The junior geometry seminar meets every Tuesday at 3:45-4:45pm central in PMA 12.166
What is the junior geometry seminar?
The organizational meeting is on Tuesday August 29 at 3:45 pm central.
Email reminders will be sent to the mailing list. Click here to be added to the mailing list.
List of past organizers (aka JG ORGANIZER HALL OF FAME )
The titles and abstracts of the talks are also available here. at the official math department seminar calendar.
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Title: Statistics via tropical geometry
Abstract: I will talk about arrangements of tropical hypersurfaces, which I will prove to be better ways to cut cakes if you want to get more pieces out of a given number of cuts than the usual hyperplane cuts. This result has an interesting interpretation in statistics. It has another implication in discrete geometry, which gives a lower bound for the number of vertices of a Minkowski sum of generic polytopes. The proof uses stratified Morse theory, which is a Morse theory for singular spaces. I will show how geometric thinking leads one to another result with statistical implications, which says every piecewise linear function is a linear combination of conewise linear functions. Based on joint work with Tran.
Title: Riemann-Roch and Division Algebras
Abstract: The Riemann-Roch theorem provides a delicate control on the number of linearly independent meromorphic functions with prescribed zeros and poles on a Riemann surface. It turns out that this forces the field of meromorphic functions on the Riemann surface to have algebraic properties of great interest. In this talk, we will go through a brief introduction to the Riemann-Roch theorem and division algebras over a field. We then give an exposition of Tsen's theorem which implies there are no nontrivial division algebras over the function field of an algebraic curve. Time permitting, we will give a brief introduction to the Brauer group and how it obstructs the Hasse principle.
Title: Virtual Fundamental Class
Abstract: I will give a conceptual introduction of virtual fundamental class following the paper Intrinsic Normal Cone by Behrend and Fantechi. If time permits, I will introduce the virtual fundamental class in the Gromov-Witten theory as an example.
Title: Group Cohomology
Abstract: So what is this mysterious group cohomology you keep hearing about? Strap in for a whirlwind tour of the world of homological algebra: projective and injective resolutions, derived functors, tor and ext, bar resolutions, and finally group cohomology. The payoff will be questionable, the befuddlement complete. Trust me, it'll be fun!
Title: Kazhdan-Lusztig conjecture
Abstract: The Kazhdan-Lusztig conjecture - proven by Beilinson-Bernstein and Brylinski-Kashiwara- gives a character formula for certain irreducible representations of complex semisimple Lie algebras. Its proof is interesting in that it solves the seemingly purely algebraic problem by geometric methods and shows us a beautiful and rapid confluence of different developments in math. In this talk, we explain the content of this conjecture and give a very brief overview of the proof, focusing on the application of perverse sheaves.
Title: Branched Covers and Riemann-Hurwitz
Abstract: Given a branched covering of a closed Riemann surface, the Riemann-Hurwitz formula gives a precise relationship between the degree of the map, the genus of the two surfaces, and the number of branch points. I will explain the formula, give some examples, and mention some applications.
Title: What is an exact triangle in the Fukaya category of a symplectic manifold?
Abstract: In this talk I will do my best to introduce the concept of an exact triangle in the Fukaya category of a symplectic manifold to a general audience. I will talk about a new proof of its existence that is the subject of my work with T. Perutz and relies -among other things- on a “intuitive” (at least I hope) stretching argument we developed. After motivating why this triangle is interesting from the point of view of Mirror Symmetry (I’m not an expert so do not expect a lot of details there), time permitting I will talk about its geometric relevance and what I am trying to do with it for my PhD thesis.
Title: Poincare to Arnol'd to Floer
Abstract: Fixed points underlied much of the study of analytical mechanics for the majority of its history. In the realm of celestial mechanics, a restricted three body problem lead to the formulation of Poincare's last geometric theorem, concerning fixed points of certain area preserving diffeomorphisms of the annulus. In this talk we discuss how Arnol'd reformulated Poincare's last geometric theorem in the language of symplectic geometry and subsequently gave the statement of what is now known as the Arnol'd conjecture. After this we will discuss Floer's breakthrough approach to the problem using an infinite dimensional version of Morse theory and many of the analytic techniques developed by Gromov relating to pseudo-holomorphic curves.
Title: Borel Anosov representations into SL(n,R)
Abstract: The theory of Anosov representations emerged from generalizing the notion of convex cocompact subgroups of the isometry group of hyperbolic space to higher rank Lie groups. These representations form a rich class of discrete and faithful representations of hyperbolic groups (e.g., surface groups and free groups), and much of the theory coming from hyperbolic geometry generalizes to this setting. In particular, Anosov representations induce quasi-isometric orbit maps into the associated symmetric space G/K, and these representations form an open subset of the representation variety Hom(Γ, G). Many well-studied representations fall into this category, such as the representations of surface groups studied by Hitchin and Fock-Goncharov. One of my favorite aspects of Anosov representations is that they induce a special boundary map that maps the Gromov boundary of Γ into the flag variety G/P. In this talk, I will introduce the notion of Anosov representations in the case when G=SL(n,R) and investigate which kinds of hyperbolic groups admit the strongest type of Anosov representation that maps the Gromov boundary into the full flag variety G/B, and hopefully show how one can use the structure of (double/opposite) Schubert cells coming from cluster algebra techniques to partially answer this.
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