Palace of Catalan Music (S. Adams/GETTY)

## Geometry, Physics, and Representation Theory Seminar

### RTG Group Seminar

#### Fall 2018, Northeastern

 Meeting weekly on Thursdays 2:50-3:50pm in 509/511 Lake Hall at Northeastern. If you are not at Northeastern, but would like to recieve announcements, join the mailing list. If you have a question or would like to speak at seminar, email

### Schedule

 Date Speaker Title Sep 6 No Seminar Sep 13 Brian Lehmann (BC) Rational curves on prime Fano threefolds. Manin's Conjecture predicts that the behavior of rational curves on a Fano variety is controlled by certain geometric invariants. We verify this conjecture for "most" Fano threefolds X of Picard rank 1 by computing the dimension and the number of components of the moduli space of rational curves. In this talk I will focus on the index 1 case and the role played by Manin's Conjecture in formulating the argument. This is joint work with Sho Tanimoto. Sep 20 Ivan Martino (Northeastern) About the motivic class of $BG$. In the beginning of the last century, Emmy Noether wondered about the rationality of the field extension $k(V)^G/k$ for any finite group $G$ and any field $k$, where $k(V)^G$ are the $G$-invariant rational functions over the regular representation $V$ of $G$. In 1969 Swan provided the first counterexample to such rationality for certain examples of cyclic groups and k being the field of rational numbers. In this talk I would like to relate this problem to the class of the classifying stack of a (finite) group, BG, in the Grothendieck ring of algebraic stack and summerize some results of Ekedahl and Totaro. In 2016, I have shown that that the motivic class of $BG$ is trivial if $G$ is a finite subgroup of $GL_3(k)$. Finally, I will present some recent work (in collaboration with R. Singh) on the underlying combinatorial structure behind these motivic problems. Sep 28 Friday 1:30pm Paolo Aluffi (FSU) Newton-Okounkov bodies and Segre classes Segre classes are a fundamental ingredient in Fulton-Macpherson intersection theory. We will present a new approach to the computation of the Segre class of a subscheme of complex projective space, based on the construction of a suitable Newton-Okounkov body. The result may be viewed as a common generalization of results of Kaveh and Khovanskii and of an earlier result on Segre classes of monomial schemes. Oct 4 Corey Bregman (Brandeis) Surface bundles and Kaehler geometry A question going back to Serre asks which groups arise as fundamental groups of smooth complex projective varieties, or more generally, compact Kaehler manifolds. These groups are called Kaehler groups. We first give a brief overview of this subject and some known results, then discuss Kaehler groups arising from surface bundles. Oct 11 2:50pm María Angélica Cueto (Ohio State) Anticanonical tropical del Pezzo cubic surfaces contain exactly 27 lines Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-know statement "any smooth surface of degree three in $P^3$ contains exactly 27 lines" is known to be false tropically. Work of Vigeland from 2007 provides examples of tropical cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in $TP^3$. In this talk I will explain how to correct this pathology by viewing the surface as a del Pezzo cubic and considering its embedding in $P^4$ via its anticanonical bundle. The combinatorics of the root system of type $E_6$ and a tropical notion of convexity will play a central role in the construction. This is joint work in progress with Anand Deopurkar. 3:50pm Ryan Grady (Montana) Joint GASC Seminar. A hierarchy of symplectic $\sigma$-models In this talk I will discuss a family of classical field theories in the Batalin-Vilkovisky formalism. These theories take as input $n$-symplectic Lie algebroids which correspond to symplectic manifolds, Poisson manifolds, and (higher) Courant algebroids. I will also discuss possible boundary conditions for these theories, e.g., Dirac structures and multiplectic structures. Finally, I will discuss aspects of the quantum theory in low dimensions. Oct 18 Junliang Shen (MIT) Gromov-Witten invariants and special subvarieties in HyperKähler varieties. By the Bogomolov-Mumford theorem, every projective K3 surface contains a rational curve. I will discuss some generalizations of this theorem to higher dimensions concerning rational curves and algebraically coisotropic subvarieties in HyperKähler varieties. I will further discuss some related open questions, examples, and the crucial role played by Gromov-Witten theory in the study of these topics. Based on joint work with Georg Oberdieck and Qizheng Yin. Oct 25 Preston Wake (IAS) Some examples around derived Hecke algebras. Venkatesh introduced a derived enrichment of the Hecke algebra of a $p$-adic group, which is a graded algebra that he shows has a graded action on the cohomology of the associated symmetric space. Much of the interesting higher derived structure comes from modular representation theory: $\mathbf k$-valued representations of a finite group don't form a semisimple category when the order of the group is zero in $\mathbf k$. In this expository talk, we'll introduce Venkatesh's construction, and discuss another situation where this non-semi-simplicity of the Hecke algebra plays a role in number theory. We'll spend the rest of the lecture discussing examples of (derived) Hecke algebras for the finite group $GL_n(\mathbf F_p)$. Nov 1 No Seminar Nov 8 Vance Blankers (Colorado State) Witten's conjecture for Mumford's kappa classes Kappa classes were introduced by Mumford as a tool to explore the intersection theory of the moduli space of curves. There is a close connection between the intersection theory of kappa classes on the moduli space of unpointed curves and the intersection theory of psi classes on all moduli spaces: we show that the potential for kappa classes is related to the Gromov-Witten potential of a point via a change of variables given by complete symmetric polynomials, rediscovering a theorem of Manin and Zokgraf from '99. In contrast to their methods, the starting point of our story is a combinatorial formula that relates intersections of kappa classes and psi classes via a graph theoretic algorithm. Further, this story is part of a large wall-crossing picture for the intersection theory of Hassett spaces, a family of birational models of the moduli space of curves. This is joint work with Renzo Cavalieri. Nov 15 Chris Elliott (IHES) The Multiplicative Hitchin System in Supersymmetric Gauge Theory Multiplicative Higgs bundles are an analogue of ordinary Higgs bundles where the Higgs field takes values in a Lie group instead of its Lie algebra. In this talk I'll discuss two contexts where multiplicative Higgs bundles appear in supersymmetric gauge theory. I'll explain how hyperkähler structures on these moduli spaces arise physically and mathematically and relate to the theory of Poisson Lie groups, and finally I'll introduce a speculative multiplicative analogue of the geometric Langlands conjecture. This is based on joint work in progress with Vasily Pestun. Nov 22 No Seminar (Thanksgiving Break) Nov 29 Kelly Jabbusch (U.Mich.,Dearborn) Towards Fujita's conjecture on projectivized toric vector bundles Let $X$ be a projective variety with mild singularities and $L$ an ample line bundle on $X$. In 1988 Fujita conjectured that for $k \geq \dim(X) + 1$, $L^k \otimes \omega_X$ is base point free, and for $k \geq \dim(X) + 2$, $L^k \otimes \omega_X$ is very ample. Fujita's conjecture is known to be true for small dimensions and for certain types of varieties, for example toric varieties. In this talk we'll focus on projectivized bundles $\mathbb{P}(\mathcal E)$, where $\mathcal{E}$ is an ample toric vector bundle on a smooth toric variety, and give evidence for Fujita's conjecture to hold for $\mathbb{P}(\mathcal{E})$ and $L = \mathcal{O}_{\mathbb{P}(\mathcal E)} (1)$. This will entail looking at the parliament of polytopes for the toric vector bundle $\mathcal{E}$ and its symmetric powers, $\text{Sym}^m \mathcal{E}$. Dec 6 Federico Scavia (U. British Columbia) Motivic classes of algebraic groups The Grothendieck ring of algebraic stacks was introduced by Ekedahl in 2009. It may be viewed as a localization of the more common Grothendieck ring of varieties. If $G$ is a finite group, then the class ${BG}$ of its classifying stack $BG$ is equal to 1 in many cases, but there are examples for which ${BG}\neq 1$. When $G$ is connected, ${BG}$ has been computed in many cases in a long series of papers, and it always turned out that ${BG}*{G}=1$. We exhibit the first example of a connected group G for which ${BG}*{G}\neq 1$. As a consequence, we produce an infinite family of non-constant finite étale group schemes A such that ${BA}\neq 1$. End of Semester (Seminar Resumes Spring 2019)

#### Previous Semesters

 Spring 2018 Fall 2017 Spring 2017 Fall 2016 Spring 2016 Fall 2015 Spring 2015