Monday, 24 April 2023

Since the first use of lattices in the Knapsack cipher system in 1982 for cryptographic purposes, lattice-based cryptography has begun to draw attention to its potential cryptographic applications. Above all, with the fact that lattice-based problems are resistant to attacks by quantum computers, lattice-based cryptosystems are considerd as one of main post quantum cryptography. Since then, Fully Homomorphic Encryption (FHE) has been also developed based on the Ideal lattice, and the entire cryptographic community has begun to pay attention to the lattice, and applications of various techniques are being studied. In this presentation, we introduce lattice-based cryptography and its applications.

The first such association was AWM, the Association for Women in Mathematics, established 50 years ago in the USA. National associations and regional associations were established over the ensuing decades: the Australian one, Women in Mathematics Special Interest Group (WIMSIG) of the Australian Mathematical Society (AustMS), WIMSIG established only in 2012. I will speak about links between these associations and the International Mathematical Union, in particular the role of the IMU in acknowledging, including, and promoting women in mathematics.

I will talk about Integrable modules of loop-toroidal Lie algebras with finite dimensional weight spaces, when a part of the center acts non-trivially on the modules.

Community detection has been extensively developed using various algorithms. One of the most powerful algorithms for undirected graphs is Walktrap, which determines the distance between vertices by employing random walk and evaluates clusters using modularity based on vertex degrees. Although several directions have been explored to extend this method to directed graphs, none of them have been effective. In this paper, we investigate the Walktrap algorithm and the spectral method and extend them to directed graphs. We propose a novel approach in which the distance between vertices is defined using hitting time, and modularity is computed based on the stationary distribution of a random walk. These definitions are highly effective, as algorithms for hitting time and stationary distribution have been developed, allowing for good computational complexity. Our proposed method is particularly useful for directed graphs, with the well-known results for undirected graphs being special cases. Additionally, we utilize the spectral method for these problems, and we have implemented our algorithms to demonstrate their plausibility and effectiveness.

Algebraic winding number

Marie-Francoise Roy, emerita professor, University of Rennes, France

Using a new algebraic definition of the winding number we can prove a fully general complex root counting result, improving a former result of [E]. This result is closely related to our previous work on the Quantitative Fundamental Theorem of Algebra |PR].

Joint work with Daniel Perrucci, University of Buenos Aires, Argentina.

[E] M. Eisermann, The fundamental theorem of algebra made effective: an elementary real-algebraic proof via Sturm chains. Amer. Math. Monthly 119 (2012), no. 9, 715–752.

[PR] D. Perrucci, M.-F. Roy, Quantitative fundamental theorem of algebra. Q. J. Math. 70 (2019), no. 3, 1009–1037.

Consider the Poincar\'e-Sobolev inequality on the hyperbolic space: for every $n \geq 3$ and $1 < p \leq \frac{n+2}{n-2},$ there exists a best constant $S {n,p, \lambda}(\mathbb{B}^{n})>0$ such that \begin{align*} S {n, p, \lambda}(\mathbb{B}^{n})\left(~\int \limits {\mathbb{B}^{n}}|u|^{{p+1}} \, {\rm d}v {\mathbb{B}^n} \right)^{\frac{2}{p+1}} \leq\int \limits {\mathbb{B}^{n}}\left(|\nabla {\mathbb{B}^{n}}u|^{2}-\lambda u^{2}\right) \, {\rm d}v {\mathbb{B}^n}, \end{align*} holds for all $u\in C c^{\infty}(\mathbb{B}^n),$ and $\lambda \leq \frac{(n-1)^2}{4},$ where $\frac{(n-1)^2}{4}$ is the bottom of the $L^2$-spectrum of $-\Delta {\mathbb{B}^n}.$ It is known from the results of Mancini and Sandeep (Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (4): 635--671, 2008) that under appropriate assumptions on $n,p$ and $\lambda$ there exists an optimizer, unique up to the hyperbolic isometries, attaining the best constant $S {n,p,\lambda}(\mathbb{B}^n).$ In this talk we;ll discuss the quantitative gradient stability of the above inequality and the corresponding Euler-Lagrange equations. Our result generalizes the sharp quantitative stability of Sobolev inequality in the Euclidean space of Bianchi and Egnell (J. Funct. Anal. 100 (1): 18--24. 1991), Ciraolo, Figalli and Maggi (Int. Math. Res. Not. IMRN (21): 6780--6797, 2018), Figalli and Glaudo (Arch. Ration. Mech. Anal, 237(1): 201--258, 2020) to the Poincar\'{e}-Sobolev inequality on the hyperbolic space.

Tuesday, 25 April 2023

Numerical studies of random plane waves, functions

$$u=\sum_{j}c_{j}e^{\frac{i}{h}\langle x,\xi_{j}\rangle}$$

where the coefficients $c_{j}$ are chosen ``at random'', have detected an apparent filament structure. The waves appear enhanced along straight lines. There has been significant difference of opinion as to whether this structure is indeed a failure to equidistribute, numerical artefact or an illusion created by the human desire to see patterns. In this talk I will present some recent results that go some way to answering the question. We study the behaviour of a random variable $G(x,\xi)=||P_{(x,\xi)}u||_{L^{2}}$ where $P_{(x,\xi)}$ is a semiclassical localiser at Planck scale around $(x,\xi)$ and show that $G(x,\xi)$ fails to equidistribute. This suggests that the observed filament structure is a configuration space reflection of the phase space concentrations.

Let G be a finite group of SL(2,C). The quotient singularity has the minimal resolution and there is a bijection between the exceptional set and irreducible representation of G. This bijection is called the McKay correspondence and it is a kind of a bridge between Geometry and Algebra. I would like to introduce it and more generalizations in this talk.

In this study, we develop an age-structured compartment model for COVID -19 with an unreported infectious class that describes the spread of COVID-19 in the National Capital Region (NCR) in the Philippines. The epidemiological compartments are divided into three age groups: namely, young (1-19 years), adults (20-64 years), and elderly (65+ years). The reporting and transmission rates are estimated by making use of the numbers of COVID-19 cases in the NCR. Optimal control theory is then employed to identify the best vaccine allocation to the three age groups. Moreover, we consider three different vaccination periods to reflect phases of vaccination priority groups wherein the first, the second, and the third period account for the inoculation of the elderly, adult and elderly, and all three age groups, respectively. This study could guide the policymakers in crafting strategies that can mitigate the spread of an infectious disease by considering age-dependent transmission and limited resources.

We establish a slowly moving target second main theorem for meromorphic functions on a non-Archimedean field, with counting functions truncated to level $1.$ As an application, we show that two meromorphic functions on a non-Archimedean field must coincide if they share $q\, (q\geq 5)\, $ distinct small functions, ignoring multiplicities.

In this talk, we consider the 2D incompressible Boussinesq equation without thermal diffusion, and aim to construct rigorous examples of small scale formations as time goes to infinity. In the viscous case, we construct examples of global-in-time smooth solutions where the H^1 norm of density grows to infinity algebraically in time. For the inviscid equation in the strip, we construct examples whose vorticity grows at least like t^3 and gradient of density grows at least like t^2 during the existence of a smooth solution. These growth results work for a broad class of initial data, where we only require certain symmetry and sign conditions. As an application, we also construct solutions to the 3D axisymmetric Euler equation whose velocity has infinite-in-time growth. (joint work with Alexander Kiselev and Jaemin Park)

Through the study of the Hecke operator, we will explore congruences of various obejcts, including the Fourier coefficients of modular functions and mock modular functions, traces of singular moduli, and the class numbers.

The zeros of the Riemann zeta-function with real part between 0 and 1 are called nontrivial zeros. We let γn denote the ordinate of the nth nontrivial zero that lies in the upper half-plane. It is a classical result of Rademacher that the sequence of fractional parts {γn} is uniformly distributed modulo one. That is, given a subinterval I of [0,1], the proportion of γn's which satisfy the condition {γn}∈I is approximately the length of the interval, |I|. This talk will be about uniform distribution of a subsequence of ({γn}) modulo one. The result has been recently established in a joint work with Steve Gonek.

For some partial differential equations modeling fluid motion, we discuss briefly the existence of solutions and control issues.

Wednesday, 26 April 2023

Multiple zeta functions have been studied at least since Euler, who found many of their algebraic properties.

In particular, they are greatly developed since the 1980s in several different contexts such as modular forms, mixed Tate motives, quantum groups, moduli spaces of vector bundles, scattering amplitudes, etc.

In this talk, we introduce a combinatorial generalization of the Euler-Zagier type multiple zeta and zeta-star functions, that we call Symmetric multiple zeta functions. These multiple zeta functions are defined as sums over combinatorial objects called semi-standard (marked shifted) Young tableaux, as an analogue of the combinatorial expression of the Symmetric functions.

We will show their properties such as determinant formulas. This is based on a joint work with W. Takeda.

Boundary value problem statement is given by definite boundary no-slip and initial conditions for cylindrical pipe in the cylindrical system of coordinates. Navier-Stokes equations and continuity equations for symmetrical fluid motion are considered. The solution in two different ways are considered. First approach could give the wall friction stress, second defines the gradient of pressure difference. As result, we could compare the two ways of solutions.

Free boundary problems for the Navier-Stokes equations have a long history. We discuss our recent results for nearly half-space. To obtain global well-posedness of the problem, we establish end-point maximal L^1-regularity for the initial-boundary value problems of the Stokes equations.

We show that relative Property (T) for the abelianization of a nilpotent normal subgroup implies relative Property (T) for the subgroup itself. We derive this as a consequence a more general theorem which we prove: If $H$ is a closed subgroup of a locally compact group $G$, and $A$ is a closed subgroup of the center of $H$, such that $A$ is normal in $G$, and $(G/A, H/A)$ has relative Property (T), then $(G, H^{(1)})$ has relative Property (T), where $H^{(1)}$ is the closure of the commutator subgroup of $H$. We also discuss relative property (T) for triple $(G,H,M)$, where $H$ is a subgroup of $G$ and $M$ is a subset (or a subgroup) of $G$. (Joint work with Indira Chatterji and Dave Witte Morris, Documenta Mathematica 23 (2018), 353-382).

``Polynomials and power series

May they forever rule the world.''

Thus begins a poem composed by Shreeram S. Abhyankar in 1970.

Polynomials are introduced at a very early stage in our studies. Yet there are many interesting fundamental problems on polynomial rings which are easy to state but difficult to approach. One such problem is the famous Abhyankar-Sathaye Conjecture which asserts that in characteristic zero, any affine hyperplane in a polynomial ring is a coordinate. An affine line is a coordinate in characteristic zero. This is the well-known Epimorphism Theorem proved by Abhyankar-Moh and Suzuki independently around 1975.

However, the next case, the affine plane case, is still open in general. Some special cases have been considered by Sathaye, Russell, Wright, Kaliman and other mathematicians. In this talk we shall mention known results on epimorphism theorems over rings due to Bhatwadekar and his school. We shall also see some partial generalisations of the results of Sathaye and Russell on linear planes to higher dimensions.

In the era of data science in the 21st century, big data technology is able to collect very large amounts of data from various sources with the aim of obtaining information and knowledge from past, present and future events. This knowledge is obtained with very high data processing speeds through machine learning with the help of artificial intelligence which is growing rapidly in various aspects of life.

Spatio-temporal modeling based on time series analysis such as the Vector Autoregressive (VAR), Space-Time Autoregressive (STAR) and Generalized Space-Time Autoregressive (GSTAR) models evolve along with big data and are applied to various field data. In this research, the GSTAR model is studied combined with the Neural Network method, called GSTAR-NN using the data analytics lifecycle methodology through six stages of research, including: discovery, data preparation, model planning, model building, communicate result and operationalize.

The results of data processing in the form of predictive analysis that produces reports visually are expected to provide recommendations for relevant agencies in making decisions.

Thursday, 27 April 2023

The theory of mixed-type equations is one of the principal parts of the general theory of PDEs. The interest in these kinds of equations arises in both theoretical and practical uses of their applications. In this work, I will present a generalization of the Tricomi problem for a loaded mixed-type equation with the Riemann-Liouville fractional differential operator. The existence and uniqueness of the solution to the problem are proved using the theory of integral equations.

We present our recent development on the structure of the class groups of global function fields. We study the Galois module structure of the class groups of Kummer function fields and Artin-schreier function fields, and we obtain the result on the class group rank of Kummer function fields and Artin-schreier function fields. As application, we prove the existence of infinitely many cyclotomic function fields with arbitrarily large $2^n$-rank for n up to 3. We also explicitly construct the infinite families of Kummer function fields and Artin-schreier function fields with guaranteed Galois module structure. (This is a joint work with Jinjoo Yoo at UNIST.)

In this talk, which is based on a joint work with Juergen Herzog, we describe the rings of invariants of the elementary abelian $p$-group $(Z/pZ)^r$ for $3$-dimensional generic representations. Furthermore we show that these rings are complete intersection rings. This proves a conjecture of Campbell-Shank-Wehlau, which they proved for $r = 3$, and later Pierron-Shank proved for $r = 4$.

A flow motion of compressible inviscid fluid such as a gas is governed by the compressible Euler system. In particular, a jump discontinuity such as a shock or a contact discontinuity occurring in a compressible flow motion can be studied as a weak solution to the Euler system. In this talk, I will present various open problems that involve appearances of transonic shocks. And, I will present recent results.

We prove some localisation estimates for solution Schrödinger equation corresponding to the sub Laplacian on the Heisenberg group.

In this talk we will associate two numbers, the geometric and homological stretch factors, to a graph map and see under what conditions they are equal. We will then upgrade these notions to automorphisms of free groups. This is joint work with Spencer Dowdall and Sam Taylor.

Given a holomorphic vector bundle over a compact Riemann surface, we characterize when the Bergman space of holomorphic sections of the restriction of the bundle to an open set is infinite dimensional. This answers a problem raised by Szoke (2020), which was motivated by the results of Carleson and Wiegerinck for planar domains. This is joint work with Anne-Katrin Gallagher and Liz Vivas.

As is well known, locally area minimizing surfaces are called minimal surfaces which appear in nature as idealized soap films. They have the property that the mean curvature is zero at every point on the surface. Analogously, there are zero mean curvature surfaces in Lorentz Minkowski space which are called maximal surfaces. We will talk about the most general solution of the PDEs called the Weirstrass Enneper representation of the surfaces. Using certain Euler-Ramanujan's identities, we will show finite decompositions of certain minimal/maximal surfaces in terms of scaled and translated versions of other minimal/maximal surfaces. We will discuss their close cousins the Born-Infeld solitons and the timelike minimal surfaces in this context.

Next we talk about the Plateau's problem and the interpolation problem for all these surfaces and recent progress we have made in this direction.

Friday, 28 April 2023

I will start by reviewing quantum groups at roots of unity and their representation theory. Then, I will explain a construction of new quantum groups using cohomology theories from topology. The construction uses the so-called cohomological Hall algebra associated to a quiver and an oriented cohomology theory. In examples, we obtain the Yangian, quantum loop algebra and elliptic quantum group, when the cohomology theories are the cohomology, K-theory, and elliptic cohomology respectively.

We will discuss the classical Chern-Simons theory and extend it to define higher order cohomological invariants. We will pose some questions and present partial answers.

The linearized Navier-Stokes equations, also known as Stokes equations, continue to be studied due to the many interesting applications and also for the rich mathematical theory associated with them. Hence a wide variety of boundary value problems pertaining to Stokes flows have been studied in science and engineering. Many of the results have been obtained for Stokes flows past spherical bodies. However in actual situations, it is more realistic to consider non-spherical shapes. But it may not be easy to find the solutions easily in such boundary value problems. We discuss the general boundary conditions applied to Stokes equations in steady and unsteady flows. In particular, we discuss some simple methods used to solve such boundary value problems in the case of Stokes flows past spherical and non-spherical particles and discuss some interesting results in each case.

We consider the high-dimensional moving average process and explore the asymptotics for eigenvalues of its sample autocovariance matrices. Under quite weak conditions, we prove, in a unified way, that the limiting spectral distribution (LSD) of any symmetric polynomial in the sample autocovariance matrices, after suitable centering and scaling, exists and is non-degenerate. We use methods from free probability in conjunction with the method of moments to establish our results. In addition, we are able to provide a general description of the limits in terms of some freely independent variables. We also establish asymptotic normality results for the traces of these matrices. We suggest statistical uses of these results in problems such as order determination of high-dimensional MA and AR processes and testing of hypotheses for coefficient matrices of such processes.

The q-analogues of classical functions have been studied in the literature for a long time. In this talk, we discuss the irrationality and linear independence of values of the q-analogue of the exponential function and certain cognate functions. Unlike the classical exponential function, not much is known in this case. The talk will be based on recent joint work with A. Dixit and V. Kumar.

The arithmetic of elliptic curves depends on the kind of reduction of the curve at a prime p. In this talk, we will review the current state of knowledge of the invariants of elliptic curves over number fields with supersingular reduction at an odd prime number p.