We will study the geometry of algebraic varieties. The course will cover affine and projective varieties, Zariski topology, dimension, smoothness, elimination via resultants etc. All necessary results and concepts used will be introduced and explained. Prerequisite: Familiarity with the basic undergraduate program of the first 3 years in math. References: 1. David Mumford "Algebraic Geometry I. Complex Projective Varieties" 2. Atiyah and I.
Macdonald "Introduction to Commutative Algebra". In the course, I will discussed relations between algebra and geometry which are useful in both directions. Newton polyhedron is a geometric generalization of the degree of a polynomial. New-ton polyhedra connect the theory of convex polyhedra with algebraic geometry. Toric varieties provide a tool for developing such connection.
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Riemann-Roch theorem for toric varieties provides a valuable information about the number of integral points in convex polyhedra and unexpected multidimensional generalization of the classical Euler-Maclaurin summation formula. Newton polyhedra allow to compute many discreet invariants of generic complete intersections. Topological version of Parshin reciprocity laws and Grothendieck residues help to study zero dimensional complete intersection. Newton-Okounkov bodies connect the theory of convex bodies not necessary polyhedra with algebraic geometry.
Tropical geometry and the theory of Grobner bases relate piecewise linear geometry and geometry of lattice with algebraic geometry.
These relations allow to describe the ring of conditions for complex torus and for other horospherical homogeneous spaces. All needed facts from algebraic geometry and will be discussed in details during the course.
Math 611: Analysis I
This course is an introduction to Siegel modular forms. Siegel modular forms were first introduced by Siegel in a paper of and nowadays often are given as a first example of holomorphic modular forms in several variables. The theory is a very important and active area in modern research; combining in many, nice ways number theory, complex analysis and algebraic geometry.
The goal of these lectures is twofold: first, we will introduce the basic concepts of the theory, like the Siegel modular group and its action on the Siegel upper half-space, reduction theory, examples of Siegel modular forms, Hecke operators and L-functions. References: H.
Kohnen, A short course on Siegel modular forms. Texts: There is no formal text. The following books are useful references. This course is divided into two parts : the first containing an introduction to the basic theory of elliptic curves, and the second covering more advanced but still accessible topics of the general of elliptic surfaces.
Starting with the definition of elliptic curves, we will turn to studying their basic geometric properties, theory of reduction, L-function, Mordell-Weil Theorem. If time allows, we will define the Picard, Selmer and Tate-Shafarevitch groups. Elliptic surfaces are omnipresent in the theory of algebraic surfaces. If time permits, we will see as well their relation with Del Pezzo surfaces in case of rational elliptic surfaces.
An intuition of projective geometry Complex analysis, basic group theory, arithmetic in finite fields. The Arithmetic of Elliptic Curves, by Silverman. A selection of topics from such areas as graph theory, combinatorial algorithms, enumeration, construction of combinatorial identities. Prerequisites: Linear algebra, elementary number theory, elementary group and field theory, elementary analysis.
The course will focus on fundamental geometric insights in Geometric Measure Theory and Geometric Analysis. Starting with the Federer-Fleming Isoperimetric inequality and its relatives we will build our way up towards Gromov's stunning proof of the systolic inequality in higher dimensions. We will explore applications of these geometric ideas to problems in minimal surface theory, string theory and quantum information. The course will be self-contained.
Some basic familiarity with Riemannian geometry is recommended. No prior knowledge of Geometric Measure Theory or theory of minimal surfaces will be assumed. This will be an introductory course in generalized geometry, with a special emphasis on Dirac, generalized complex and Kahler geometry. Dirac geometry is based on the idea of unifying the geometry of a Poisson structure with that of a closed 2-form, whereas generalized complex geometry unifies complex and symplectic geometry.
For this reason, the latter is intimately related to the ideas of mirror. The main references for this class are the published papers on generalized complex and Kahler geometry, but we will also draw from more recent developments in the physics literature. A very basic familiarity with complex and symplectic manifolds will be assumed; here is a list of topics which will be covered in the lecture course:.
Prerequisite: A basic familiarity with smooth manifolds, complex structures, and ideally symplectic structures. References: The main texts are all drawn from the literature in generalized geometry over the past 10 years. This includes the main papers by Hitchin, Gualtieri, Cavalcanti, Goto et al.
Texts: 1 V. Buchstaber and T. Panov 3 Toric Topology proceedings, Osaka Some of the most basic objects of study in Connes's non-commutative geometryfor instance, the non-commutative toriwill be considered from an elementary point of view.
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In particular, various aspects of the structure and classification of these objects will be studied, and a comparison made between the properties of these objects and the properties of the underlying geometrical systems. Some indication will be given of their use in index theory. References: M. Connes, Noncommutative Geometry, Academic Press, Gracia-Bondia, J. Varilly, and H. Figueora, Elements of Noncommutative Geometry, Birkhauser, Kawahigashi and D. Rordam, F.
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Larsen, and N. Pedersen, Analysis Now, Springer, The topics include: Riemannian metrics, Levi-Civita connection, geodesics, isometric embeddings and the Gauss formula, complete manifolds, variation of energy. It will cover chapters of the "Riemannian Geometry" book by Do Carmo. Set theory and its relations with other branches of mathematics.
ZFC axioms. Ordinal and cardinal numbers. Reflection principle. Constructible sets and the continuum hypothesis. Introduction to independence proofs.
Topics from large cardinals, infinitary combinatorics and descriptive set theory. The goals of the course include techniques for teaching large classes, sensitivity to possible problems, and developing an ability to criticize one's own teaching and correct problems. Assignments will include such things as preparing sample classes, tests, assignments, course outlines, designs for new courses, instructions for teaching assistants, identifying and dealing with various types of problems, dealing with administrative requirements, etc.
The course will also include teaching a few classes in a large course under the supervision of the instructor. A video camera will be available to enable students to tape their teaching for later private assessment. General Relativity is a geometric theory proposed by Einstein in as a unified theory of space, time and gravitation.