# This Week @ the SCGP

Tuesday, 3-28-2017

Turbulent and laminar flows in two dimensions: Brad Marston, Brown University

Time: 11:00 AM - 1:00 PM

Location: SCGP 313

Description: Speaker: Brad Marston, Brown University
Title: El NiĆ±o as a Topological Insulator: A Surprising Connection Between Geo-, and Quantum, Physics

Wednesday, 3-29-2017

Math Seminar / Math of Gauge Fields: Paul Feehan

Time: 11:00 AM - 12:00 PM

Location: SCGP 313

Description: Title: SO(3)-monopoles and relations between Donaldson and Seiberg-Witten invariants
Abstract: In this series of lectures we shall describe the SO(3)-monopole cobordism approach to proving two results concerning gauge-theoretic invariants of closed, four-dimensional, smooth manifolds. First, we shall explain how the SO(3)-monopole cobordism are used to prove that all four-manifolds with Seiberg-Witten simple type satisfy the superconformal simple type condition defined by Marino, Moore, and Peradze (1999). This result implies a lower bound, conjectured by Fintushel and Stern (2001), on the number of Seiberg-Witten basic classes in terms of topological data. Second, we shall explain how the SO(3)-monopole cobordism and the superconformal simple type property are used to prove Witten's Conjecture (1994) relating the Donaldson and Seiberg-Witten invariants. In the first lecture, we shall give an introduction to SO(3) monopoles and overview of how the SO(3)-monopole cobordism may be used to prove the Mari{\~n}o-Moore-Peradze and Witten conjectures. In the second lecture, we shall discuss the SO(3)-monopole cobordism, its compactification, Kuranishi-style gluing models, and explain how the cobordism may be used to prove the SO(3)-monopole link-pairing formula, which gives a very general (though non-explicit) relationship between Donaldson and Seiberg-Witten invariants. In the third lecture, we shall explain how a combination of blow-up formulae for Donaldson and Seiberg-Witten invariants, key examples of four-dimensional manifolds, and the SO(3)-monopole link-pairing formula can be used to prove the Marino-Moore-Peradze and Witten conjectures. Our lectures are primarily based on our articles arXiv:1408.5307 and arXiv:1408.5085 and book arXiv:math/0203047 (to appear in Memoirs of the American Mathematical Society), all joint with Thomas Leness.

Physics Seminar: Leonardo Giusti (CERN, University of Milano Bicocca, INFN), "New perspectives in lattice QCD"

Time: 1:15 PM - 2:15 PM

Location: SCGP 102

Description: Speaker: Leonardo Giusti (CERN, University of Milano Bicocca, INFN)
Title: New perspectives in lattice QCD
Abstract: Lattice gauge theory is the theoretical framework where the dynamics of strongly interacting particles can be investigated starting from the fundamental equations and keeping track of all systematic and statistical errors. After a brief introduction to the subject, I will review the recent progress on the precise quantitative understanding of the spontaneous and the anomalous breaking of chiral symmetry in Quantum Chromodynamics. The last part of the talk will be devoted to discuss a recent proposal for factorizing the fermion determinant in lattice theories with fermions. This paves the way for multilevel Monte Carlo integration in the presence of fermions, opening new perspectives in lattice QCD.

Thursday, 3-30-2017

Math Seminar / Math of Gauge Fields: Paul Feehan

Time: 11:00 AM - 12:00 PM

Location: SCGP 313

Description: Title: SO(3)-monopole cobordism formula
Abstract: In this series of lectures we shall describe the SO(3)-monopole cobordism approach to proving two results concerning gauge-theoretic invariants of closed, four-dimensional, smooth manifolds. First, we shall explain how the SO(3)-monopole cobordism are used to prove that all four-manifolds with Seiberg-Witten simple type satisfy the superconformal simple type condition defined by Marino, Moore, and Peradze (1999). This result implies a lower bound, conjectured by Fintushel and Stern (2001), on the number of Seiberg-Witten basic classes in terms of topological data. Second, we shall explain how the SO(3)-monopole cobordism and the superconformal simple type property are used to prove Witten's Conjecture (1994) relating the Donaldson and Seiberg-Witten invariants. In the first lecture, we shall give an introduction to SO(3) monopoles and overview of how the SO(3)-monopole cobordism may be used to prove the Mari{\~n}o-Moore-Peradze and Witten conjectures. In the second lecture, we shall discuss the SO(3)-monopole cobordism, its compactification, Kuranishi-style gluing models, and explain how the cobordism may be used to prove the SO(3)-monopole link-pairing formula, which gives a very general (though non-explicit) relationship between Donaldson and Seiberg-Witten invariants. In the third lecture, we shall explain how a combination of blow-up formulae for Donaldson and seiberg-Witten invariants, key examples of four-dimensional manifolds, and the SO(3)-monopole link-pairing formula can be used to prove the Marino-Moore-Peradze and Witten conjectures. Our lectures are primarily based on our articles arXiv:1408.5307 and arXiv:1408.5085 and book arXiv:math/0203047 (to appear in Memoirs of the American Mathematical Society), all joint with Thomas Leness.

Math Seminar / Math of Gauge Fields: Paul Feehan

Time: 1:00 PM - 2:00 PM

Location: SCGP 313

Description: Title: Superconformal simple type and Witten's conjecture
Abstract: In this series of lectures we shall describe the SO(3)-monopole cobordism approach to proving two results concerning gauge-theoretic invariants of closed, four-dimensional, smooth manifolds. First, we shall explain how the SO(3)-monopole cobordism are used to prove that all four-manifolds with Seiberg-Witten simple type satisfy the superconformal simple type condition defined by Marino, Moore, and Peradze (1999). This result implies a lower bound, conjectured by Fintushel and Stern (2001), on the number of Seiberg-Witten basic classes in terms of topological data. Second, we shall explain how the SO(3)-monopole cobordism and the superconformal simple type property are used to prove Witten's Conjecture (1994) relating the Donaldson and Seiberg-Witten invariants. In the first lecture, we shall give an introduction to SO(3) monopoles and overview of how the SO(3)-monopole cobordism may be used to prove the Mari{\~n}o-Moore-Peradze and Witten conjectures. In the second lecture, we shall discuss the SO(3)-monopole cobordism, its compactification, Kuranishi-style gluing models, and explain how the cobordism may be used to prove the SO(3)-monopole link-pairing formula, which gives a very general (though non-explicit) relationship between Donaldson and Seiberg-Witten invariants. In the third lecture, we shall explain how a combination of blow-up formulae for Donaldson and Seiberg-Witten invariants, key examples of four-dimensional manifolds, and the SO(3)-monopole link-pairing formula can be used to prove the Marino-Moore-Peradze and Witten conjectures. Our lectures are primarily based on our articles arXiv:1408.5307 and arXiv:1408.5085 and book arXiv:math/0203047 (to appear in Memoirs of the American Mathematical Society), all joint with Thomas Leness.

**YITP Event:**YITP seminar, Peter Adshead (UIUC) [cosmology]

Time: 2:30 PM - 3:30 PM

Location: YITP seminar room

Description: Title: Asymmetric reheating and chilly dark sectors
Abstract:
In a broad class of theories, the relic abundance of dark matter is determined by interactions internal to a thermalized dark sector, with no direct involvement of the Standard Model. These theories raise an immediate cosmological question: how was the dark sector initially populated in the early universe? I will discuss one possibility, asymmetric reheating, which can populate a thermal dark sector that never reaches thermal equilibrium with the SM.

**Math Event:**Math Colloquium: Ken Ono - Cant you just feel the Moonshine?

Time: 4:00 PM - 5:00 PM

Location: SCGP 102

Description: Title: Cant you just feel the Moonshine?

Speaker: Ken Ono [Emory]

Abstract: Richard Borcherds won the Fields medal in 1998 for his proof of the Monstrous Moonshine Conjecture. Formulated in 1979 by John Conway and Simon Norton, 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. 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 the last remaining problem raised by Conway and Norton in their groundbreaking 1979 paper. The most recent Moonshine (announced here) yields unexpected applications to the arithmetic of 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.

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Speaker: Ken Ono [Emory]

Abstract: Richard Borcherds won the Fields medal in 1998 for his proof of the Monstrous Moonshine Conjecture. Formulated in 1979 by John Conway and Simon Norton, 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. 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 the last remaining problem raised by Conway and Norton in their groundbreaking 1979 paper. The most recent Moonshine (announced here) yields unexpected applications to the arithmetic of 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.

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Friday, 3-31-2017

Math of Gauge fields: Tsuyoshi Kato: Higher degree of the covering monopole map in non commutative geometry

Time: 1:00 PM - 2:00 PM

Location: 313

Description: Speaker Tsuyoshi Kato, Kyoto University
Title: Higher degree of the covering monopole map in non commutative geometry
Abstract: I will introduce a monopole map over the universal covering space of a compact four manifold.
In particular we formulate higher degree of the covering monopole map when the linearized map is isomorphic, which induces a homomorphism between K theory of group C^* algebras.
As an application we propose an aspherical inequality on compact aspherical four manifolds.
This presents a stronger version to 10/8 inequality by Furuta, in the aspherical class of four manifolds.
This holds for many cases which include aspherical spin with residually finite fundamental groups.
Technically the construction of the covering monopole map requires non linear estimates in Sobolev spaces and will motivate L^p analysis on non compact manifolds.