First and Second Year Student Seminar

from Thursday
June 01, 2017 to Sunday
December 31, 2017
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Tuesday
August 29, 2017

5:30 PM - 6:30 PM
Math Tower 5-127
organizational meeting, Stony Brook University
First and Second Year Student Seminar

organizational meeting, pizza will be served


Tuesday
September 12, 2017

5:30 PM - 6:30 PM
Math Tower 5-127
Robert Lazarsfeld, Stony Brook University
Robert's rules of graduate study


Tuesday
September 26, 2017

5:30 PM - 6:30 PM
Math Tower 5-127
Oleg Viro, Stony Brook University
Topology between real and complex

A geometric object may admit complexification. The presence of complexification dignifies geometric and topological characteristics and properties. A parity elevates to an integer. Non-invariant turns to invariant and new conservation laws emerge. Some phenomena of this kind are well-known, yet many new ones are still to be discovered.


Tuesday
October 10, 2017

5:30 PM - 6:30 PM
Math Tower 5-127
Professor Dennis Sullivan, Stony Brook University
Topology and Related Fields

Algebraic Topology provides transitions from compact spaces and geometrical objects like manifolds to algebra, such as finitely generated fundamental groups and finitely generated cohomology rings.

Finite and infinite aspects:
One can compartmentalise this information into the finite part and the infinite part.
Finite part : to an Abelian or non abelian group one may associate the system of its finite quotients. The inverse limit group is called the profinite completion of the group and one writes: G---> G^.
There is an analogue for the cohomology rings. This information is part but not all of the full finitistic aspect of the algebraic topology of a space.
Infinite part: One may tensor with the rational numbers.
First, the tower of nilpotent quotients of the fundamental group defined by the lower central series.
Second the cohomology ring A ---> AoQ.
This information is also illustrative of the full infinite part of the algebraic topology of a space.

For abelian groups, the exact sequence
0---> A ---> AoQ + A^ ----> quotient---> 0
illustrates the possibility to reconstruct the original information from these two aspects, finite and infinite. Here the quotient is constructed by Andre' Weil & called the finite adeles.

Related fields
Part I] Algebraic Varieties, Galois coverings and the finite part.
Part II] Geometry, Analysis and the infinite part.

Part I] It turns out that the profinite fundamental group defined by unbranched finite coverings makes sense for any algebraic variety defined over any field. Doing this for complements of subvarieties with a Cech-like combination shows the remarkable theory of Grothendieck : the finite part of the full classical algebraic topology of an algebraic variety over the complex numbers can be defined purely algebraically in a way that makes sense for varieties over fields of any characteristic.
[compare chapter V of the The 1970 MIT Notes]

Part II] The de Rham differential forms A with its differential d describes the cohomology ring A of a space tensor the real numbers; but there is a remarkable aspect . A,d is a graded commutative associative differential algebra. Thanks to Whitney's book on topology and analysis "Geometric Integration" there is a version of differential forms for any space. One can
adapt Whitney to a Q version of differential forms and then do DGA constructions & construct the infinite part of the algebraic topology in terms that are not alien to geometry and analysis.[compare "Infinitesimal computations in topology" ]

Thus we have two machines for the two parts of the information: algebraic geometry and Galois theory for the finite aspect AND geometry and analysis for the infinite part.
A corollary of the first was the Adams Conjecture for vector bundles. A corollary of the second was the determination of the infinite part for Kaehler manifolds.

The two citations are numbers 6 and 39 online : Dennis Sullivan Publications. (http://www.math.stonybrook.edu/~dennis/publications/)


Tuesday
October 24, 2017

5:25 PM - 6:30 PM
Math Tower 5-127
Panel discussion, Stony Brook University
Prepare qual/comps

We will hold a discussion session with Ben Wu, Tobias Shin, Liz Hernandez Vazquez, and Aleksandar Milivojevic about preparing/taking qual/comps.


Tuesday
November 07, 2017

5:30 PM - 6:30 PM
Math Tower 5-127
Professor Marco Martens, Stony Brook University
Some questions in Dynamics


Tuesday
November 14, 2017

5:30 PM - 6:30 PM
Math Tower 5-127
Radu Laza, Stony Brook University
Moduli and periods

I will give a brief overview of moduli, and then discuss the period map as a tool for studying moduli spaces. I will close with some recent developments in the field, and mention some open questions.


Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars