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Texas A&M University
Mathematics

Geometry Seminar

Fall 2018

 

Date:August 31, 2018
Time:4:00pm
Location:BLOC 628
Speaker:Frank Sottile, Texas A&M University
Title:Galois Groups for Systems of Sparse Polynomials
Abstract:Camille Jordan observed that Galois groups arise in enumerative geometry, and we now also understand them as monodromy groups. A study of this question in the Schubert calculus has determined many such Galois groups, all known Schubert Galois groups are either the full symmetric group or are imprimitive. Recently, Esterov considered this question for systems of sparse polynomials and proved this dichotomy in that setting. While this classification identifies polynomial systems with imprimitive Galois groups, it does not identify the groups.

I will sketch the background, before explaining Esterov's classification and ongoing work identifying some of the imprimitive Galois groups for polynomial systems.


Date:September 3, 2018
Time:3:00pm
Location:BLOC 628
Speaker:A. Conner, TAMU
Title:Tensors with large symmetry groups
Abstract:I will describe two new paths using algebraic geometry and representation theory to prove upper bounds on the exponent of matrix multiplication. The first approach aims to apply the laser method of Strassen to previously unstudied tensors which uniquely share certain geometric properties with the Coppersmith-Winograd tensor. The second approach for upper bounds transforms the problem into that of finding a certain sequence of finite groups and associated representations. The first approach is joint work with Fulvio Gesmundo, JM Landsberg, and Emanuele Ventura.

Date:September 7, 2018
Time:4:00pm
Location:BLOC 628
Speaker:Rafael Oliveria, U. Toronto
Title:Scaling algorithms, applications and the null-cone problem
Abstract:Scaling problems have a rich and diverse history, and thereby have found numerous applications in several fields of science and engineering. For instance, the matrix scaling problem has had applications ranging from theoretical computer science to telephone forecasting, economics, statistics, optimization, among many other fields. Recently, a generalization of matrix scaling known as operator scaling has found applications in non-commutative algebra, invariant theory, combinatorics and algebraic complexity; and a further generalization (tensor scaling) has found more applications in quantum information theory, geometric complexity theory and invariant theory. In this talk, we will describe in detail the scaling problems mentioned above, showing how alternate minimization algorithms naturally arise in this setting, and we shall present a general (3-step) framework to rigorously analyze such algorithms. We will also present a more general perspective on scaling algorithms, connecting it to the null-cone problem in invariant theory. This framework is based on concepts from invariant theory, which we will define. No prior background on Invariant Theory will be needed. Talk based on joint works with Peter Buergisser, Ankit Garg, Leonid Gurvits, Michael Walter and Avi Wigderson.

Date:September 10, 2018
Time:3:00pm
Location:BLOC 628
Speaker:JM Landsberg, TAMU
Title:Several astounding conjectures on the asymptotic geometry of tensors.
Abstract:Many computer scientists believe the astounding conjecture that asymptotically (as n goes to infinity), it becomes almost as easy to multiply matrices as to add them. Since progress on this conjecture stalled around 1989, Strassen made an even more astounding conjecture that would imply the matrix multiplication conjecture. Later Burgisser-Clausen-Shokrollahi made an even more astounding generalization to the effect that all tensors "asymptotically look the same" in a way I'll explain precisely. In this talk (joint work with A. Conner, F. Gesmundo, Y. Wang and E. Ventura), I will discuss these conjectures and insight geometry can provide us.

Date:October 1, 2018
Time:3:00pm
Location:BLOC 628
Speaker:Guangbo Xu, Simons Center of Geometry and Physics of Stony Brook
Title:Bershadsky-Cecotti-Ooguri-Vafa torsion of Landau-Ginzburg Models
Abstract:In their seminal work in 1994, Bershadsky-Cecotti-Ooguri-Vafa introduced a particular Ray-Singer analytic torsion of Calabi-Yau manifolds which coincides with the genus one topological string partition function. They also proved a holomorphic anomaly formula for this torsion which is related to the variation of Hodge structure and the Weil-Petersson geometry of deformation spaces. In this joint work with Shu Shen and Jianqing Yu, we consider the similar object in Landau-Ginzburg models. We prove an index theorem for the associated Dirac operator and rigorously define the BCOV torsion. We also obtain a partial result towards proving a holomorphic anomaly formula.

Date:October 12, 2018
Time:4:00pm
Location:BLOC 628
Speaker:B. Ullery, Harvard
Title:The gonality of complete intersection curves (Postponed)
Abstract:The gonality of a smooth projective curve is the smallest degree of a map from the curve to the projective line. If a curve is embedded in projective space, it is natural to ask whether the gonality is related to the embedding. In my talk, I will discuss recent work with James Hotchkiss. Our main result is that, under mild degree hypotheses, the gonality of a general complete intersection curve in projective space is computed by projection from a codimension 2 linear space, and any minimal degree branched covering of P^1 arises in this way.

Date:October 15, 2018
Time:3:00pm
Location:BLOC 628
Speaker:Sam Raskin, UT Austin
Title:An overview of local geometric Langlands
Abstract:Abstract: The (arithmetic) Langlands program is a cornerstone of modern representation theory and number theory. It has two incarnations: local and global. The former conjectures the existence of certain "local terms," and the latter predicts remarkable interactions between these local terms. By necessity, the global story is predicated on the local. Geometric Langlands attempts to find similar patterns in the geometry of curves. However, the scope of the subject has been limited by a meager local theory, which has not been adequately developed. The subject of this talk is a part of a larger investigation into local geometric Langlands. We will give an elementary overview of the expectations of this theory, discuss a certain concrete conjecture in the area (on "temperedness"), and provide evidence for this conjecture. One application of our results is a proof of Beilinson-Bernstein localization for the affine Grassmannian for GL_2, which was previously conjectured by Frenkel-Gaitsgory. (Note: the talk will have slides. I will post the slides online before the talk, so feel free to bring a laptop if you prefer to follow along on your own computer.)

Date:October 19, 2018
Time:4:00pm
Location:BLOC 628
Speaker:Yue Ren, MPI MiS Leipzig
Title:TBA

Date:October 26, 2018
Time:4:00pm
Location:BLOC 628
Speaker:Dylan Allegretti , University of Sheffield
Title:The monodromy of meromorphic projective structures
Abstract:A projective structure on an oriented surface S is an atlas of charts mapping open subsets of S into the Riemann sphere. There is a natural map from the space of projective structures to the PGL(2,C) character variety of S which sends a projective structure to its monodromy representation. In this talk, I will describe a meromorphic analog of this construction. I will introduce a moduli space parametrizing projective structures with poles at a discrete set of points. I will explain how, in this setting, the object parametrizing monodromy data is a type of cluster variety. This is joint work with Tom Bridgeland.

Date:October 29, 2018
Time:3:00pm
Location:BLOC 628
Speaker:C. Ikenmeyer, Simons Inst. and Saarbruchen
Title:On Algebraic Branching Programs of Small Width
Abstract:In 1979, Valiant showed that the complexity class VF of families with polynomially bounded formula size is contained in the class VBP of families that have algebraic branching programs (ABPs) of polynomially bounded size. Motivated by the problem of separating these classes, we study the topological closure of VF, i.e., the class of polynomials that can be approximated arbitrarily closely by polynomials in VF. We describe this closure using the well-known continuant polynomial (in characteristic different from 2). Further understanding this polynomial seems to be a promising route to new formula size lower bounds. Our methods are rooted in the study of ABPs of small constant width. In 1992, Ben-Or and Cleve showed that formula size is polynomially equivalent to width-3 ABP size. We extend their result (in characteristic different from 2) by showing that approximate formula size is polynomially equivalent to approximate width-2 ABP size. This is surprising because in 2011 Allender and Wang gave explicit polynomials that cannot be computed by width-2 ABPs at all! The details of our construction lead to the aforementioned characterization of VF. This is joint work with Bringmann and Zuiddam.

Date:November 2, 2018
Time:4:00pm
Location:BLOC 628
Speaker:Sebastian Casalaina-Martin, University of Colorado at Boulder.
Title:Distinguished models of intermediate Jacobians
Abstract:In this talk I will discuss joint work with J. Achter and C. Vial showing that the image of the Abel-Jacobi map on algebraically trivial cycles descends to the field of definition for smooth projective varieties defined over subfields of the complex numbers. The main focus will be on applications to topics such as: descending cohomology geometrically, a conjecture of Orlov regarding the derived category and Hodge theory, and motivated admissible normal functions.

Date:November 3, 2018
Time:10:30am
Location:BLOC 149
Title:Texas Algebraic Geometry Symposium, Fall Workshop
Abstract:10:30-11: Registration
11-12: Alicia Harper, Weak Factorization for Deligne-Mumford Stacks.
12-1:30: Lunch
1:30-2:30: Sebastian Casalaina-Martin, Geometry and topology of moduli space.
2:30-3: Tea.
3-4: Emily Witt, Frobenius powers of ideals.
4-4:20: Break.
4:20-5:20: Benjamin Schmidt, The Halphen Problem.
Abstracts of the talks are available from the link below

Date:November 4, 2018
Time:10:00am
Location:BLOC 149
Title:Texas Algebraic Geometry Symposium, Fall Workshop
Abstract:10-11: Souvik Goswami, Height Pairings
11-11:20: Break.
11:20-12:20: Daniel Hast, Rational Points and Unipotent Fundamental Groups.

Date:November 16, 2018
Time:4:00pm
Location:BLOC 628
Speaker:Giulio Belletti, Scuola Normale Superiore, Pisa
Title:Asymptotics of Turaev-Viro invariants and volume
Abstract:The basic building block of many quantum invariants of 3-manifolds and links is the quantum 6j-symbol. In this talk, I will introduce this object and show how it can produce the Turaev-Viro invariants of 3-manifolds. Furthermore, I will talk about a recent joint work with Detcherry, Kalfagianni and Yang giving an asymptotically sharp upper bound on the 6j-symbol, implying the Turaev-Viro volume conjecture for an interesting infinite family of hyperbolic 3-manifolds. If time permits, I will also briefly discuss some applications of these results to the study of quantum invariants.

Date:November 19, 2018
Time:3:00pm
Location:BLOC 628
Speaker:Shuang Ming, UC Davis
Title:On TQFT representations of mapping class groups with boundary
Abstract:(2+1)-dimensional topological quantum field theories provide many interesting finite-dimensional representations of mapping class groups of surfaces. In this talk, I will discuss the irreducibility and denseness of those representations. This is joint work with Greg Kuperberg.

Date:November 26, 2018
Time:3:00pm
Location:BLOC 628
Speaker:F. Gesmundo, U. Copenhagen
Title:Rank of forms and partial derivatives
Abstract:The polynomial Waring problem consists in determining a decomposition of a (homogeneous) polynomial as sum of powers of linear forms; the length of a minimal decomposition of this type is called Waring rank. A classical generalization considers a number of homogeneous polynomial and attempts to determine a simultaneous decomposition of all of them. In recent work with A. Oneto and E. Ventura, we established connections between the simultaneous Waring rank of the partial derivatives of a polynomial and its (partially symmetric) tensor rank. In this seminar, I will introduce Sylvester's classical apolarity theory, which is the most used tool in this study, and I present some of the results.

Date:November 30, 2018
Time:2:00pm
Location:BLOC 624
Speaker:S. Gong, UCLA
Title:Results on Spectral Sequences for Singular Instanton Floer Homology
Abstract:We introduce a version of Khovanov homology for alternating links with marking data, $\omega$, inspired by instanton theory. We show that the analogue of the spectral sequence from Khovanov homology to singular instanton homology (Kronheimer and Mrowka, \textit{Khovanov homology is an unknot-detector}) collapses on the $E_2$ page for alternating links. We moreover show that the Khovanov homology we introduce for alternating links does not depend on $\omega$; thus, the instanton homology also does not depend on $\omega$ for alternating links.

Date:December 7, 2018
Time:4:00pm
Location:BLOC 628
Speaker:Y. Qi, U. Chicago
Title:TBA