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Talks and seminars, possibly part of a conference or series.

The Beauville-Voisin conjecture for Hilb(K3) and the Virasoro algebra

Posted in
Speaker: 
Andrei Negut
Affiliation: 
MIT
Date: 
Thu, 2019-07-04 10:30 - 12:00
Location: 
MPIM Lecture Hall

We give a geometric representation theory proof of a mild version of the Beauville-Voisin Conjecture for Hilbert schemes of K3 surfaces, namely the injectivity of the cycle map restricted to the subring of Chow generated by tautological classes. Our approach involves lifting formulas of Lehn and Li-Qin-Wang from cohomology to Chow groups, and using them to solve the problem by invoking the irreducibility criteria of Virasoro algebra modules, due to Feigin-Fuchs. Joint work with Davesh Maulik.

DAHA approach to algebraic knots and links

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Speaker: 
Ivan Cherednik
Affiliation: 
(ETH-ITS/UNC at Chapel Hill)
Date: 
Wed, 2019-07-24 14:00 - 15:00
Location: 
MPIM Seminar Room
Parent event: 
Extra talk

DAHA generally provide refined invariants of colored iterated links, which generalize the
WRT-invariants and HOMFLY-PT polynomials. In the uncolored case and for iterated knots,
they are conjectured to coincide with the stable reduced Khovanov-Rozansky polynomials (the most
powerful numerical invariants we have). The "intrinsic" DAHA conjectures are mostly verified
at the moment; these properties are generally difficult to check topologically. The DAHA super-
duality is an important example (a theorem for DAHA, but far from obvious in topology).
Its conjectural coincidence with the functional equation in the motivic approach (my next talk),
can be a fundamental development.  We will focus on the DAHA construction in this talk, with
some explicit calculations (for trefoil and beyond). 

Vertex Operator Algebras and Modular Forms

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Speaker: 
Don Zagier
Affiliation: 
MPIM
Date: 
Tue, 2019-07-02 16:30 - 18:00
Location: 
MPIM Lecture Hall

Time: Tuesdays, 4.30 - 6 pm
Place: MPIM Lecture Hall, Vivatsgasse 7
First lecture: on April 2, 2019, end on July 2

Recent developments in Quantum Topology

Posted in
Speaker: 
Stavros Garoufalidis
Affiliation: 
MPIM
Date: 
Tue, 2019-07-02 12:30 - 14:00
Location: 
MPIM Lecture Hall

We will review the basics of quantum topology such as the colored Jones polynomial of a knot, its standard conjectures relating to asymptotics, arithmeticity and modularity, as well as the recent quantum hyperbolic invariants of Kashaev et al, their state-integrals and their structural properties. The course is aimed to be accessible by graduate students and young researchers.

The singular (co)chains and rational homotopy theory, II

Posted in
Speaker: 
Felix Wierstra
Affiliation: 
MPIM
Date: 
Mon, 2019-07-01 16:30 - 17:30
Location: 
MPIM Lecture Hall
Parent event: 
MPIM Topology Seminar

The central theme of this talk is the question: How much of the rational homotopy type of a space can we deduce from the (co)chains on the space? More precisely, in this talk we explain what the relationship is between the singular (co)chains and various approaches to the rational homotopy theory. 

 
Bousfield and Kan showed that there are two possible approaches to non-simply-connected homotopy theory. Both their approaches are given by a completion on the level of spaces. The first goal of this talk is to explain the relationship between these completions and the singular chains. On the singular chains with rational coefficients of a space there are several different notions of equivalence, the most important ones are given by quasi-isomorphisms and $\Omega$-quasi-isomorphisms, which are maps that become a quasi-isomorphisms after applying the cobar construction. In the first part of the talk we show that these two notions of weak equivalence correspond to the two completions of Bousfield and Kan.
 
The second goal of the talk focuses on the cochains on a simply-connected space. A famous theorem by Sullivan shows that two simply-connected spaces are rationally equivalent if and only if their commutative algebras of polynomial de Rham forms can be connected by a zig-zag of quasi-isomorphisms of 
commutative algebras. Since the polynomial de Rham forms are quasi-isomorphic to the singular cochains as associative algebras it is a natural question to ask whether the cochains seen as an associative algebra also determine the rational homotopy type of a space. We will show that this is indeed the case, by showing that the much more general statement that two commutative algebras can be connected by a zig-zag of commutive quasi-isomorphisms if and only if they can be connected by a zig-zag of associative quasi-isomorphisms.
 
This talk is based on a combination of two projects, one joint with Manuel Rivera and Mahmoud Zeinalian and one joint with Ricardo Campos, Dan Petersen and Daniel Robert-Nicoud.

 

The singular (co)chains and rational homotopy theory, I

Posted in
Speaker: 
Felix Wierstra
Affiliation: 
MPIM
Date: 
Mon, 2019-07-01 15:00 - 16:00
Location: 
MPIM Lecture Hall
Parent event: 
MPIM Topology Seminar

The central theme of this talk is the question: How much of the rational homotopy type of a space can we deduce from the (co)chains on the space? More precisely, in this talk we explain what the relationship is between the singular (co)chains and various approaches to the rational homotopy theory. 

 
Bousfield and Kan showed that there are two possible approaches to non-simply-connected homotopy theory. Both their approaches are given by a completion on the level of spaces. The first goal of this talk is to explain the relationship between these completions and the singular chains. On the singular chains with rational coefficients of a space there are several different notions of equivalence, the most important ones are given by quasi-isomorphisms and $\Omega$-quasi-isomorphisms, which are maps that become a quasi-isomorphisms after applying the cobar construction. In the first part of the talk we show that these two notions of weak equivalence correspond to the two completions of Bousfield and Kan.
 
The second goal of the talk focuses on the cochains on a simply-connected space. A famous theorem by Sullivan shows that two simply-connected spaces are rationally equivalent if and only if their commutative algebras of polynomial de Rham forms can be connected by a zig-zag of quasi-isomorphisms of 
commutative algebras. Since the polynomial de Rham forms are quasi-isomorphic to the singular cochains as associative algebras it is a natural question to ask whether the cochains seen as an associative algebra also determine the rational homotopy type of a space. We will show that this is indeed the case, by showing that the much more general statement that two commutative algebras can be connected by a zig-zag of commutive quasi-isomorphisms if and only if they can be connected by a zig-zag of associative quasi-isomorphisms.
 
This talk is based on a combination of two projects, one joint with Manuel Rivera and Mahmoud Zeinalian and one joint with Ricardo Campos, Dan Petersen and Daniel Robert-Nicoud.

 

On Graphs of Hecke operators and Hall algebras

Posted in
Speaker: 
Roberto Alvarenga
Affiliation: 
University of Sao Paolo, Sao Carlos/University of California, Irvine
Date: 
Thu, 2019-06-27 16:30 - 17:30
Location: 
MPIM Lecture Hall
Parent event: 
Extra talk

In Bombay 1979, Don Zagier observes that if the kernel of certain
operators on automorphic forms turns out to be an unitarizable
representation, over the field of rational numbers $Q$, a formula of Hecke implies the Riemann hypothesis.

Zagier calls the elements of this kernel toroidal automorphic forms.
Moreover, Zagier asks what happens if $Q$ is replaced by a global function field and remarks that  the space of unramified toroidal automorphic forms can be expected to be finite dimensional.

Motivated by these questions, Oliver Lorscheid  introduces, in 2012,
the  *graphs of Hecke operators* for global function fields.

This theory allowed him to prove, among other things,  that the space
of unramified toroidal automorphic forms for a global function field
is indeed, finite dimensional. The graphs of Hecke operators
introduced by Lorscheid encode the action of Hecke operators on
automorphic forms.

On the other hand, Ringel(1990), Kapranov (1997), Burban and
Schiffmann (2012) et al. have been developing the theory of  *Hall
algebras * of coherent sheaves over a smooth geometric irreducible projective
curve over a finite field (in general for a finitary category).

For this talk we discuss the connection between graphs of Hecke
operators and Hall algebras.

In the elliptic case, Atiyah's work on vector bundles (1957) allow us
to describe (explicitly) these graphs.

Deformations of VB-groupoids

Posted in
Speaker: 
Pier Paolo La Pastina
Affiliation: 
La Sapienza, Rome
Date: 
Thu, 2019-07-04 13:30 - 15:00
Location: 
MPIM Lecture Hall

VB-groupoids can be understood as vector bundles in the category of Lie groupoids. They encompass several classical objects, such as Lie group representations, 2-vector spaces, Lie group actions on vector bundles; moreover, they provide geometric pictures for 2-term representations up to homotopy of Lie groupoids, in particular the adjoint representation. In this talk, I will attach to every VB-groupoid a cochain complex controlling its deformations and discuss some features, such as Morita invariance, as well as some examples and applications. I will also compare it to its infinitesimal counterpart, the linear deformation complex of a VB-algebroid, via a Van Est map. This is joint work with Luca Vitagliano.

Hochschild cohomology and deformation quantization of affine toric varieties

Posted in
Speaker: 
Matej Filip
Affiliation: 
Johannes Gutenberg-Universität Mainz
Date: 
Thu, 2019-06-27 09:15 - 10:15
Location: 
MPIM Lecture Hall
Parent event: 
Extra talk

For an affine toric variety we give a convex geometric description of the Hodge decomposition of its Hochschild cohomology. Moreover, we show that every Poisson structure on a possibly singular affine toric variety can be quantized in the sense of deformation quantization.

Semistability and Approximate Hermitian-Einstein Structures

Posted in
Speaker: 
Johannes Schäfer
Affiliation: 
MPIM
Date: 
Fri, 2019-06-28 10:15 - 11:45
Location: 
MPIM Seminar Room

tba

Posted in
Speaker: 
Rafael Herrera
Affiliation: 
CIMAT, Mexico
Date: 
Wed, 2019-08-07 10:30 - 12:00
Location: 
MPIM Lecture Hall

Frobenius lifts and correspondences on GL_n

Posted in
Speaker: 
Alexandru Buium
Affiliation: 
University of New Mexico/MPIM
Date: 
Wed, 2019-07-03 14:30 - 15:30
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Let GL_n be the general linear group scheme over the integers.We show that the p-adic completion of GL_n possessescertain remarkable Frobenius lifts attached to integral symmetric matrices; these Frobenius liftscan be viewed as  arithmetic analogues, for Spec Z,  of the Levi-Civita connection attached to a metric on a manifold. We then show how these Frobeniuslifts admit algebraizations by correspondences on GL_n; the commutatorsof these correspondences can be viewed as an arithmetic analogue, for Spec Z, of Riemannian curvature. 
 

tba

Posted in
Speaker: 
Ralph Kaufmann
Affiliation: 
Purdue University West Lafayette
Date: 
Tue, 2019-07-09 13:50 - 14:50
Location: 
MPIM Lecture Hall

Vertex Operator Algebras and Modular Forms

Posted in
Speaker: 
Don Zagier
Affiliation: 
MPIM
Date: 
Tue, 2019-06-25 16:30 - 18:00
Location: 
MPIM Lecture Hall

Time: Tuesdays, 4.30 - 6 pm
Place: MPIM Lecture Hall, Vivatsgasse 7
First lecture: on April 2, 2019, end on July 2

Recent developments in Quantum Topology

Posted in
Speaker: 
Stavros Garoufalidis
Affiliation: 
MPIM
Date: 
Tue, 2019-06-25 12:30 - 14:00
Location: 
MPIM Lecture Hall

We will review the basics of quantum topology such as the colored Jones polynomial of a knot, its standard conjectures relating to asymptotics, arithmeticity and modularity, as well as the recent quantum hyperbolic invariants of Kashaev et al, their state-integrals and their structural properties. The course is aimed to be accessible by graduate students and young researchers.

Recent developments in Quantum Topology

Posted in
Speaker: 
Stavros Garoufalidis
Affiliation: 
MPIM
Date: 
Tue, 2019-06-18 12:30 - 14:00
Location: 
MPIM Lecture Hall

We will review the basics of quantum topology such as the colored Jones polynomial of a knot, its standard conjectures relating to asymptotics, arithmeticity and modularity, as well as the recent quantum hyperbolic invariants of Kashaev et al, their state-integrals and their structural properties. The course is aimed to be accessible by graduate students and young researchers.

Vertex Operator Algebras and Modular Forms

Posted in
Speaker: 
Don Zagier
Affiliation: 
MPIM
Date: 
Tue, 2019-06-18 16:30 - 18:00
Location: 
MPIM Lecture Hall

Time: Tuesdays, 4.30 - 6 pm
Place: MPIM Lecture Hall, Vivatsgasse 7
First lecture: on April 2, 2019, end on July 2

Registration

Posted in
Date: 
Mon, 2019-07-08 08:30 - 09:30

Program discussion

Posted in
Date: 
Tue, 2019-07-09 12:00 - 12:30
Location: 
MPIM Lecture Hall

Program discussion

Posted in
Date: 
Mon, 2019-07-08 12:00 - 12:30
Location: 
MPIM Lecture Hall
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