- Djordje Baralic (Mathematical Institute SASA, Belgrade, Serbia)
- Mai Hoàng Biên (HCMUS)
- Nguyễn Thanh Bình (HCMUS)
- William Cherry (University of North Texas, USA)
- Dương Hoàng Dũng (Kyushu University, Japan)
- Lý Kim Hà (HCMUS)
- Đặng Tuấn Hiệp (National Center for Theoretical Sciences, Taiwan)
- Nguyễn Quốc Hưng (Scuola Normale Superiore di Pisa, Italy)
- Nguyễn Tiến Khải (North Carolina State University, USA)
- Lê Hải Khôi (Nanyang Technological University, Singapore)
- Nguyễn Minh Quân (International University, VNU-HCM)
- Nguyễn Công Phúc (Lousiana State University, USA)
- Phan Thanh Toàn (Tôn Đức Thắng University)

## Titles and abstracts

- Djordje Baralic,
**Colorful KKM and the Lebesgue theorems**

We extend the Lebesgue theorem (on covers of cubes) and the Knaster–Kuratowski–Mazurkiewicz theorem (on covers of simplices) to different classes of convex polytopes. We also show that the \(n\)-dimensional Hex theorem admits a generalization where the \(n\)-dimensional cube is replaced by a \(n\)-colorable simple polytope. Our methods assume the use of specially designed quasitoric manifolds from toric topology, with easily computable cohomology rings and the cohomological cup-length and they offers a great flexibility and versatility in applications. - Mai Hoàng Biên,
**Subnormal subgroups in division rings with generalized power central group identities**

Let \(D\) be a division ring with center \(C\) and \(c\in D^*=D\backslash \{0\}\). In 1978, Herstein conjectured that if for every \(a\in D^*,\) there exists a positive integer \(n_a\) such that \((cac^{-1}a^{-1})^{n_a}\in C\), then \(c\in C\). In this talk, we provide some positive supports for the conjecture. References:

M. H. Bien,*Subnormal subgroups in division rings with generalized power central group identities*, Arch. Math. 106 (2016), 315--321. - Nguyễn Thanh Bình,
**Orthogonal Regularization for Training Recurrent Neural Networks**

Recurrent Neural Networks (RNNs) are difficult to train due to the well-known problem of exploding and vanishing gradient. Recent approaches have attempted to enforce the orthogonal or unitary transition matrix in order to stabilise the gradient. These methods often restrict the recurrent matrix to be unitary or orthogonal at each training iteration, which can be achieved either by doing a parametrization or a direct optimisation to enforce the constraint. Unlike the existing approaches, in this work, we propose an orthogonal regularisation scheme which relaxes the strict orthogonal constraint during RNN training. In particular, instead of restricting the recurrent matrix to be exactly orthogonal at every training iteration, the orthogonal regularization is able to keep the recurrent matrix being close to orthogonal to prevent the exploding and vanishing gradient without doing computationally intensive parametrization or extra optimization effort at every iteration. We show that the weak orthogonal regularisation is sufficient to address the exploding and vanishing gradient problem. We also investigate the generalization error when using the weak orthogonal condition on training general deep networks, and show that the generalisation error of a network is bounded based on the distance to the orthogonality of its parameters. Finally, we conduct an extensive set of experiments, in which our promising empirical results validate that the orthogonal regularizer can effectively address the vanishing and exploding gradient during RNN training, which not only benefits training better simple RNN, but also improves a range of RNN variants. - William Cherry,
**Non-Archimedean Function Theory**

I will survey non-Archimedean analogs of classical complex function theory, including Nevanlinna's theory of value distribution and Benedetto's analogs of the Ahlfors Island Theorems. - Dương Hoàng Dũng,
**On lattice subfield attack against NTRU**

In this talk, I will first introduce about post-quantum cryptography, specially lattice-based cryptography. Next, I will introduce about lattice subfield attack against NTRU developed recently. - Đặng Tuấn Hiệp,
**Schubert calculus on the Lagrangian Grassmannian**

Let \(LG\) be a complex Lagrangian Grassmannian parametrizing Lagrangian (i.e. maximal isotropic) subspaces in a complex symplectic vector space of dimension \(2n\). This talk is mainly devoted to the geometry of \( LG\) . More concretely, I will recall the definition of Schubert classes on \( LG\) and some basic results which are similar to the classical Pieri and Giambelli rules. A presentation of the cohomology ring of \( LG\) will be discussed. Finally, I will discuss recent results related to quantum cohomology of \( LG\). - Lý Kim Hà,
**Global Lipschitz continuity of Bergman projections in a class of convex domains in \(C^2\)** - Nguyễn Quốc Hưng,
**Gradient estimates for singular quasilinear elliptic equations with measure data** - Nguyễn Tiến Khải,
**Conservation laws and some applications to traffic flows** - Lê Hải Khôi,
**Complex symmetric weighted composition operators on the Fock space**

The general study of the complex symmetry was commenced by Garcia and Putinar (2006-2007). Thereafter, a number of the papers is devoted to the topic. The results show that the bounded complex symmetric operators are quite diverse. It includes the Volterra integration operators, normal operators, compressed Toeplitz operators, etc.A question about whether there exist the complex symmetric weighted composition operators was analyzed recently by Garcia-Hammond and Jung-Kim-Ko-Lee (2014), for Hardy spaces in the unit disk. Independently, they discovered the complex symmetric structure when the conjugation is of the form \(C f(z)=\overline{f(\bar{z})}\). In this talk, we study general weighted composition conjugations of the form \(\left(C_{\xi,\eta}f\right)(z)=\xi(z)\overline{f(\overline{\eta(z)})}\) on the Fock space \({\mathcal F}^2(\mathbb C)\) and study the conditions under which weighted composition operators are complex symmetric.The results are based on joint works with P.V. Hai. - Nguyễn Minh Quân,
**Soliton-like behavior in pulse collisions in perturbed linear systems of coupled-PDEs**In this talk, we present the soliton-like behavior in fast two-pulse collisions of pulses of weakly perturbed linear systems of coupld-PDEs. The behavior is demonstrated for linear systems of coupled-PDEs with weak cubic loss and for systems described by linear diffusion-advection models with weak quadratic loss. We show that in both systems, the expressions for the collision-induced amplitude shifts due to the nonlinear loss have the same form as the expression for the amplitude shift in a fast collision between two optical solitons in the presence of weak cubic loss. Our work shows that conclusions drawn from analysis of fast two-soliton collisions in the presence of dissipation can be applied for understanding the dynamics of fast two-pulse collisions in a large class of weakly perturbed linear physical systems. This is a joint work with Avner Peleg and Toan Huynh.

- Nguyễn Công Phúc,
**Quasilinear equations with gradient natural growth and distributional data** - Phan Thanh Toàn, De
**dekind-Mertens lemma and content formulas for polynomials and power series**Let \(R[X]\) and \(R[[X]]\) be the polynomial ring and the power series ring respectively over a commutative ring $R$ with identity. For \(f\in R[[X]]\), denote by \(A_f\) the content ideal of \(f\), i.e., the ideal of \(R\) generated by the coefficients of \(f\). The central question is “what is the relationship between the content ideal of the product \(fg\) and those of \(f\) and \(g\)?” Dedekind-Mertens lemma tells us that if \(f,g\in R[X]\), then

\[ (1)\qquad A_{f}^{k+1}A_{g}=A_{f}^{k}A_{fg} \]

for some positive integer \(k\) (depending only on \(g\)). Recently Epstein and Shapiro have successfully extended Dedekind-Mertens formula (1) to power series over Noetherian rings. They showed that if \(R\) is a Noetherian ring and if \(f,g\in R[[X]]\), then\[ A_{f}^{k+1}A_{g}=A_{f}^{k}A_{fg} \]

for some positive integer \(k\) (depending only on \(g\)). In this talk we introduce our recent results in the case \(R\) is non-Noetherian. We further present some other content formulas relating \(A_{f}\), \(A_{g}\), and \(A_{fg}\) for \(f,g\in R[[X]]\). This is a joint work with Byung Gyun Kang and Mi Hee Park.