An algebro-geometric approach to S-duality and T-duality and proof of modularity conjectures in BPS counting theories
In String theory S-duality and T-duality each correspond to certain relation between two quantum field theories. The former corresponds to a duality between the two string theories with coupling constants which are inverse of each other and the latter describes a certain symmetry between two theories associated to different geometries. The talk is a brief report on algebro-geometric constructions of these dualities and exploiting them in proving modularity conjectures for certain BPS counting theories such as D4-D2-D0 and D6-D2-D0 supersymmetric BPS theories.
Topos Theory
Representations of the fundamental group
Topology of algebraic varieties and perverse sheaves II
Kirillov-Reshetikhin crystals, Macdonald polynomials, affine Demazure characters, and combinatorial models
In recent work with S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, we show that in all untwisted affine types the specialization of a Macdonald polynomial at t=0 is the graded character of a tensor product of one-column Kirillov-Reshetikhin (KR) modules. We also obtain two uniform models for the corresponding KR crystals, namely a generalization of the Lakshmibai-Seshadri (canonical Littelmann) paths based on the so-called parabolic quantum Bruhat graph, and the quantum alcove model of myself and A. Lubovsky. I will also mention other closely related topics: affine Demazure crystals (extending the work of Ion and Fourier-Littelmann), expressing the energy function, and a uniform realization of the combinatorial R-matrix, which commutes factors in a tensor product of KR crystals (with A. Lubovsky).
From quadratic polynomials and continued fractions to modular forms
New guests at the MPIM
Threshold functions for systems of equations on random sets
I will present a unified framework to deal with threshold
functions for the existence of certain combinatorial structures in random
sets. More precisely, let M·x=0 be a linear system defining our structure
(k-arithmetic progressions, k-sums, B_h[g] sets or Hilbert cubes, for
example), and A be a random set on {1,...,n} where each element is chosen
independently with the same probability.
I will show that, under certain natural conditions, there exists a
threshold function for the property "A^m contains a non-trivial solution of
Mx=0" which only depends on the dimensions of M. I will focus on the
behavior of the limiting distribution for the number of non-trivial
solutions in the threshold scale, and show that it follows a Poisson
distribution in terms of volumes of certain convex polytopes arising from
the linear system under study.
(Joint work with J. Rué)
Supersymmetric quantum mechanics on vortex moduli spaces
I will report on an ongoing project aiming at uncovering fundamental
features of N=(2,2) supersymmetric quantum mechanics on moduli spaces
of vortices on compact Riemann surfaces, in analogy with the spectrum
of quantum dyon-monopole bound states that emerged in connection with
Sen's S-duality conjectures in the 1990s. My focus in this talk will
be on the geometry underlying the coupling of waveforms to local
systems in effective theories for supersymmetric two-dimensional sigma-models
with toric gauge symmetry. I will consider models with
both linear and nonlinear targets; the corresponding ground states
can be investigated by means of the theory of L2-invariants. I shall
explain why the quanta of such abelian gauge theories can nontrivially
realize nonabelian statistics, and motivate a conjecture regarding a
nonlinear superposition principle for the ground states.
tba; Bachelor/Master-Seminar in Darstellungstheorie
Cycles on Jacobians
Yang-Mills theory
Topology of algebraic varieties and perverse sheaves I
The topology of the gauge group
Bing shrinking and triangulations of manifolds
On Drinfeld type automorphic forms and Rankin triple product L-functions over function fields
Automorphic forms of Drinfeld type can be viewed as function field analogue of weight 2 modular forms. After a brief review of basic facts about Drinfeld type cusp forms, I will discuss the Rankin triple product L-functions associated to these forms. From the Garrett-type integral representation, we obtain the functional equation of these L-functions. When the "root number" is positive, I present an analogue of the Gross-Kudla formula for the central critical values. Two examples will be shown at the end.
Finite Quadratic Modules and Weil Representations over Number Fields
In the study of Hilbert, Jacobi and orthogonal modular forms
of low weight over number fields it is essential to understand the
representations of Hilbert modular groups or of certain two-fold
central extensions. In the case of the field of natural numbers it is
known that the key to the study of all representations of the modular
group $SL(2,Z)$ which are interesting in the mentioned context are
the Weil representations associated to finite quadratic modules. In
analogy to the case of the field of rational numbers we developed a
theory of finite quadratic modules and their associated Weil
representations over arbitrary number fields. In this talk we report
about the main features of this new theory, about interesting new
phenomena arising in the general theory over arbitrary number fields,
and we indicate applications to the explicit construction of automorphic
forms over number fields.
The (co)tangent complex, derived intersections and cobordisms
Low dimensional collapsing theory and soul theorem for Alexandrov spaces
In this talk we will give an overview of the collapsing theory for 3-dimensional manifolds with a lower sectional curvature bound. This theory first developed by Shioya and Yamaguchi served as the completion of the last claim in Perelman's Proof of Geometrization Conjecture. The basic techniques used in the study will be presented with emphasis on Perelman's Fibration Theorem and Soul Theorem. Collapsing theory with bounded sectional curvature will also be quickly reviewed so that we can see the differences/parallel structures between these two theory.
Topos Theory
English
Deutsch