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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 313Abstract: 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 Aigners 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.

Title: Lattice structures of torsion classes

Speaker: Shijie Zhu (Northeastern University)

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

Place: MONT 313Abstract: Torsion classes of finitely generated modules over a finite dimensional algebra form a lattice. We characterize the notion of cover relations, complete join irreducible elements and canonical join representations in the context of the lattice of torsion classes. As an application, we study the algebra $RA_n$ whose lattice of torsion classes is isomorphic to the weak order of the Weyl group of type $A_n$. This is a joint work with Emily Barnard and Andrew Carrol.

Title: Equations of Kalman Varieties

Speaker: Hang Huang (University of Wisconsin)

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

Place: MONT 313Abstract: Given a subspace $L$ of a vector space $V$, the Kalman variety consists of all matrices of $V$ that have a nonzero eigenvector in $L$. We will discuss how to apply Kempf Vanishing technique with some more explicit constructions to get a long exact sequence involving coordinate ring of Kalman variety, its normalization and some other related varieties in characteristic zero. Time permitting we will also discuss how to extract more information from the long exact sequence including the minimal defining equations for Kalman varieties.

Title: Images of Iterated Polynomials over Finite Fields

Speaker: Jamie Juul (Amherst College)

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

Place: MONT 313Abstract: We discuss how to bound the size of the image of the $n$-th iterate of a polynomial over a finite field using results about arboreal Galois representations. The main term in this bound involves the fixed point proportion of the Galois group of the field extension of $\mathbb{F}_q(t)$ obtained by adjoining all pre-images of the transcendental $t$ under the $n$-th iterate of the polynomial. We give explicit bounds on the fixed point proportion of the group in a general case.

Title: Rank parity in families of nonordinary modular forms

Speaker: Jeffrey Hatley (Union College)

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

Place: MONT 313Abstract: Given a pair of elliptic curves, there is a well-defined notion of "congruence mod $p$" between these curves. When two curves are congruent mod $p$, one can often use arithmetic information about one curve to deduce information about the other. This talk will discuss recent results which show how to compare the ranks of two elliptic curves (or, more generally, modular forms) which are congruent mod $p$. We will be especially interested in the non-ordinary case, where the $p$-th Fourier coefficients of our modular forms are not $p$-adic units. This is joint work with Antonio Lei of Laval University.

Title: Bounds of the rank of the Mordell--Weil group of Jacobians of Hyperelliptic Curves

Speaker: Erik Wallace (University of Connecticut)

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

Place: MONT 313Abstract: In a recent article written in collaboration with Alvaro Lozano-Robledo and Harris Daniels, we extend work of Lehmer, Shanks, and Washington on cyclic extensions, and elliptic curves associated to the simplest cubic fields. In particular, we give families of examples of hyperelliptic curves $C: y^2=f(x)$ defined over $\mathbb{Q}$, with $f(x)$ of degree $p$, where $p$ is a Sophie Germain prime, such that the rank of the Mordell--Weil group of the jacobian $J/\mathbb{Q}$ of $C$ is bounded by the genus of $C$ and the $2$-rank of the class group of the (cyclic) field defined by $f(x)$, and exhibit examples where this bound is sharp.

Title: Prime divisors in orbits and Galois groups of iterates

Speaker: Wade Hindes (City University of New York)

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

Place: MONT 313Abstract: Given a global field $K$ and a polynomial $\phi \in K[x]$, we study two finiteness questions related to iteration of $\phi$: whether all but finitely many terms of an orbit of $\phi$ must possess a primitive prime divisor, and whether the Galois groups of iterates of $\phi$ must have finite index in their natural overgroup $\mathrm{Aut}(T_d)$, where $T_d$ is the infinite tree of iterated preimages of $0$ under $\phi$. We focus particularly on the case where $K$ is a finite extension of $\mathbb{F}_p(t)$, where far less is known. We resolve the first question in the affirmative under relatively weak hypotheses; interestingly, the main step in our proof is to rule out ``Riccati differential equations" in backwards orbits. We then apply our result on primitive prime divisors to produce a family of polynomials for which the second question has an affirmative answer; these are the first non-isotrivial examples of such polynomials.

Title: Recursion formuals for modular traces of weak Maass forms of weight zero

Speaker: Chang Heon Kim (Sungkyunkwan University)

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

Place: MONT 313Abstract: In this talk I will explain how to derive recursion formulas satisfied by modular traces of weakly weakly holomorphic modular functions and more generally modular traces of certain weak Maass forms of weight zero. This is a joint work with Soyoung Choi.

Title: Castelnuovo-Mumford Regularity of Vector Bundles on Abelian Varieties

Speaker: Yusuf Mustopa (Tufts University)

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

Place: MONT 313Abstract: The CM (Castelnuovo-Mumford) regularity of a vector bundle F on a subvariety of projective space measures the homological complexity of its corresponding graded module. It also provides a measure of positivity, since the twist F(m) is ample and globally generated whenever m exceeds the CM-regularity of F. Though positivity can be quite subtle even in the line bundle case, there is a clean conjectural picture for the positivity of adjoint bundles due to Fujita and Mukai. Following Kollars suggestion that the right statements of Fujita-Mukai type should depend on intersection numbers, it is natural to ask when the CM-regularity of an adjoint bundle depends only on intersection numbers. Vector bundles on abelian varieties form a fertile testing ground for this question, due to their robust positivity theory. Here the correct notion is that of continuous CM-regularity, which I previously introduced in arXiv:1607.06550. In this talk I will present a sharp bound for the continuous CM-regularity of vector bundles on abelian varieties, as well as more detailed results for semihomogeneous bundles. This is joint work with Alex Kuronya.

Organizer: Liang Xiao