Northwestern University, winter 2020

**Time:** Wednesdays 5:00pm-7:00pm

**Location:** Lunt 107 (5pm-6pm), Lunt 105 (6pm-7pm)

**Remark:** We started on Jan 5 and the final talk was on March 11.

**Talks:**

- introduction (Viktor Burghardt)
- the (un)stable motivic homotopy categories (Stephan Snegirov)
- The motivic Adams Spectral Sequence (Eva Belmont)
- Refinded enumerative geometry (Catherine Ray)
- motivic cohomology and Voevodsky motives (Carlos Cortez)
- Voevodsky's slice filtration (Eivind Hjelle; notes)
- any further talks canceled due to COVID-19

- references: [1][2][3][4][5]

- references: [14][15]

- references: [13][2]

- references: [3][7][11][21][22][23]

- references: [6][7][8][9][10][11][12]

**Future topics:**

- The Bloch-Kato conjecture, aka the norm residue isomorphism theorem ([16][17])
- MGL, algebraic cobordism ([1][18][19])
- The zero-line of the stable homotopy groups of the motivic sphere and MW-K-theory ([1][20])
- Milnor-Witt cohomology, MW-motives ([11][24][25][26][27])
- The motivic Atiyah-Hirzebruch Spectral Sequence ([28][29][30][8])
- The motivic Steenrod algebra ([22][31][32])
- Motivic infinite loop space theory ([33][34])
- Classical stable theory embeds into motivic stable theory ([35][36])
- A1-homotopy classification of vector bundles, principal bundles, homogeneous spaces ([20][37][38][39])
- (shifted) connectivity theorems ([40][41][42][43])
- Motivic and real étale stable homotopy theory ([44])
- Framed correspondences ([45][46][47])
- ...

**References:**

- [1] An introduction to A1-homotopy theory - Morel
- [2] Unstable motivic homotopy theory - Wickelgren, Williams
- [3] Lecture notes on motivic cohomology, Galatius (lecture 14 and onward)
- [4] On the motivic stable $\pi_0$ of the sphere spectrum - Morel
- [5] Motivic homotopy theory - Lectures at a Summer School in Nordfjordeid
- [6] The motivic slice spectral sequence - ECHT lecture by Röndigs
- [7] open problems in motivic stable homotopy theory I - Voevodsky
- [8] a possible new approach to the motivic ss for alg. k-theory - Voevodsky
- [9] motivic Postnikov towers - Levine
- [10] Motivic slices and colored operads - Gutiérrez, Röndigs, Spitzweck, Østvær
- [11] The Generalized Slices of Hermitian K-Theory - Bachmann
- [12] convergence of Voevodsky's slice tower - Levine
- [13] Toward an enumerative geometry with quadratic forms - Levine
- [14] The motivic ASS - Dugger, Isaksen
- [15] Convergence of the motivic ASS - Hu, Kriz, Ormsby
- [16] Bloch-Kato conjecture and motivic cohomology with finite coefficients - Suslin, Voevodsky

- [17] The Norm Residue Theorem in Motivic Cohomology - Haesemeyer, Weibel
- [18] A1-homotopy theory - Voevodsky at ICM
- [19] From algebraic cobordism to motivic cohomology- Hoyois
- [20] A1-algebraic topology over a field - Morel
- [21] Lecture notes on motivic cohomology - Mazza, Weibel, Voevodsky
- [22] Stony Brook Mathematics Colloquium Video, Motivic cohomology and Steenrod operations - Marc Hoyois
- [23] Notes from a seminar on motivic cohomology by Marc Hoyois and Clark Barwick
- [24] Finite Chow-Witt correspondences - Calmès, Fasel
- [25] MW-motivic complexes - Déglise, Fasel
- [26] A comparison theorem for MW-motivic cohomology - Déglise, Fasel
- [27] On the effectivity of spectra representing motivic cohomology theories - Bachmann, Fasel
- [28] THE SPECTRAL SEQUENCE RELATINGALGEBRAICK-THEORY TO MOTIVIC COHOMOLOGY - Friedlander, Suslin
- [29] The Motivic Spectral Sequence - Grayson
- [30] THE ATIYAH-HIRZEBRUCH SPECTRAL SEQUENCE FOR ALGEBRAIC K-THEORY - Barwick
- [31] Reduced power operations in motivic cohomology - Voevodsky
- [32] The motivic Steenrod algebra in positive characteristic - Hoyois, Kelly, Østvær

- [33] motivic infinite loop spaces - Elmanto, Hoyois, Khan, Sosnilo, Yakerson
- [34] Framed motivic Gamma-spaces - Garkusha, Panin, Østvær
- [35] A comparison of motivic and classical homotopy theories - Levine
- [36] The stable Galois correspondence for real closed fields - Heller, Ormsby
- [37] Affine representability results in A1-homotopy theory I: vector bundles - Asok, Hoyois, Wendt
- [38] Affine representability results in A1-homotopy theory II: principal bundles and homogeneous spaces - Asok, Hoyois, Wendt
- [39] Affine representability results in A1-homotopy theory III: finite fields and complements - Asok, Hoyois, Wendt
- [40] The stable A1-connectivity theorems - Morel
- [41] Stable A1-connectivity over Dedekind schemes- Schmidt, Strunk
- [42] On Morel's stable A1-connectivity theorem - Elmanto
- [43] stable connectivity over a base - Druzhinin
- [44] Motivic and Real Etale Stable Homotopy Theory - Bachmann
- [45] Notes on framed correspondences - Voevodsky
- [46] Framed correspondences and the Milnor-Witt K-theory - Neshitov
- [47] Motivic stable homotopy groups via framed correspondences - Iakerson

organized by: Viktor Burghardt