand larg
e values of zeta. This is based on joint work with Louis-Pierre Arguin.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daodao Yang (TU Graz)
DTSTART;VALUE=DATE-TIME:20221107T170000Z
DTEND;VALUE=DATE-TIME:20221107T180000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/9
DESCRIPTION:Title: Large values of derivatives of the Riemann zeta function and relate
d problems\nby Daodao Yang (TU Graz) as part of FRG Grad Seminar (Aver
ages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nLarge
values of the Riemann zeta function and L-functions are classical topics
in analytic number theory\, which can be dated back to a result of Bohr an
d Landau in 1910. Resonance methods are modern tools to produce large valu
es of zeta and L-functions. GCD sums are one of important ingredients\, wh
ich naturally appears in a Diophantine approximation problem considered by
Hardy and Littlewood in 1922. I will talk on producing large values of de
rivatives of zeta and L-functions via resonance methods. On the other hand
\, I will talk on conditional upper bounds and asymptotic formulas when as
suming RH (GRH) and a conjecture of Granville-Soundararajan on character s
ums. If time permits\, the log-type GCD sums and related spectral norms wi
ll be discussed.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jakob Streipel (University of Maine)
DTSTART;VALUE=DATE-TIME:20221114T160000Z
DTEND;VALUE=DATE-TIME:20221114T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/10
DESCRIPTION:Title: Using second moments to count zeros\nby Jakob Streipel (Univer
sity of Maine) as part of FRG Grad Seminar (Averages of of L-functions and
Arithmetic Stratification)\n\n\nAbstract\nUsing Selberg's somewhat strang
e looking version of the argument principle\, it is possible to count zero
s of families of L-functions using upper bounds on second moments. We will
explore this argument principle\, how one uses it\, and some applications
of it to various zero counting problems\, old and new.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Dunn (Caltech)
DTSTART;VALUE=DATE-TIME:20221121T170000Z
DTEND;VALUE=DATE-TIME:20221121T180000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/11
DESCRIPTION:Title: Bias in cubic Gauss sums: Patterson's conjecture\nby Alexander
Dunn (Caltech) as part of FRG Grad Seminar (Averages of of L-functions an
d Arithmetic Stratification)\n\n\nAbstract\nWe prove\, in this joint work
with Maksym Radziwill\, a 1978 conjecture of S. Patterson (conditional on
the Generalised Riemann hypothesis) concerning the bias of cubic Gauss su
ms. This explains a well-known numerical bias in the distribution of cubic
Gauss sums first observed by Kummer in 1846.\n\nOne important byproduct
of our proof is that we show Heath-Brown's cubic large sieve is sharp unde
r GRH. This disproves the popular belief that the cubic large sieve can b
e improved.\n\nAn important ingredient in our proof is a dispersion estima
te for cubic Gauss sums. It can be interpreted as a cubic large sieve wit
h correction by a non-trivial asymptotic main term.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aled Walker (King's College\, London)
DTSTART;VALUE=DATE-TIME:20221128T170000Z
DTEND;VALUE=DATE-TIME:20221128T180000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/12
DESCRIPTION:Title: Correlations of sieve weights and distributions of zeros\nby A
led Walker (King's College\, London) as part of FRG Grad Seminar (Averages
of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nIn this t
alk\, we will briefly review Montgomery's pair correlation conjecture for
the zeros of the Riemann zeta function\, before discussing a (conditional
) partial lower bound on the Fourier transform of this pair correlation fu
nction: the so-called 'form factor' $F_T(x)$. The methods\, based in part
on ideas of Goldston and Gonek\, utilise some new correlation estimates fo
r Selberg sieve weights.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Asif Zaman (University of Toronto)
DTSTART;VALUE=DATE-TIME:20221205T170000Z
DTEND;VALUE=DATE-TIME:20221205T180000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/13
DESCRIPTION:Title: Random multiplicative functions and a simplified model\nby Asi
f Zaman (University of Toronto) as part of FRG Grad Seminar (Averages of o
f L-functions and Arithmetic Stratification)\n\n\nAbstract\nOver the past
few years\, there has been a lot of interest in random multiplicative func
tions and their partial sums. This subject has many intriguing questions a
nd connections to other areas of number theory and probability. In joint w
ork with Soundararajan\, we have introduced a simplified model of partial
sums of random multiplicative functions and established a result parallel
to Harper’s breakthrough on better-than-squareroot cancellation. In this
expository talk\, I will review some of the history of random multiplicat
ive functions\, and illustrate how random multiplicative functions connect
to our simplified model.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Dobner (University of Michigan)
DTSTART;VALUE=DATE-TIME:20221212T170000Z
DTEND;VALUE=DATE-TIME:20221212T180000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/14
DESCRIPTION:Title: Optimization and moment methods in number theory\nby Alexander
Dobner (University of Michigan) as part of FRG Grad Seminar (Averages of
of L-functions and Arithmetic Stratification)\n\n\nAbstract\nA common tech
nique in analytic number theory is to turn a number theoretic problem into
some sort of optimization problem which is hopefully more tractable. A we
ll known example is the Selberg sieve method which turns classical sieving
problems into a quadratic optimization problem. This technique also appe
ars in conjunction with the so-called moment method from probability theor
y. In this talk I'll summarize several instances of this including finding
primes in bounded intervals\, finding small/large gaps between zeta zeros
\, and finding large values of Dirichlet series.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathan Ng (University of Lethbridge)
DTSTART;VALUE=DATE-TIME:20230123T170000Z
DTEND;VALUE=DATE-TIME:20230123T180000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/15
DESCRIPTION:Title: The eighth moment of the Riemann zeta function\nby Nathan Ng (
University of Lethbridge) as part of FRG Grad Seminar (Averages of of L-fu
nctions and Arithmetic Stratification)\n\n\nAbstract\nIn recent work (http
s://arxiv.org/abs/2204.13891)\, Quanli Shen\, Peng-Jie Wong\, and I have s
hown that the Riemann hypothesis and a conjecture for quaternary additive
divisor sums implies the conjectured asymptotic for the eighth moment of t
he Riemann zeta function. This builds on earlier work on the sixth moment
of the Riemann zeta function (Ng\, Discrete Analysis\, 2021). One key dif
ference is that sharp bounds for shifted moments of the zeta function on t
he critical line are required. In this talk\, I will discuss some of the
ideas that go into the proof.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Farmer (American Institute of Mathematics)
DTSTART;VALUE=DATE-TIME:20230130T170000Z
DTEND;VALUE=DATE-TIME:20230130T180000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/16
DESCRIPTION:Title: The zeta function when it is particularly large\nby David Farm
er (American Institute of Mathematics) as part of FRG Grad Seminar (Averag
es of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nWhat do
es the zeta function look like in a neighborhood of its largest values? N
obody knows for sure\, because particularly large values have never been c
omputed. We will give a plausible answer by combining theorems from analy
tic number theory\, first principles reasoning\, and examples of random ch
aracteristic polynomials.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matilde Lalín (Université de Montréal)
DTSTART;VALUE=DATE-TIME:20230417T160000Z
DTEND;VALUE=DATE-TIME:20230417T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/17
DESCRIPTION:by Matilde Lalín (Université de Montréal) as part of FRG Gr
ad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\nA
bstract: TBA\n
LOCATION:https://researchseminars.org/talk/frggradseminar/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrés Chirre (University of Rochester)
DTSTART;VALUE=DATE-TIME:20230612T160000Z
DTEND;VALUE=DATE-TIME:20230612T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/18
DESCRIPTION:by Andrés Chirre (University of Rochester) as part of FRG Gra
d Seminar (Averages of of L-functions and Arithmetic Stratification)\n\nAb
stract: TBA\n
LOCATION:https://researchseminars.org/talk/frggradseminar/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Keshav Aggarwal (Alfréd Rényi Institute of Mathematics)
DTSTART;VALUE=DATE-TIME:20230206T170000Z
DTEND;VALUE=DATE-TIME:20230206T180000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/19
DESCRIPTION:Title: Bound for the existence of prime gap graphs\nby Keshav Aggarwa
l (Alfréd Rényi Institute of Mathematics) as part of FRG Grad Seminar (A
verages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nGi
ven a sequence $\\mathbf{D}$ of non-negative integers\, it is interesting
to know whether there exists a graph with vertices of degrees equaling the
integers in $\\mathbf{D}$. If that happens\, we say $\\mathbf{D}$ is grap
hic. Clearly\, if the sequence is graphic\, then the sum of its members mu
st be even. However\, it is not self-evident whether a given sequence is g
raphic. There are exponentially many different realizations for almost eve
ry graphic degree sequence. At the same time\, the number of all graphic d
egree sequences is infinitesimal compared to the number of integer partiti
ons of the sum of the degrees. Therefore it is incredibly hard to come up
with an interesting (or non-trivial) graphic degree sequence.\n\nLet us ca
ll a simple graph on $n>2$ vertices a prime gap graph if its vertex degree
s are $1$ and the first $n-1$ prime gaps. Recently\, Erdős-Harcos-Kharel-
Maga-Mezei-Toroczkai showed that the prime gap\nsequence is graphic for la
rge enough $n$. In a joint work with Robin Frot\, we make their work effec
tive.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rizwanur Khan (University of Mississippi)
DTSTART;VALUE=DATE-TIME:20230213T170000Z
DTEND;VALUE=DATE-TIME:20230213T180000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/20
DESCRIPTION:by Rizwanur Khan (University of Mississippi) as part of FRG Gr
ad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\nA
bstract: TBA\n
LOCATION:https://researchseminars.org/talk/frggradseminar/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katy Woo (Princeton University)
DTSTART;VALUE=DATE-TIME:20230220T170000Z
DTEND;VALUE=DATE-TIME:20230220T180000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/21
DESCRIPTION:by Katy Woo (Princeton University) as part of FRG Grad Seminar
(Averages of of L-functions and Arithmetic Stratification)\n\nAbstract: T
BA\n
LOCATION:https://researchseminars.org/talk/frggradseminar/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ian Whitehead (Swarthmore College)
DTSTART;VALUE=DATE-TIME:20230313T160000Z
DTEND;VALUE=DATE-TIME:20230313T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/22
DESCRIPTION:by Ian Whitehead (Swarthmore College) as part of FRG Grad Semi
nar (Averages of of L-functions and Arithmetic Stratification)\n\nAbstract
: TBA\n
LOCATION:https://researchseminars.org/talk/frggradseminar/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Zenz (Brown University)
DTSTART;VALUE=DATE-TIME:20230320T160000Z
DTEND;VALUE=DATE-TIME:20230320T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/23
DESCRIPTION:by Peter Zenz (Brown University) as part of FRG Grad Seminar (
Averages of of L-functions and Arithmetic Stratification)\n\nAbstract: TBA
\n
LOCATION:https://researchseminars.org/talk/frggradseminar/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aleksander Simonič (UNSW Canberra)
DTSTART;VALUE=DATE-TIME:20230410T160000Z
DTEND;VALUE=DATE-TIME:20230410T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/24
DESCRIPTION:by Aleksander Simonič (UNSW Canberra) as part of FRG Grad Sem
inar (Averages of of L-functions and Arithmetic Stratification)\n\nAbstrac
t: TBA\n
LOCATION:https://researchseminars.org/talk/frggradseminar/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eun Hye Lee (Stony Brook University)
DTSTART;VALUE=DATE-TIME:20230424T160000Z
DTEND;VALUE=DATE-TIME:20230424T170000Z
DTSTAMP;VALUE=DATE-TIME:20230205T210154Z
UID:frggradseminar/25
DESCRIPTION:by Eun Hye Lee (Stony Brook University) as part of FRG Grad Se
minar (Averages of of L-functions and Arithmetic Stratification)\n\nAbstra
ct: TBA\n
LOCATION:https://researchseminars.org/talk/frggradseminar/25/
END:VEVENT
END:VCALENDAR