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Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

Online research seminar : Algebra, Geometry & Physics

Tuesdays 2pm

Organisers: Gaëtan Borot (HU Berlin), Yuri Manin (MPIM)

New connection link: https://hu-berlin.zoom.us/j/61339297016

For HU students in Maths -- or Physics P27 or P28 -- this is 2SWS and you can get credits by regular attendance and writing at least one report on a chosen talk during the term.

Archive of talks

 

Upcoming talks


19 Jan. 2021

Alice Guionnet (ENS Lyon)
Topological expansions, random matrices and operator algebras theory
Slides

In this lecture, I will discuss the remarkable connection between random matrices and the enumeration of maps and some applications to operator algebras and physics. Part of my talk will be dedicated to my collaboration with Vaughan Jones on this subject.


26 Jan. 2021

Bruno Klingler (HU Berlin)
Tame geometry and Hodge theory
Slides

Originating in Grothendieck's "Esquisse d'un programme", tame geometry has been developed by model theorists under the name "o-minimal structures". It studies structures where every definable set has a finite geometry complexity. It has for prototype real semi-algebraic geometry, but is much richer. After recalling its basic features, I will describe its recent applications to Hodge theory and period maps.


2 Feb. 2021

Tomáš Procházka (LMU München)
W algebras and integrable structures
Slides

I will review basic properties of W-algebras, in particular the WN family and the associated Winfinity algebra. These algebraic structures show up at many places in mathematical physics. Winfinity admits two different descriptions: the traditional description starts from the Virasoro algebra of 2d conformal field theory and extends it by local conserved currents of higher spin. The description discovered more recently is the Yangian description manifesting the integrable structure of the algebra. The map between these two pictures is non-trivial, but can be understood by using the Maulik-Okounkov 'instanton' R-matrix as a bridge between these two pictures.

9 Feb. 2021

Karim Adiprasito (Hebrew University of Jerusalem)
Combinatorics, Lefschetz theorems beyond positivity and transversality of primes

I will survey applications of Hodge Theory to combinatorics, and, quite suprisingly, how Hodge-Riemann relations and Lefschetz type theorems can be proven using combinatorial methods, in settings that are beyond classical algebraic geometry, at least as long as some notion of positivity is available.

I then go one step further, and ask how many triangles a PL embedded simplicial complex in R4 can have (the answer, generalizing a result of Euler and Descartes, is at most 4 times the number of edges). I discuss how to reduce this problem to a Lefschetz property beyond projective structure.

The main part of the talk is devoted to provide an indication how the proof works, explain the notion of transversal primes as well as Hall matching theorems for spaces of linear operators, and their connection in the Hall-Laman relations which replace our knowledge of the signature of the Hodge-Riemann relations in the Kähler case with nondegeneracy at monomial ideals.


16 Feb. 2021

Vivek Shende (U. Berkeley/SDU Odense)
Holomorphic curves, boundaries, skeins, and recursion
Slides

I will explain how to define all genus open Gromov-Witten invariants for Calabi-Yau 3-folds.  The key idea is to count curves by their boundary in the skein modules of Lagrangians. Then I will prove the assertion of Ooguri and Vafa that the colored HOMFLYPT polynomials of a knot are exactly the counts of holomorphic curves in the resolved conifold with boundary on a Lagrangian associated to the knot. In the process we will see the geometric origin of recursion relations for colored knot invariants. This talk presents joint work with Tobias Ekholm. 

23 Feb. 2021

Vasily Golyshev (MCCME, Moscow)
Non-abelian Abel's theorems and quaternionic rotation

I will talk on a subject which has evolved over the years in discussions with M. Kontsevich, A. Mellit, V. Roubtsov, and D. van Straten. I will explain how the kernels of non-abelian Abel's theorems can be seen as a low-technology alternative to geometric Langlands, and produce a Clausen-type lift from the Markov to a quaternionic local system on a punctured genus 1 curve


2 Mar. 2021

Jean-Louis Colliot-Thélène (CNRS & Université Paris Saclay)
Hilbert's irreducibility theorem and jumps in the rank of the Mordell-Weil group
Slides

Let k be a number field and U a smooth integral k-variety. Let X → U be an abelian scheme of relative dimension at least one. We consider the set U(k)+ ⊂ U(k) of k-rational points m ∈ U(k) such that the Mordell-Weil rank of the fibre Xm at m, which is an abelian variety over k, is strictly bigger than the Mordell-Weil rank of the generic fibre Xk(U) over the function field k(U). We prove: if the k-variety X is k-unirational, then U(k)+ is dense for the Zariski topology on U. If the k-variety X is k-rational, then U(k)+ is not a thin set in U. The second result leads us to a discussion of varieties over which Hilbert's irreducibility theorem holds.


9 Mar. 2021

Thomas Willwacher (ETH Zürich)
Embeddings of manifolds in euclidean space and Feynman diagrams

The spaces of embeddings of manifolds M in Rn are classical objects in topology. I will discuss the computation of their rational homotopy type, for fairly general M, based on recent joint work with B. Fresse and V. Turchin. The answer can be formulated in terms of combinatorial (Feynman) diagrams.


16 Mar. 2021

Pedro Tamaroff (Trinity College Dublin)
Poincaré-Birkhoff-Witt theorems: homotopical and effective computational methods for universal envelopes.

In joint work with V. Dotsenko, we developed a categorical framework for Poincaré-Birkhoff-Witt type theorems about universal enveloping algebras of various algebraic structures, and used methods of term rewriting for operads to obtain new PBW theorems, in particular answering an open question of J.-L. Loday. Later, in joint work with A. Khoroshkin, we developed a formalism to study Poincaré–Birkhoff–Witt type theorems for universal envelopes of algebras over differential graded operads, motivated by the problem of computing the universal enveloping algebra functor on dg Lie algebras in the homotopy category. Our formalism allows us, among other things, to obtain a homotopy invariant version of the classical Poincaré–Birkhoff–Witt theorem for universal envelopes of Lie algebras, and extend Quillen's quasi-isomorphism C(g) → BU(g) to homotopy Lie algebras. I will survey and explain the role homological algebra, homotopical algebra, and effective computational methods play in the main results obtained with both V. Dotsenko (1804.06485) and A. Khoroshikin (2003.06055) and, if time allows, explain a new direction in which these methods can be used to study certain operads as universal envelopes of pre-Lie algebras.


23 Mar. 2021

Gereon Quick (NTNU Oslo)
From rational points to étale homotopy fixed points

To decide whether integral or rational solutions to polynomial equations exist is a classical problem in mathematics. Such solutions correspond to what we now call rational points on algebraic varieties. To detect such points is still a notoriously difficult task. In this talk we will build a bridge from rational points to homotopy theory and discuss how étale homotopy fixed points under the Galois action can be used to define obstructions for the existence of rational points. Along the way we will review ideas from étale homotopy theory and review the difference between fixed and homotopy fixed points under group actions.

30. Mar. 2021

Mark Gross (Cambridge Univ.)
Intrinsic mirror symmetry

I will talk about the program developed jointly with Bernd Siebert to understand mirror symmetry. In particular, I will focus on the general construction of mirrors. Associated to any log Calabi-Yau pair (X,D) with maximally degenerate boundary D or to any maximally unipotent degeneration of Calabi-Yau manifolds X→S, we associate a mirror object as either the Spec or Proj of a ring. This ring, analogous to the degree 0 part of symplectic homology of X\D, is constructed using a flavor of Gromov-Witten invariants called punctured invariants, developed jointly with Abramovich, Chen and Siebert.


6 Apr. 2021

Yuri Manin (MPIM Bonn)
TBA


13 Apr. 2021

Hülya Argüz (Université Versailles St-Quentin)
TBA


20 Apr. 2021

Anna Wienhard (Heidelberg Universität)
TBA


27 Apr. 2021

Yuri B. Suris (TU Berlin)
TBA


4 May 2021

TBA


11 May 2021

TBA


18 May 2021

TBA


25 May 2021

TBA