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SUMMARY:Yuuji Tanaka (Oxford)
DTSTART;VALUE=DATE-TIME:20200423T120000Z
DTEND;VALUE=DATE-TIME:20200423T140000Z
DTSTAMP;VALUE=DATE-TIME:20200705T043444Z
UID:AGNTISTA/1
DESCRIPTION:Title: Vafa-Witten invariants on projective surfaces\nby Yuuji
Tanaka (Oxford) as part of Algebraic Geometry and Number Theory seminar -
ISTA\n\n\nAbstract\nThe first half of this talk will be a gentle introduc
tion to the theory of Vafa-Witten invariants\, especially on projective su
rfaces. In the second half part - after a break- we focus more on computat
ional results. This talk is based on joint work with Richard Thomas.\n
END:VEVENT
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SUMMARY:Daniel Fiorilli (Université Paris-Sud)
DTSTART;VALUE=DATE-TIME:20200430T120000Z
DTEND;VALUE=DATE-TIME:20200430T140000Z
DTSTAMP;VALUE=DATE-TIME:20200705T043444Z
UID:AGNTISTA/2
DESCRIPTION:Title: On the distribution of the error term in Chebotarev's d
ensity theorem and applications\nby Daniel Fiorilli (Université Paris-Sud
) as part of Algebraic Geometry and Number Theory seminar - ISTA\n\n\nAbst
ract\nWe will discuss both extreme and generic values of the error term in
Chebotarev's density theorem. This will allow us to deduce applications o
n a conjecture of K. Murty on the least unramified prime ideal in a given
Frobenius set as well as on asymptotic properties of Chebyshev's bias.\n
END:VEVENT
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SUMMARY:Sergei Gukov (Caltech / MPI Bonn)
DTSTART;VALUE=DATE-TIME:20200521T160000Z
DTEND;VALUE=DATE-TIME:20200521T180000Z
DTSTAMP;VALUE=DATE-TIME:20200705T043444Z
UID:AGNTISTA/5
DESCRIPTION:Title: From the Generalized Volume Conjecture to Turaev and Ra
manujan\nby Sergei Gukov (Caltech / MPI Bonn) as part of Algebraic Geometr
y and Number Theory seminar - ISTA\n\n\nAbstract\nIn this talk\, intended
for a broad audience\, we will survey the development of a new 3-manifold
invariant that provides an answer to questions like this: What do Dedekind
eta and Alexander polynomial have in common? In fact\, illustrated by thi
s question is perhaps the most attractive feature of this new invariant: i
t provides new and often unexpected connections between different areas of
mathematics. Originating from complex Chern-Simons theory and quantizatio
n of $\\operatorname{SL}(2\,\\mathbb{C})$ character varieties\, it evaluat
es to $q$-series expressions that are more commonly seen in the theory of
mock modular forms and in logarithmic Vertex Operator Algebras (VOAs). The
goal of the talk will be to survey these relations using least amount of
technical details.\n
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SUMMARY:Peter Humphries (University College London)
DTSTART;VALUE=DATE-TIME:20200528T120000Z
DTEND;VALUE=DATE-TIME:20200528T130000Z
DTSTAMP;VALUE=DATE-TIME:20200705T043444Z
UID:AGNTISTA/6
DESCRIPTION:Title: Small scale equidistribution of lattice points on the s
phere\nby Peter Humphries (University College London) as part of Algebraic
Geometry and Number Theory seminar - ISTA\n\n\nAbstract\nConsider the pro
jection onto the unit sphere in $\\mathbb{R}^3$ of the set of lattice poin
ts $(x_1\, x_2\, x_3) \\in \\mathbb{Z}^3$ lying on the sphere of radius $\
\sqrt{n}$. Duke and Schulze-Pillot showed in 1990 that these points equidi
stribute on the sphere as $n \\to \\infty$. We study a small scale refinem
ent of this theorem\, where one asks whether these points equidistribute i
n subsets of the sphere whose surface area shrinks as $n$ grows. A particu
lar case of this is a conjecture of Linnik\, which states that for all $\\
delta > 0$\, the equation $x_1^2 + x_2^2 + x_3^2 = n$ has a solution with
$|x_3| < n^{\\delta}$ for all sufficiently large $n$. We make nontrivial p
rogress towards this\, as well as proving an averaged form of this conject
ure. This is joint work with Maksym Radziwiłł.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Junliang Shen (MIT)
DTSTART;VALUE=DATE-TIME:20200618T120000Z
DTEND;VALUE=DATE-TIME:20200618T140000Z
DTSTAMP;VALUE=DATE-TIME:20200705T043444Z
UID:AGNTISTA/8
DESCRIPTION:Title: On the topology of Hitchin fibrations\nby Junliang Shen
(MIT) as part of Algebraic Geometry and Number Theory seminar - ISTA\n\n\
nAbstract\nThe topology of the Hitchin fibrations plays crucial role in ge
ometry\, math physics\, and representation theory. In this talk\, we will
discuss two questions arised naturally in the study of Hitchin fibrations
in view of the P=W conjecture: (a) How to locate the tautological classes
in the perverse filtration? (b) Is the perverse filtration multiplicative
for Hitchin fibrations? \nI will explain how connnections to the geometry
of some special algebraic varieties (Hilbert schemes\, abelian surfaces\,
hyper-Kahler manifolds) lead to progress to answering these questions.\nB
ased on joint work with Mark de Cataldo\, Davesh Maulik\, Qizheng Yin\, Zi
li Zhang.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dimitri Wyss (L'Ecole polytechnique fédérale de Lausanne (EPFL))
DTSTART;VALUE=DATE-TIME:20200611T120000Z
DTEND;VALUE=DATE-TIME:20200611T140000Z
DTSTAMP;VALUE=DATE-TIME:20200705T043444Z
UID:AGNTISTA/9
DESCRIPTION:Title: P-adic integration\, geometry and Higgs bundles\nby Dim
itri Wyss (L'Ecole polytechnique fédérale de Lausanne (EPFL)) as part of
Algebraic Geometry and Number Theory seminar - ISTA\n\n\nAbstract\nIntegr
ation with respect to the Haar measure over a non-archimedean local field
F shares many formal properties with integration over the reals while at t
he same time being closely related to the arithmetic and geometry over the
residue field of F. In the first part I will give an overview of the theo
ry and explain two classical applications\, namely rationality of Igusa's
local zeta functions and Batyrev's proof of the equality of Hodge numbers
for smooth projective birational Calabi-Yau varieties.\n\nIn the second pa
rt I explain joint work with Michael Groechenig and Paul Ziegler\, where w
e apply these ideas to the moduli space of G-Higgs bundles. In quite gener
al situations we can relate p-adic volumes of Higgs spaces for Langlands-d
ual groups\, from which we derive two results: the topological mirror symm
etry conjecture of Hausel-Thaddeus\, which relates Hodge numbers for SL_n
and PGL_n Higgs spaces\, and the geometric stabilization theorem for aniso
tropic Hitchin fibers of Ngô. If time permits I will also discuss recent
ideas on how to effectively compute the p-adic volumes appearing in our ar
gument.\n
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