for more information please visit: http://scgp.stonybrook.edu/archives/18377

1:00pm Am.Math.Soc. (AMS) Chapter Seminar: Alex Ginory - Double Affine Weyl Groups and Modular Invariance

**Where:** Math Tower P-131**When:** Mon, Mar 27 1:00pm — 2:00pm

**Title:** Double Affine Weyl Groups and Modular Invariance

**Speaker:** Alex Ginory [Rutgers University]

**Abstract:** The modular invariance property of characters of certain affine Kac-Moody Lie algebra modules (at a fixed level) is well-known and can be studied through the use of theta functions. Another approach involves the affinization of their Weyl groups, which can be used to introduce the action of the modular group in a natural fashion. We will discuss how this comes about and how this gives a natural construction of the modular action on the space of characters for the same class of affine Kac-Moody Lie algebras and an action of congruence subgroups for the remaining ones.

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4:00pm Analysis Student Seminar: Silvia Ghinassi - Doubling Measures and Quasisymmetric maps

**Where:** Math Tower 5-127**When:** Mon, Mar 27 4:00pm — 5:00pm

**Title:** Doubling Measures and Quasisymmetric maps

**Speaker:** Silvia Ghinassi [Stony Brook]

**Abstract:** When does a metric space embed quasisymmetrically into some Euclidean space? We provide a complete answer to this question (i.e. iff the space is metrically doubling). Moreover we will investigate relationship between doubling spaces and doubling measures.

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5:30pm First and Second Year Student Seminar: Song Sun - Introduction to Differential Geometry

**Where:** Math Tower P-131**When:** Mon, Mar 27 5:30pm — 6:30pm

**Title:** Introduction to Differential Geometry

**Speaker:** Song Sun [Stony Brook]

**Abstract:** Professor Sun works in Differential Geometry. He will introduce us with his work and Differential Geometry group at Stony Brook.

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1:00pm **SCGP:** SCGP Weekly Talk: Neal Weiner

**Where:** SCGP 102**When:** Tue, Mar 28 1:00pm — 2:00pm

**Title:** Dark Matter: Past, Present, and Future

Hashtag: #workshop

Hashtag: #workshop

4:00pm Geometry/Topology Seminar: Simon Donaldson - TBA

**Where:** Math Tower P-131**When:** Tue, Mar 28 4:00pm — 5:00pm

**Title:** TBA

**Speaker:** Simon Donaldson [SCGP ]

**Abstract:**

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11:00am **SCGP:** Math Seminar / Math of Gauge Fields: Paul Feehan

**Where:** SCGP 313**When:** Wed, Mar 29 11:00am — 12:00pm

**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.

11:30am RTG Student Geometry Seminar: Michael Albanese - Pointed Riemann surfaces and the moduli space of J-holomorphic curves

**Where:** Math Tower 5-127**When:** Wed, Mar 29 11:30am — 12:30pm

**Title:** Pointed Riemann surfaces and the moduli space of J-holomorphic curves

**Speaker:** Michael Albanese [Stony Brook]

**Abstract:** We will introduce the moduli space of J-holomorphic curves of a symplectic manifold. In order to understand the local structure of this space, we will need to study the moduli space of genus g Riemann surfaces with m marked points. In particular, we will discuss the classification of such objects.

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1:00pm Graduate Student Seminar: Fangyu Zou - A Topological Proof of Abel's Theorem

**Where:** Math Tower P-131**When:** Wed, Mar 29 1:00pm — 2:00pm

**Title:** A Topological Proof of Abel's Theorem

**Speaker:** Fangyu Zou [Stony Brook]

**Abstract:** I will start by defining the associated branched covering and monodromy group of an algebraic function. Then I will use this idea to give anther proof of the Abel's theorem (i.e. the general algebraic equations of degree >= 5 do not have radical formulas, as the case of degree 2, 3 and 4.) The basic idea is to show that the monodromy group of a general type is not solvable while the monodromy group of functions representable by radicals are solvable. Reference can be found in the little booklet:

"Abels Theorem in Problems and Solutions", by V.B. Alexeev

https://www.mathcamp.org/2015/abel/abel.pdf

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"Abels Theorem in Problems and Solutions", by V.B. Alexeev

https://www.mathcamp.org/2015/abel/abel.pdf

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1:15pm **SCGP:** Physics Seminar: Leonardo Giusti (CERN, University of Milano Bicocca, INFN), "New perspectives in lattice QCD"

**Where:** SCGP 102**When:** Wed, Mar 29 1:15pm — 2:15pm

**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.

1:30pm Equivalence Method and Exterior Differential Systems Seminar: Lorenzo Foscolo - The Cartan-Kaehler Theorem (II)

**Where:** Physics Tower, P-117**When:** Wed, Mar 29 1:30pm — 2:30pm

**Title:** The Cartan-Kaehler Theorem (II)

**Speaker:** Lorenzo Foscolo [Stony Brook]

**Abstract:** The Cartan-Kaehler Theorem establishes sufficient conditions for the existence of integral manifolds of real analytic Exterior Differential Systems. The theorem is one of the fundamental results in the field and is based on successive extensions of p-dimensional integral submanifolds to (p+1)-dimensional integral submanifolds by solving initial value problems using the Cauchy-Kowalewski Theorem.

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4:00pm Algebraic geometry seminar: Dima Arinkin - Oper structures on connections on Riemann surfaces

**Where:** Math Tower P-131**When:** Wed, Mar 29 4:00pm — 5:30pm

**Title:** Oper structures on connections on Riemann surfaces

**Speaker:** Dima Arinkin [U. Wisconsin, Madison]

**Abstract:** There is a classical correspondence between systems of n linear ordinary differential equations (ODEs) of order one and linear ODEs of order n; this may be viewed as a kind of canonical normal form for systems of ODEs. Restated geometrically, the claim is that if E is a vector bundle on a Riemann surface, and $\nabla$ is a connection on E (with arbitrary singularities), then there is a rational basis of E such that $\nabla$ is in the canonical normal form.

The statement makes sense for arbitrary reductive Lie group G (with the case of systems of ODEs corresponding to G=GL(n)): given a connection on a G-bundle on a Riemann surface, we can look for a rational gauge change that brings the connection to the `canonical normal form' (also known as a rational oper structure). This generalized statement turns out to be significantly harder. In my talk, I will sketch a proof of this claim and discuss its conjectural generalizations that are important for the geometric Langlands program.

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The statement makes sense for arbitrary reductive Lie group G (with the case of systems of ODEs corresponding to G=GL(n)): given a connection on a G-bundle on a Riemann surface, we can look for a rational gauge change that brings the connection to the `canonical normal form' (also known as a rational oper structure). This generalized statement turns out to be significantly harder. In my talk, I will sketch a proof of this claim and discuss its conjectural generalizations that are important for the geometric Langlands program.

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11:00am **SCGP:** Math Seminar / Math of Gauge Fields: Paul Feehan

**Where:** SCGP 313**When:** Thu, Mar 30 11:00am — 12:00pm

**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.

1:00pm **SCGP:** Math Seminar / Math of Gauge Fields: Paul Feehan

**Where:** SCGP 313**When:** Thu, Mar 30 1:00pm — 2:00pm

**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.

2:30pm **YITP:** YITP seminar, Peter Adshead (UIUC) [cosmology]

**Where:** YITP seminar room**When:** Thu, Mar 30 2:30pm — 3:30pm

**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.

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.

4:00pm Colloquium: Ken Ono - Cant you just feel the Moonshine?

**Where:** Math Tower P-131**When:** Thu, Mar 30 4:00pm — 5:00pm

**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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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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1:00pm **SCGP:** Math of Gauge fields: Tsuyoshi Kato: Higher degree of the covering monopole map in non commutative geometry

**Where:** 313**When:** Fri, Mar 31 1:00pm — 2:00pm

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.

Speaker Tsuyoshi Kato, Kyoto University

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.

1:00pm Grad / Postdoc Professional Development Seminar: Ninth meeting

**Where:** Math Tower P-131**When:** Fri, Mar 31 1:00pm — 2:30pm

**Title:** Ninth meeting

**Speaker:** TBA [Stony Brook]

**Abstract:**

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2:30pm Dynamical Systems Seminar: Giulio Tiozzo - TBA

**Where:** Math Tower P-131**When:** Fri, Mar 31 2:30pm — 3:30pm

**Title:** TBA

**Speaker:** Giulio Tiozzo [University of Toronto]

**Abstract:** TBA

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4:00pm Geometric Analysis Learning Seminar: Jean-Francois Arbour - TBA

**Where:** P-131 Math Tower**When:** Fri, Mar 31 4:00pm — 6:00pm

**Title:** TBA

**Speaker:** Jean-Francois Arbour [Stony Brook]

**Abstract:** TBA

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4:00pm Analysis Student Seminar: Conformal Gauges

**Where:** Math Tower 5-127**When:** Mon, Apr 3 4:00pm — 5:00pm

**Title:** Conformal Gauges

**Speaker:** TBA [Stony Brook]

**Abstract:**

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