September 27: Yoav Rieck (University of Arkansas)
Title: The unbearable hardness of unknotting Abstract: While much is known about existence of algorithms in the study of 3-manifolds and knot theory, much less is known about lower bounds on their complexity (hardness results). In this talk we will discuss hardness of several problems. We prove: Theorem 1: given a 2- or 3-dimensional complex X, deciding if X embeds in R^{3} in NP-hard. We also prove that certain link invariants that are defined using 4-dimensional topology give rise to NP-hard problems; for example: Theorem 2: deciding if a link in the 3-sphere bounds a smooth surface of non-negative Euler characteristic in the 4-ball is NP-hard. For the main event we turn our attention to knots. The unknot recognition problem (solved by Haken in the 60's) is known to be in NP and co-NP, and as such, is not expected to be hard. Lackenby proved a polynomial bound on the number of Reidemeister moves needed to untangle an unknot diagram. In light of these two facts, one might hope for an efficient algorithm that find this optimal untangling. Unfortunately this is unlikely to happen since we prove: Theorem 3: given an unknot diagram D and a positive integer n, deciding if D can be untangled using n Reidemeister moves is NP-hard. This is joint work with Arnaud de Mesmay, Eric Sedgwick, and Martin Tancer. |
October 4: Alain Hénaut (Université de Bordeaux)
Title: On planar web geometry Abstract:Web geometry deals with foliations in general position. In the planar case and the complex setting, a d-web is given by the generic family of integral curves of an analytic or an algebraic differential equation F(x,y,y')=0 with y'-degree d. Invariants of these configurations as abelian relations (related to Abel's addition theorem), Lie symmetries or Godbillon-Vey sequences are investigated. This viewpoint enlarges the qualitative study of differential equations and their moduli. In the nonsingular case and through the singularities, Cartan-Spencer and meromorphic connections methods will be used. Basic examples will be given from different domains including classic algebraic geometry and WDVV-equations. Standard results and open problems will be mentioned. Illustration of the interplay between differential and algebraic geometry, new results will be presented. |
October 4: Michael McQuillan (University of Rome, Tor Vergata)
Title: TBA Abstract:TBA. |
October 5: Southern California Algebraic Geometry Seminar, at USC. |
October 18: Laure Flapan (MIT)
Title: Algebraic Hecke characters and Hodge/Tate classes on self-products Abstract:We examine the relationship between having an algebraic Hecke character attached to the cohomology of a smooth projective variety X equipped with a finite-order automorphism and the algebraicity of some Hodge/Tate classes on the product X^{n}. As a consequence, we deduce the Hodge and Tate conjectures for some self-products of varieties, including some self-products of K3 surfaces. |
October 25: Justin Lacini (UC San Diego)
Title: On pluricanonical maps of varieties of general type Abstract:Hacon and M^{c}Kernan have proved that there exist integers r_{n} such that if X is a smooth variety of general type and dimension n, then the pluricanonical maps |rK_{X}| are birational for all r≥r_{n}. These values are typically very large: for example r_{3}≥27 and r_{4}≥94. In this talk we will show that the r-th canonical maps of smooth threefolds and fourfolds of general type have birationally bounded fibers for r≥2 and r≥4 respectively. Furthermore, we will generalize these results to higher dimensions in terms of the constants r≥n and we will discuss recent progress on a conjecture of Chen and Jiang. |
October 25: Iacopo Brivio (UC San Diego)
Title: On algebraic invariance of plurigenera Abstract:A famous theorem of Y. T. Siu states that plurigenera of projective complex manifolds are invariant under deformation. The only known proof of this result uses deep techniques from complex analysis, which are not available in the algebraic category. In this talk, we will illustrate some recent progress toward an algebraic proof of Siu's result, and explain how these methods can be used to prove analogous results in positive and mixed characteristic. |
November 8: Jake Levinson (University of Washington)
Title: A topological proof of the Shapiro-Shapiro Conjecture Abstract:Consider a rational curve, described by a map f : P^{1} → P^{n}. The Shapiro-Shapiro conjecture says the following: if all the inflection points of the curve (the roots of the Wronskian of f) are real, then the curve itself is defined by real polynomials (up to change of coordinates). An equivalent statement is that certain real Schubert varieties in the Grassmannian intersect transversely - a fact with broad combinatorial and topological consequences. The conjecture, made in the 90s, was proven by Mukhin-Tarasov-Varchenko in '05/'09 using methods from quantum mechanics. I will present a generalization of the Shapiro-Shapiro conjecture, joint with Kevin Purbhoo, where we allow the Wronskian to have complex conjugate pairs of roots. We decompose the real Schubert cell according to the number of such roots and define an orientation of each connected component. For each part of this decomposition, we prove that the topological degree of the restricted Wronski map is given by a symmetric group character. In the case where all the roots are real, this implies that the restricted Wronski map is a topologically trivial covering map; in particular, this gives a new proof of the Shapiro-Shapiro conjecture. |
Colloquium
November 8: Benjamin Bakker (UGA) Title: Hodge theory and o-minimality Abstract:The cohomology groups of complex algebraic varieties come equipped with a powerful but intrinsically analytic invariant called a Hodge structure. The fact that Hodge structures of certain very special algebraic varieties are nonetheless parametrized by algebraic varieties has led to many important applications in algebraic and arithmetic geometry. While this fails in general, recent joint work with Y. Brunebarbe, B. Klingler, and J. Tsimerman shows that parameter spaces of Hodge structures always admit a "tame" analytic structure in a sense made precise using ideas from model theory. A salient feature of the tame analytic category is that it allows for the local flexibility of the full analytic category while preserving the global behavior of the algebraic category. In this talk I will explain this perspective as well as some important applications, including an easy proof of a celebrated theorem of Cattani-Deligne-Kaplan on the algebraicity of Hodge loci and the resolution of a longstanding conjecture of Griffiths on the quasiprojectivity of the images of period maps. |
November 15: Xiaolei Zhao (UC Santa Barbara)
Title: Stability conditions on Gushel-Mukai fourfolds Abstract:An ordinary Gushel-Mukai fourfold X is a smooth quadric section of a linear section of the Grassmannian G(2,5). Kuznetsov and Perry proved that the bounded derived category of X admits a semiorthogonal decomposition whose non-trivial component is a subcategory of K3 type. In this talk I will report on a joint work in progress with Alex Perry and Laura Pertusi, in which we construct Bridgeland stability conditions on the K3 subcategory of X. Then I will explain some applications concerning the existence of a homological associated K3 surface, and related algebraic constructions in hyperkaehler geometry. |
Colloquium
November 15: Harold Williams (UCD) Title: Geometric representation theory through the lens of physics Abstract:Ideas from theoretical physics have had a profound impact on geometry, topology, and representation theory over the last several decades. An early high point of this interaction was Witten's quantum field theoretic interpretation of the celebrated Donaldson invariants, which in turn opened the door to his discovery of the even-more-celebrated Seiberg-Witten invariants. In this talk, we'll explain how more recently this interaction has made possible dramatic advances in geometric representation theory, with a focus on joint work with Sabin Cautis revealing the structure of the coherent Satake category of a complex Lie group. This is an intricate cousin of the constructible Satake category appearing in the geometric Satake equivalence, a cornerstone of the geometric Langlands program. The coherent Satake category turns out to have rich connections to the Fomin-Zelevinsky theory of cluster algebras, as well as to the representation theory of quantum groups and quiver Hecke algebras. However, while these connections can be stated in purely mathematical terms, their discovery hinged crucially on first understanding how to interpret the coherent Satake category in terms of physics - in fact, the very same physics (4d N=2 supersymmetric Yang-Mills theory) behind the Donaldson and Seiberg-Witten invariants. |
Novermber 22: Harold Blum (University of Utah)
Title: Openness of K-stability for Fano varieties Abstract:Until recently, it was unclear if there was a natural way to construct (compactified) moduli spaces of Fano varieties. One approach to solving this problem is the K-moduli Conjecture, which predicts that K-polystable Fano varieties of fixed dimension and volume are parametrized by a projective good moduli space. In this talk, I will survey recent progress on this conjecture and discuss a result with Yuchen Liu and Chenyang Xu proving the openness of K-stability (a step in constructing K-moduli spaces). |
December 6: Ben Wormleighton (UC Berkeley)
Title: McKay correspondence and walls for G-Hilb Abstract:The McKay correspondence takes many guises but at its core connects the geometry of minimal resolutions for quotient singularities C^{n}/G to the representation theory of the group G. When G is an abelian subgroup of SL(3), Craw-Ishii showed that every minimal resolution can be realised as a moduli space of stable quiver representations naturally associated to G, although the chamber structure for the stability parameter and associated wall-crossing behaviour is poorly understood. I will describe my recent work giving explicit representation-theoretic descriptions of the walls and wall-crossing behaviour for the chamber corresponding to a particular minimal resolution called the G-Hilbert scheme. Time permitting, I will also discuss ongoing work with Yukari Ito (IPMU) and Tom Ducat (Bristol) to better understand the geometry, chambers, and corresponding representation theory for other minimal resolutions. |
Organizers: Elham Izadi, James McKernan and Dragos Oprea
This seminar is supported in part by grants from the NSF. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Past quarters: Fall 2013, Winter 2014, Spring 2014, Fall 2014, Fall 2017, Winter 2018, Spring 2018, Fall 2018, Winter 2019.
The design of this webpage is copied shamelessly from the MIT Number Theory seminar site. Contact Iacopo Brivio at ibrivio@ucsd.edu about problems with the website or posters.