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Biao Ma

Publications and source records attributed to Biao Ma.

27 records · Page 2Linked to original sources
arXiv2022

Boundary representations of mapping class groups

Let $S = S_g$ be a closed orientable surface of genus $g \geq 2$ and $Mod(S)$ be the mapping class group of $S$. In this paper, we show that the boundary representation of $Mod(S)$ is ergodic using statistical hyperbolicity, which generalizes the classical result of Masur on ergodicity of the action of $Mod(S)$ on the projective measured foliation space $\mathcal{PMF}(S).$ As a corollary, we show that the boundary representation of $Mod(S)$ is irreducible.

math.GT↗
arXiv2022

Curve shortening flow on Riemann surfaces with possible ambient conic singularities

In this paper, we study the curve shortening flow (CSF) on Riemann surfaces. We generalize Huisken's comparison function to Riemann surfaces and surfaces with conic singularities. We reprove the Gage-Hamilton-Grayson theorem on surfaces. We also prove that for embedded simple closed curves, CSF can not touch conic singularities with cone angles $\leq π$.

math.DG↗
arXiv2021

On a family of unitary representations of mapping class groups

For a compact surface $S = S_{g,n}$ with $3g + n \geq 4$, we introduce a family of unitary representations of the mapping class group Mod($S$) based on the space of measured foliations. For this family of representations, we show that none of them has almost invariant vectors. As one application, we obtain an inequality concerning the action of Mod($S$) on the Teichmüller space of $S$. Moreover, using the same method plus recent results about weak equivalence, we also give a classification, up to weak equivalence, for the unitary quasi-representations with respect to geometrical subgroups.

math.GT↗
arXiv2019

Combinatorics of faithfully balanced modules

We study and classify faithfully balanced modules for the algebra of lower triangular $n$ by $n$ matrices. The theory extends known results about tilting modules, which are classified by binary trees, and counted with the Catalan numbers. The number of faithfully balanced modules is a $2$-factorial number. Among them are $n!$ modules with $n$ indecomposable summands, which can be classified by interleaved binary trees or by increasing binary trees.

math.RT↗
arXiv2019

On faithfully balanced modules, F-cotilting and F-Auslander algebras

We revisit faithfully balanced modules. These are faithful modules having the double centralizer property. For finite-dimensional algebras our main tool is the category ${\rm cogen}^1(M)$ of modules with a copresentation by summands of finite sums of $M$ on which ${\rm Hom}(-,M)$ is exact. For a faithfully balanced module $M$ the functor ${\rm Hom}(-,M)$ is a duality on these categories - for cotilting modules this is the Brenner-Butler theorem. We also study new classes of faithfully balanced modules combining cogenerators and cotilting modules. Then we turn to relative homological algebra in the sense of Auslander-Solberg and define a relative version of faithfully balancedness which we call $1$-$\mathbf{F}$-faithful. We find relative versions of the best known classes of faithfully balanced modules (including (co)generators ,(co)tilting and cluster tilting modules). Here we characterize the corresponding modules over the endomorphism ring of the faithfully balanced module - this is what we call a \emph{correspondence}. Two highlights are the relative (higher) Auslander correspondence and the relative cotilting correspondence - the second is a generalization of a relative cotilting correspondence of Auslander-Solberg to an involution (as the usual cotilting correspondence is).

math.RA↗
arXiv2014

Applications of shuffle product to restricted decomposition formulas for multiple zeta values

In this paper we obtain a recursive formula for the shuffle product and apply it to derive two restricted decomposition formulas for multiple zeta values (MZVs). The first formula generalizes the decomposition formula of Euler and is similar to the restricted formula of Eie and Wei for MZVs with one strings of 1's. The second formula generalizes the previous results to the product of two MZVs with one and two strings of 1's respectively.

math.NT↗