## Modification of vector bundles

## Extremal entropy rigidities for Riemannian metric measure spaces

We extend the work of Ledrappier-Wang and Besson-Courtois-Gallot 's barycenter technique to apply it to RCD measure metric spaces.

We prove minimal and maximal volume entropy rigidity results and express them with a new notion of homotopic degree suitable for RCD spaces.

For example, we show that an RCD(-(N-1), N) space homotopic to a hyperbolic manifold M has total measure bounded below by M's hyperbolic

volume, and equality occurs if and only if the space is isometric to M.

Joint with Dai, Connell, Perales, Núñez Zimbrón, and Wei.

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## Multivariable polynomials of knots from Nichols algebras

Using finite dimensional Nichols algebras with automorphisms, we construct R-matrices with entries polynomials in the structure constants of the Nichols algebras. The R-matrices satisfy the Yang-Baxter equation and can be used to define multivariable polynomials of knots that generalize the colored Jones polynomial. For a Nichols algebra of rank 2, we define a sequence of polynomials $V_n(t,q)$ in two variables, whose first knot invariant is the Links-Gould polynomial and the second one $V_2$ is a new polynomial invariant that gives bounds for the Seifert-genus of a knot. Joint work with Rinat Kashaev.

## CAT(0) polygonal complexes are 2 median

A (1-)median space is a space in which for every three points the intersection of the three intervals between them is a unique point.

Having this in mind, in the talk I will define a 2-median space which is a 2 dimensional variation of the median space, explain the title

and present the idea of the proof of the theorem in the title.

This talk is based on my Master Thesis done under the supervision of Nir Lazarovich.

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## Towards non-perturbative quantization and the mass gap problem for the Yang-Mills Field

In 2000 Jaffe and Witten stated the long standing problem of exact quantization for the 4D Yang-Mills field on the Minkowski space and the existence of a mass gap in the theory as one on the Clay Institute Millenium problems. In this talk I shall explain how to reduce the problem of quantization of the Yang-Mills field Hamiltonian to a problem for defining a probability measure on an infinite-dimensional space of gauge equivalence classes of connections, the so called Yang-Mills measure associated to a 3D Yang-Mills theory on a Euclidean space. A formally self-adjoint expression for the quantized Yang-Mills Hamiltonian as an operator on the corresponding Lebesgue L_2-space will be presented.

In the case when the Yang-Mills field is associated to the abelian group U(1) the Yang-Mills measure will be defined. This measure is Gaussian and depends on a real parameter m>0. The quantized Hamiltonian, which is the corresponding Ornstein–Uhlenbeck operator, can be realized a self-adjoint operator in a Fock space. Its spectrum has a gap separating the rest of the spectrum from the ground state zero eigenvalue. This yields a non-standard quantization of the Hamiltonian of the electromagnetic field.

Recently, based on Hairer's theory of regularity structures, Chandra, Chevyrev, Hairer and Shen developed an approach for constructing Yang-Mills measures for Euclidean Yang-Mills theories in 2D and 3D associated to arbitrary compact Lie groups. Using the corresponding Langevin equation they defined a Markov process on a space of gauge orbits for which the Yang-Mills measure should be defined as the invariant measure provided that the process converges. This result is proved in 2D and the problem is still open in the physically important 3D case. In this framework the quantized Yang-Mills Hamiltonian is the generator of the stochastic process the convergence of which is equivalent to the existence of the mass gap.

The presentation in this talk will be self-contained and requires no special background from the audience.

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