Emmy Noether Seminar / Geometry Seminar
at Bar-Ilan University

Abstract: In this talk, I will present some standard theorems as well as some of my own results (joint work with some others) about the mapping class group of a surface, which is the group of homeomorphisms of that surface modulo those isotopic to the identity. Mapping class groups have been extensively studied, so I will focus only on one topological/dynamical part and one algebraic part of the theory: the Nielsen-Thurston classification of mapping classes and the action on the fundamental group of the surface with particular attention to the lower central series of the fundamental group. My results describe some relations between the two parts of the theory.

Abstract: According to the Merkujev-Susslin theorem, every central simple algebra is Brauer equivalent to a tensor product of cyclic algebras whose degrees are equal to the exponent of that algebra. Consequently, cyclic algebras and their tensor products are of major importance in the theory of central simple algebras. Chain lemmas form in this context a tool for studying the different symbol presentations of such algebras. We will discuss specifically the chain lemma for biquaternion algebras, both in characteristic 2 and characteristic other than 2.

The characteristic not 2 case is based on a joint work with Prof. Uzi Vishne.

Abstract: We outline the construction of the derived category D(M) of an abelian category M. We then define left and right derived functors. We introduce K-projective, K-injective and K-flat resolutions, and prove existence of some derived functors when M is either the category of modules over a ring, or the category of sheaves of modules over a sheaf of rings. DG algebras and their derived categories will also be mentioned.

Next we discuss some more specialized topics: dualizing complexes (commutative and noncommutative), two-sided tilting complexes, derived Morita theory, and rigid dualizing complexes.

Lecture notes are at http://www.math.bgu.ac.il/~amyekut/lectures/intro-der-cats/notes.pdf

Abstract: Relative symplectic embedding obstructions provide a way to measure the "size" of a Lagrangian submanifold in a symplectic manifold. We will describe the blow-up and blow-down process for points on Lagrangian submanifolds of a symplectic manfiold, and apply this to determine relative packing obstructions for the real locus of a familty of real symplectic 4-manifolds.

Abstract: These will be survey talks about asymptotic dimension. We will review the definition of dimension dim from classical dimension theory in topology, and compare it to the definition of asymptotic dimension asdim in metric coarse geometry. We will then see some properties of asdim, and some examples and applications to countable groups.

Abstract: We are going to describe the construction of the fundamental group scheme of a given scheme X over a field k, pointed at a k-rational point x. We will recall its relations with the étale fundamental group of X and, when comparable, with the usual fundamental group. The definitions of group scheme and torsor will be recalled.

Abstract: When a sequence of smooth embedded complex curves (C_n) in the complex plane degenerates to a curve C_0 with a singular point p, one can identify a braid B_p on C_0 around p. On this braid, one computes the Milnor number \mu_p which gives us g(C_n)-g(C_0), i.e. how much topology we have lost going from C_n to C_0.

Complex curves in the complex plane generalize to minimal surfaces in Riemannian 4-manifolds. Can we define a Milnor number for smooth minimal surfaces C_n converging to a minimal surface C_0 with a branch point? It turns out that we have to define not one, but two or three Milnor numbers (which coincide in the complex case). We will discuss these quantities and the questions they raise.

Abstract: This will be an introduction to knot theory via several combinatorial models that can be used to handle knots. This talk will focus more on the models rather than the invariants that can be obtained using these models, but applications to these will be highlighted. Models include knot diagrams, the (signed) Tait graph and its spanning trees, grid diagrams (on a torus), one-vertex dessin d'enfants (giving a special type of chord diagram), and the new perfect matching model which is one of my main research interests.

No prior knowledge of knot theory will be assumed, although some more complicated aspects will be glossed over.

Abstract: The roots of symplectic topology lie in the classical formulation of Hamiltonian mechanics. While classical symplectic geometry had a very dynamical flavor, using techniques like calculus of variations and generating functions, many of the early problems in the field were eventually solved using the topological methods which dominate the modern field. I will describe the progression from the dynamics problems to the topological solutions, and illustrate how those methods can tell us more about the Hamiltonians on a given manifold.

No prior knowledge of symplectic geometry will be required.

Abstract: I will introduce an approach to study compactifications of C^2 via studying families of curves which intersect the curves at infinity transversally. A possible way to encode such families is via corresponding Puiseux series. I will show how a study of these Puiseux series immediately gives conceptually simple proofs of Jung's theorem on autmorphisms of C[x,y] and Remmert-Van de Ven's theorem that CP^2 is the only non-singular analytic compactification of C^2 with Picard number one. The main tool is a basic result on valuations of C(x,y) which says that any valuation can be encoded via a Puiseux series. The talk is based on arxiv:1110.6905v2.

Abstract: We present "residual periodicity", a relatively new concept in arithmetic dynamics, as defined by Bandman, Grunewald and Kunyavskii. A rational self-map of a quasiprojective variety defined over a number field is strongly residually periodic if its minimal periods are bounded modulo almost every prime. We discuss some interesting examples, and present results about residual periodicity on cubic surfaces.

Abstract: One way to study the structure of a given division algebra is through special elements it contains, such as the p-central elements, i.e. elements who become central when taken to the power of some prime p. If a set of such elements forms a vector space over the centre, one can take its corresponding Clifford algebra and through its structure conclude something about the structure of the original algebra. In this lecture we discuss several recent results on Clifford algebras of two dimensional spaces for primes greater than 3, and the newly introduced Clifford algebra of a degree p-space.

Abstract: Say you you have a stable map from a closed (but not necessarily connected) surface F to the 3-sphere. Stable meaning that only have singularities and intersections that cannot be solved by slight changes - in other words regular intersections of two or three planes and cross-caps.

Such a map will create a graph of intersections in F (pre images of all intersections and singularities) and that will divide the surface into a collection of surfaces with boundaries - which I call 2-components. The map image will also cut the 3-sphere into different 3-manifolds with a boundary (a single boundary component if i(F) is connected) which I call 3-components.

A result by Ballesteros and Saeki from 1999 indicates that that 3-components can be coloured by two colours, such that adjacent components will have opposed colours, I found that there is a similar colouring for the 2-components i F is orientable. These two colourings can actually be used instead of orientations, as one can deduce these form these given some conventions.

Three very elementary questions you can ask yourself are what intersection graphs and collections of 2 and 3 components may arise from such maps. In this talk I will attempt to answer these questions to some extant, using colourings as a tool to both deduce result about, and construct, stable maps from orientable surface to the 3-sphere.

For more details and some great figures, see this extended abstract.

Abstract: Dessin d'Enfant (literally "children's drawings") come in many names for many different contexts in math and physics: graphs embedded on surfaces, ribbon graphs, combinatorial maps, rotation systems. Last week we heard about some of the algebraic settings; this week I introduce some of the more combinatorial settings, including graph theory and topology.

Then I will return to the algebraic setting to show in detail how the graphs are obtained directly from the equation of an algebraic curve.

From here we discuss graph theoretic properties.

Abstract: A Belyi pair is a smooth connected algebraic curve together with a non-constant meromorphic function on it with no more than 3 critical values. Belyi pairs are closely related with tamely embedded graphs on Riemann surfaces, so-called Grothendieck dessins d'enfants. An introduction to the theory will be given, including modern applications. In particular, we will discuss the generalized Chebyshev polynomials and their geometry, visualization of the Galois group action and its invariants, and Grothendieck dessins d'enfants on reducible curves.

Abstract: In recent decades, great progress was made in the research of the geometry of irregular varieties. The representative works are Chen-Hacon and Pareschi-Popa's studies of the adjoint linear systems and the pluricanonical maps on irregular varieties with maximal Albanese dimension. These results show that irregular varieties behave like curves. This talk would like to introduce my recent work with Jiang which generalises some results of Chen-Hacon.

Let X be a smooth projective variety of Albanese fiber dimension 1 and of general type. We prove that the translates through 0 of all components of V^0(\omega_X) generate \Pic^0(X). We then study the pluricanonical maps of X. We show that |4K_X| induces a birational map.

Abstract: Kac-Moody geometry is the study of infinite dimensional objects, whose symmetries are governed by affine Kac-Moody groups. It emerged during the last 20 years in the work of Chuu-Lian Terng, Ernst Heintze and many others as an infinite dimensional counterpart to the geometry, governed by semi-simple Lie groups, hence symmetric spaces, polar actions, isoparametric submanifolds and spherical buildings.

In this talk we first review the finite dimensional theory and describe the basic objects of Kac-Moody geometry: Kac-Moody symmetric spaces, isoparametric submanifolds in Hilbert space, polar actions on Hilbert spaces, twin cities and the relations among them.

Abstract: Hyperbolic geometry didn’t come about just to make things difficult. Rather, it arose because the universe itself, according to Einstein’s special theory of relativity, is essentially regulated by hyperbolic geometry. The mere mention of hyperbolic geometry is enough to strike fear in the heart of the undergraduatemathematics and physics student. Some regard themselves as excluded from the profound insights of hyperbolic geometry so that this enormous portion of human achievement is a closed door to them. The mission of this seminar is to open that door by making the hyperbolic geometry of Bolyai and Lobachevsky, as well as the special relativity theory of Einstein that it regulates, accessible to a wider audience in terms of novel analogies that the modern and unknown share with the classical and familiar.

Specifically, in this seminar we will extend the concepts of groups and vector spaces into gyrogroups and gyrovector spaces, discovering that gyrovector spaces form the algebraic setting for hyperbolic geometry just as vector spaces form the algebraic setting for Euclidean geometry.

The seminar is based on the lecturer’s six books listed below. As a mathematical prerequisite, it is assumed familiarity with Euclidean geometry from the point of view of vectors. In particular, prior acquaintance with either the hyperbolic geom- etry of Bolyai and Lobachevsky or the special relativity theory of Einstein is not assumed.

1. A. A. Ungar, Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces. Dordrecht: Kluwer Acad. Publ., 2001.
2. A. A. Ungar, Analytic Hyperbolic Geometry: Mathematical Foundations and Applications. Singapore: World Scientific, 2005.
3. A. A. Ungar, Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativ- ity. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.
4. Abraham A. Ungar. A gyrovector space approach to hyperbolic geometry. Morgan & Claypool Pub., San Rafael, California, 2009.
5. Abraham A. Ungar. Hyperbolic triangle centers: The special relativistic approach. Springer- Verlag, New York, 2010.
6. Abraham A. Ungar. Barycentric calculus in Euclidean and hyperbolic geometry: A compar- ative introduction. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010.

Abstract: In my talks I would like to recall the GAGA (géométrie algébrique = géométrie analytique) principle, which has been introduced by Serre in 1956, the fundamental Kodaira embedding theorem and Chow's theorem, then introduce the notions of complex super manifolds, super algebraic varieties and super projective spaces. Finally I would like to discuss how the before mentionned classical results generalize to the case of super manifolds.

Abstract: In his text on knot theory, Kauffman uses states on the shadow of a knot to give, amongst other things, the ubiquitous Alexander polynomial. Kauffman creates a lattice of these states with an operation called the clock move. A recent paper by Abe on the arXiv defines the clock number to be the minimum over all possible shadows of the knot of the height of this lattice, and relates this to the crossing number of the knot, achieving a sharp bound if and only if the knot is in the class of two-bridge knots. In recent work by Cohen-Dasbach-Russell on the Alexander polynomial, these states are reconsidered as perfect matchings of a bipartite graph. This context gives better insight into the questions introduced by Abe: specifically upper bounds on both the clock number and the bridge number of a knot coming from the lattice associated to the shadow. The strategy is to count the number of concentric cycles within the bipartite graph. Preliminary results, conjectures, and future directions will be discussed.

Abstract: We present the recent idea of "Tropical Cryptography" by Dima Grigoriev and Vladimir Shpilrain (see http://www.sci.ccny.cuny.edu/~shpil/tropical.pdf). They employ tropical algebras as platforms for several cryptographic schemes that would be vulnerable to linear algebra attacks were they based on "usual" algebras as platforms. In particular they propose "tropical variants" of slight generalisations of a key establishment protocol by Stickel and of a public key cryptosystem by Tzuong-Tsieng Moh. They make a case for using tropical algebras as platforms by using, among other things, the fact that in the "tropical" case, even solving systems of linear equations is computationally infeasible in general.

Abstract: The equation |z^2_1|+|z^2_2|=1 in two complex variables defines the standard three-dimensional sphere, S^3. The equation z^p_1=z^q_2 cuts in this sphere a knot.

The equations |z^2_1|+..+|z^2_5|=1, z^2_1+z^2_2+z^2_3+z^3_4+z^5_5=0 define a smooth subset of S^9. This subset is homeomorphic to S^7 but not diffeomorphic to the standard S^7. It is an exotic sphere.

In both examples we have the link, i.e. the intersection of a singular hypersurface with the unit sphere centered at the origin. Another related object is the Milnor fibre: the smoothing of the hypersurface.

The link and the Milnor fibre are very strong invariants, they capture many topological and geometrical properties of the local hypersurface singularity.

Vice versa, the topology and geometry of the link and Milnor fibre are determined by the local singularity. Hence allowing purely algebraic constructions of interesting topological objects.

I will give a sketchy introduction the topic. No preliminary knowledge beyond the standard undergraduate courses is assumed.

Abstract: In 1929, Zariski has found that the branch curve of a smooth cubic surface in P^3 (over a filed of char=0) is a sextic plane curve with 6 cusps, all of them lying on a conic. A year later, Segre generalized this, proving a similar theorem on smooth surfaces of any degree in P^3. Explicitly, he proved that there are two curves of unexpectedly low degree, passing through the nodes and the cusps of the branch curve of this surface (these two curves are called adjoint curves).
I will discuss these theorems, and also their generalizations to any surface in P^N.

This lecture is based on a joint work with R. Lehman, M. Leyenson and M. Teicher.

Abstract: The braid monodromy factorization (BMF) of the branch curve of a surface of general type is known to be an invariant that completely determines the diffeomorphism type of the surface. Calculating this factorization is of high technical complexity; computing the BMF of branch curves of surfaces uncovers new facts and invariants of the surfaces. Since finding the branch curve of a surface is very difficult, we degenerate the surface into a union of planes. Thus, we can find the braid monodromy of the branch curve of the degenerated surface, which is a union of lines. The regeneration of the singularities of the branch curve, studied locally, leads us to find the global BMF of the branch curve of the original surface. So far, the regenerations of the BMF of 3,4,5 and 6-point have been computed.

In this talk, we present two surfaces with '8-point degenerations'. Finding the BMF of each branch curve, we can compute the related fundamental groups, whose two quotients are isomorhpic.

This is a joint work with Garber-Shwartz-Teicher.

Abstract: In this talk we will discuss the following question:
Given a triple of integers (k,m,n), which finite simple groups are quotients of the triangle group T(k,m,n)?
This question, originally arising in group theory, has found applications in the classification of certain algebraic surfaces, known as Beauville surfaces, providing solutions to conjectures of Bauer, Catanese and Grunewald.

This talk is accessible to first year mathematics graduate students.

Abstract: A knot is a closed one-dim'l manifold embedded in three-space, but it can be treated as a combinatorial object, as well. A central question in Knot Theory is to distinguish knots by means of invariants like the Alexander and Jones polynomials. This talk will introduce a new, more accessible way of achieving these polynomials:

From a given knot, construct a bipartite graph whose edges are labelled by certain weights and signs. Multiply the weights and signs of each edge in a perfect matching, and then sum over all perfect matchings. This operation is in fact the determinant of the adjacency matrix of the graph. This so called "dimer model" is a concept from statistical physics.

A dimer model for the Alexander polynomial is extended to the "twisted" Alexander polynomial, which incorporates the additional information of a representation of the knot group. Tutte's notion of activity is used to produce a dimer model for the Jones polynomial for the class of pretzel knots as well as some other constructions. Due to a computational complexity result, the Jones polynomial cannot be produced in this way for an arbitrary knot.

Portions of this talk are joint work with Oliver Dasbach and Heather Russell.