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Abstracts for Seminar on Algebra, Geometry and Physics

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Eigenvalues and planar networks

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Speaker: 
Maria Podkopaeva
Date: 
Tue, 2012-05-15 14:00 - 14:50
Location: 
MPIM Lecture Hall
Parent event: 
Seminar on Algebra, Geometry and Physics

The Gelfand--Zetlin integrable system is defined by the set of
generalized eigenvalues of hermitian matrices. I will construct a
"tropical analog" of this set using the combinatorics of some weighted
planar graphs. The same methods can be used to give a description of the
Horn inequalities on the eigenvalues of sums of hermitian matrices.

Universal R-matrix and functional relations

Posted in
Speaker: 
A. V. Razumov
Affiliation: 
Institute for High Energy Physics, Protvino
Date: 
Tue, 2012-05-15 15:00 - 15:50
Location: 
MPIM Lecture Hall
Parent event: 
Seminar on Algebra, Geometry and Physics

It seems that the most productive, although not comprehensive, modern approach to the theory of quantum integrable systems is the approach based on the concept of quantum group invented by Drinfeld and Jimbo. In this approach all the objects describing the model and related to its integrability originate from the universal R-matrix and the functional relations, principal for the integration procedure, are consequences of the properties of the appropriate representations of the quantum group.

The Topos Approach to Quantum Theory And Generalised Gelfand Spectra of Noncommutative Operator Algebras

Posted in
Speaker: 
Andreas Döring
Affiliation: 
Dept. of Computer Science, U of Oxford
Date: 
Tue, 2012-05-29 14:00 - 15:00
Location: 
MPIM Lecture Hall
Parent event: 
Seminar on Algebra, Geometry and Physics

I will present an outline of the so-called topos approach to quantum theory. This approach provides
a new mathematical formulation of algebraic quantum theory based on structures in suitable presheaf
and sheaf topoi. In particular, to each noncommutative unital C*-algebra or von Neumann algebra,
a spectral presheaf is assigned which generalises the Gelfand spectrum of a commutative algebra.
This assignment is shown to be functorial and contravariant. Physically, the spectral presheaf plays

Birational Geometry of moduli spaces of sheaves on K3 surfaces via Bridgeland stability

Posted in
Speaker: 
Arend Bayer
Date: 
Tue, 2012-06-19 14:00 - 15:00
Location: 
MPIM Lecture Hall
Parent event: 
Seminar on Algebra, Geometry and Physics

We use wall-crossing for Bridgeland stability conditions to systematically study the birational
geometry of a moduli space M of Gieseker-stable sheaves on a K3 surface. In particular, we show:
- Any K-equivalent birational model of M appears as a moduli of Bridgeland stable objects, such that
the birational transformation is induced by wall-crossing.
- We complete Markman's proof the Kawamata-Morrison cone conjecture on the moveable cone of M.
- We establish the Hassett-Tschinkel/Huybrechts/Sawon conjecture on the existence of birational

Enumerative geometry of knot homologies

Posted in
Speaker: 
Sergei Gukov
Date: 
Tue, 2012-06-26 14:00 - 15:00
Location: 
MPIM Lecture Hall
Parent event: 
Seminar on Algebra, Geometry and Physics

I will explain an intriguing connection between homological invariants of knots and enumerative geometry of
toric Calabi-Yau 3-folds. This connection (motivated from physics) allows to formulate many existent knot
homologies (such as Khovanov homology, knot Floer homology, etc.) in a unified framework based on
counting supersymmetric configurations, called refined BPS states in the physics literature or motivic
Donaldson-Thomas invariants in the math literature. In the opposite direction, it implies certain integrality

Differential equations satisfied by integers and the field with one element

Posted in
Speaker: 
A. Buium
Affiliation: 
U. of New Mexico/MPI
Date: 
Tue, 2012-07-17 14:00 - 15:00
Location: 
MPIM Lecture Hall
Parent event: 
Seminar on Algebra, Geometry and Physics

One can view the Fermat quotient operations on (algebraic) integers as arithmetic analogues of partial
differential operators acting on functions. This leads to a ``differential calculus over the field with one element".
When the machinery  is applied to arithmetic curves one can  obtain diophantine applications  (e.g. to Heegner points).
When the machinery is applied to rings of Witt vectors one obtains an arithmetic analogue $\Gamma$ for the integers

Power structure over the Grothendieck ring of complex quasi-projective varieties and its applications

Posted in
Speaker: 
S. Gusein-Zade
Affiliation: 
Moscow Lomonosov St. U./MPI
Date: 
Tue, 2012-07-24 14:00 - 15:00
Location: 
MPIM Lecture Hall
Parent event: 
Seminar on Algebra, Geometry and Physics

A power structure over a ring is a method to give sense to an expression
of the form $(1 + a_1 t + a_2 t^2 + ...)^m$ where $a_i$ and $m$ belong to
the ring. A natural power structure over the Grothendieck ring of complex
quasi-projective varieties has a geometric description (due to the author
with A.Melle and I.Luengo). It can be used both for writing a number of
statements in a short form and for proving new ones. For example one can

Schrödinger equation, deformation theory and $tt^*$ geometry

Posted in
Speaker: 
Huijun Fan
Affiliation: 
Peking U.
Date: 
Tue, 2012-07-31 14:00 - 15:00
Location: 
MPIM Lecture Hall
Parent event: 
Seminar on Algebra, Geometry and Physics

Abstract: I will explain my recent work on the deformation theory of
Schr\"odinger operator related to a strongly tame section-bundle system
$(M,g,f)$. This is a differential geometric description of
Landau-Ginzburg B model. We can construct the Hodge theory after proving
a key spectrum theorem of the form Schr\"odinger operators, then prove
the stability theorem, and finally we can construct the $tt^*$ geometry
structure on the Hodge bundle of the moduli space. As one application, we
can get Frobenius manifold structure via primitive vector which is given

Ribbon Graphs and Mirror Symmetry

Posted in
Speaker: 
Eric Zaslow
Affiliation: 
(Northwestern U, Weinberg College of Arts and Sciences/MPI
Date: 
Tue, 2012-08-07 14:00 - 15:00
Location: 
MPIM Lecture Hall
Parent event: 
Seminar on Algebra, Geometry and Physics 
Starting from a ribbon graph, I will
define a category which serves as a stand-in
for the Fukaya category of the associated punctured
Riemann surface, thought of as a large-volume
limit.  When the ribbon graph has a combinatorial
version of a torus fibration with section, a
mirror "large complex" limit exists,
a singular algebraic curve.  In this
case, our category is equivalent to vector bundles
on the algebraic curve.

Positive G-laminations and surface affine Grassmannian

Posted in
Speaker: 
A. Goncharov
Affiliation: 
Yale U/MPI
Date: 
Tue, 2012-08-14 14:00 - 15:00
Location: 
MPIM Lecture Hall
Parent event: 
Seminar on Algebra, Geometry and Physics

This is a joint work with Linhui Shen (Yale).

Let  G be a split reductive group and S a topological surface with a finite set of points on
the boundary  modulo isotopy. Our goal is to define a canonical basis in the space of regular
functions on the space of G^L-local systems on S. Here G^L is the Langlands dual  group.

We define a set of positive G-lamination on S. It is the set of positive integral tropical points
of positive moduli space with potential. We introduce a surface afffine Grassmannian related

 

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