Talks in mathematical physics

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Spring Semester 2017

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Date / Time Speaker Title Location
23 February 2017
Wellington Galleas
ETH Zurich
Partition function of the elliptic gl2 SoS model as a single determinant  HG G 43 
Abstract: Partition functions of vertex models with domain-wall boundaries have appeared in a variety of contexts ranging from enumerative combinatorics to the study of gauge theories. The six-vertex model is the prototype model where such boundary conditions were adopted and, in that case, the model's partition function admits a compact representation in terms of a single determinant. Such representation has led to a number of developments in the field but seemed to be restricted to the six-vertex model. The next natural candidate for this study is the elliptic gl2 Solid-on-Solid (SoS) model but, despite all efforts, the results suggested single determinantal representations did not exist. This problem has only been recently solved with the help of special functional equations originating from the dynamical version of the Yang-Baxter algebra. In this talk I will discuss this problem and show how families of single determinantal representations for the elliptic SoS model's partition function can be derived in a constructive manner.
2 March 2017
Alexander Veselov
Loughborough University
Lyapunov spectrum for Markov dynamics and hyperbolic structures  HG G 43 
Abstract: We study the Lyapunov exponents Λ(x) for Markov dynamics as a function of path determined by x ∈ RP1 on a binary planar tree, describing the growth of Markov triples and their ``tropical" version - Euclid triples. We show that the corresponding Lyapunov spectrum is [0, ln φ], where φ is the golden ratio, and prove that on the set X of the most irrational numbers the corresponding function ΛX is convex and strictly monotonic. The key step is using the relation of Markov numbers with hyperbolic structures on punctured torus, going back to D. Gorshkov and H. Cohn, and, more precisely, the recent result by V. Fock, who combined this with Thurston’s lamination ideas. The talk is based on joint work with K. Spalding.
* 9 March 2017
Ping Xu
Penn State University
Formality theorem and Kontsevich-Duflo theorem for Lie pairs  HG G 19.1 
Abstract: A Lie pair (L,A) consists of a Lie algebra (or more generally, a Lie algebroid) L together with a Lie subalgebra (or Lie subalgebroid) A. A wide range of geometric situations can be described in terms of Lie pairs including complex manifolds, foliations, and manifolds equipped with Lie group actions. To each Lie pair (L,A) are associated two L-infinity algebras, which play roles similar to the spaces of polyvector fields and polydifferential operators. We establish the formality theorem for Lie pairs. As an application, we obtain Kontsevich-Duflo type theorem for Lie pairs. Besides using Kontsevich formality theorem, our approach is based on the construction of a dg manifold (L[1] + L/A, Q) together with a dg foliation, called the Fedosov dg Lie algebroid. This is a joint work with Hsuan-Yi Liao and Mathieu Stienon.
16 March 2017
Ben Kohli
Université de Bourgogne
The Links-Gould invariants seen as generalizations of the Alexander polynomial  HG G 43 
Abstract: The Links-Gould invariants are a family of two variable quantum link isotopy invariants. They are derived from Hopf superalgebras U_q(gl(m|n)). However we now know that the Alexander invariant can be obtained through several different evaluations from the Links-Gould polynomial,using work by De Wit-Ishii-Links, and more recently joint work with Bertrand Patureau-Mirand. Therefore all the information contained in the Alexander polynomial can be found inside the Links-Gould invariants, including the homological information: lower bound for the genus, criterion for fiberedness of knots, ... So we can wonder whether the Links-Gould invariants generalize any classical properties of the Alexander invariant. This seems to be true, which suggests that some classical construction should account for these quantum invariants. I will mainly focus on the two different techniques that were used to obtain the evaluations.
* 23 March 2017
Jens Hoppe
KTH, Stockholm
Quantized minimal surfaces HG G 19.1 
30 March 2017
Robert Seiringer
IST Austria
Stability of quantum many-body systems with point interactions  HG G 43 
Abstract: We present a proof that a system of N fermions interacting with an additional particle via point interactions is stable if the ratio of the mass of the additional particle to the one of the fermions is larger than some critical m∗. The value of m∗ is independent of N and turns out to be less than 1. This fact is of relevance for the stability of fermionic gases in the unitary limit. We also present a rigorous version of the Tan relations valid for all wave functions in the domain of the Hamiltonian of this model.
6 April 2017
Dev Sinha
University of Oregon
A program to construct integer-valued finite-type knot invariants  HG G 43 
Abstract: Using techniques closely related to his celebrated work in deformation theory, Kontsevich constructed an explicit, geometric universal Vassiliev invariant over the real numbers (or by deep results, over the rational numbers). Vassiliev invariants over the integers are far from settled. We outline a program to investigate them using the Goodwillie-Weiss tower for the embedding functor. We recently showed this tower has needed structure and provided spectral sequence evidence for the conjecture that the tower serves as a universal finite-type invariant over the integers. We have also developed Hopf invariants for homotopy groups and now more general mapping sets, which have the potential to detect components of the tower. We describe these two lines of progress and then ideas for next steps.

Archive: SS 17  AS 16  SS 16  AS 15  SS 15  AS 14  SS 14  AS 13  SS 13  AS 12  SS 12  AS 11 

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