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Title: Seshadri constants for curve classes

Speaker: Mihai Fulger (University of Connecticut)

Time: Wednesday, September 6, 2017 at 11:15 am

Place: MONT 113Abstract: For a pair $(D,x)$ of a divisor and a point on a projective variety $X$, its Seshadri constant measures the local positivity of $D$ near $x$. Seshadri gave a characterization of the ampleness of $D$ in terms of these constants. The locus where the Seshadri constants vanish coincides with an important invariant of $D$, its augmented base locus. Demailly interprets Seshadri constants as asymptotic measures of jet separation and uses this to prove results relating to an important conjecture of Fujita. We construct a theory of Seshadri constants for pairs $(C,x)$, were $C$ is a curve on $X$ and $x$ is a point. We obtain analogues of the above results.

Title: Characterizations of projective space and Seshadri constants in positive characteristic

Speaker: Takumi Murayama (University of Michigan)

Time: Wednesday, September 13, 2017 at 11:15 am

Place: MONT 113Abstract: Projective spaces are, in some sense, the simplest algebraic varieties. It is therefore useful to know when a given variety is actually projective space. A famous result in this direction is due to Mori, who invented bend and break techniques to show that when a variety has a "positive" tangent bundle, it is in fact projective space. A stronger result is known in characteristic zero, and is due to Cho, Miyaoka, and Shepherd-Barron. We will present some progress toward this stronger result in positive characteristic using Seshadri constants and the Frobenius morphism. The key ingredient is a positive-characteristic proof of Demailly's criterion for separation of higher-order jets by adjoint bundles.

Title: Slopes of Hilbert modular forms

Speaker: Christopher Birkbeck (University College London)

Time: Wednesday, September 20, 2017 at 11:15 am

Place: MONT 113Abstract: Work of Buzzard and Kilford (among others) on slopes (the p-adic valuation of the Up eigenvalues) of overconvergent modular forms gave us great insights into the geometry of the associated eigenvarieties and are the basis of many conjectures. This is an active area of research and in many cases these conjectures are now known, yet not much is known in the case of Hilbert modular forms. In my talk I will discuss how one computes slopes in the Hilbert case, what they suggest about the geometry of the associated eigenvarieties and what we can prove in this setting.

Title: Markov Number Ordering Conjectures

Speaker: Michelle Rabideau (University of Connecticut)

Time: Wednesday, September 27, 2017 at 11:15 am

Place: MONT 214Abstract: A Markov number is a number in the triple $(x,y,z)$ of positive integer solutions to the Diophantine equation $x^2+y^2+z^2 = 3xyz$. Markov numbers are a classical topic in number theory related to many areas of mathematics such as combinatorics and cluster algebras. Markov numbers are related to cluster algebras by Markov snake graphs, where a Markov snake graph is the snake graph of a cluster variable of the once punctured torus. The number of perfect matchings of a Markov snake graph, given by the numerator of the associated continued fraction, is a Markov number. In this talk, we discuss three conjectures given in Martin AignerÂ’s book [A] that provide an ordering on the Markov numbers $m_{p/q}$ for a fixed numerator $p$, fixed denominator $q$ and a fixed sum $p+q$. \ [A] M. Aigner, Markov's theorem and 100 years of the uniqueness conjecture, Springer 2010

Title: Derived Category of Moduli of Pointed Curves

Speaker: Jenia Tevelev (UMass at Amherst)

Time: Wednesday, October 4, 2017 at 11:15 am

Place: MONT 313Abstract: I will report on the project, joint with A.-M. Castravet, devoted to derived category of moduli spaces of curves of genus 0 with marked points in the direction of conjectures of Orlov and Merkurjev-Panov. We develop several approaches to describe derived category equivariantly with respect to the action of the finite group. As an application, we construct an equivariant full exceptional collection on the Losev-Manin space which categorifies derangements. Combining our results with the method of windows in derived categories, we construct an equivariant full exceptional collection on the GIT quotient (or its Kirwan resolution) birational contraction of the Losev-Manin space.

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Speaker: Shijie Zhu (Northeastern University)

Time: Wednesday, October 11, 2017 at 11:15 am

Place: MONT 214Abstract: TBA

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Speaker: Hang Huang (University of Wisconsin)

Time: Wednesday, October 18, 2017 at 11:15 am

Place: MONT 214Abstract: TBA

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Speaker: Jamie Juul (Amherst College)

Time: Wednesday, October 25, 2017 at 11:15 am

Place: MONT 214Abstract: TBA

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Speaker: Jeffrey Hatley (Union College)

Time: Wednesday, November 1, 2017 at 11:15 am

Place: MONT 214Abstract: TBA

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Speaker: Erik Wallace (University of Connecticut)

Time: Wednesday, November 8, 2017 at 11:15 am

Place: MONT 214Abstract: TBA

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Speaker: Wade Hindes (City University of New York)

Time: Wednesday, November 15, 2017 at 11:15 am

Place: MONT 214Abstract: TBA

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Speaker: Chang Heon Kim (Sungkyunkwan University)

Time: Wednesday, November 29, 2017 at 11:15 am

Place: MONT 214Abstract: TBA

Organizer: Liang Xiao