Algebra and Number Theory Seminar
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Posted January 15, 2017

Last modified March 22, 2017

Peter Jorgensen, Newcastle University

Thick subcategories of d-abelian categories

Let d be a positive integer. The notion of d-abelian categories was introduced by Jasso. Such a category does not have kernels and cokernels, but rather d-kernels and d-cokernels which are longer complexes with weaker universal properties. Canonical examples of d-abelian categories are d-cluster tilting subcategories of abelian categories. We introduce the notion of thick subcategories of d-abelian categories. We show that functorially finite thick subcategories of d-cluster tilting subcategories are in bijection to so-called d-rigid epimorphisms. This generalises a classic result by Geigle and Lenzing. We apply this to show a classification of the thick subcategories of a family of d-abelian categories associated to quivers of type A_n. This is a report on joint work with Martin Herschend and Laertis Vaso.

Probability Seminar
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Posted March 9, 2017

Last modified March 13, 2017

Hui-Hsiung Kuo, Mathematics Department, LSU

Ito's formula for adapted and instantly independent stochastic processes

Topology Seminar
Seminar website

Posted February 3, 2017

Last modified March 21, 2017

Francesco Lin, Princeton University

TBD

Postponed until Fall 2017

Colloquium
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Posted October 25, 2016

Last modified March 6, 2017

Qing Xiang, University of Delaware

Applications of Linear Algebraic Methods in Combinatorics and Finite Geometry

Abstract: Most combinatorial objects can be described by incidence, adjacency, or some other (0,1)-matrices. So one basic approach in combinatorics is to investigate combinatorial objects by using linear algebraic parameters (ranks over various fields, spectrum, Smith normal forms, etc.) of their corresponding matrices. In this talk, we will look at some successful examples of this approach; some examples are old, and some are new. In particular, we will talk about the recent bounds on the size of partial spreads of H(2d-1,q^2) and on the size of partial ovoids of the Ree-Tits octagon.

Algebra and Number Theory Seminar
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Posted January 15, 2017

Last modified March 16, 2017

Dima Arinkin, University of Wisconsin

TBD

Topology Seminar
Seminar website

Posted March 8, 2017

3:30 pm - 4:20 pm Lockett 233
Gregor Masbaum, CNRS, Institut de Mathematiques de Jussieu, Paris, France

An application of TQFT to modular representation theory

Colloquium
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Posted March 7, 2017

Last modified March 22, 2017

Peter Jorgensen, Newcastle University

SL_2-tilings, infinite triangulations, and continuous cluster categories

Abstract: An SL_2-tiling is an infinite grid of positive integers such that each adjacent 2x2-submatrix has determinant 1. These tilings were introduced by Assem, Reutenauer, and Smith for combinatorial purposes. We will show that each SL_2-tiling can be obtained by a procedure called Conway--Coxeter counting from certain infinite triangulations of the circle with four accumulation points. We will see how properties of the tilings are reflected in the triangulations. For instance, the entry 1 of a tiling always gives an arc of the corresponding triangulation, and 1 can occur infinitely often in a tiling. On the other hand, if a tiling has no entry equal to 1, then the minimal entry of the tiling is unique, and the minimal entry can be seen as a more complex pattern in the triangulation. The infinite triangulations also give rise to cluster tilting subcategories in a certain cluster category with infinite clusters related to the continuous cluster categories of Igusa and Todorov. The SL_2-tilings can be viewed as the corresponding cluster characters. This is a report on joint work with Christine Bessenrodt and Thorsten Holm.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted December 12, 2016

Last modified February 3, 2017

Ken Ono, Emory University

Gems of Ramanujan and their Lasting Impact on Mathematics

Abstract: Ramanujan's work has has a truly transformative effect on modern mathematics, and continues to do so as we understand further lines from his letters and notebooks. In this lecture, some of the studies of Ramanujan that are most accessible to the general public will be presented and how Ramanujan's findings fundamentally changed modern mathematics, and also influenced the lecturer's work, will be discussed. The speaker is an Associate Producer of the film *The Man Who Knew Infinity* (starring Dev Patel and Jeremy Irons) about Ramanujan. He will share several clips from the film in the lecture.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted December 12, 2016

Last modified February 3, 2017

Ken Ono, Emory University

Cool Theorems Proved by Undergraduates

Abstract. The speaker has been organizing summer research programs for undergraduate students for many years. This lecture will give a sample of their accomplishments. The speaker will talk about partitioning integers, prime numbers, number fields, and generalizations of classical theorems of Euler, Gauss, and Jacobi.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted December 12, 2016

Last modified February 4, 2017

Ken Ono, Emory University

Can't you just feel the Moonshine?

Borcherds won the Fields medal in 1998 for his proof of the Monstrous Moonshine Conjecture. Loosely speaking, the conjecture asserts that the representation theory of the Monster, the largest sporadic finite simple group, is dictated by the Fourier expansions of a distinguished set of modular functions. This conjecture arose from astonishing coincidences noticed by finite group theorists and arithmetic geometers in the 1970s. Recently, mathematical physicists have revisited moonshine, and they discovered evidence of undiscovered moonshine which some believe will have applications to string theory and 3d quantum gravity. The speaker and his collaborators have been developing the mathematical facets of this theory, and have proved the conjectures which have been formulated. These results include a proof of the Umbral Moonshine Conjecture, and Moonshine for the first sporadic finite simple group which does not occur as a subgroup or subquotient of the Monster. The most recent Moonshine (announced here) yields unexpected applications to the arithmetic elliptic curves thanks to theorems related to the Birch and Swinnerton-Dyer Conjecture and the Main Conjectures of Iwasawa theory for modular forms. This is joint work with John Duncan, Michael Griffin and Michael Mertens.

Algebra and Number Theory Seminar
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Posted January 23, 2017

Last modified March 17, 2017

Jie Zhou, Perimeter institute

Periods and Gromov-Witten invariants

Abstract: The mirror symmetry conjectures asserts that the generating series of Gromov-Witten invariants (curve counting) of a Calabi-Yau variety are identical to some "universal" differential polynomials of period integrals of its mirror Calabi-Yau variety. I will explain in detail how these "universal" polynomials can be read off from the Picard-Fuchs system of the mirror Calabi-Yau variety, for the genus zero and one cases which are so far the only cases proved rigorously in mathematics. I will also discuss some nice ingredients (e.g., generating series of point counting, polylogarithms, Feynman diagrams and manipulation on Picard-Fuchs equations) which seem to have a motivic nature. A particularly interesting example of Calabi-Yau 3-fold will be emphasized, in which modular forms arise naturally.

Topology Seminar
Seminar website

Posted February 3, 2017

3:30 pm - 4:20 pm Lockett 233
Jose Ceniceros, Louisiana State University

TBD

Colloquium
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Posted February 23, 2017

Last modified February 28, 2017

Earl Taft, Rutgers University and UC Berkeley

Left quantum groups

Abstract: Quantum groups are Hopf algebras, and possess an antipode, which is the analogue of inversion in a group. The antipode satisfies a left and a right condition. In the interest of breaking symmetry in physics, S. Rodriguez and E. Taft constructed a version of quantum SL_2 which has a left antipode but not a right antipode. This was not of interest to physicists (we will explain why not), but turned out to be of interest to combinatorists, as it was used to give a quantum version of the MacMahon master theorem(also known as the boson-fermion correspondence). A. Lauve and E. Taft extended the Rodriguez-Taft construction to a version of quantum SL_n. Using continuous duals, we relate our work to the quantum universal enveloping algebra of the Lie algebra sl_2, and discuss an open question of whether or not there is a left quantum group containing (U_q)(sl(2)).