MenuGeometry Seminar
Spring 2018
| Date: | January 22, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Frank Sottile, TAMU |
| Title: | Newton-Okounkov Bodies for Applications |
| Abstract: | Newton-Okounkov bodies were introduced by Kaveh-Khovanskii and Lazarsfeld-Mustata to extend the theory of Newton polytopes to functions more general than Laurent polynomials. This theory has at least two implications for applications. First is that Newton-Okounkov bodies provide an approach to counting the number of solutions to systems of equations that arise in applications. Another is that when the Newton-Okounkov body is an integer polytope (there is a Khovanskii basis), there is a degeneration to a toric variety which in principal should give a numerical homotopy algorithm for computing the solutions. This talk will sketch both applications. |
| Date: | February 9, 2018 |
| Time: | 4:00pm |
| Location: | BLOC 628 |
| Speaker: | Tri Lai, University of Nebraska - Lincoln |
| Title: | Tilings and More |
| Abstract: | The field of enumeration of tilings dates back to the early 1900s when MacMahon proved his classical theorem on plane partitions. The enumeration of tilings has since taken on a life of its own as a subfield of combinatorics with connections and applications to diverse areas of mathematics, including representation theory, linear algebra, group theory, mathematical physics, graph theory, probability, and cluster algebra, just to name a few. In this talk, we focus on an interesting connection between tilings, linear algebra, and a mathematical model of electrical networks. In particular, we will go over the proof of a conjecture of Kenyon and Wilson on `tiling-representation' of semi-contiguous minors. |
| Date: | February 16, 2018 |
| Time: | 4:00pm |
| Location: | BLOC 628 |
| Speaker: | Sara Maloni, University of Virginia |
| Title: | The geometry of quasi-Hitchin symplectic Anosov representations |
| Abstract: | In this talk we will focus on our joint work in progress with Daniele Alessandrini and Anna Wienhard about quasi-Hitchin representations in Sp(4,C), which are deformations of Fuchsian representations which remain Anosov. These representations acts on the space Lag(C^4) of complex lagrangian subspaces of C^4. We will show that the quotient of the domain of discontinuity for this action is a fiber bundle over the surface and we will describe the fiber. In particular, we will describe how the projection map comes from an interesting parametrization of Lag(C^4) as the space of regular ideal hyperbolic tetrahedra and their degenerations. |
| Date: | February 19, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Francis Bonahon, USC |
| Title: | The relation (X+Y)^n = X^n + Y^n, and miraculous cancellations in quantum SL_2 |
| Abstract: | The convenient formula (X+Y)^n = X^n + Y^n is (unfortunately) frequently used by our calculus students. Our more advanced students know that this relation does hold in some special cases, for instance in prime characteristic n or when YX=qXY with q a primitive n-root of unity. I will discuss similar ``miraculous cancellations`` for 2-by-2 matrices, in the context of the quantum group U_q(sl_2). |
| Date: | February 26, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Ron Rosenthal, Technion |
| Title: | Random Steiner complexes |
| Abstract: | We will discuss a new model for random d-dimensional simplicial complexes, for d ≥ 2, whose (d − 1)-cells have bounded degrees. The construction relies on Keevash's results on the existence of Steiner systems which are generalizations of regular graphs. We will show that with high probability, complexes sampled according to this model are high-dimensional expanders. This gives a full solution to a question raised by Dotterrer and Kahle, which was solved in the two-dimensional case by Lubotzky and Meshulam. In addition, we will discuss the limits of their spectral measures and their relation to the spectral measure of certain high-dimensional regular trees. Based on a joint work with Alex Lubotzky and Zur Luria and a work in progress with Yuval Peled. |
| Date: | March 2, 2018 |
| Time: | 4:00pm |
| Location: | BLOC 628 |
| Speaker: | J. Weyman, U. Conn. |
| Title: | Resonance varieties |
| Abstract: | I will discuss the Koszul modules introduced by Papadima and Suciu and their relation to Resonance Varieties and Alexander type invariants of finitely generated groups. In special case related to representations S_g(C^2) of SL_2 we get nilpotent modules whose nilpotency degree is related to Green conjecture for canonical curves of genus g. The talk is based on forthcoming work joint with Aprodu, Farkas, Papadima and Raicu. |
| Date: | March 5, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Frank Sottile, Texas A&M University |
| Title: | Intersection Theory in Numerical Algebraic Geometry |
| Abstract: | I will describe how some ideas from intersection theory are useful in numerical algebraic geometry. The fundamental data structure in numerical algebraic geometry is that of a witness set, which is considered to be an instantiation of Weil’s notion of a generic point. Reinterpreting a witness set in terms of duality of the intersection pairing in intersection theory leads to a generalization of the notion that makes sense on many spaces and leads to a general notion of a witness set. I will also describe how rational equivalence is linked to homotopy methods. These notions are most productive for homogenous spaces, such as projective spaces, Grassmannians, and their products. After explaining the general theory, I will sketch what this means for the Grassmannian. This is joint work with Bates, Hauenstein, and Leykin. |
| Date: | March 26, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Cris Negron, MIT |
| Title: | Cohomology for Drinfeld doubles of finite group scheme |
| Abstract: | In the mid 2000’s Etingof and Ostrik conjectured that the cohomology H*(A,F) of any finite dimensional Hopf algebra A over an arbitrary field F is itself a finitely generated algebra, under the standard (Yoneda) product. This conjecture was motivated, in part, by fantastic work of Friedlander and Suslin from the 90’s, in which they showed that any finite group scheme in characteristic p has finitely generated cohomology. I will discuss joint work with E. Friedlander, where we return to the finite characteristic setting to provide a strong analysis of cohomology for so-called Drinfeld doubles of finite group schemes. I will discuss the central role such doubles play in the more general theory of finite tensor categories, and explain how the cohomology of such doubles can be understood via “classical” data. |
| Date: | April 2, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Gregory Pearlstein, Texas A&M |
| Title: | Hodge theory, Gromov-Witten theory and representation theory |
| Abstract: | In advance of the Texas Algebraic Geometry Symposium, this talk will introduce hyperkahler manifolds, Gromov-Witten theory and certain aspects of infinite dimensional representation theory in terms of Hodge theory. |
| Date: | April 5, 2018 |
| Time: | 4:00pm |
| Location: | BLOC 628- Note s |
| Speaker: | N. Ressayre, U. Lyon |
| Title: | On the tensor semigroup of affine Kac-Moody Lie algebras. |
| Abstract: | In this talk, we are interested in the decomposition of the tensor product of two representations of a symmetrizable Kac-Moody Lie algebra g. Let P + be the set of dominant integral weights. For λ ∈ P + , L(λ) denotes the irreducible, integrable, highest weight representation of g with highest weight λ. Consider the tensor cone Γ(g) := {(λ 1 , λ 2 , μ) ∈ P + 3 | ∃N > 1 L(N μ) ⊂ L(N λ 1 ) ⊗ L(N λ 2 )}. If g is finite dimensional, Γ(g) is a polyhedral convex cone described by Belkale-Kumar by an explicit finite list of inequalities. In general, Γ(g) is nor polyhedral, nor closed. We will describe the closure of Γ(g) by an explicit countable family of linear inequalities, when g is untwisted affine. This solves a Brown-Kumar’s conjecture in this case. |
| Date: | April 6, 2018 |
| Time: | 4:00pm |
| Location: | |
| Speaker: | Texas Algebraic Geometry Seminar |
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| Date: | April 7, 2018 |
| Time: | 09:00am |
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| Speaker: | Texas Algebraic Geometry Seminar |
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| Date: | April 8, 2018 |
| Time: | 09:00am |
| Location: | |
| Speaker: | Texas Algebraic Geometry Seminar |
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| Date: | April 9, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Zhiwei Zheng, Tsinghua University |
| Title: | Moduli of Symmetric Cubic Fourfolds |
| Abstract: | The period map is a powerful tool to study moduli spaces of many kinds of objects related to K3 surfaces and cubic fourfolds, thanks to the global Torelli theorems. In this spirit, Allcock-Carlson-Toledo (2003) realized the moduli of smooth cubic threefolds as an arrangement complement in a 10-dimensional arithmetic ball quotient and studied its compactifications (both GIT and Satake-Baily-Borel) and recently, Laza-Pearlstein-Zhang studied the moduli of pairs consisting of a cubic threefold and a hyperplane section. I will talk about a joint work with Chenglong Yu about the moduli space of cubic fourfolds with automorphism group specified, and its compactification. As examples, we recover some of the works by Allcock-Carlson-Toledo and Laza-Pearlstein-Zhang mentioned above. |
| Date: | April 13, 2018 |
| Time: | 4:00pm |
| Location: | BLOC 628 |
| Speaker: | Renaud Detcherry, Michigan State Universeity |
| Title: | Quantum representations and monodromies of fibered links |
| Abstract: | According to a conjecture of Andersen, Masbaum and Ueno, the Witten-Reshetikhin-Turaev quantum representations of mapping class groups send pseudo-Anosov mapping classes to infinite order elements, when the level is big enough. We relate this conjecture to a properties about the growth rate of Turaev-Viro invariants, and derive infinite families of pseudo-Anosov mapping classes that satisfy the conjecture, in all surfaces with n boundary components and genus g>n>=2. These families are obtained as monodromies of fibered links containing some specific sublinks. |
| Date: | April 16, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Christine Lee, UT Austin |
| Title: | A knot with no tail |
| Abstract: | In this talk, we will discuss the stability behavior of the U_q(sl(2))-colored Jones polynomial, a quantum link invariant that assigns to a link K in S^3 a sequence of Laurent polynomials {J_K^n(q)} from n=2 to infinity. The colored Jones polynomial is said to have a tail if there is a power series whose coefficients encode the asymptotic behavior of the coefficients of J_K^n(q) for large n. Since Armond and Garoufalidis-Le proved the existence of a tail for the colored Jones polynomial of an adequate knot, first conjectured by Dasbach-Lin, it has been conjectured that multiple tails exist for all knots. Moreover, the stable coefficients of the tail have been shown to relate to the topology and the geometry of the alternating link complement, prompting the Coarse Volume Conjecture by Futer-Kalfagianni-Purcell. I will talk about an unexpected example of a knot, recently discovered in joint work with Roland van der Veen, where the colored Jones polynomial does not admit a tail, and discuss potential ways to view this example in the context of the categorification of the polynomial, the aforementioned Coarse Volume Conjecture, and a general conjecture made by Garoufalidis-Vuong concerning the stability of the colored Jones polynomial colored by irreducible representations of Lie algebras different from U_q(sl(2)). |
| Date: | April 20, 2018 |
| Time: | 4:00pm |
| Location: | BLOC 628 |
| Speaker: | F. Gesmundo, U. Cophenhagen |
| Title: | Cactus rank and multihomogeneous polynomials |
| Abstract: | The standard notion of matrix rank has several generalizations in algebraic geometry. Classical examples are Waring rank for homogeneous polynomials, tensor rank and in general X-rank with respect to an algebraic variety X. One additional generalization, of a more algebraic nature, is cactus rank, defined for every (smooth) algebraic variety and studied in the recent years in the settings of homogeneous polynomials and tensors. In this seminar, I will introduce cactus rank and present some of its features. In particular, we will see that whereas cactus rank presents a strong barrier in the study of other notions of rank, some of its characteristics are of great help in determining Waring rank and more generally partially symmetric rank in the tensor setting. |
| Date: | April 23, 2018 |
| Time: | 4:00pm |
| Location: | BLOC 220 |
| Speaker: | Shamgar Gurevich, University of Wisconsin |
| Title: | A look on Representations of SL(2,q) through the Lens of Size |
| Abstract: | How to study a nice function f of the real line? A physically motivated technique (called Harmonic analysis/Fourier theory) is to expand f in the basis of exponentials (also called frequencies) and study the meaningful terms in the expansion. Now, suppose f lives on a finite non-commutative group G, and is invariant under conjugation. There is a well-known analog of Fourier analysis, using the irreducible characters of G. This can be applied to many functions f that express interesting properties of G. To study f we want to know: Question: Which characters contributes most for the sum? I will describe for you the G=SL(2,Fq) case of the theory we are developing with Roger Howe (Yale/Texas A&M), which attempts to answer the above question. Remark: The irreducible representations of SL(2,Fq) are “well known” for a very long time and are a prototype example in many introductory course on the subject. So, it is nice that we can say something new about them. In particular, it turns out that the representations that people classify as “anomalies” in the old theory are the building blocks of our new theory. |
| Date: | April 27, 2018 |
| Time: | 4:00pm |
| Location: | BLOC 628 |
| Speaker: | Jen Berg, Rice |
| Title: | Odd order transcendental obstructions to the Hasse principle on general K3 surfaces |
| Abstract: | Varieties that fail to have rational points despite having local points for each prime are said to fail the Hasse principle. A systematic tool accounting for these failures is called the Brauer-Manin obstruction, which uses [subsets of] the Brauer group, Br X, to preclude the existence of rational points on a variety X. After fixing numerical invariants such as dimension, it is natural to ask which birational classes of varieties fail the Hasse principle, and moreover whether the Brauer group (or certain distinguished subsets) explains this failure. In this talk, we will focus on K3 surfaces, which are relatively simple surfaces in terms of geometric complexity, but whose arithmetic is more mysterious. For example, in 2014 it was asked whether any odd torsion in the Brauer group of a K3 surface could obstruct the Hasse principle. We answer this question in the affirmative; we exhibit a general degree 2 K3 surface Y over the rationals in which an order 3 transcendental Brauer class A obstructs. Motivated by Hodge theory, the pair (Y,A) is constructed from a special cubic fourfold X which admits a fibration into sextic del Pezzo surfaces over the projective plane. Notably, our construction does not rely on the presence of a central simple algebra representative for A. Instead, we prove that a sufficient condition for such a Brauer class to obstruct is insolubility of the fourfold X at 3 and local solubility at all other primes. This is joint work with Tony Varilly-Alvarado. |
| Date: | April 30, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Christopher O'Neill, UC Davis |
| Title: | Random numerical semigroups |
| Abstract: | A numerical semigroup is a subset of the natural numbers that is closed under addition. Consider a numerical semigroup S selected via the following random process: fix a probability p and a positive integer M, and select a generating set for S from the integers 1,...,M where each potential generator has probability p of being selected. What properties can we expect the numerical semigroup S to have? For instance, how many minimal generators do we expect S to have? In this talk, we answer several such questions, and describe some surprisingly deep geometric and combinatorial structures that arise naturally in the process. No familiarity with numerical semigroups or probability will be assumed for this talk. |
