All Seminars

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Title: An arithmetic count of the lines on a cubic surface.
Seminar: Algebra
Speaker: Kirsten Wickelgren of Georgia Institute of Technology
Contact: John Duncan, john.duncan@emory.edu
Date: 2017-11-14 at 4:00PM
Venue: W306
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Abstract:
A celebrated 19th century result of Cayley and Salmon is that a smooth cubic surface over the complex numbers contains exactly 27 lines. Over the real numbers, it is a lovely observation of Finashin–Kharlamov and Okonek–Teleman that while the number of real lines depends on the surface, a certain signed count of lines is always 3. We extend this count to an arbitrary field k using an Euler number in A1-homotopy theory. The resulting count is valued in the Grothendieck-Witt group of non-degenerate symmetric bilinear forms. This is joint work with Jesse Kass.
Title: Packing nearly optimal Ramsey $R(3,t)$ graphs
Seminar: Combinatorics
Speaker: Lutz Warnke of Georgia Institute of Technology
Contact: Dwight Duffus, dwight@mathcs.emory.edu
Date: 2017-10-23 at 4:00PM
Venue: W302
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Abstract:
TBA
Title: The rank of the Eisenstein ideal
Seminar: Algebra
Speaker: Preston Wake of UCLA
Contact: John Duncan, john.duncan@emory.edu
Date: 2017-10-03 at 4:00PM
Venue: W306
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Abstract:
In his landmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. We use deformation theory of pseudorepresentations to study the corresponding Hecke algebra. We will discuss how this method can be used to refine Mazur's results, quantifying the number of Eisenstein congruences. Time permitting, we'll also discuss some partial results in the composite-level case. This is joint work with Carl Wang-Erickson.
Title: Stabilizing Spectral Functors of Exact Categories
Seminar: Algebra
Speaker: Juan Villeta-Garcia of Emory University
Contact: John Duncan, john.duncan@emory.edu
Date: 2017-09-26 at 4:00PM
Venue: W306
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Abstract:
Algebraic K-Theory is often thought of as “the” universal additive invariant of rings (or more generally, exact categories). Often, however, functors on exact categories don’t satisfy additivity. We will describe a procedure due to McCarthy that constructs a functor’s universal additive approximation, and apply it to different local coefficient systems, recovering known invariants of rings (K-Theory, THH, etc.). We will talk about what happens when we push these constructions to the world of spectra, and tie in work of Lindenstrauss and McCarthy on the Taylor tower of Algebraic K-Theory.
Title: Counting restricted orientations of random graphs
Seminar: Combinatorics
Speaker: Yoshi Kohayakawa of University of Sao Paulo
Contact: Dwight Duffus, dwight@mathcs.emory.edu
Date: 2017-09-25 at 4:00PM
Venue: W302
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Abstract:
Following a suggestion of Erdos (1974), Alon and Yuster (2006) investigated the maximum number of orientations graphs of a given order admit if we forbid copies of a fixed tournament. We discuss the analogous problem in which certain restricted orientations of typical graphs of a given order and a given number of edges are considered.\\ \\This is joint work with M. Collares (Belo Horizonte), R. Morris (Rio de Janeiro) and G. O. Mota (Sao Paulo).
Title: A new tensor framework - theory and applications
Seminar: Numerical Analysis and Scientific Computing
Speaker: Dr. Misha Kilmer of Tufts University
Contact: James Nagy, jnagy@emory.edu
Date: 2017-09-22 at 2:00PM
Venue: W301
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Abstract:
Tensors (aka multiway arrays) can be instrumental in revealing latent correlations residing in high dimensional spaces. Despite their applicability to a broad range of applications in machine learning, speech recognition, and imaging, inconsistencies between tensor and matrix algebra have been complicating their broader utility. Researchers seeking to overcome those discrepancies have introduced several different candidate extensions, each introducing unique advantages and challenges. In this talk, we review some of the common tensor definitions, discuss their limitations, and introduce our tensor product framework which permits the elegant extension of linear algebraic concepts and algorithms to tensors. Following introduction of fundamental tensor operations, we discuss in further depth tensor decompositions and in particular the tensor SVD (t-SVD) and its randomized variant, which can be computed efficiently in parallel. We present details of the t-SVD, theoretical results, and provide numerical results that show the promise of our approach for compression and analysis of operators and datasets, highlighting examples such as facial recognition and model reduction.
Title: Unifying relaxed notions of modular forms
Seminar: Algebra
Speaker: Martin Raum of Chalmers Technical University, Gothenburg, Sweden
Contact: John Duncan, john.duncan@emory.edu
Date: 2017-09-21 at 4:00PM
Venue: W306
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Abstract:
Elliptic modular forms are functions on the complex upper half plane that are invariant under a certain action of the special linear group with integer entries. Their history comprises close to two centuries of amazing discoveries and application: The proof of Fermat's Last Theorem is probably the most famous; The theory of theta functions is among its most frequently employed parts.\\ \\During the past decade it has been à la mode to study relaxed notions of modularity. Relevant keywords that we will discuss are mock modular forms and higher order modular forms. We have witnessed their application, equally stunning as surprising, to conformal field theory, string theory, combinatorics, and many more areas.\\ \\In this talk, we suggest a change of perspective on such generalizations. Most of the novel variants of modular forms (with one prominent exception) can be viewed as components of vector-valued modular forms. This unification draws its charm from the past and the future. On the one hand, we integrate results by Kuga and Shimura that hitherto seemed almost forgotten. On the other hand, we can point out connections, for example, between mock modular forms and so-called iterated integrals that have not yet been noticed. Experts will be pleased to have in the future a ``Petersson pairing'' for mixed mock modular forms at their disposal.\\ \\This is joint work with Michael Mertens
Title: On the Birkhoff--von Neumann decomposition and its use in solving sparse linear systems
Seminar: Numerical Analysis and Scientific Computing
Speaker: Dr. Bora Ucar of CNRS and ENS Lyon, France (visiting GaTech this year)
Contact: Michele Benzi, benzi@mathcs.emory.edu
Date: 2017-09-15 at 2:00PM
Venue: W301
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Abstract:
The Birkhoff--von Neumann decomposition expresses a doubly stochastic matrix as a convex combination of permutation matrices. This talk will be an introduction to this decomposition. We are going to see its use in solving sparse linear systems, and investigate some algorithmic and combinatorial problems associated with it. This talk contains results from joint work with Michele Benzi (Emory Univ., Atlanta), Fanny Dufosse (Inria, France), Kamer Kaya (Sabanci Univ, Turkey), and Ioannis Panagiotas (ENS Lyon, France).
Title: Linked Fields of Characteristic 2 and their u-Invariant.
Seminar: Algebra
Speaker: Dr. Adam Chapman of Tel-Hai Academic College, Israel.
Contact: Dr. John Duncan, john.duncan@emory.edu
Date: 2017-09-14 at 4:00PM
Venue: W306
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Abstract:
The u-invariant of a field is the maximal dimension of a nonsingular anisotropic quadratic form over that field, whose order in the Witt group of the field is finite. By a classical theorem of Elman and Lam, the u-invariant of a linked field of characteristic different from 2 can be either 0,1,2,4 or 8. The analogous question in the case of characteristic 2 remained open for a long time.. We will discuss the proof of the equivalent statement in characteristic 2, recently obtained in a joint work by Andrew Dolphin and the speaker.
Title: Spectrum of singularities, exponential sums and the irreducibility of polynomials in two variables
Seminar: Algebra
Speaker: Jorge Jimenez Urroz of U. Politecnica, Catalunya
Contact: John Duncan, john.duncan@emory.edu
Date: 2017-08-29 at 4:00PM
Venue: W306
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Abstract:
In order to control the spectrum of singularities of the generalized Riemann function, we need to find sharp bounds for certain kind of Gauss sums with frequencies on polynomials. This can be achieved by Weil bounds on completely irreducible algebraic curves, which lead us to prove some theorems on irreducibility of polynomials in two variables. We will prove a general theorem in this field. An example of the theorem is the absolute irreducibility of p(x)-p(y)+1, for any p(x) with integer coefficients.