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We give a conjugacy relation on certain type of Frobenius manifold structures using the theory of flat pencils of metrics. It leads to a geometric interpretation for the inversion symmetry of solutions to Witten-Dijkgraaf-Verlinde-Verlinde…

We discuss a construction that gives counterexamples to various questions of unique determination of convex bodies.

In this paper we give an example to show Clemens' conjecture is not a first order deformation problem.

We explain a connection between the combinatorial Kashiwara-Vergne conjecture and the Kontsevich formula for quantization of Poisson manifolds

In this article, we prove a weighted version of Saitoh's conjecture. As an application, we prove a weighted version of Saitoh's conjecture for higher derivatives.

In this short note we prove the convexity of minimizers of some variational problem in the Gauss space. This proof is based on a geometric version of an older argument due to Korevaar.

One of the aims of this article is to provide a class of polynomial mappings for which the Jacobian conjecture is true. Also, we state and prove several global univalence theorems and present a couple of applications of them.

We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that…

Some conjectures and open problems in convex geometry are presented, and their physical origin, meaning, and importance, for quantum theory and generic statistical theories, are briefly discussed.

This is the first in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global confor- mal invariants"; these are defined to be conformally invariant integrals of geometric scalars.…

We settle the Hadwiger-Boltyanski Illumination Conjecture for all 1-unconditional convex bodies in ${\mathbb R}^3$ and in ${\mathbb R}^4$. Moreover, we settle the conjecture for those higher-dimensional 1-unconditional convex bodies which…

This is a survey of recent advances in commutative algebra, especially in mixed characteristic, obtained by using the theory of perfectoid spaces. An explanation of these techniques and a short account of the author's proof of the direct…

We give a counterexample to a recently conjectured variant of the Penrose inequality.

We establish a conjecture of Mumford characterizing rationally connected complex projective manifolds in several cases.

We provide a proof of a variant of the Landau-Siegel Zeros conjecture.

The equivariant coarse Novikov conjecture provides an algorithm for determining nonvanishing of equivariant higher index of elliptic differential operators on noncompact manifolds. In this article, we prove the equivariant coarse Novikov…

In the paper we complete a case by case proof of Reeder's Conjecture started in our previous work, proving the conjecture for simple Lie algebras of type $D$ and for the exceptional cases.

In this paper, we establish a general discrete Fourier restriction theorem. As an application, we make some progress on the discrete Fourier restriction associated with KdV equation.

In this paper, we prove the Geometric Arveson-Douglas Conjecture for a special case which allow some singularity on $\partial{\mathbb{B}_n}$. More precisely, we show that if a variety can be decomposed into two varieties, each having nice…

In this paper, we proved the normal scalar curvature conjecture and the Bottcher-Wenzel conjecture.