Future Activities and Research Plans
We plan to continue with the following "engines":
(1) ENI Research Projects
(2) Post-doctoral Fellowships (ENI Fellows)
(3) Emmy Noether Lectures
(4) Workshops, Conferences
(5) Exchange Program
Within the original scientific scope of the Institute – Algebra, Geometry and Function Theory, we will prioritize algebraic geometry, related fields, and applications. We have developed an international leadership position in Topology and the Geometry of Algebraic Varieties of Low Dimension, braid group techniques, dynamics of Linear groups, and more. Moreover, we actively apply pure mathematics to engineering applications (electro optics, nanotechnology, agriculture, medical devises, neuro computation). We have applied Algebraic Geometry to Computer Vision, Cryptography and Robotics and plan to continue in these directions.
A few conferences are already planned for the coming years: a conference on Geometrical Methods in Computational Neuroscience (June 2021) an ERCOM meeting hosted by ENI, March 2022, a conference on Complex Dynamics (joint with Ort Braude), May 2022
We hope to have a targeted grant to be able plan to host leading mathematicians, mainly in the above fields, for extended periods, and to invite more young researchers. Our experience has shown that hosting top mathematicians leads to continued collaborations with local researchers, inspires young people, and enables postdoctoral fellows to connect with these leading mathematicians. We will choose our visitors and postdocs (preferably postdocs in similar fields as the Visiting Professors) with the help of the Advisory Board. We will also plan small related workshops. The focus of interest is influenced by the Visiting Professors, and we will have a dynamic, open environment, with the likelihood of interesting innovative applications inside and outside of mathematics. We plan to enhance the existing exchange agreements (MPI-Bonn, MPI-Leipzig, SNS, Banach Institute, Tata Institute, ECNU, NYU Abu-Dhabi), and will develop and encourage new exchange agreements.
For 2020, we will focus on six topics and we will host visiting professors and postdoctoral fellows in those topics, who will form thinking groups with the prospect to create ongoing future collaborations that will have a significant impact. Each group will be coordinated by a local researcher, who is a leading mathematician, and he will be responsible for assembling the group, organizing workshops, etc.
Group 1: Topology vs Combinatorics of Line Arrangements and Applications Coordinator: Prof. Mina Teicher
Postdoctoral Fellows: Dr. Evgeny Mayanskiy, Dr. Eran Liberman
ENI Guests and Collaborators: Dr. Meirav. Amram, Prof. Ciro. Ciliberto, Dr. D. Garber, Dr. Robert. Shwartz, Prof. Sheng.-Li. Tan
One of the most important aims in classifying line arrangements is to compare its topology with its combinatorics. For the topology, we shall study the singularities of the arrangement, use Moishezon-Teicher braid monodromy techniques, and apply the Zariski-van Kampen Theorem to compute the fundamental group of its complement in CP2. For the combinatorics, we write down defining equations involving parameters for a given intersection lattice. The problem is solved (via multiple international efforts) for arrangements of up to 9 lines and almost completely solved for 10 lines. The fundamental groups of the complements of line arrangements of 11 (and 12) projective lines with a quintuple point and with a quadruple point, still needs study. Comparison of Topology and Combinatorics relates to the existence of a Zariski Pair. For line arrangements with 13 lines, there is a theorem regarding the existence of a Zariski pair but no explicit example. In order to find new Zariski pairs, we shall study the moduli space of arrangements (space of parameters), since if it is irreducible, then there are no Zariski pairs. We shall also work on this problem in the real case.
We can use regeneration of braid monodromy of line arrangements in order to compute fundamental groups related to branch curves of algebraic surfaces that have a degeneration to union of planes. The examples that we have already computed for such surfaces have given us almost solvable groups, and this indicates that we can use that structure to construct new discrete invariants of surfaces. We also aim to classify such surfaces.
Group 2: Braid Groups and Combinatorics for Cryptography Coordinators: Prof. Mina Teicher, Prof. Nathan Keller
Local Researchers: Dr. Boaz Tsaban, Prof. Yuval Roichman, Prof. Tahl Nowik
Postdoctoral Fellows: Dr. Shalom Bronner, Dr. Eran Liberman, Dr. Gary Vinokur
ENI Guests and Collaborators: Dr. Arkadius Kalka, Dr. David Garber, Karene Chu
Only a few public-key schemes are considered secure. Nearly all of them are based on commutative groups, and can be broken in subexponential time using standard computers, and in polynomial time using quantum computers. About 10 years ago, an investigation began on basing a public-key scheme on non-commutative algebraic structures that has the potential for solving the mentioned problems.
Braid groups are ideal candidates for non-commutative group-based cryptosystems, as they have an infinite cyclic center and a very difficult conjugacy problem. With the knowledge accumulated in the ENI on the braid group (for applications to Algebraic Geometry) and algorithms for its basic problems (WORD, Conjugacy and Hurwitz problems), ENI was a major leader in the last decade in the new frontier of braids and cryptography. In these fields, we foresee a few future directions that shape our goals (with international collaborations):
A. New Algorithms for Base Problems in the Braid Group
B. New Crypto Algorithms
C. Improve Efficiency and Complexity of algorithms
D. Other Non-commutative Crypto Systems
Group 3: Dynamics on Finite and Infinite Linear Groups
Coordinator: Prof. Boris Kunyavskii
Local Researchers: Prof. Eugene Plotkin, Prof. Gregory Soifer
ENI Guests and Collaborators: Prof Efim Zalmanov, Prof. Tatiana Bandman, Prof. Nikolai Gordeev, Prof. Jun Morita, Prof. Elena Klimenko, Prof. Andreas Thom
With the expertise of the expected Visiting Professors, we foresee two directions of study and research goals:
A. Verbal properties of linear algebraic groups, with a focus on the study of the image of word maps and generalizing some dominance theorems, in the spirit of the seminal Borel theorem. Promising prospects are to the infinite-dimensional set-up, namely, to the setting of Kac-Moody groups (as in J. Morita and E. Plotkin), to the Lie-algebraic set-up (as in Bandman, Gordeev, E. Plotkin), to the setting of Kac-Moody algebras. Images of word maps have already been thoroughly studied by Prof. Andreas Thom.
B. Study of minimal models of representations of finite groups. The main goal is to describe, in terms of the representation under consideration, the obstructions to the existence of such a model. The main tools are Galois cohomology and the theory of fields of moduli, coming from similar problems on curves, covers of curves and dynamical systems. Rational and integer models of representations of finite groups have been extensively studied by W. Plesken, G. Nebe and J. Jongen.
Group 4: Automorphic Forms
Coordinator: Prof. Andre Reznikov
ENI Guests and Collaborators: Prof. Valentin Bloomer, Bernhard Kroetz, Prof. Joseph Bernstein
The main theme of the project revolves around bounds on periods of automorphic representations and subconvexity for L-functions. The goal is to develop representation theoretic methods. This topic has a great deal of interactions with works of V. Bloomer and of B. Kroetz.
Group 5: Geometry in Neural Computation
Coordinator: Prof. Mina Teicher
ENI Guests and Collaborators: Prof. Moshe Abeles (BIU), Prof. Jurgen Jost (MPI), Prof. Ed Atstein (Freiburf), Yael Eisenberg (Cornell), Anette Stawski (Cornell).
Young Researchers: Dr Vered Moscovich, Ahmed Soleman, Amir Kleks, Zohar Noy, Miriam Dagan, Erez Nachmias,
Contemporary advanced mathematical research is essential to understand how the brain works. We plan to assemble an international research group for the application of mathematical methods (some developed at the ENI), such as geometric data mining, tropical algorithms, moduli space of movements in space, as well as dynamical systems and stochastics - to contribute to the understanding the mechanism of brain activity, and in particular to establish the longstanding conjecture of the synfire chains. In parallel, we will assemble an international group interested in different medical disorders (e.g., sleep disorders and epilepsy). Bar-Ilan University owns a MEG device (the only one in Israel), that detects the magnetic field induced by the electrical activity of the brain. We plan to use MEG recording for innovative multidisciplinary projects on mathematical talents - to distinguish between talent for algebra vs geometry and then ensure that teaching mathematics is more effective and the resources are better used.
Group 6: Geometry, Graphs and Networks in Complex Systems
Coordinator: Dr. Baruch Barzel
Postdoctoral fellows: Dr. Suman Acharyya, Dr. Chandrakala Meena, Dr. Aradhana Singh, Dr. Nir Schreiber,
ENI Guests and Collaborators: Dr. Chittaranjan Hens, Eran Reches, Alon Eldan, Gilad Reti, Sagi Buaron
Graph structures are at the heart of our most pertinent and crucial systems, from sub-cellular biology to our critical infrastructure, the environment and our globally connected society. The past two decades have seen spectacular advances in mapping the structure of these graphs, finding that despite their diversity in scale and purpose, they follow common universal rules. This can potentially lay the grounds for a unified theory; however, the observed structural universality does not naturally translate to the dynamics of these systems, casting severe limits on our ability to leverage graph topology, into systematic predictions on dynamics. The challenges are substantial, as the relevant systems exhibit highly disordered and heterogeneous structures combined with diverse, nonlinear and often unknown internal mechanisms. Thus, most advances are restricted to specific domains or to low dimensional systems, lacking the generality and broad applicability of a Statistical Physics framework. Our overall goals are:
- To break these theoretical barriers, and develop systematic theoretical tools
in order to translate known topological features into their observable dynamic outcomes.
B. To understand the inner mechanisms of complex system dynamics, to predict their behavior, and ultimately, Influencing them towards a desired dynamic outcome.
