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Title: Using lattices for computational number theory Speaker: Jean-François Biasse, University of Waterloo Waterloo, ON Time: 3:00pm‐4:00pm Place: CMC 130
We begin this talk with a computational problem attributed to Archimedes. It can be solved by computing the unit group of a number field, which is a fundamental task in number theory.
Then I will introduce the state-of-the-art methods for computing the ideal class group and the unit group of a number field. We will see how these methods allow us to check if an ideal in the ring of integers of a number field is principal, and if so, compute a generator. This fundamental task has recently received a lot of attention from the cryptographic community because a series of lattice-based schemes rely on its difficulty.
Finally, I will mention the state of the art methods for solving these problems with quantum computers. These algorithms are typical examples of cases where quantum computers can drastically improve on the theoretical complexity. We will conclude by mentioning the relevance of these results to what is called “post-quantum cryptography”.
Title: The index approach to permutation polynomials over finite fields Speaker: Steve Wang, Carleton University Ottawa, Ontario, Canada Time: 3:00pm‐4:00pm Place: CMC 130
A permutation polynomial (PP) \(f\) over a finite field is a nonzero polynomial that acts as a permutation of the elements of the field, i.e., the map \(x\rightarrow f(x)\) is one-to-one. Equivalently, the size of the value set of \(f\) is the extreme case. The study of permutation polynomials over a finite field goes back to 19th century when Hermite and later Dickson pioneered this area of research. In recent years, permutation polynomials have attracted a lot of attention due to their applications in many areas of science and engineering. As a consequence, many interesting discoveries have been made recently by various researchers.
However, the classification of PPs is still out of reach. Most current research concentrate on constructions, distribution, and enumeration of PPs in terms of parameters of finite fields and parameters of polynomials such as coefficients and exponents (including the degree). In this talk I will report some general results on PPs, in particular, an index approach to study PPs over finite fields. We note that index is another parameter for any polynomial over a finite field, which was newly introduced by Akbary, Ghioca and the speaker. This parameter turns out to be also very useful in studying several problems on polynomials over finite fields.
Title: From exponential bases to the discrete Hilbert transform Speaker: Laura De Carli, Florida International University Time: 3:00pm‐4:00pm Place: CMC 130 Sponsor: Dmitry Khavinson
The discrete Hilbert transform was first studied by D. Hilbert and H. Weil in 1908 and has generated interest among the mathematician ever since.
In this talk I will show how a seemingly simple problem on exponential bases on \(L^2\) lead to the investigation of a one-parameter semigroup of operators on \(l^2\) whose infinitesimal generator is the discrete Hilbert transform. If time allows, I will also present other families of discrete operator that appear in connection with problems on exponential bases. Part of this work-in-progress is joint with my student Shaikh Gohin Samar, who is completing his Master's at FIU. The talk will be accessible to graduate and advanced undergraduate students in Mathematics
Title: Determinantal representations of polynomials Speaker: Greg Knese, Washington University Time: 3:00pm‐4:00pm Place: CMC 130
When can a polynomial be written as a determinant of a linear combination of matrices? Why would we want to do such a thing? These questions get really interesting when we restrict the polynomial or restrict the matrices under consideration. For example, one can consider real polynomials with no zeros on a product of upper half planes (a generalization of a real-zero polynomial in one variable) and representations using self-adjoint matrices. Such polynomials occur naturally in analysis, combinatorics, and optimization. We will discuss some of these connections as well as other variations on the polynomials and matrices we consider.
Title: Groups Arising at the Transition to Asphericity Speaker: William Bogley, Oregon State University Time: 3:00pm‐4:00pm Place: CMC 109
The homotopy type of an aspherical simplicial or CW complex is uniquely determined by its fundamental group, so homotopy invariants of an aspherical space are invariants of its fundamental group. I will describe asphericity as it relates to relative group presentations and present applications to the theory of cyclically presented groups. Through work of several authors over the past two decades, details are emerging of an interface where aspherical relative presentations transition to non-aspherical ones. Focusing on this interface, I will describe joint work with Gerald Williams in which we discover infinite families of efficient finite groups that admit short presentations and whose orders involve all Mersenne numbers, as well as other conjecturally infinite families of rational primes.
Title: Stochastic discrete two-dimensional growth Speaker: Mark Mineev-Weinstein, New Mexico Consortium US & International Institute of Physics, Brazil Time: 4:00pm‐5:00pm Place: CMC 130
We consider a stochastic discrete growth process in two dimensions, in which randomly-distributed sources issue an arbitrary number of diffusing particles that aggregate upon contact with clusters. We calculate the probability of obtaining various shapes for clusters with the same area. After performing functional integration in a quasi-classical limit (when the number of aggregated particles goes to infinity, while the area deposited by a single random walker goes to zero), we obtain that the probability to obtain a given domain is given by the tau-function of the dispersionless integrable Today hierarchy, minus the term corresponding to the Coulomb self-energy of the cluster.
We will conclude with a new approach to pattern selection in light of this specific model, relate it to the entropy production, and discuss the derivation of classical Laplacian growth as the extremal of a generalized stochastic functional.
Title: Finite Type Enhancements Speaker: Sam Nelson, Claremont McKenna College, CA Time: 3:00pm‐4:00pm Place: CMC 130
Finite type invariants, also known as Vassiliev invariants, are a type of integer-valued knot and link invariant. In this talk we will see a method for using finite type invariants of biquandle-colored knots to enhance the quandle counting invariant.
Title: A geometric approach to understanding neural codes in recurrent networks Speaker: Carina Curto, Pennsylvania State University Time: 2:00pm‐3:00pm Place: CMC 108
Synapses in many cortical areas of the brain are dominated by local, recurrent connections. It has long been suggested, therefore, that cortical networks may serve to restore a noisy or incomplete signal by evolving it towards a stored pattern of activity. These “preferred” activity patterns are constrained by the network's connections, and are typically modeled as stable fixed points of the dynamics. In this talk I will briefly review the permitted and forbidden sets model for cortical networks, first introduced by Hahnloser et al. (Nature, 2000), and then present some recent results that provide a geometric handle on permitted sets. Specifically, I will show how questions about fixed points can be translated to questions in classical distance geometry. Finally, I will use the geometric description of fixed points to show that these networks can perform error correction and pattern completion for a wide range of connectivities.
Title: Categorification in Applied Mathematics Speaker: Robert Ghrist, University of Pennsylvania Time: 1:30pm‐2:30pm Place: CMC 130
One of the organizing principles in Mathematics is that of categorification — the systematic lifting of numerical equalities to isomorphisms of higher algebraic objects: e.g., much of algebraic topology consists of categorification of numerical counts. This talk will be a gentle survey of several ways in which categorification lurks in applied mathematics, with classical and modern ideas alike having enrichments to algebraic structures that reveal richer relationships than numerical equality.
Title: A topological approach for investigating the intrinsic structure of neural activity Speaker: Vladimir Itskov, Pennsylvania State University Time: 3:00pm‐4:00pm Place: CMC 130
Experimental neuroscience is achieving rapid progress in the ability to collect neural activity and connectivity data. This holds promise to directly test many theoretical ideas, and thus advance our understanding of “how the brain works”. Detecting meaningful structure in this data is challenging because of unknown nonlinearities, where measured quantities are related to more “fundamental” variables by an unknown nonlinear transformation. We find that combinatorial topology can be used to obtain meaningful answers to questions about the structure of neural activity and introduce an approach that extracts features of the data invariant under arbitrary nonlinear monotone transformations. These features can be used to distinguish random and geometric structure, and depend only on the relative ordering of matrix entries. We apply our technique to neural activity in rat hippocampus, and find that the intrinsic pattern of correlations possesses a geometric organization in both spatial and non-spatial behaviors.
Title: Orthogonal polynomials for normal matrix models with discrete rotational symmetries Speaker: Ferenc Balogh, Concordia University Canada Time: 4:00pm‐5:00pm Place: CMC 130
Statistical observables of random unitary invariant normal matrix models can be expressed in terms of the joint probability distribution of the matrix eigenvalues. This leads to a logarithmic Coulomb gas model in which the eigenvalues are thought of as charged particles in the complex plane under the influence of an external potential.
The joint density of the eigenvalues can be written as a determinant with entries assembled from planar orthogonal polynomials associated to the given background potential. As the matrix size goes to infinity, the asymptotics of eigenvalue statistics, up to leading term, are encoded into the equilibrium measure, the solution of the continuum limit of the Coulomb gas variational problem. To obtain more refined scaling limits, one needs to study the strong asymptotics of the corresponding orthogonal polynomials. After a brief review of the known results to date, I will introduce a special one-parameter normal matrix model with a discrete rotational symmetry for which the equilibrium measure can be found explicitly for all values of the parameter, including a critical value where a non-trivial topological transition of the support is observed. It will be shown how the corresponding orthogonal polynomials can be analyzed using nonlinear steepest descent techniques, based on a trick of writing two-dimensional orthogonality relations in terms of contour integrals, leading to a Riemann-Hilbert problem. In particular, The results confirm a conjectured relation between the limiting zero distribution of the orthogonal polynomials and the equilibrium measure via a balayage procedure.
The talk is based on joint works with T. Grava and D. Merzi.
Title: Why the oracle may not exist: ergodic families of Jacobi matrices, absolute continuity without almost periodicity Speaker: Alexander Volberg, Michigan State University Time: 3:00pm‐4:00pm Place: CMC 130
We will explain the recent solution of Kotani's problem pertinent to the existence/non-existence of “oracle” (almost periodicity) for the ergodic families of Jacobi matrices (discrete Schröedinger operators). Kotani suggested that such families are subject to the following implication: if family has a non-trivial absolutely continuous spectrum (this happens almost surely) then almost surely it consists of almost periodic matrices (hence the possibility to predict the future by the past). Kotani proved an important positive result of this sort. Recently independently Artur Avila and Peter Yuditskii—myself disproved this conjecture of Kotani (by two different approaches). We will show the hidden singularity that defines when such Kotani's oracle exists or not.
Title: Conformal mapping and random Laplacian growth Speaker: Alan Sola, University of Cambridge Time: 3:35pm‐4:45pm Place: CMC 109
In my talk, I will survey recent advances in the conformal mapping approach to Laplacian random growth in the plane, where aggregating particles are represented by simple conformal maps, and growth of aggregates is encoded through composition of random copies of such maps.
The random planar sets one obtains in this way exhibit rich and fascinating structures, but the analysis of the large-scale geometry and microscopic features of these so-called clusters present a formidable challenge to mathematicians, with many basic questions remaining wide open.
Computer simulations will be used to illustrate the results obtained, and also to formulate a number of open problems.
Title: Asymptotic regimes of complex orthogonal polynomials with varying quartic weight: Global behaviour critical points Alexander Tovbis Speaker: Alexander Tovbis, University of Central Florida Time: 3:00pm‐4:00pm Place: CMC 130
We study the asymptotics of recurrence coefficients for monic orthogonal polynomials \(\pi_n(z)\) with the quartic exponential weight \(\exp\left[-N\left(\frac{z^2}2+\frac{tz^4}4\right)\right]\), where \(t\in\mathbb{C}\) and \(N\in\mathbb{N}\), \(N\to\infty\). Our goal is to describe the regions of different asymptotic behaviour globally in \(t\in\mathbb{C}\) as well as behaviour near the critical points.
Title: The Homotopy Analysis Method for Differential Equations, and Hierarchies of Integrable Systems Speaker: Mathew Baxter, University of Central Florida Time: 2:00pm‐3:00pm Place: CMC 130
The Homotopy Analysis Method is an innovative new way (Liao, 1992) to get analytical solutions to nonlinear di erential equations. We begin with a brief introduction to the concept of homotopy from topology. From there, the Homotopy Analysis Method is discussed in detail. We describe how the idea of homotopy is applied to introduce a parameter into ordinary/partial differential equations that do not have one to begin with. The homotopy between a linear operator and a nonlinear operator allows us to use perturbation on this parameter to obtain analytical solutions to these equations with small error. The idea is applied to an equation governing the nonlinear evolution of a vector potential of an electromagnetic pulse propagating in an arbitrary pair plasma with temperature asymmetry. We also look at the Hasegawa-Mima equation, a very difficult PDE that governs the electric potential due to a drift wave in a plasma. Future work is discussed.
After this, hierarchies of integrable systems will be discussed. In particular we look at the integrability of the Zakharov-Ito hierarchy due to its zero-curvature representation. We find the bi-Hamiltonian structure, and show the Hamiltonians are involute in pairs under a well-defined Poisson bracket, implying that the equations in our hierarchy are Liouville integrable. Future work is discussed.
Title: Augmented eigenfunctions: a new spectral object appearing in the integral representation of the solution of linear initial-boundary value problems Speaker: David Smith, University of Cincinnati Time: 3:45pm‐4:45pm Place: CMC 109
We study initial-boundary value problems for linear, constant-coefficient partial differential equations of arbitrary order, on a finite or semi-infinite domain, with arbitrary boundary conditions. It has been shown that the recent Unified Transform Method of Fokas can be used to solve all such classically well-posed problems. The solution thus obtained is expressed as an integral, which represents a new kind of spectral transform. We compare the new method, and its solution representation, with classical Fourier transform techniques, and their resulting solution representations. In doing so, we discover a new species of spectral object, encoded by the spectral transforms of the Unified Method.
Title: Quantum invariants of knots, hyperbolic geometry and \(q\)-series Speaker: Mustafa Hajij, Louisiana State University Time: 4:00pm‐5:00pm Place: CMC 130
The discovery of the Jones polynomial lead to a vast family of invariants called the quantum invariants. Quantum invariants deeply connect many domains of mathematics such as quantum groups, hyperbolic geometry, knot theory and number theory. In this talk I will talk about quantum invariants and some of their connections with the geometry of the knot complement. Furthermore, I will describe some recent connections with number theory.
Title: Random Matrices and Potential Theory Speaker: Thomas Bloom, University of Toronto Time: 3:00pm‐4:00pm Place: CMC 130
I will introduce the Gaussian Unitary Ensemble and other unitary ensembles of random matrices.
I will discuss aspects of these ensembles which can be studied and established via potential theory.
Specifically, I will deal with the convergence of the empirical measure of the eigenvalues and large deviation principles.
Title: Algebraic Geometry and Approximation Theory Speaker: Michael DiPasquale, University of Illinois at Urbana-Champaign Time: 4:30pm‐5:30pm Place: CHE 302
Piecewise polynomial functions, also known as splines, are a cornerstone of approximation theory today. A question of fundamental interest in spline theory is to determine the dimension of (and a basis for) the vector space of splines of degree at most \(\mathbf{d}\) over a polytopal complex. We give a survey of some ways in which this question may be approached from the perspective of commutative algebra and algebraic geometry. Key players in this story are the Hilbert function, the Hilbert polynomial, and Castelnuovo-Mumford regularity.
Title: \(q\)-polynomial invariant of rooted trees; state sum invariant of graphs Speaker: Jozef Przytycki, George Washington University Time: 3:30pm‐4:30pm Place: ISA 3050
We describe in this note a new invariant of rooted trees and following up state sum invariant of pointed graphs.
We argue that the invariant is interesting on it own, and that it has connections to knot theory and homological algebra. Another reason that we propose this invariant is that we deal here with an elementary, interesting an new mathematics and after the Colloquium everybody can take part in developing the topic inventing new results and connections to other disciplines of mathematics (and likely statistical mechanics and combinatorial biology).
Title: Pattern formation and bifurcations in reaction-diffusion-advection ecological models Speaker: Junping Shi, College of William and Mary Williamsburg, VA Time: 3:00pm‐4:00pm Place: CMC 130
Spatial-temporal patterns appear often in historical ecosystem data, and the cause of the patterns can be attributed to various internal or external forces. We demonstrate that in spatial ecological models, spatial-temporal patterns can arise as a result of self-organization of the ecosystem. By using bifurcation theory, we show that the spatial-temporal patterns are generated with the effect of diffusion, advection, chemotaxis or time delay.
Title: Planar functions and their importance in cryptography Speaker: Stephen M. Gagola, University of Witwatersrand Time: 2:00pm‐3:00pm Place: CMC 130
Planar functions were first introduced by Dembowski and Ostrom. Since 1991 such functions have attracted interest in cryptography as fuctions with optimal resistance to differential cryptanalysis. They were first used in this way by Nyberg where they were given another name “perfect nonlinear” which describes their important cryptographic property of being as far from linear as possible.
Now planar functions have applications in classical cryptographic systems, quantum cryptographic systems, wireless communication, and coding theory. Commutative semifields are equivalent to those planar functions that are known as Dembowski-Ostrom polynomials (DO polynomials). Here I will introduce how Joanne Hall and I have developed methods of constructing families of planar functions and commutative semifields of order \(p^{2r}\) for any odd prime \(p\) and any positive integer \(r\). These families yield a more general construction which includes some other families of known planar functions while at the same time creates new classes of planar functions. Subsequently these were used to construct mutually unbiased bases, a structure of importance in quantum information theory.
Title: Combining Riesz bases Speaker: Shahaf Nitzan, Kent State University Time: 4:00pm‐5:00pm Place: CMC 130
Orthonormal bases (ONB) are used throughout mathematics and its applications. However, in many settings such bases are not easy to come by. For example, it is known that even the union of as few as two intervals may not admit an ONB of exponentials. In cases where there is no ONB, the next best option is a Riesz basis (i.e. the image of an ONB under a bounded invertible operator).
In this talk I will discuss the following question: Does every finite union of rectangles in \(R^d\), with edges parallel to the axes, admit a Riesz basis of exponentials? In particular, does every finite union of intervals in \(R\) admit such a basis?
This is joint work with Gady Kozma.
Title: Mathematics in Cryptography: Today's Applications and Tomorrow's Foundation Speaker: Rainer Steinwandt, Florida Atlantic University Time: 3:00pm‐4:00pm Place: CMC 130
The first part of the talk will show how mathematical techniques are applied in the design of modern cryptographic protocols. Taking the task of establishing a secret key among \(n\ge2\) users over an insecure network as example, we discuss how computational assumptions enable the derivation of an efficient solution with strong provable guarantees.
Regrettably, some of the most common assumptions needed for today's cryptographic solutions are no longer justifiable in a so-called post-quantum scenario. In particular, popular constructions involving elliptic curves are not available in this setting. Post-quantum cryptography is of interest when cryptographic solutions are expected to guarantee security for many years. The cryptographic community is currently trying to identify mathematical platforms for efficient post-quantum solutions of basic cryptographic tasks like public-key encryption or digital signatures. The second part of the talk will discuss some of the current approaches, including in particular attempts that invoke tools from group theory.
Title: Self-similar groups and expanding graphs Speaker: Ievgen Bondarenko, Kyiv National University Time: 3:00pm‐4:00pm Place: CMC 130
Expanding graphs are highly connected sparse graphs which have numerous applications in mathematics and theoretical computer science. Reingold, Vadhan, and Wigderson (2002) discovered a simple combinatorial construction of expanding graphs. This construction was based on the new operation on regular graphs — the zig-zag product, which is closely related to the replacement product of graphs.
In this talk I will describe how to model iterated replacement product and zig-zag product of graphs by finite automata. Also I will explain a simple construction of self-similar groups whose action graphs produce a family of expanding graphs.
Title: New Challenges in Cryptography — Securing the Cloud and Users' Devices Speaker: Feng-Hao Liu, University of Maryland Time: 3:00pm‐4:00pm Place: CMC 130
In recent years, we have seen the emergence of cloud computing, where a service provider (the cloud) offers storage and/or computation to individuals, and those individuals can later access their data from various devices. For example, users store documents on Dropbox and later retrieve them via smartphones, tablets, or laptops. This scenario has numerous advantages: it makes it more convenient for individuals to access and share their data, and it can amortize the cost of maintaining a large storage infrastructure. However, as the data may often contain personal or sensitive information, security concerns, such as information leakage, integrity breach, etc., have been a major barrier for individuals, businesses, and organizations in fully adopting the new computing paradigm. To achieve the full power of such paradigm, we must tackle these challenges.
I will talk about my research that explores new cryptographic techniques for ensuring security in the above scenario. I consider emerging threats on two fronts: the remote cloud, and users local devices. With respect to the cloud, I seek techniques to ensure that a compromised provider cannot access users personal information, or return a wrong answer to a request for some computation. With respect to users local devices, one critical security issue I have explored is defending against non-traditional physical attacks implemented by side-channel leakage and/or malicious tampering. To guarantee security, we not only need to introduce new models and definitions of security, but must also develop new algorithmic and analytical techniques to defend against these new classes of attacks.