College of Liberal Arts and Sciences

# Department of Mathematics

## Algebra Seminar

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Title: Organization meeting
Speaker:
Time: Wednesday, August 29, 2018 at 11:15 am
Place: MONT 313Abstract: A good time to suggest speakers. Earlier is better. -Mihai

Title: Frieze vectors and unitary friezes
Speaker: Emily Gunawan (University of Connecticut)
Time: Wednesday, September 5, 2018 at 11:15 am
Place: MONT 313Abstract: Let Q be a quiver without loops and 2-cycles, let A(Q) be the corresponding cluster algebra and let x be a cluster. We introduce a new class of integer vectors which we call frieze vectors relative to x. These frieze vectors are defined as solutions of certain Diophantine equations given by the cluster variables in the cluster algebra. We show that every cluster gives rise to a frieze vector and that the frieze vector determines the cluster. We also study friezes of type Q as homomorphisms from the cluster algebra to an arbitrary integral domain. In particular, we show that every positive integral frieze of affine Dynkin type A is unitary, which means it is obtained by specializing each cluster variable in one cluster to the constant 1. This completes the answer to the question of unitarity for all positive integral friezes of Dynkin and affine Dynkin types. This is based on https://arxiv.org/abs/1806.00940 which is joint work with Ralf Schiffler. Comments are welcome.

Title: Galois representations attached to elliptic curves with complex multiplication
Speaker: Alvaro Lozano-Robledo
Time: Wednesday, September 12, 2018 at 11:15 am
Place: MONT 313Abstract: Let $F$ be a number field, and let $E/F$ be an elliptic curve. For each $n>1$, one can construct a Galois representation $BREAK ho_{E,n}\colon \text{Gal}(\overline{F}/F) \to \text{GL}(2,\mathbb{Z}/n\mathbb{Z})$ induced by the natural action of Galois on $n$-torsion points. In this talk, we will describe for each $n$ (and up to conjugation) all the possible Galois representations that may arise from elliptic curves with complex multiplication (CM, i.e., for elliptic curves with extra endomorphisms). We will also review what is known about Galois representations for elliptic curves without CM.

Title: Polynomials of high strength
Speaker: Steven Sam (UCSD)
Time: Wednesday, September 19, 2018 at 11:15 am
Place: MONT 313Abstract: A polynomial has high strength if it cannot be decomposed into a small sum of a product of lower degree polynomials. Inspired by a conjecture of Stillman, Ananyan and Hochster recently proved that polynomials of a fixed degree and sufficiently high strength acquire very good properties mimicking algebraically independent variables. I will make this precise and outline one way this can be proven. This is based on joint work with Daniel Erman and Andrew Snowden.

Title: The supersingular locus of some unitary Shimura varieties
Speaker: Sungyoon Cho (Northwestern University)
Time: Wednesday, September 26, 2018 at 11:15 am
Place: MONT 313Abstract: A version of the Arithmetic Gan-Gross-Prasad conjecture predicts a relation between the intersection number of a certain arithmetic cycle in unitary Shimura variety to the non-vanishing of the central derivative of a certain L-function. Here, the supersingular locus plays an important role. In this talk, I will explain the geometric structure of the supersingular locus of the relevant unitary Shimura variety at a place with bad reduction.

Title: Symmetric and skew symmetric degeneracy loci and constructions of hyperkahler manifolds
Speaker: Kristian Ranestad (University of Oslo)
Time: Friday, September 28, 2018 at 10:00 am
Place: MONT 214Abstract: Minors of matrices are defining equations for many classical varieties, which are therefore called degeneracy loci. Starting with Kummer surfaces, I shall explain a relation between symmetric and skew symmetric degeneracy loci. I shall go on to show how hyperkahler manifolds can be constructed from such degeneracy loci, in reporting on work with A. Iliev, G. Kapustka and M. Kapustka.

Title: Cohomology of the cotangent bundle to a Grassmannian and puzzles
Speaker: Voula Collins (University of Connecticut)
Time: Wednesday, October 3, 2018 at 11:15 am
Place: MONT 313Abstract: Maulik and Okounkov described a stable basis for the $T$-equivariant cohomology ring $H^*_{T\times \mathbb{C}^{\times}}(T^*Gr_k(\mathbb{C}^n))$ of the cotangent bundle to a Grassmannian. A natural idea to consider is the product structure of these basis elements. In this talk I will give a way to compute the the structure constants for the projective case, as well as a conjectural positive formula using puzzles. I will also discuss ideas for modifying this formula to apply to a general Grassmannian using R-matrices, based on work done by Knutson and Zinn-Justin.

Title: Equivariant D-modules and applications
Speaker: Andras Lorincz (Purdue University)
Time: Friday, October 5, 2018 at 10:00 am
Place: MONT 214Abstract: Let X be an algebraic variety equipped with the action of an algebraic group G. In this talk I will discuss some results on G-equivariant D-modules on X, focusing on the case when G acts on X with finitely many orbits. In this setting, the category of equivariant coherent D-modules is equivalent to the category of finite-dimensional representations of a quiver with relations. We describe explicitly these categories in some special cases, when the quivers turn out to be of finite or tame representation type. We apply these results to local cohomology modules supported in some orbit closures by describing their explicit D-module structure. In particular, we determine the Lyubeznik numbers of all determinantal rings, thus answering a question of M. Hochster.

Title: No occurrence obstructions in geometric complexity theory
Speaker: Peter Burgisser (Technical University of Berlin)
Time: Wednesday, October 10, 2018 at 11:15 am
Place: MONT 313Abstract: The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes VP and VNP. Mulmuley and Sohoni suggested to study a strengthened version of this conjecture over the complex numbers that amounts to separating the orbit closures of the determinant and padded permanent polynomials. In that paper it was also proposed to separate these orbit closures by exhibiting occurrence obstructions, which are irreducible representations of GL_{n^2}(C), which occur in one coordinate ring of the orbit closure, but not in the other. Recently it was shown by us that this approach is impossible. On the positive side, this research has led to new and surprising insights about Kronecker coefficients and plethysm coefficients.

Title: Counting Solvable Extensions of Number Fields
Speaker: Brandon Alberts (University of Connecticut)
Time: Wednesday, October 17, 2018 at 11:15 am
Place: MONT 313Abstract: Fix a finite group $G$ and a number field $K$. How many $G$-extensions $L/K$ are there with $disc(L/K) < X$, taken as $X$ tends towards infinity? This is in general a difficult question, at least as hard as the inverse Galois problem. In this talk, I will outline the proof of an upper bound for this quantity when $G$ is solvable, which is conditional on the size of the $\ell$-torsion of class groups of number fields with fixed degree. The new conditional bounds give evidence in support of Malle's conjecture, and can be used to prove unconditional bounds which improve on previously known results when $G$ is 'nearly nilpotent'.

Title: Converse theorems and the grand simplicity hypothesis
Speaker: Thomas Oliver (Oxford University)
Time: Wednesday, October 24, 2018 at 11:15 am
Place: MONT 313Abstract: In this talk, we will be interested in two manifestations of the so-called grand simplicity hypothesis for the zeros of automorphic L-functions. Specifically, we will see how the simplicity and independence of zeros can be related to the characterisation of automorphic L-functions in terms of analytic data. We will state two converse theorems in low degree and outline their proof in terms of the asymptotics of hypergeometric functions.

Title: Measures of irrationality for hypersurfaces of large degree
Speaker: Ruijie Yang (Stony Brook University)
Time: Wednesday, November 7, 2018 at 11:15 am
Place: MONT 313Abstract: A classical question in algebraic geometry is to determine whether or not a variety is rational. In this talk, I will focus on its opposite side: how 'irrational' can a variety be? In particular, I will discuss my work on certain measures of irrationality of hypersurfaces of large degree in projective spaces.

Title: Equations for point configurations to lie on a rational normal curve
Speaker: Luca Schaffler (Univeriversity of Massachusetts Amherst )
Time: Wednesday, November 14, 2018 at 11:15 am
Place: MONT 313Abstract: Let $V_{d,n}\subseteq(\mathbb{P}^d)^n$ be the Zariski closure of the set of $n$-tuples of points lying on a rational normal curve. The variety $V_{d,n}$ was introduced because it provides interesting birational models of $\overline{M}_{0,n}$: namely, the GIT quotients $V_{d,n}/ /SL_{d+1}$. In this talk our goal is to find the defining equations of $V_{d,n}$. In the case $d=2$ we have a complete answer. For twisted cubics, we use the Gale transform to find equations defining $V_{3,n}$ union the locus of degenerate point configurations. We prove a similar result for $d\geq 4$ and $n=d+4$. This is joint work with Alessio Caminata, Noah Giansiracusa, and Han-Bom Moon.

Title: Thanksgiving recess
Speaker:
Time: Wednesday, November 21, 2018 at 11:15 am
Place: MONT 313

Title: Torsion Subgroups of Elliptic Curves over Function Fields
Speaker: Robert McDonald (University of Connecticut)
Time: Wednesday, November 28, 2018 at 11:15 am
Place: MONT 313Abstract: Let $\mathbb F_q$ be a finite field of characteristic $p$, and $C/\mathbb F_q$ be a smooth, projective, absolutely irreducible curve. Let $K=\mathbb F_q(C)$ be the function field of $C$. When the genus of $C$ is $0$, and $p e 2,3$, Cox and Parry provide a minimal list of prime-to-$p$ torsion subgroups that can appear for an elliptic curve $E/K$. In this talk, we extend this result by determining the complete list of full torsion subgroups possible for an elliptic curve $E/K$ for any prime $p$ when the genus of $C$ is $0$ or $1$.

Title: Quotients of symmetric polynomial rings deforming the cohomology of the Grassmannian
Speaker: Darij Grinberg (University of Minnesota)
Place: MONT 313Abstract: The cohomology ring of a Grassmannian $Gr(k, n)$ is long known to be a quotient of the ring $S$ of symmetric polynomials in k variables. More precisely, it is the quotient of S by the ideal generated by the $k$ consecutive complete homogeneous symmetric polynomials $h_{n-k,1},\ h_{n-k,2},\ldots,\ h_n$. We propose and begin to study a deformation of this quotient, in which the ideal is replaced by the ideal generated by $h_{n-k,1} - a_1, h_{n-k,2} - a_2, ..., h_n - a_k$ for some $k$ fixed elements $a_1, a_2, ..., a_k$ of the base ring. This generalizes both the classical and the quantum cohomology rings of $Gr(k, n)$. We find two bases for the new quotient, as well as an $S_3$-symmetry of its structure constants and a fairly complicated Pieri rule. We conjecture that the structure constants are nonnegative in an appropriate sense (treating the a_i as signed variables), which suggests a geometric or combinatorial meaning for the quotient.